Subgroups — C₆₇×#33187 #30712438

Order 17152.

1168 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#496×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#469×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#468×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#466×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#465×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2620	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2619	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1973	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1972	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1970	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#1968	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₇×#1958	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₇×#1957	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#1847	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
29	#496×C₆₇	—	2	4	no	—
30	#469×C₆₇	—	2	4	no	—
31	#468×C₆₇	—	2	4	no	—
32	#466×C₆₇	—	2	4	no	—
33	#465×C₆₇	—	2	4	no	—
34	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
43	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
46	#405×C₆₇	—	2	4	no	—
47	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
48	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
49	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
50	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
51	#396×C₆₇	—	2	2	no	—
52	#394×C₆₇	—	2	2	no	—
53	#394×C₆₇	—	2	2	no	—
54	#392×C₆₇	—	2	2	no	—
55	#391×C₆₇	C₂²	1	1	yes	characteristic
56	#390×C₆₇	C₂²	1	1	yes	characteristic
57	#390×C₆₇	C₂²	1	1	yes	characteristic
58	#390×C₆₇	C₂²	1	1	yes	characteristic
59	#384×C₆₇	C₂²	1	1	yes	characteristic
60	#382×C₆₇	C₂²	1	1	yes	characteristic
61	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂²×D₄×C₆₇	—	2	1	no	—
63	C₂²×D₄×C₆₇	—	2	1	no	—
64	C₂²×D₄×C₆₇	—	2	1	no	—
65	C₂²×D₄×C₆₇	—	2	2	no	—
66	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×SD₁₆×C₆₇	—	2	2	no	—
71	C₂×SD₁₆×C₆₇	—	2	2	no	—
72	C₂×SD₁₆×C₆₇	—	2	2	no	—
73	C₂×SD₁₆×C₆₇	—	2	2	no	—
74	C₂×SD₁₆×C₆₇	—	2	2	no	—
75	C₂×SD₁₆×C₆₇	—	2	2	no	—
76	C₂×D₈×C₆₇	—	2	2	no	—
77	C₂×D₈×C₆₇	—	2	2	no	—
78	C₂×D₈×C₆₇	—	2	2	no	—
79	C₂×D₈×C₆₇	—	2	2	no	—
80	C₂×D₈×C₆₇	—	2	2	no	—
81	C₂×D₈×C₆₇	—	2	2	no	—
82	C₂×M₁₆×C₆₇	—	2	2	no	—
83	C₂×M₁₆×C₆₇	—	2	2	no	—
84	C₂×M₁₆×C₆₇	D₄	1	4	yes	complemented
85	C₂²×C₈×C₆₇	—	2	2	no	—
86	C₂²×C₈×C₆₇	—	2	2	no	—
87	C₂²×C₈×C₆₇	D₄	1	4	yes	complemented
88	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
89	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
90	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
91	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
92	#127×C₆₇	—	2	2	no	—
93	#127×C₆₇	—	2	2	no	—
94	#124×C₆₇	—	2	2	no	—
95	#124×C₆₇	—	2	2	no	—
96	C₂²⋊Q₈×C₆₇	—	2	2	no	—
97	C₂²⋊Q₈×C₆₇	—	2	2	no	—
98	#121×C₆₇	—	2	2	no	—
99	#121×C₆₇	—	2	2	no	—
100	#121×C₆₇	—	2	2	no	—
101	#121×C₆₇	—	2	2	no	—
102	#121×C₆₇	—	2	2	no	—
103	#121×C₆₇	—	2	2	no	—
104	#117×C₆₇	—	2	2	no	—
105	#117×C₆₇	—	2	2	no	—
106	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
107	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
108	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
113	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
114	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
115	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
116	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
117	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
118	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
119	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
120	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
121	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
122	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
124	#107×C₆₇	—	2	2	no	—
125	#107×C₆₇	—	2	2	no	—
126	#106×C₆₇	—	2	2	no	—
127	#106×C₆₇	—	2	2	no	—
128	#103×C₆₇	—	2	2	no	—
129	#103×C₆₇	—	2	2	no	—
130	#102×C₆₇	—	2	2	no	—
131	#102×C₆₇	—	2	2	no	—
132	#102×C₆₇	—	2	2	no	—
133	#102×C₆₇	—	2	2	no	—
134	#102×C₆₇	—	2	2	no	—
135	#102×C₆₇	—	2	2	no	—
136	#97×C₆₇	—	2	2	no	—
137	#97×C₆₇	D₄	1	4	yes	complemented
138	C₄×C₈×C₆₇	—	2	2	no	—
139	C₄×C₈×C₆₇	—	2	2	no	—
140	#95×C₆₇	—	2	2	no	—
141	#95×C₆₇	—	2	2	no	—
142	C₂⁴×C₆₇	—	2	1	no	—
143	C₂⁴×C₆₇	—	2	1	no	—
144	C₂⁴×C₆₇	—	2	1	no	—
145	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
146	C₂×Q₈×C₆₇	—	2	1	no	—
147	C₂×Q₈×C₆₇	—	2	1	no	—
148	C₂×Q₈×C₆₇	—	2	1	no	—
149	C₂×Q₈×C₆₇	—	2	1	no	—
150	C₂×Q₈×C₆₇	—	2	1	no	—
151	C₂×Q₈×C₆₇	—	2	1	no	—
152	C₂×D₄×C₆₇	—	2	1	no	—
153	C₂×D₄×C₆₇	—	2	1	no	—
154	C₂×D₄×C₆₇	—	2	1	no	—
155	C₂×D₄×C₆₇	—	2	1	no	—
156	C₂×D₄×C₆₇	—	2	1	no	—
157	C₂×D₄×C₆₇	—	2	1	no	—
158	C₂×D₄×C₆₇	—	2	1	no	—
159	C₂×D₄×C₆₇	—	2	1	no	—
160	C₂×D₄×C₆₇	—	2	1	no	—
161	C₂×D₄×C₆₇	—	2	1	no	—
162	C₂×D₄×C₆₇	—	2	1	no	—
163	C₂×D₄×C₆₇	—	2	1	no	—
164	C₂×D₄×C₆₇	—	2	1	no	—
165	C₂×D₄×C₆₇	—	2	1	no	—
166	C₂×D₄×C₆₇	—	2	1	no	—
167	C₂×D₄×C₆₇	—	2	1	no	—
168	C₂×D₄×C₆₇	—	2	1	no	—
169	C₂×D₄×C₆₇	—	2	1	no	—
170	C₂×D₄×C₆₇	—	4	2	no	—
171	C₂×D₄×C₆₇	—	4	2	no	—
172	C₂×D₄×C₆₇	—	4	2	no	—
173	C₂×D₄×C₆₇	—	4	4	no	—
174	C₂×D₄×C₆₇	—	8	2	no	—
175	C₂×D₄×C₆₇	—	8	2	no	—
176	C₂×D₄×C₆₇	—	8	2	no	—
177	C₂×D₄×C₆₇	—	8	2	no	—
178	C₂×D₄×C₆₇	—	8	2	no	—
179	C₂²×C₄×C₆₇	—	2	1	no	—
180	C₂²×C₄×C₆₇	—	2	1	no	—
181	C₂²×C₄×C₆₇	—	2	1	no	—
182	C₂²×C₄×C₆₇	—	2	1	no	—
183	C₂²×C₄×C₆₇	—	2	1	no	—
184	C₂²×C₄×C₆₇	—	2	1	no	—
185	C₂²×C₄×C₆₇	—	2	1	no	—
186	C₂²×C₄×C₆₇	—	2	1	no	—
187	C₂²×C₄×C₆₇	—	2	1	no	—
188	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
189	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
190	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
191	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
192	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
193	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
194	SD₁₆×C₆₇	—	4	4	no	—
195	SD₁₆×C₆₇	—	4	4	no	—
196	D₈×C₆₇	—	4	4	no	—
197	D₈×C₆₇	—	4	4	no	—
198	M₁₆×C₆₇	—	2	8	no	—
199	C₂×C₈×C₆₇	—	2	1	no	—
200	C₂×C₈×C₆₇	—	2	1	no	—
201	C₂×C₈×C₆₇	—	2	1	no	—
202	C₂×C₈×C₆₇	—	2	1	no	—
203	C₂×C₈×C₆₇	—	2	8	no	—
204	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
205	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
206	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
207	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
208	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
209	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
210	C₂×C₈×C₆₇	#41	1	2	yes	—
211	C₂×C₈×C₆₇	#41	1	2	yes	—
212	C₄⋊C₄×C₆₇	—	2	4	no	—
213	C₄⋊C₄×C₆₇	—	2	1	no	—
214	C₄⋊C₄×C₆₇	—	2	1	no	—
215	C₄⋊C₄×C₆₇	—	2	1	no	—
216	C₄⋊C₄×C₆₇	—	2	1	no	—
217	C₄⋊C₄×C₆₇	—	2	1	no	—
218	C₄⋊C₄×C₆₇	—	2	1	no	—
219	C₄⋊C₄×C₆₇	—	2	1	no	—
220	C₄⋊C₄×C₆₇	—	2	1	no	—
221	C₄⋊C₄×C₆₇	—	4	2	no	—
222	C₄⋊C₄×C₆₇	—	8	2	no	—
223	C₂²⋊C₄×C₆₇	—	2	4	no	—
224	C₂²⋊C₄×C₆₇	—	4	2	no	—
225	C₂²⋊C₄×C₆₇	—	4	4	no	—
226	C₂²⋊C₄×C₆₇	—	4	2	no	—
227	C₂²⋊C₄×C₆₇	—	4	2	no	—
228	C₂²⋊C₄×C₆₇	—	4	2	no	—
229	C₂²⋊C₄×C₆₇	—	8	2	no	—
230	C₂²⋊C₄×C₆₇	—	8	2	no	—
231	C₂²⋊C₄×C₆₇	—	8	2	no	—
232	C₄²×C₆₇	—	2	1	no	—
233	C₄²×C₆₇	—	2	1	no	—
234	C₄²×C₆₇	C₂×D₄	1	2	yes	—
235	C₄²×C₆₇	#41	1	2	yes	—
236	C₂³×C₆₇	—	2	1	no	—
237	C₂³×C₆₇	—	2	1	no	—
238	C₂³×C₆₇	—	2	2	no	—
239	C₂³×C₆₇	—	2	2	no	—
240	C₂³×C₆₇	—	4	1	no	—
241	C₂³×C₆₇	—	4	1	no	—
242	C₂³×C₆₇	—	4	1	no	—
243	C₂³×C₆₇	—	4	1	no	—
244	C₂³×C₆₇	—	4	1	no	—
245	C₂³×C₆₇	—	4	1	no	—
246	C₂³×C₆₇	—	4	1	no	—
247	C₂³×C₆₇	—	4	1	no	—
248	C₂³×C₆₇	—	4	1	no	—
249	C₂³×C₆₇	—	4	2	no	—
250	C₂³×C₆₇	—	4	2	no	—
251	C₂³×C₆₇	—	4	2	no	—
252	C₂³×C₆₇	—	4	2	no	—
253	C₂³×C₆₇	—	4	2	no	—
254	C₂³×C₆₇	—	4	2	no	—
255	C₂³×C₆₇	#127	1	2	yes	—
256	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
257	Q₈×C₆₇	—	4	1	no	—
258	Q₈×C₆₇	—	4	1	no	—
259	Q₈×C₆₇	—	4	1	no	—
260	Q₈×C₆₇	—	4	1	no	—
261	D₄×C₆₇	—	4	1	no	—
262	D₄×C₆₇	—	4	1	no	—
263	D₄×C₆₇	—	4	1	no	—
264	D₄×C₆₇	—	4	1	no	—
265	D₄×C₆₇	—	4	1	no	—
266	D₄×C₆₇	—	4	1	no	—
267	D₄×C₆₇	—	4	1	no	—
268	D₄×C₆₇	—	4	1	no	—
269	D₄×C₆₇	—	4	1	no	—
270	D₄×C₆₇	—	4	1	no	—
271	D₄×C₆₇	—	4	1	no	—
272	D₄×C₆₇	—	4	1	no	—
273	D₄×C₆₇	—	8	8	no	—
274	D₄×C₆₇	—	8	4	no	—
275	D₄×C₆₇	—	8	4	no	—
276	D₄×C₆₇	—	8	4	no	—
277	C₂×C₄×C₆₇	—	2	1	no	—
278	C₂×C₄×C₆₇	—	2	1	no	—
279	C₂×C₄×C₆₇	—	2	2	no	—
280	C₂×C₄×C₆₇	—	2	2	no	—
281	C₂×C₄×C₆₇	—	2	2	no	—
282	C₂×C₄×C₆₇	—	2	2	no	—
283	C₂×C₄×C₆₇	—	2	2	no	—
284	C₂×C₄×C₆₇	—	2	2	no	—
285	C₂×C₄×C₆₇	—	2	2	no	—
286	C₂×C₄×C₆₇	—	2	2	no	—
287	C₂×C₄×C₆₇	—	2	2	no	—
288	C₂×C₄×C₆₇	—	4	1	no	—
289	C₂×C₄×C₆₇	—	4	1	no	—
290	C₂×C₄×C₆₇	—	4	1	no	—
291	C₂×C₄×C₆₇	—	4	1	no	—
292	C₂×C₄×C₆₇	—	4	1	no	—
293	C₂×C₄×C₆₇	—	4	1	no	—
294	C₂×C₄×C₆₇	—	4	1	no	—
295	C₂×C₄×C₆₇	—	4	2	no	—
296	C₂×C₄×C₆₇	—	4	2	no	—
297	C₂×C₄×C₆₇	—	4	2	no	—
298	C₂×C₄×C₆₇	—	4	2	no	—
299	C₂×C₄×C₆₇	#121	1	4	yes	—
300	C₂×C₄×C₆₇	#127	1	2	yes	—
301	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
302	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
303	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
304	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
305	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
306	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
307	C₈×C₆₇	—	2	2	no	—
308	C₈×C₆₇	—	2	2	no	—
309	C₈×C₆₇	#121	1	4	yes	—
310	C₈×C₆₇	#121	1	4	yes	—
311	C₂²×C₆₇	—	2	2	no	—
312	C₂²×C₆₇	—	2	2	no	—
313	C₂²×C₆₇	—	2	2	no	—
314	C₂²×C₆₇	—	4	2	no	—
315	C₂²×C₆₇	—	4	2	no	—
316	C₂²×C₆₇	—	8	1	no	—
317	C₂²×C₆₇	—	8	1	no	—
318	C₂²×C₆₇	—	8	1	no	—
319	C₂²×C₆₇	—	8	1	no	—
320	C₂²×C₆₇	—	8	1	no	—
321	C₂²×C₆₇	—	8	1	no	—
322	C₂²×C₆₇	—	8	1	no	—
323	C₂²×C₆₇	—	8	1	no	—
324	C₂²×C₆₇	—	8	1	no	—
325	C₂²×C₆₇	—	8	2	no	—
326	C₂²×C₆₇	—	8	2	no	—
327	C₂²×C₆₇	—	8	2	no	—
328	C₂²×C₆₇	—	8	2	no	—
329	C₂²×C₆₇	—	8	2	no	—
330	C₂²×C₆₇	—	8	2	no	—
331	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
332	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
333	C₂²×C₆₇	#532	1	1	yes	characteristic
334	C₂²×C₆₇	#576	1	2	yes	—
335	C₂²×C₆₇	#577	1	2	yes	—
336	C₄×C₆₇	—	2	2	no	—
337	C₄×C₆₇	—	2	4	no	—
338	C₄×C₆₇	—	4	2	no	—
339	C₄×C₆₇	—	8	1	no	—
340	C₄×C₆₇	—	8	1	no	—
341	C₄×C₆₇	—	8	1	no	—
342	C₄×C₆₇	—	8	1	no	—
343	C₄×C₆₇	—	8	1	no	—
344	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
345	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
346	C₄×C₆₇	#532	1	1	yes	characteristic
347	C₄×C₆₇	#532	1	1	yes	characteristic
348	#33187	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
349	C₂×C₆₇	—	4	2	no	—
350	C₂×C₆₇	—	16	1	no	—
351	C₂×C₆₇	—	16	1	no	—
352	C₂×C₆₇	—	16	1	no	—
353	C₂×C₆₇	#2856	1	1	yes	characteristic
354	C₂×C₆₇	#3062	1	2	yes	—
355	C₂×C₆₇	#3154	1	1	yes	characteristic
356	C₂×C₆₇	#3155	1	1	yes	characteristic
357	C₂×C₆₇	#3179	1	2	yes	—
358	C₂×#496	C₂×C₆₇	1	1	yes	characteristic, complemented
359	C₂×#469	C₂×C₆₇	1	1	yes	characteristic, complemented
360	C₂×#468	C₂×C₆₇	1	1	yes	characteristic, complemented
361	C₂×#466	C₂×C₆₇	1	1	yes	characteristic, complemented
362	C₂×#465	C₂×C₆₇	1	1	yes	characteristic, complemented
363	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
364	#2620	C₂×C₆₇	1	1	yes	characteristic, complemented
365	#2619	C₂×C₆₇	1	1	yes	characteristic, complemented
366	#1973	C₂×C₆₇	1	1	yes	characteristic, complemented
367	#1972	C₂×C₆₇	1	1	yes	characteristic, complemented
368	#1970	C₂×C₆₇	1	1	yes	characteristic, complemented
369	#1968	C₂×C₆₇	1	1	yes	characteristic, complemented
370	#1958	C₂×C₆₇	1	1	yes	characteristic, complemented
371	#1957	C₂×C₆₇	1	1	yes	characteristic, complemented
372	#1847	C₂×C₆₇	1	1	yes	characteristic, complemented
373	C₆₇	#33187	1	1	yes	characteristic, direct product factor, Sylow
374	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
375	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
376	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
377	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
378	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
379	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
380	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
381	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
382	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
383	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
384	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
385	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
386	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
387	#496	—	2	4	no	—
388	#469	—	2	4	no	—
389	#468	—	2	4	no	—
390	#466	—	2	4	no	—
391	#465	—	2	4	no	—
392	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
393	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
394	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
395	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
396	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
397	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
398	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
399	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
400	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
401	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
402	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
403	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
404	#405	—	2	4	no	—
405	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
406	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
407	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
408	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
409	#396	—	2	2	no	—
410	#394	—	2	2	no	—
411	#394	—	2	2	no	—
412	#392	—	2	2	no	—
413	#391	C₂²×C₆₇	1	1	yes	characteristic
414	#390	C₂²×C₆₇	1	1	yes	characteristic
415	#390	C₂²×C₆₇	1	1	yes	characteristic
416	#390	C₂²×C₆₇	1	1	yes	characteristic
417	#384	C₂²×C₆₇	1	1	yes	characteristic
418	#382	C₂²×C₆₇	1	1	yes	characteristic
419	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
420	C₂²×D₄	—	2	1	no	—
421	C₂²×D₄	—	2	1	no	—
422	C₂²×D₄	—	2	1	no	—
423	C₂²×D₄	—	2	2	no	—
424	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
425	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
426	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
427	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
428	C₂×SD₁₆	—	2	2	no	—
429	C₂×SD₁₆	—	2	2	no	—
430	C₂×SD₁₆	—	2	2	no	—
431	C₂×SD₁₆	—	2	2	no	—
432	C₂×SD₁₆	—	2	2	no	—
433	C₂×SD₁₆	—	2	2	no	—
434	C₂×D₈	—	2	2	no	—
435	C₂×D₈	—	2	2	no	—
436	C₂×D₈	—	2	2	no	—
437	C₂×D₈	—	2	2	no	—
438	C₂×D₈	—	2	2	no	—
439	C₂×D₈	—	2	2	no	—
440	C₂×M₁₆	—	2	2	no	—
441	C₂×M₁₆	—	2	2	no	—
442	C₂×M₁₆	D₄×C₆₇	1	4	yes	complemented
443	C₂²×C₈	—	2	2	no	—
444	C₂²×C₈	—	2	2	no	—
445	C₂²×C₈	D₄×C₆₇	1	4	yes	complemented
446	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
447	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
448	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
449	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
450	#127	—	2	2	no	—
451	#127	—	2	2	no	—
452	#124	—	2	2	no	—
453	#124	—	2	2	no	—
454	C₂²⋊Q₈	—	2	2	no	—
455	C₂²⋊Q₈	—	2	2	no	—
456	#121	—	2	2	no	—
457	#121	—	2	2	no	—
458	#121	—	2	2	no	—
459	#121	—	2	2	no	—
460	#121	—	2	2	no	—
461	#121	—	2	2	no	—
462	#117	—	2	2	no	—
463	#117	—	2	2	no	—
464	C₂×C₄⋊C₄	—	2	1	no	—
465	C₂×C₄⋊C₄	—	2	2	no	—
466	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
467	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
468	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
469	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
470	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
471	C₂×C₂²⋊C₄	—	2	1	no	—
472	C₂×C₂²⋊C₄	—	2	1	no	—
473	C₂×C₂²⋊C₄	—	2	1	no	—
474	C₂×C₂²⋊C₄	—	2	1	no	—
475	C₂×C₂²⋊C₄	—	2	2	no	—
476	C₂×C₂²⋊C₄	—	2	2	no	—
477	C₂×C₂²⋊C₄	—	2	2	no	—
478	C₂×C₂²⋊C₄	—	2	2	no	—
479	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
480	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
481	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
482	#107	—	2	2	no	—
483	#107	—	2	2	no	—
484	#106	—	2	2	no	—
485	#106	—	2	2	no	—
486	#103	—	2	2	no	—
487	#103	—	2	2	no	—
488	#102	—	2	2	no	—
489	#102	—	2	2	no	—
490	#102	—	2	2	no	—
491	#102	—	2	2	no	—
492	#102	—	2	2	no	—
493	#102	—	2	2	no	—
494	#97	—	2	2	no	—
495	#97	D₄×C₆₇	1	4	yes	complemented
496	C₄×C₈	—	2	2	no	—
497	C₄×C₈	—	2	2	no	—
498	#95	—	2	2	no	—
499	#95	—	2	2	no	—
500	C₂⁴	—	2	1	no	—
501	C₂⁴	—	2	1	no	—
502	C₂⁴	—	2	1	no	—
503	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
504	C₂×Q₈	—	2	1	no	—
505	C₂×Q₈	—	2	1	no	—
506	C₂×Q₈	—	2	1	no	—
507	C₂×Q₈	—	2	1	no	—
508	C₂×Q₈	—	2	1	no	—
509	C₂×Q₈	—	2	1	no	—
510	C₂×D₄	—	2	1	no	—
511	C₂×D₄	—	2	1	no	—
512	C₂×D₄	—	2	1	no	—
513	C₂×D₄	—	2	1	no	—
514	C₂×D₄	—	2	1	no	—
515	C₂×D₄	—	2	1	no	—
516	C₂×D₄	—	2	1	no	—
517	C₂×D₄	—	2	1	no	—
518	C₂×D₄	—	2	1	no	—
519	C₂×D₄	—	2	1	no	—
520	C₂×D₄	—	2	1	no	—
521	C₂×D₄	—	2	1	no	—
522	C₂×D₄	—	2	1	no	—
523	C₂×D₄	—	2	1	no	—
524	C₂×D₄	—	2	1	no	—
525	C₂×D₄	—	2	1	no	—
526	C₂×D₄	—	2	1	no	—
527	C₂×D₄	—	2	1	no	—
528	C₂×D₄	—	4	2	no	—
529	C₂×D₄	—	4	2	no	—
530	C₂×D₄	—	4	2	no	—
531	C₂×D₄	—	4	4	no	—
532	C₂×D₄	—	8	2	no	—
533	C₂×D₄	—	8	2	no	—
534	C₂×D₄	—	8	2	no	—
535	C₂×D₄	—	8	2	no	—
536	C₂×D₄	—	8	2	no	—
537	C₂²×C₄	—	2	1	no	—
538	C₂²×C₄	—	2	1	no	—
539	C₂²×C₄	—	2	1	no	—
540	C₂²×C₄	—	2	1	no	—
541	C₂²×C₄	—	2	1	no	—
542	C₂²×C₄	—	2	1	no	—
543	C₂²×C₄	—	2	1	no	—
544	C₂²×C₄	—	2	1	no	—
545	C₂²×C₄	—	2	1	no	—
546	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
547	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
548	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
549	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
550	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
551	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
552	SD₁₆	—	4	4	no	—
553	SD₁₆	—	4	4	no	—
554	D₈	—	4	4	no	—
555	D₈	—	4	4	no	—
556	M₁₆	—	2	8	no	—
557	C₂×C₈	—	2	1	no	—
558	C₂×C₈	—	2	1	no	—
559	C₂×C₈	—	2	1	no	—
560	C₂×C₈	—	2	1	no	—
561	C₂×C₈	—	2	8	no	—
562	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
563	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
564	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
565	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
566	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
567	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
568	C₂×C₈	#41×C₆₇	1	2	yes	—
569	C₂×C₈	#41×C₆₇	1	2	yes	—
570	C₄⋊C₄	—	2	4	no	—
571	C₄⋊C₄	—	2	1	no	—
572	C₄⋊C₄	—	2	1	no	—
573	C₄⋊C₄	—	2	1	no	—
574	C₄⋊C₄	—	2	1	no	—
575	C₄⋊C₄	—	2	1	no	—
576	C₄⋊C₄	—	2	1	no	—
577	C₄⋊C₄	—	2	1	no	—
578	C₄⋊C₄	—	2	1	no	—
579	C₄⋊C₄	—	4	2	no	—
580	C₄⋊C₄	—	8	2	no	—
581	C₂²⋊C₄	—	2	4	no	—
582	C₂²⋊C₄	—	4	2	no	—
583	C₂²⋊C₄	—	4	4	no	—
584	C₂²⋊C₄	—	4	2	no	—
585	C₂²⋊C₄	—	4	2	no	—
586	C₂²⋊C₄	—	4	2	no	—
587	C₂²⋊C₄	—	8	2	no	—
588	C₂²⋊C₄	—	8	2	no	—
589	C₂²⋊C₄	—	8	2	no	—
590	C₄²	—	2	1	no	—
591	C₄²	—	2	1	no	—
592	C₄²	C₂×D₄×C₆₇	1	2	yes	—
593	C₄²	#41×C₆₇	1	2	yes	—
594	C₂³	—	2	1	no	—
595	C₂³	—	2	1	no	—
596	C₂³	—	2	2	no	—
597	C₂³	—	2	2	no	—
598	C₂³	—	4	1	no	—
599	C₂³	—	4	1	no	—
600	C₂³	—	4	1	no	—
601	C₂³	—	4	1	no	—
602	C₂³	—	4	1	no	—
603	C₂³	—	4	1	no	—
604	C₂³	—	4	1	no	—
605	C₂³	—	4	1	no	—
606	C₂³	—	4	1	no	—
607	C₂³	—	4	2	no	—
608	C₂³	—	4	2	no	—
609	C₂³	—	4	2	no	—
610	C₂³	—	4	2	no	—
611	C₂³	—	4	2	no	—
612	C₂³	—	4	2	no	—
613	C₂³	#127×C₆₇	1	2	yes	—
614	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
615	Q₈	—	4	1	no	—
616	Q₈	—	4	1	no	—
617	Q₈	—	4	1	no	—
618	Q₈	—	4	1	no	—
619	D₄	—	4	1	no	—
620	D₄	—	4	1	no	—
621	D₄	—	4	1	no	—
622	D₄	—	4	1	no	—
623	D₄	—	4	1	no	—
624	D₄	—	4	1	no	—
625	D₄	—	4	1	no	—
626	D₄	—	4	1	no	—
627	D₄	—	4	1	no	—
628	D₄	—	4	1	no	—
629	D₄	—	4	1	no	—
630	D₄	—	4	1	no	—
631	D₄	—	8	8	no	—
632	D₄	—	8	4	no	—
633	D₄	—	8	4	no	—
634	D₄	—	8	4	no	—
635	C₂×C₄	—	2	1	no	—
636	C₂×C₄	—	2	1	no	—
637	C₂×C₄	—	2	2	no	—
638	C₂×C₄	—	2	2	no	—
639	C₂×C₄	—	2	2	no	—
640	C₂×C₄	—	2	2	no	—
641	C₂×C₄	—	2	2	no	—
642	C₂×C₄	—	2	2	no	—
643	C₂×C₄	—	2	2	no	—
644	C₂×C₄	—	2	2	no	—
645	C₂×C₄	—	2	2	no	—
646	C₂×C₄	—	4	1	no	—
647	C₂×C₄	—	4	1	no	—
648	C₂×C₄	—	4	1	no	—
649	C₂×C₄	—	4	1	no	—
650	C₂×C₄	—	4	1	no	—
651	C₂×C₄	—	4	1	no	—
652	C₂×C₄	—	4	1	no	—
653	C₂×C₄	—	4	2	no	—
654	C₂×C₄	—	4	2	no	—
655	C₂×C₄	—	4	2	no	—
656	C₂×C₄	—	4	2	no	—
657	C₂×C₄	#121×C₆₇	1	4	yes	—
658	C₂×C₄	#127×C₆₇	1	2	yes	—
659	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
660	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
661	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
662	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
663	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
664	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
665	C₈	—	2	2	no	—
666	C₈	—	2	2	no	—
667	C₈	#121×C₆₇	1	4	yes	—
668	C₈	#121×C₆₇	1	4	yes	—
669	C₂²	—	2	2	no	—
670	C₂²	—	2	2	no	—
671	C₂²	—	2	2	no	—
672	C₂²	—	4	2	no	—
673	C₂²	—	4	2	no	—
674	C₂²	—	8	2	no	—
675	C₂²	—	8	2	no	—
676	C₂²	—	8	2	no	—
677	C₂²	—	8	1	no	—
678	C₂²	—	8	1	no	—
679	C₂²	—	8	1	no	—
680	C₂²	—	8	1	no	—
681	C₂²	—	8	1	no	—
682	C₂²	—	8	1	no	—
683	C₂²	—	8	1	no	—
684	C₂²	—	8	1	no	—
685	C₂²	—	8	1	no	—
686	C₂²	—	8	2	no	—
687	C₂²	—	8	2	no	—
688	C₂²	—	8	2	no	—
689	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
690	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
691	C₂²	#532×C₆₇	1	1	yes	characteristic
692	C₂²	#576×C₆₇	1	2	yes	—
693	C₂²	#577×C₆₇	1	2	yes	—
694	C₄	—	2	4	no	—
695	C₄	—	2	2	no	—
696	C₄	—	4	2	no	—
697	C₄	—	8	1	no	—
698	C₄	—	8	1	no	—
699	C₄	—	8	1	no	—
700	C₄	—	8	1	no	—
701	C₄	—	8	1	no	—
702	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
703	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
704	C₄	#532×C₆₇	1	1	yes	characteristic
705	C₄	#532×C₆₇	1	1	yes	characteristic
706	C₂	—	4	2	no	—
707	C₂	—	16	1	no	—
708	C₂	—	16	1	no	—
709	C₂	—	16	1	no	—
710	C₂	C₆₇×#2856	1	1	yes	characteristic
711	C₂	C₆₇×#3062	1	2	yes	—
712	C₂	C₆₇×#3154	1	1	yes	characteristic
713	C₂	C₆₇×#3155	1	1	yes	characteristic
714	C₂	C₆₇×#3179	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#496×C₆₇	C₂	semidirect
2. C₂×#469×C₆₇	C₂	semidirect
3. C₂×#468×C₆₇	C₂	semidirect
4. C₂×#466×C₆₇	C₂	semidirect
5. C₂×#465×C₆₇	C₂	semidirect
6. C₂×#405×C₆₇	C₂	semidirect
7. C₆₇×#2620	C₂	semidirect
8. C₆₇×#2619	C₂	semidirect
9. C₆₇×#1973	C₂	semidirect
10. C₆₇×#1972	C₂	semidirect
11. C₆₇×#1970	C₂	semidirect
12. C₆₇×#1968	C₂	semidirect
13. C₆₇×#1958	C₂	semidirect
14. C₆₇×#1957	C₂	semidirect
15. C₆₇×#1847	C₂	semidirect
84. C₂×M₁₆×C₆₇	D₄	semidirect
87. C₂²×C₈×C₆₇	D₄	semidirect
137. #97×C₆₇	D₄	semidirect
348. #33187	C₆₇	direct (N × H)
358. C₂×#496	C₂×C₆₇	semidirect
359. C₂×#469	C₂×C₆₇	semidirect
360. C₂×#468	C₂×C₆₇	semidirect
361. C₂×#466	C₂×C₆₇	semidirect
362. C₂×#465	C₂×C₆₇	semidirect
363. C₂×#405	C₂×C₆₇	semidirect
364. #2620	C₂×C₆₇	semidirect
365. #2619	C₂×C₆₇	semidirect
366. #1973	C₂×C₆₇	semidirect
367. #1972	C₂×C₆₇	semidirect
368. #1970	C₂×C₆₇	semidirect
369. #1968	C₂×C₆₇	semidirect
370. #1958	C₂×C₆₇	semidirect
371. #1957	C₂×C₆₇	semidirect
372. #1847	C₂×C₆₇	semidirect
373. C₆₇	#33187	direct (N × H)
442. C₂×M₁₆	D₄×C₆₇	semidirect
445. C₂²×C₈	D₄×C₆₇	semidirect
495. #97	D₄×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#496×C₆₇	2
C₂×#469×C₆₇	2
C₂×#468×C₆₇	2
C₂×#466×C₆₇	2
C₂×#465×C₆₇	2
C₂×#405×C₆₇	2
C₆₇×#2620	2
C₆₇×#2619	2
C₆₇×#1973	2
C₆₇×#1972	2
C₆₇×#1970	2
C₆₇×#1968	2
C₆₇×#1958	2
C₆₇×#1957	2
C₆₇×#1847	2
#33187	67

Subgroups — C67×#33187 #30712438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33187 #30712438