Subgroups — C₆₇×#33188 #30712439

Order 17152.

1068 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#497×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#470×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#468×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#467×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#465×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2621	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2620	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1973	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1971	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1969	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×#1956	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#1847	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×SD₁₆×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×C₄⋊Q₈×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×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊Q₈×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	#497×C₆₇	—	2	4	no	—
30	#470×C₆₇	—	2	4	no	—
31	#468×C₆₇	—	2	4	no	—
32	#467×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×#103×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
43	C₂×#103×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	#398×C₆₇	—	2	2	no	—
52	#397×C₆₇	—	2	2	no	—
53	#396×C₆₇	—	2	2	no	—
54	#394×C₆₇	—	2	2	no	—
55	#391×C₆₇	C₂²	1	1	yes	characteristic
56	#391×C₆₇	C₂²	1	1	yes	characteristic
57	#391×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×Q₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₆₇	—	2	1	no	—
65	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₂×Q₁₆×C₆₇	—	2	2	no	—
68	C₂×Q₁₆×C₆₇	—	2	2	no	—
69	C₂×Q₁₆×C₆₇	—	2	2	no	—
70	C₂×Q₁₆×C₆₇	—	2	2	no	—
71	C₂×Q₁₆×C₆₇	—	2	2	no	—
72	C₂×Q₁₆×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×SD₁₆×C₆₇	—	2	2	no	—
77	C₂×SD₁₆×C₆₇	—	2	2	no	—
78	C₂×SD₁₆×C₆₇	—	2	2	no	—
79	C₂×M₁₆×C₆₇	—	2	2	no	—
80	C₂×M₁₆×C₆₇	—	2	2	no	—
81	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
82	C₂²×C₈×C₆₇	—	2	2	no	—
83	C₂²×C₈×C₆₇	—	2	2	no	—
84	C₂²×C₈×C₆₇	D₄	1	4	yes	—
85	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
87	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
88	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
89	C₄⋊Q₈×C₆₇	—	2	2	no	—
90	C₄⋊Q₈×C₆₇	—	2	2	no	—
91	#124×C₆₇	—	2	2	no	—
92	#124×C₆₇	—	2	2	no	—
93	C₂²⋊Q₈×C₆₇	—	2	2	no	—
94	C₂²⋊Q₈×C₆₇	—	2	2	no	—
95	C₂²⋊Q₈×C₆₇	—	2	2	no	—
96	C₂²⋊Q₈×C₆₇	—	2	2	no	—
97	C₂²⋊Q₈×C₆₇	—	2	2	no	—
98	C₂²⋊Q₈×C₆₇	—	2	2	no	—
99	#121×C₆₇	—	2	2	no	—
100	#121×C₆₇	—	2	2	no	—
101	#117×C₆₇	—	2	2	no	—
102	#117×C₆₇	—	2	2	no	—
103	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
104	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
105	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
106	C₂×C₄⋊C₄×C₆₇	—	2	2	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	D₄	1	2	yes	—
120	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
122	#107×C₆₇	—	2	2	no	—
123	#107×C₆₇	—	2	2	no	—
124	#106×C₆₇	—	2	2	no	—
125	#106×C₆₇	—	2	2	no	—
126	#103×C₆₇	—	2	2	no	—
127	#103×C₆₇	—	2	2	no	—
128	#103×C₆₇	—	2	2	no	—
129	#103×C₆₇	—	2	2	no	—
130	#103×C₆₇	—	2	2	no	—
131	#103×C₆₇	—	2	2	no	—
132	#102×C₆₇	—	2	2	no	—
133	#102×C₆₇	—	2	2	no	—
134	#97×C₆₇	—	2	2	no	—
135	#97×C₆₇	D₄	1	4	yes	complemented
136	C₄×C₈×C₆₇	—	2	2	no	—
137	C₄×C₈×C₆₇	—	2	2	no	—
138	#95×C₆₇	—	2	2	no	—
139	#95×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	C₂×D₄	1	1	yes	characteristic
144	C₂×Q₈×C₆₇	—	2	1	no	—
145	C₂×Q₈×C₆₇	—	2	1	no	—
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×Q₈×C₆₇	—	2	1	no	—
153	C₂×Q₈×C₆₇	—	2	1	no	—
154	C₂×Q₈×C₆₇	—	2	1	no	—
155	C₂×Q₈×C₆₇	—	2	1	no	—
156	C₂×Q₈×C₆₇	—	2	1	no	—
157	C₂×Q₈×C₆₇	—	2	1	no	—
158	C₂×Q₈×C₆₇	—	2	1	no	—
159	C₂×Q₈×C₆₇	—	2	1	no	—
160	C₂×Q₈×C₆₇	—	2	1	no	—
161	C₂×Q₈×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	—	4	2	no	—
169	C₂×D₄×C₆₇	—	8	2	no	—
170	C₂²×C₄×C₆₇	—	2	1	no	—
171	C₂²×C₄×C₆₇	—	2	1	no	—
172	C₂²×C₄×C₆₇	—	2	1	no	—
173	C₂²×C₄×C₆₇	—	2	1	no	—
174	C₂²×C₄×C₆₇	—	2	1	no	—
175	C₂²×C₄×C₆₇	—	2	1	no	—
176	C₂²×C₄×C₆₇	—	2	1	no	—
177	C₂²×C₄×C₆₇	—	2	1	no	—
178	C₂²×C₄×C₆₇	—	2	1	no	—
179	C₂²×C₄×C₆₇	—	2	1	no	—
180	C₂²×C₄×C₆₇	—	2	1	no	—
181	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
182	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
183	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
184	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
185	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
186	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
187	Q₁₆×C₆₇	—	4	4	no	—
188	Q₁₆×C₆₇	—	4	4	no	—
189	SD₁₆×C₆₇	—	4	4	no	—
190	SD₁₆×C₆₇	—	4	4	no	—
191	M₁₆×C₆₇	—	2	8	no	—
192	C₂×C₈×C₆₇	—	2	1	no	—
193	C₂×C₈×C₆₇	—	2	1	no	—
194	C₂×C₈×C₆₇	—	2	1	no	—
195	C₂×C₈×C₆₇	—	2	1	no	—
196	C₂×C₈×C₆₇	—	2	8	no	—
197	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
198	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
199	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
200	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
201	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
202	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
203	C₂×C₈×C₆₇	#41	1	2	yes	—
204	C₂×C₈×C₆₇	#41	1	2	yes	—
205	C₄⋊C₄×C₆₇	—	2	4	no	—
206	C₄⋊C₄×C₆₇	—	2	1	no	—
207	C₄⋊C₄×C₆₇	—	2	1	no	—
208	C₄⋊C₄×C₆₇	—	2	1	no	—
209	C₄⋊C₄×C₆₇	—	2	1	no	—
210	C₄⋊C₄×C₆₇	—	2	1	no	—
211	C₄⋊C₄×C₆₇	—	2	1	no	—
212	C₄⋊C₄×C₆₇	—	2	1	no	—
213	C₄⋊C₄×C₆₇	—	2	1	no	—
214	C₄⋊C₄×C₆₇	—	4	4	no	—
215	C₄⋊C₄×C₆₇	—	4	2	no	—
216	C₄⋊C₄×C₆₇	—	4	2	no	—
217	C₄⋊C₄×C₆₇	—	4	2	no	—
218	C₄⋊C₄×C₆₇	—	8	2	no	—
219	C₂²⋊C₄×C₆₇	—	2	4	no	—
220	C₂²⋊C₄×C₆₇	—	4	2	no	—
221	C₂²⋊C₄×C₆₇	—	4	2	no	—
222	C₂²⋊C₄×C₆₇	—	4	2	no	—
223	C₂²⋊C₄×C₆₇	—	4	4	no	—
224	C₂²⋊C₄×C₆₇	—	4	2	no	—
225	C₂²⋊C₄×C₆₇	—	8	2	no	—
226	C₄²×C₆₇	—	2	1	no	—
227	C₄²×C₆₇	—	2	1	no	—
228	C₄²×C₆₇	C₂×D₄	1	2	yes	—
229	C₄²×C₆₇	#41	1	2	yes	—
230	C₂³×C₆₇	—	2	1	no	—
231	C₂³×C₆₇	—	2	1	no	—
232	C₂³×C₆₇	—	2	2	no	—
233	C₂³×C₆₇	—	2	2	no	—
234	C₂³×C₆₇	—	4	1	no	—
235	C₂³×C₆₇	—	4	1	no	—
236	C₂³×C₆₇	—	4	1	no	—
237	C₂³×C₆₇	—	4	2	no	—
238	C₂³×C₆₇	—	4	2	no	—
239	C₂³×C₆₇	#127	1	2	yes	—
240	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
241	Q₈×C₆₇	—	4	1	no	—
242	Q₈×C₆₇	—	4	1	no	—
243	Q₈×C₆₇	—	4	1	no	—
244	Q₈×C₆₇	—	4	1	no	—
245	Q₈×C₆₇	—	4	1	no	—
246	Q₈×C₆₇	—	4	1	no	—
247	Q₈×C₆₇	—	4	1	no	—
248	Q₈×C₆₇	—	4	1	no	—
249	Q₈×C₆₇	—	4	1	no	—
250	Q₈×C₆₇	—	4	1	no	—
251	Q₈×C₆₇	—	4	1	no	—
252	Q₈×C₆₇	—	4	1	no	—
253	D₄×C₆₇	—	4	1	no	—
254	D₄×C₆₇	—	4	1	no	—
255	D₄×C₆₇	—	4	1	no	—
256	D₄×C₆₇	—	4	1	no	—
257	D₄×C₆₇	—	8	4	no	—
258	C₂×C₄×C₆₇	—	2	1	no	—
259	C₂×C₄×C₆₇	—	2	1	no	—
260	C₂×C₄×C₆₇	—	2	2	no	—
261	C₂×C₄×C₆₇	—	2	2	no	—
262	C₂×C₄×C₆₇	—	2	2	no	—
263	C₂×C₄×C₆₇	—	2	2	no	—
264	C₂×C₄×C₆₇	—	2	2	no	—
265	C₂×C₄×C₆₇	—	2	2	no	—
266	C₂×C₄×C₆₇	—	2	2	no	—
267	C₂×C₄×C₆₇	—	2	2	no	—
268	C₂×C₄×C₆₇	—	2	2	no	—
269	C₂×C₄×C₆₇	—	4	1	no	—
270	C₂×C₄×C₆₇	—	4	1	no	—
271	C₂×C₄×C₆₇	—	4	1	no	—
272	C₂×C₄×C₆₇	—	4	1	no	—
273	C₂×C₄×C₆₇	—	4	1	no	—
274	C₂×C₄×C₆₇	—	4	1	no	—
275	C₂×C₄×C₆₇	—	4	1	no	—
276	C₂×C₄×C₆₇	—	4	1	no	—
277	C₂×C₄×C₆₇	—	4	1	no	—
278	C₂×C₄×C₆₇	—	4	1	no	—
279	C₂×C₄×C₆₇	—	4	1	no	—
280	C₂×C₄×C₆₇	—	4	1	no	—
281	C₂×C₄×C₆₇	—	4	1	no	—
282	C₂×C₄×C₆₇	—	4	2	no	—
283	C₂×C₄×C₆₇	—	4	2	no	—
284	C₂×C₄×C₆₇	—	4	2	no	—
285	C₂×C₄×C₆₇	—	4	2	no	—
286	C₂×C₄×C₆₇	#121	1	4	yes	—
287	C₂×C₄×C₆₇	#127	1	2	yes	—
288	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
289	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
290	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
291	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
292	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
293	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
294	C₈×C₆₇	—	2	2	no	—
295	C₈×C₆₇	—	2	2	no	—
296	C₈×C₆₇	#121	1	4	yes	—
297	C₈×C₆₇	#121	1	4	yes	—
298	C₂²×C₆₇	—	2	2	no	—
299	C₂²×C₆₇	—	2	2	no	—
300	C₂²×C₆₇	—	2	2	no	—
301	C₂²×C₆₇	—	4	2	no	—
302	C₂²×C₆₇	—	4	2	no	—
303	C₂²×C₆₇	—	8	1	no	—
304	C₂²×C₆₇	—	8	1	no	—
305	C₂²×C₆₇	—	8	1	no	—
306	C₂²×C₆₇	—	8	2	no	—
307	C₂²×C₆₇	—	8	2	no	—
308	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
309	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
310	C₂²×C₆₇	#532	1	1	yes	characteristic
311	C₂²×C₆₇	#577	1	2	yes	—
312	C₂²×C₆₇	#578	1	2	yes	—
313	C₄×C₆₇	—	2	2	no	—
314	C₄×C₆₇	—	2	4	no	—
315	C₄×C₆₇	—	4	2	no	—
316	C₄×C₆₇	—	8	1	no	—
317	C₄×C₆₇	—	8	1	no	—
318	C₄×C₆₇	—	8	1	no	—
319	C₄×C₆₇	—	8	1	no	—
320	C₄×C₆₇	—	8	1	no	—
321	C₄×C₆₇	—	8	1	no	—
322	C₄×C₆₇	—	8	1	no	—
323	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
324	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
325	C₄×C₆₇	#532	1	1	yes	characteristic
326	C₄×C₆₇	#532	1	1	yes	characteristic
327	#33188	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
328	C₂×C₆₇	—	4	2	no	—
329	C₂×C₆₇	—	16	1	no	—
330	C₂×C₆₇	#2856	1	1	yes	characteristic
331	C₂×C₆₇	#3063	1	2	yes	—
332	C₂×C₆₇	#3154	1	1	yes	characteristic
333	C₂×C₆₇	#3156	1	1	yes	characteristic
334	C₂×C₆₇	#3178	1	2	yes	—
335	C₂×#497	C₂×C₆₇	1	1	yes	characteristic, complemented
336	C₂×#470	C₂×C₆₇	1	1	yes	characteristic, complemented
337	C₂×#468	C₂×C₆₇	1	1	yes	characteristic
338	C₂×#467	C₂×C₆₇	1	1	yes	characteristic, complemented
339	C₂×#465	C₂×C₆₇	1	1	yes	characteristic
340	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
341	#2621	C₂×C₆₇	1	1	yes	characteristic, complemented
342	#2620	C₂×C₆₇	1	1	yes	characteristic
343	#1973	C₂×C₆₇	1	1	yes	characteristic, complemented
344	#1971	C₂×C₆₇	1	1	yes	characteristic, complemented
345	#1969	C₂×C₆₇	1	1	yes	characteristic, complemented
346	#1968	C₂×C₆₇	1	1	yes	characteristic, complemented
347	#1958	C₂×C₆₇	1	1	yes	characteristic, complemented
348	#1956	C₂×C₆₇	1	1	yes	characteristic, complemented
349	#1847	C₂×C₆₇	1	1	yes	characteristic, complemented
350	C₆₇	#33188	1	1	yes	characteristic, direct product factor, Sylow
351	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
352	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
353	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
354	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
355	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
356	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
357	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
358	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
359	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
360	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
361	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
362	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
363	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
364	#497	—	2	4	no	—
365	#470	—	2	4	no	—
366	#468	—	2	4	no	—
367	#467	—	2	4	no	—
368	#465	—	2	4	no	—
369	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
370	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
371	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
372	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
373	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
374	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
375	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
376	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
377	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
378	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
379	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
380	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
381	#405	—	2	4	no	—
382	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
383	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
384	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
385	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
386	#398	—	2	2	no	—
387	#397	—	2	2	no	—
388	#396	—	2	2	no	—
389	#394	—	2	2	no	—
390	#391	C₂²×C₆₇	1	1	yes	characteristic
391	#391	C₂²×C₆₇	1	1	yes	characteristic
392	#391	C₂²×C₆₇	1	1	yes	characteristic
393	#390	C₂²×C₆₇	1	1	yes	characteristic
394	#384	C₂²×C₆₇	1	1	yes	characteristic
395	#382	C₂²×C₆₇	1	1	yes	characteristic
396	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
397	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
398	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
399	C₂²×D₄	—	2	1	no	—
400	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
401	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
402	C₂×Q₁₆	—	2	2	no	—
403	C₂×Q₁₆	—	2	2	no	—
404	C₂×Q₁₆	—	2	2	no	—
405	C₂×Q₁₆	—	2	2	no	—
406	C₂×Q₁₆	—	2	2	no	—
407	C₂×Q₁₆	—	2	2	no	—
408	C₂×SD₁₆	—	2	2	no	—
409	C₂×SD₁₆	—	2	2	no	—
410	C₂×SD₁₆	—	2	2	no	—
411	C₂×SD₁₆	—	2	2	no	—
412	C₂×SD₁₆	—	2	2	no	—
413	C₂×SD₁₆	—	2	2	no	—
414	C₂×M₁₆	—	2	2	no	—
415	C₂×M₁₆	—	2	2	no	—
416	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
417	C₂²×C₈	—	2	2	no	—
418	C₂²×C₈	—	2	2	no	—
419	C₂²×C₈	D₄×C₆₇	1	4	yes	—
420	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
421	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
422	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
423	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
424	C₄⋊Q₈	—	2	2	no	—
425	C₄⋊Q₈	—	2	2	no	—
426	#124	—	2	2	no	—
427	#124	—	2	2	no	—
428	C₂²⋊Q₈	—	2	2	no	—
429	C₂²⋊Q₈	—	2	2	no	—
430	C₂²⋊Q₈	—	2	2	no	—
431	C₂²⋊Q₈	—	2	2	no	—
432	C₂²⋊Q₈	—	2	2	no	—
433	C₂²⋊Q₈	—	2	2	no	—
434	#121	—	2	2	no	—
435	#121	—	2	2	no	—
436	#117	—	2	2	no	—
437	#117	—	2	2	no	—
438	C₂×C₄⋊C₄	—	2	1	no	—
439	C₂×C₄⋊C₄	—	2	1	no	—
440	C₂×C₄⋊C₄	—	2	1	no	—
441	C₂×C₄⋊C₄	—	2	2	no	—
442	C₂×C₄⋊C₄	—	2	2	no	—
443	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
444	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
445	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
446	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
447	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
448	C₂×C₂²⋊C₄	—	2	1	no	—
449	C₂×C₂²⋊C₄	—	2	1	no	—
450	C₂×C₂²⋊C₄	—	2	1	no	—
451	C₂×C₂²⋊C₄	—	2	1	no	—
452	C₂×C₂²⋊C₄	—	2	2	no	—
453	C₂×C₂²⋊C₄	—	2	2	no	—
454	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
455	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
456	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
457	#107	—	2	2	no	—
458	#107	—	2	2	no	—
459	#106	—	2	2	no	—
460	#106	—	2	2	no	—
461	#103	—	2	2	no	—
462	#103	—	2	2	no	—
463	#103	—	2	2	no	—
464	#103	—	2	2	no	—
465	#103	—	2	2	no	—
466	#103	—	2	2	no	—
467	#102	—	2	2	no	—
468	#102	—	2	2	no	—
469	#97	—	2	2	no	—
470	#97	D₄×C₆₇	1	4	yes	complemented
471	C₄×C₈	—	2	2	no	—
472	C₄×C₈	—	2	2	no	—
473	#95	—	2	2	no	—
474	#95	—	2	2	no	—
475	#95	—	2	2	no	—
476	#95	—	2	2	no	—
477	C₂⁴	—	2	1	no	—
478	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
479	C₂×Q₈	—	2	1	no	—
480	C₂×Q₈	—	2	1	no	—
481	C₂×Q₈	—	2	1	no	—
482	C₂×Q₈	—	2	1	no	—
483	C₂×Q₈	—	2	1	no	—
484	C₂×Q₈	—	2	1	no	—
485	C₂×Q₈	—	2	1	no	—
486	C₂×Q₈	—	2	1	no	—
487	C₂×Q₈	—	2	1	no	—
488	C₂×Q₈	—	2	1	no	—
489	C₂×Q₈	—	2	1	no	—
490	C₂×Q₈	—	2	1	no	—
491	C₂×Q₈	—	2	1	no	—
492	C₂×Q₈	—	2	1	no	—
493	C₂×Q₈	—	2	1	no	—
494	C₂×Q₈	—	2	1	no	—
495	C₂×Q₈	—	2	1	no	—
496	C₂×Q₈	—	2	1	no	—
497	C₂×D₄	—	2	1	no	—
498	C₂×D₄	—	2	1	no	—
499	C₂×D₄	—	2	1	no	—
500	C₂×D₄	—	2	1	no	—
501	C₂×D₄	—	2	1	no	—
502	C₂×D₄	—	2	1	no	—
503	C₂×D₄	—	4	2	no	—
504	C₂×D₄	—	8	2	no	—
505	C₂²×C₄	—	2	1	no	—
506	C₂²×C₄	—	2	1	no	—
507	C₂²×C₄	—	2	1	no	—
508	C₂²×C₄	—	2	1	no	—
509	C₂²×C₄	—	2	1	no	—
510	C₂²×C₄	—	2	1	no	—
511	C₂²×C₄	—	2	1	no	—
512	C₂²×C₄	—	2	1	no	—
513	C₂²×C₄	—	2	1	no	—
514	C₂²×C₄	—	2	1	no	—
515	C₂²×C₄	—	2	1	no	—
516	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
517	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
518	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
519	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
520	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
521	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
522	Q₁₆	—	4	4	no	—
523	Q₁₆	—	4	4	no	—
524	SD₁₆	—	4	4	no	—
525	SD₁₆	—	4	4	no	—
526	M₁₆	—	2	8	no	—
527	C₂×C₈	—	2	1	no	—
528	C₂×C₈	—	2	1	no	—
529	C₂×C₈	—	2	1	no	—
530	C₂×C₈	—	2	1	no	—
531	C₂×C₈	—	2	8	no	—
532	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
533	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
534	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
535	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
536	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
537	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
538	C₂×C₈	#41×C₆₇	1	2	yes	—
539	C₂×C₈	#41×C₆₇	1	2	yes	—
540	C₄⋊C₄	—	2	4	no	—
541	C₄⋊C₄	—	2	1	no	—
542	C₄⋊C₄	—	2	1	no	—
543	C₄⋊C₄	—	2	1	no	—
544	C₄⋊C₄	—	2	1	no	—
545	C₄⋊C₄	—	2	1	no	—
546	C₄⋊C₄	—	2	1	no	—
547	C₄⋊C₄	—	2	1	no	—
548	C₄⋊C₄	—	2	1	no	—
549	C₄⋊C₄	—	4	4	no	—
550	C₄⋊C₄	—	4	2	no	—
551	C₄⋊C₄	—	4	2	no	—
552	C₄⋊C₄	—	4	2	no	—
553	C₄⋊C₄	—	8	2	no	—
554	C₂²⋊C₄	—	2	4	no	—
555	C₂²⋊C₄	—	4	2	no	—
556	C₂²⋊C₄	—	4	2	no	—
557	C₂²⋊C₄	—	4	2	no	—
558	C₂²⋊C₄	—	4	4	no	—
559	C₂²⋊C₄	—	4	2	no	—
560	C₂²⋊C₄	—	8	2	no	—
561	C₄²	—	2	1	no	—
562	C₄²	—	2	1	no	—
563	C₄²	C₂×D₄×C₆₇	1	2	yes	—
564	C₄²	#41×C₆₇	1	2	yes	—
565	C₂³	—	2	1	no	—
566	C₂³	—	2	1	no	—
567	C₂³	—	2	2	no	—
568	C₂³	—	2	2	no	—
569	C₂³	—	4	1	no	—
570	C₂³	—	4	1	no	—
571	C₂³	—	4	1	no	—
572	C₂³	—	4	2	no	—
573	C₂³	—	4	2	no	—
574	C₂³	#127×C₆₇	1	2	yes	—
575	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
576	Q₈	—	4	1	no	—
577	Q₈	—	4	1	no	—
578	Q₈	—	4	1	no	—
579	Q₈	—	4	1	no	—
580	Q₈	—	4	1	no	—
581	Q₈	—	4	1	no	—
582	Q₈	—	4	1	no	—
583	Q₈	—	4	1	no	—
584	Q₈	—	4	1	no	—
585	Q₈	—	4	1	no	—
586	Q₈	—	4	1	no	—
587	Q₈	—	4	1	no	—
588	D₄	—	4	1	no	—
589	D₄	—	4	1	no	—
590	D₄	—	4	1	no	—
591	D₄	—	4	1	no	—
592	D₄	—	8	4	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	2	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	4	1	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	4	1	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	4	1	no	—
611	C₂×C₄	—	4	1	no	—
612	C₂×C₄	—	4	1	no	—
613	C₂×C₄	—	4	1	no	—
614	C₂×C₄	—	4	1	no	—
615	C₂×C₄	—	4	1	no	—
616	C₂×C₄	—	4	1	no	—
617	C₂×C₄	—	4	2	no	—
618	C₂×C₄	—	4	2	no	—
619	C₂×C₄	—	4	2	no	—
620	C₂×C₄	—	4	2	no	—
621	C₂×C₄	#121×C₆₇	1	4	yes	—
622	C₂×C₄	#127×C₆₇	1	2	yes	—
623	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
624	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
625	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
626	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
627	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
628	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
629	C₈	—	2	2	no	—
630	C₈	—	2	2	no	—
631	C₈	#121×C₆₇	1	4	yes	—
632	C₈	#121×C₆₇	1	4	yes	—
633	C₂²	—	2	2	no	—
634	C₂²	—	2	2	no	—
635	C₂²	—	2	2	no	—
636	C₂²	—	4	2	no	—
637	C₂²	—	4	2	no	—
638	C₂²	—	8	2	no	—
639	C₂²	—	8	1	no	—
640	C₂²	—	8	1	no	—
641	C₂²	—	8	1	no	—
642	C₂²	—	8	2	no	—
643	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
644	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
645	C₂²	#532×C₆₇	1	1	yes	characteristic
646	C₂²	#577×C₆₇	1	2	yes	—
647	C₂²	#578×C₆₇	1	2	yes	—
648	C₄	—	2	4	no	—
649	C₄	—	2	2	no	—
650	C₄	—	4	2	no	—
651	C₄	—	8	1	no	—
652	C₄	—	8	1	no	—
653	C₄	—	8	1	no	—
654	C₄	—	8	1	no	—
655	C₄	—	8	1	no	—
656	C₄	—	8	1	no	—
657	C₄	—	8	1	no	—
658	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
659	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
660	C₄	#532×C₆₇	1	1	yes	characteristic
661	C₄	#532×C₆₇	1	1	yes	characteristic
662	C₂	—	4	2	no	—
663	C₂	—	16	1	no	—
664	C₂	C₆₇×#2856	1	1	yes	characteristic
665	C₂	C₆₇×#3063	1	2	yes	—
666	C₂	C₆₇×#3154	1	1	yes	characteristic
667	C₂	C₆₇×#3156	1	1	yes	characteristic
668	C₂	C₆₇×#3178	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#121×C₆₇	C₂²
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×#497×C₆₇	C₂	semidirect
2. C₂×#470×C₆₇	C₂	semidirect
4. C₂×#467×C₆₇	C₂	semidirect
6. C₂×#405×C₆₇	C₂	semidirect
7. C₆₇×#2621	C₂	semidirect
9. C₆₇×#1973	C₂	semidirect
10. C₆₇×#1971	C₂	semidirect
11. C₆₇×#1969	C₂	semidirect
12. C₆₇×#1968	C₂	semidirect
13. C₆₇×#1958	C₂	semidirect
14. C₆₇×#1956	C₂	semidirect
15. C₆₇×#1847	C₂	semidirect
135. #97×C₆₇	D₄	semidirect
327. #33188	C₆₇	direct (N × H)
335. C₂×#497	C₂×C₆₇	semidirect
336. C₂×#470	C₂×C₆₇	semidirect
338. C₂×#467	C₂×C₆₇	semidirect
340. C₂×#405	C₂×C₆₇	semidirect
341. #2621	C₂×C₆₇	semidirect
343. #1973	C₂×C₆₇	semidirect
344. #1971	C₂×C₆₇	semidirect
345. #1969	C₂×C₆₇	semidirect
346. #1968	C₂×C₆₇	semidirect
347. #1958	C₂×C₆₇	semidirect
348. #1956	C₂×C₆₇	semidirect
349. #1847	C₂×C₆₇	semidirect
350. C₆₇	#33188	direct (N × H)
470. #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×#497×C₆₇	2
C₂×#470×C₆₇	2
C₂×#468×C₆₇	2
C₂×#467×C₆₇	2
C₂×#465×C₆₇	2
C₂×#405×C₆₇	2
C₆₇×#2621	2
C₆₇×#2620	2
C₆₇×#1973	2
C₆₇×#1971	2
C₆₇×#1969	2
C₆₇×#1968	2
C₆₇×#1958	2
C₆₇×#1956	2
C₆₇×#1847	2
#33188	67

Subgroups — C67×#33188 #30712439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33188 #30712439