Subgroups — C₆₇×#33183 #30712434

Order 17152.

1268 conjugacy classes of subgroups.

The subgroup lattice has 510 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×#466×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#465×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2856	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2850	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1973	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1972	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1961	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1959	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1847	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
13	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
14	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#121×C₆₇	—	2	2	no	—
19	C₂×#121×C₆₇	C₂²	1	2	yes	—
20	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
22	#496×C₆₇	—	2	4	no	—
23	#466×C₆₇	—	2	8	no	—
24	#465×C₆₇	—	2	8	no	—
25	C₂×#107×C₆₇	C₂²	1	2	yes	—
26	C₂×#106×C₆₇	C₂²	1	2	yes	—
27	C₂×#103×C₆₇	C₂²	1	2	yes	—
28	C₂×#102×C₆₇	C₂²	1	2	yes	—
29	C₂×#102×C₆₇	C₂²	1	2	yes	—
30	C₂×#102×C₆₇	C₂²	1	2	yes	—
31	C₂×#97×C₆₇	—	2	4	no	—
32	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×#97×C₆₇	C₂²	1	2	yes	—
34	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
35	#391×C₆₇	C₂²	1	1	yes	characteristic
36	#390×C₆₇	—	2	2	no	—
37	#390×C₆₇	C₂²	1	1	yes	characteristic
38	#390×C₆₇	C₂²	1	2	yes	—
39	#388×C₆₇	—	2	2	no	—
40	#386×C₆₇	—	2	2	no	—
41	#384×C₆₇	C₂²	1	1	yes	characteristic
42	#382×C₆₇	C₂²	1	1	yes	characteristic
43	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₆₇	—	2	2	no	—
45	C₂²×D₄×C₆₇	—	2	1	no	—
46	C₂²×D₄×C₆₇	—	2	2	no	—
47	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
48	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
49	C₂³×C₄×C₆₇	D₄	1	2	yes	—
50	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₂×SD₁₆×C₆₇	—	2	4	no	—
52	C₂×SD₁₆×C₆₇	—	2	4	no	—
53	C₂×SD₁₆×C₆₇	—	2	4	no	—
54	C₂×D₈×C₆₇	—	2	4	no	—
55	C₂×D₈×C₆₇	—	2	4	no	—
56	C₂×D₈×C₆₇	—	2	4	no	—
57	C₂²×C₈×C₆₇	—	2	4	no	—
58	C₂²×C₈×C₆₇	—	2	4	no	—
59	C₂²×C₈×C₆₇	D₄	1	8	yes	complemented
60	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
61	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
62	#127×C₆₇	—	2	2	no	—
63	#127×C₆₇	—	2	2	no	—
64	#124×C₆₇	—	2	2	no	—
65	#124×C₆₇	—	2	2	no	—
66	C₂²⋊Q₈×C₆₇	—	2	4	no	—
67	#121×C₆₇	—	2	4	no	—
68	#121×C₆₇	—	2	4	no	—
69	#121×C₆₇	—	2	4	no	—
70	#121×C₆₇	—	4	8	no	—
71	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
72	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
73	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
76	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
77	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
78	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
79	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
80	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
81	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
82	C₂×C₄²×C₆₇	—	2	2	no	—
83	C₂×C₄²×C₆₇	—	2	2	no	—
84	C₂×C₄²×C₆₇	D₄	1	2	yes	—
85	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
87	#107×C₆₇	—	2	4	no	—
88	#106×C₆₇	—	2	4	no	—
89	#103×C₆₇	—	2	4	no	—
90	#102×C₆₇	—	2	4	no	—
91	#102×C₆₇	—	2	4	no	—
92	#102×C₆₇	—	2	4	no	—
93	#97×C₆₇	—	2	4	no	—
94	#97×C₆₇	—	2	2	no	—
95	#97×C₆₇	D₄	1	4	yes	complemented
96	#95×C₆₇	—	2	2	no	—
97	#95×C₆₇	—	2	2	no	—
98	C₂⁴×C₆₇	—	2	1	no	—
99	C₂⁴×C₆₇	—	2	2	no	—
100	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂×Q₈×C₆₇	—	2	2	no	—
102	C₂×Q₈×C₆₇	—	2	1	no	—
103	C₂×Q₈×C₆₇	—	2	1	no	—
104	C₂×Q₈×C₆₇	—	2	2	no	—
105	C₂×D₄×C₆₇	—	2	2	no	—
106	C₂×D₄×C₆₇	—	2	2	no	—
107	C₂×D₄×C₆₇	—	2	2	no	—
108	C₂×D₄×C₆₇	—	2	1	no	—
109	C₂×D₄×C₆₇	—	2	2	no	—
110	C₂×D₄×C₆₇	—	2	1	no	—
111	C₂×D₄×C₆₇	—	2	2	no	—
112	C₂×D₄×C₆₇	—	2	2	no	—
113	C₂×D₄×C₆₇	—	2	2	no	—
114	C₂×D₄×C₆₇	—	2	2	no	—
115	C₂×D₄×C₆₇	—	4	4	no	—
116	C₂×D₄×C₆₇	—	4	4	no	—
117	C₂×D₄×C₆₇	—	4	4	no	—
118	C₂×D₄×C₆₇	—	4	2	no	—
119	C₂×D₄×C₆₇	—	4	8	no	—
120	C₂×D₄×C₆₇	—	4	4	no	—
121	C₂×D₄×C₆₇	—	4	4	no	—
122	C₂²×C₄×C₆₇	—	2	1	no	—
123	C₂²×C₄×C₆₇	—	2	1	no	—
124	C₂²×C₄×C₆₇	—	2	2	no	—
125	C₂²×C₄×C₆₇	—	2	4	no	—
126	C₂²×C₄×C₆₇	—	2	4	no	—
127	C₂²×C₄×C₆₇	—	2	2	no	—
128	C₂²×C₄×C₆₇	—	2	1	no	—
129	C₂²×C₄×C₆₇	—	2	1	no	—
130	C₂²×C₄×C₆₇	—	2	1	no	—
131	C₂²×C₄×C₆₇	—	4	2	no	—
132	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
136	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
137	SD₁₆×C₆₇	—	4	8	no	—
138	D₈×C₆₇	—	4	8	no	—
139	C₂×C₈×C₆₇	—	2	2	no	—
140	C₂×C₈×C₆₇	—	2	2	no	—
141	C₂×C₈×C₆₇	—	2	16	no	—
142	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
143	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
144	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
145	C₂×C₈×C₆₇	#41	1	4	yes	—
146	C₄⋊C₄×C₆₇	—	2	2	no	—
147	C₄⋊C₄×C₆₇	—	2	2	no	—
148	C₄⋊C₄×C₆₇	—	2	2	no	—
149	C₄⋊C₄×C₆₇	—	2	2	no	—
150	C₄⋊C₄×C₆₇	—	4	4	no	—
151	C₄⋊C₄×C₆₇	—	4	2	no	—
152	C₂²⋊C₄×C₆₇	—	4	4	no	—
153	C₂²⋊C₄×C₆₇	—	4	8	no	—
154	C₂²⋊C₄×C₆₇	—	4	2	no	—
155	C₂²⋊C₄×C₆₇	—	4	4	no	—
156	C₂²⋊C₄×C₆₇	—	4	2	no	—
157	C₂²⋊C₄×C₆₇	—	4	4	no	—
158	C₄²×C₆₇	—	2	4	no	—
159	C₄²×C₆₇	—	2	1	no	—
160	C₄²×C₆₇	—	2	1	no	—
161	C₄²×C₆₇	C₂×D₄	1	2	yes	—
162	C₄²×C₆₇	#41	1	2	yes	—
163	C₂³×C₆₇	—	2	1	no	—
164	C₂³×C₆₇	—	2	1	no	—
165	C₂³×C₆₇	—	2	2	no	—
166	C₂³×C₆₇	—	4	1	no	—
167	C₂³×C₆₇	—	4	2	no	—
168	C₂³×C₆₇	—	4	1	no	—
169	C₂³×C₆₇	—	4	2	no	—
170	C₂³×C₆₇	—	4	1	no	—
171	C₂³×C₆₇	—	4	2	no	—
172	C₂³×C₆₇	—	4	4	no	—
173	C₂³×C₆₇	—	4	2	no	—
174	C₂³×C₆₇	—	4	2	no	—
175	C₂³×C₆₇	—	4	4	no	—
176	C₂³×C₆₇	#127	1	2	yes	—
177	C₂³×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
178	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
179	Q₈×C₆₇	—	4	2	no	—
180	Q₈×C₆₇	—	4	2	no	—
181	D₄×C₆₇	—	4	2	no	—
182	D₄×C₆₇	—	4	2	no	—
183	D₄×C₆₇	—	4	2	no	—
184	D₄×C₆₇	—	4	2	no	—
185	D₄×C₆₇	—	4	2	no	—
186	D₄×C₆₇	—	4	2	no	—
187	D₄×C₆₇	—	8	8	no	—
188	D₄×C₆₇	—	8	8	no	—
189	D₄×C₆₇	—	8	4	no	—
190	C₂×C₄×C₆₇	—	2	1	no	—
191	C₂×C₄×C₆₇	—	2	1	no	—
192	C₂×C₄×C₆₇	—	2	2	no	—
193	C₂×C₄×C₆₇	—	2	2	no	—
194	C₂×C₄×C₆₇	—	2	4	no	—
195	C₂×C₄×C₆₇	—	2	4	no	—
196	C₂×C₄×C₆₇	—	2	2	no	—
197	C₂×C₄×C₆₇	—	2	2	no	—
198	C₂×C₄×C₆₇	—	2	2	no	—
199	C₂×C₄×C₆₇	—	4	8	no	—
200	C₂×C₄×C₆₇	—	4	1	no	—
201	C₂×C₄×C₆₇	—	4	1	no	—
202	C₂×C₄×C₆₇	—	4	1	no	—
203	C₂×C₄×C₆₇	—	4	2	no	—
204	C₂×C₄×C₆₇	—	4	1	no	—
205	C₂×C₄×C₆₇	—	4	1	no	—
206	C₂×C₄×C₆₇	—	4	4	no	—
207	C₂×C₄×C₆₇	—	4	2	no	—
208	C₂×C₄×C₆₇	—	4	2	no	—
209	C₂×C₄×C₆₇	#121	1	4	yes	—
210	C₂×C₄×C₆₇	#127	1	2	yes	—
211	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
213	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
214	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
215	C₈×C₆₇	—	2	4	no	—
216	C₈×C₆₇	#121	1	8	yes	complemented
217	C₂²×C₆₇	—	2	2	no	—
218	C₂²×C₆₇	—	2	2	no	—
219	C₂²×C₆₇	—	2	4	no	—
220	C₂²×C₆₇	—	2	4	no	—
221	C₂²×C₆₇	—	8	1	no	—
222	C₂²×C₆₇	—	8	2	no	—
223	C₂²×C₆₇	—	8	2	no	—
224	C₂²×C₆₇	—	8	1	no	—
225	C₂²×C₆₇	—	8	1	no	—
226	C₂²×C₆₇	—	8	2	no	—
227	C₂²×C₆₇	—	8	4	no	—
228	C₂²×C₆₇	—	8	2	no	—
229	C₂²×C₆₇	—	8	4	no	—
230	C₂²×C₆₇	—	8	2	no	—
231	C₂²×C₆₇	#496	1	4	yes	complemented
232	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
233	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
234	C₂²×C₆₇	#532	1	1	yes	characteristic
235	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
236	C₂²×C₆₇	#575	1	2	yes	—
237	C₄×C₆₇	—	2	2	no	—
238	C₄×C₆₇	—	2	4	no	—
239	C₄×C₆₇	—	4	2	no	—
240	C₄×C₆₇	—	8	1	no	—
241	C₄×C₆₇	—	8	2	no	—
242	C₄×C₆₇	—	8	1	no	—
243	C₄×C₆₇	—	8	1	no	—
244	C₄×C₆₇	C₂×#121	1	2	yes	—
245	C₄×C₆₇	#532	1	2	yes	—
246	#33183	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
247	C₂×C₆₇	—	2	4	no	—
248	C₂×C₆₇	—	16	2	no	—
249	C₂×C₆₇	—	16	1	no	—
250	C₂×C₆₇	#2856	1	1	yes	characteristic
251	C₂×C₆₇	#3055	1	2	yes	—
252	C₂×C₆₇	C₂×#496	1	1	yes	characteristic
253	C₂×C₆₇	#3150	1	1	yes	characteristic
254	C₂×C₆₇	#3180	1	2	yes	—
255	C₂×#496	C₂×C₆₇	1	1	yes	characteristic, complemented
256	C₂×#466	C₂×C₆₇	1	2	yes	complemented
257	C₂×#465	C₂×C₆₇	1	2	yes	complemented
258	C₂²×#97	C₂×C₆₇	1	1	yes	characteristic, complemented
259	#2856	C₂×C₆₇	1	1	yes	characteristic, complemented
260	#2850	C₂×C₆₇	1	1	yes	characteristic, complemented
261	#1973	C₂×C₆₇	1	2	yes	complemented
262	#1972	C₂×C₆₇	1	2	yes	complemented
263	#1961	C₂×C₆₇	1	1	yes	characteristic, complemented
264	#1959	C₂×C₆₇	1	1	yes	characteristic, complemented
265	#1847	C₂×C₆₇	1	1	yes	characteristic, complemented
266	C₆₇	#33183	1	1	yes	characteristic, direct product factor, Sylow
267	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
268	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
269	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
270	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
271	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
272	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
273	C₂×#121	—	2	2	no	—
274	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
275	C₂×#121	C₂²×C₆₇	1	2	yes	—
276	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
277	#496	—	2	4	no	—
278	#466	—	2	8	no	—
279	#465	—	2	8	no	—
280	C₂×#107	C₂²×C₆₇	1	2	yes	—
281	C₂×#106	C₂²×C₆₇	1	2	yes	—
282	C₂×#103	C₂²×C₆₇	1	2	yes	—
283	C₂×#102	C₂²×C₆₇	1	2	yes	—
284	C₂×#102	C₂²×C₆₇	1	2	yes	—
285	C₂×#102	C₂²×C₆₇	1	2	yes	—
286	C₂×#97	—	2	4	no	—
287	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
288	C₂×#97	C₂²×C₆₇	1	2	yes	—
289	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
290	#391	C₂²×C₆₇	1	1	yes	characteristic
291	#390	—	2	2	no	—
292	#390	C₂²×C₆₇	1	1	yes	characteristic
293	#390	C₂²×C₆₇	1	2	yes	—
294	#388	—	2	2	no	—
295	#386	—	2	2	no	—
296	#384	C₂²×C₆₇	1	1	yes	characteristic
297	#382	C₂²×C₆₇	1	1	yes	characteristic
298	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
299	C₂²×D₄	—	2	2	no	—
300	C₂²×D₄	—	2	1	no	—
301	C₂²×D₄	—	2	2	no	—
302	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
303	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
304	C₂³×C₄	D₄×C₆₇	1	2	yes	—
305	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
306	C₂×SD₁₆	—	2	4	no	—
307	C₂×SD₁₆	—	2	4	no	—
308	C₂×SD₁₆	—	2	4	no	—
309	C₂×D₈	—	2	4	no	—
310	C₂×D₈	—	2	4	no	—
311	C₂×D₈	—	2	4	no	—
312	C₂²×C₈	—	2	4	no	—
313	C₂²×C₈	—	2	4	no	—
314	C₂²×C₈	D₄×C₆₇	1	8	yes	complemented
315	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
316	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
317	#127	—	2	2	no	—
318	#127	—	2	2	no	—
319	#124	—	2	2	no	—
320	#124	—	2	2	no	—
321	C₂²⋊Q₈	—	2	4	no	—
322	#121	—	2	4	no	—
323	#121	—	2	4	no	—
324	#121	—	2	4	no	—
325	#121	—	4	8	no	—
326	C₂×C₄⋊C₄	—	2	2	no	—
327	C₂×C₄⋊C₄	—	2	1	no	—
328	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
329	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
330	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
331	C₂×C₂²⋊C₄	—	2	1	no	—
332	C₂×C₂²⋊C₄	—	2	4	no	—
333	C₂×C₂²⋊C₄	—	2	1	no	—
334	C₂×C₂²⋊C₄	—	2	2	no	—
335	C₂×C₂²⋊C₄	—	2	2	no	—
336	C₂×C₂²⋊C₄	—	2	2	no	—
337	C₂×C₄²	—	2	2	no	—
338	C₂×C₄²	—	2	2	no	—
339	C₂×C₄²	D₄×C₆₇	1	2	yes	—
340	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
341	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
342	#107	—	2	4	no	—
343	#106	—	2	4	no	—
344	#103	—	2	4	no	—
345	#102	—	2	4	no	—
346	#102	—	2	4	no	—
347	#102	—	2	4	no	—
348	#97	—	2	4	no	—
349	#97	—	2	2	no	—
350	#97	D₄×C₆₇	1	4	yes	complemented
351	#95	—	2	2	no	—
352	#95	—	2	2	no	—
353	C₂⁴	—	2	1	no	—
354	C₂⁴	—	2	2	no	—
355	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
356	C₂×Q₈	—	2	2	no	—
357	C₂×Q₈	—	2	1	no	—
358	C₂×Q₈	—	2	1	no	—
359	C₂×Q₈	—	2	2	no	—
360	C₂×D₄	—	2	2	no	—
361	C₂×D₄	—	2	2	no	—
362	C₂×D₄	—	2	2	no	—
363	C₂×D₄	—	2	1	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	—	2	1	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	2	2	no	—
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	—	4	4	no	—
371	C₂×D₄	—	4	4	no	—
372	C₂×D₄	—	4	4	no	—
373	C₂×D₄	—	4	2	no	—
374	C₂×D₄	—	4	8	no	—
375	C₂×D₄	—	4	4	no	—
376	C₂×D₄	—	4	4	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	4	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	4	2	no	—
387	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
388	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
389	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
390	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
391	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
392	SD₁₆	—	4	8	no	—
393	D₈	—	4	8	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	16	no	—
397	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
398	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
399	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
400	C₂×C₈	#41×C₆₇	1	4	yes	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	2	2	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	4	4	no	—
406	C₄⋊C₄	—	4	2	no	—
407	C₂²⋊C₄	—	4	8	no	—
408	C₂²⋊C₄	—	4	4	no	—
409	C₂²⋊C₄	—	4	2	no	—
410	C₂²⋊C₄	—	4	4	no	—
411	C₂²⋊C₄	—	4	2	no	—
412	C₂²⋊C₄	—	4	4	no	—
413	C₄²	—	2	4	no	—
414	C₄²	—	2	1	no	—
415	C₄²	—	2	1	no	—
416	C₄²	C₂×D₄×C₆₇	1	2	yes	—
417	C₄²	#41×C₆₇	1	2	yes	—
418	C₂³	—	2	1	no	—
419	C₂³	—	2	1	no	—
420	C₂³	—	2	2	no	—
421	C₂³	—	4	1	no	—
422	C₂³	—	4	2	no	—
423	C₂³	—	4	1	no	—
424	C₂³	—	4	2	no	—
425	C₂³	—	4	1	no	—
426	C₂³	—	4	2	no	—
427	C₂³	—	4	4	no	—
428	C₂³	—	4	2	no	—
429	C₂³	—	4	2	no	—
430	C₂³	—	4	4	no	—
431	C₂³	#127×C₆₇	1	2	yes	—
432	C₂³	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
433	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
434	Q₈	—	4	2	no	—
435	Q₈	—	4	2	no	—
436	D₄	—	4	2	no	—
437	D₄	—	4	2	no	—
438	D₄	—	4	2	no	—
439	D₄	—	4	2	no	—
440	D₄	—	4	2	no	—
441	D₄	—	4	2	no	—
442	D₄	—	8	8	no	—
443	D₄	—	8	8	no	—
444	D₄	—	8	4	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	4	8	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	4	1	no	—
457	C₂×C₄	—	4	1	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	1	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	4	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	#121×C₆₇	1	4	yes	—
465	C₂×C₄	#127×C₆₇	1	2	yes	—
466	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
467	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
468	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
469	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
470	C₈	—	2	4	no	—
471	C₈	#121×C₆₇	1	8	yes	complemented
472	C₂²	—	2	4	no	—
473	C₂²	—	2	2	no	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	4	no	—
476	C₂²	—	8	4	no	—
477	C₂²	—	8	2	no	—
478	C₂²	—	8	1	no	—
479	C₂²	—	8	2	no	—
480	C₂²	—	8	2	no	—
481	C₂²	—	8	1	no	—
482	C₂²	—	8	1	no	—
483	C₂²	—	8	2	no	—
484	C₂²	—	8	4	no	—
485	C₂²	—	8	2	no	—
486	C₂²	#496×C₆₇	1	4	yes	complemented
487	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
488	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
489	C₂²	#532×C₆₇	1	1	yes	characteristic
490	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
491	C₂²	#575×C₆₇	1	2	yes	—
492	C₄	—	2	4	no	—
493	C₄	—	2	2	no	—
494	C₄	—	4	2	no	—
495	C₄	—	8	1	no	—
496	C₄	—	8	2	no	—
497	C₄	—	8	1	no	—
498	C₄	—	8	1	no	—
499	C₄	C₂×#121×C₆₇	1	2	yes	—
500	C₄	#532×C₆₇	1	2	yes	—
501	C₂	—	2	4	no	—
502	C₂	—	16	2	no	—
503	C₂	—	16	1	no	—
504	C₂	C₆₇×#2856	1	1	yes	characteristic
505	C₂	C₆₇×#3055	1	2	yes	—
506	C₂	C₂×#496×C₆₇	1	1	yes	characteristic
507	C₂	C₆₇×#3150	1	1	yes	characteristic
508	C₂	C₆₇×#3180	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 ◁ C₂²×C₄²×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×#466×C₆₇	C₂	semidirect
3. C₂×#465×C₆₇	C₂	semidirect
4. C₂²×#97×C₆₇	C₂	semidirect
5. C₆₇×#2856	C₂	semidirect
6. C₆₇×#2850	C₂	semidirect
7. C₆₇×#1973	C₂	semidirect
8. C₆₇×#1972	C₂	semidirect
9. C₆₇×#1961	C₂	semidirect
10. C₆₇×#1959	C₂	semidirect
11. C₆₇×#1847	C₂	semidirect
59. C₂²×C₈×C₆₇	D₄	semidirect
95. #97×C₆₇	D₄	semidirect
216. C₈×C₆₇	#121	semidirect
231. C₂²×C₆₇	#496	semidirect
246. #33183	C₆₇	direct (N × H)
255. C₂×#496	C₂×C₆₇	semidirect
256. C₂×#466	C₂×C₆₇	semidirect
257. C₂×#465	C₂×C₆₇	semidirect
258. C₂²×#97	C₂×C₆₇	semidirect
259. #2856	C₂×C₆₇	semidirect
260. #2850	C₂×C₆₇	semidirect
261. #1973	C₂×C₆₇	semidirect
262. #1972	C₂×C₆₇	semidirect
263. #1961	C₂×C₆₇	semidirect
264. #1959	C₂×C₆₇	semidirect
265. #1847	C₂×C₆₇	semidirect
266. C₆₇	#33183	direct (N × H)
314. C₂²×C₈	D₄×C₆₇	semidirect
350. #97	D₄×C₆₇	semidirect
471. C₈	#121×C₆₇	semidirect
486. C₂²	#496×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×#466×C₆₇	2
C₂×#465×C₆₇	2
C₂²×#97×C₆₇	2
C₆₇×#2856	2
C₆₇×#2850	2
C₆₇×#1973	2
C₆₇×#1972	2
C₆₇×#1961	2
C₆₇×#1959	2
C₆₇×#1847	2
#33183	67

Subgroups — C67×#33183 #30712434

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33183 #30712434