Subgroups — C₆₇×#32935 #30712186

Order 17152.

1580 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#468×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#465×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2848	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2847	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1895	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1894	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1893	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1892	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1809	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1808	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
14	C₂²×M₁₆×C₆₇	—	2	8	no	—
15	C₂²×M₁₆×C₆₇	—	2	4	no	—
16	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
17	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
18	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
22	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#121×C₆₇	C₂²	1	2	yes	—
24	C₂×#120×C₆₇	—	2	1	no	—
25	#468×C₆₇	—	2	8	no	—
26	#465×C₆₇	—	2	4	no	—
27	C₂×#107×C₆₇	C₂²	1	2	yes	complemented
28	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic, complemented
29	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic, complemented
30	C₂×#103×C₆₇	C₂²	1	2	yes	complemented
31	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic, complemented
32	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
33	#386×C₆₇	—	2	2	no	—
34	#386×C₆₇	—	2	1	no	—
35	#385×C₆₇	—	2	2	no	—
36	#385×C₆₇	—	2	1	no	—
37	#380×C₆₇	C₂²	1	1	yes	characteristic
38	#380×C₆₇	C₂²	1	1	yes	characteristic
39	#380×C₆₇	C₂²	1	2	yes	—
40	#380×C₆₇	C₂²	1	2	yes	—
41	#379×C₆₇	—	2	1	no	—
42	#340×C₆₇	C₂²	1	2	yes	complemented
43	#340×C₆₇	C₂²	1	2	yes	complemented
44	#340×C₆₇	C₂²	1	2	yes	complemented
45	C₂⁵×C₆₇	D₄	1	1	yes	characteristic
46	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×D₄×C₆₇	—	2	2	no	—
48	C₂²×D₄×C₆₇	—	2	1	no	—
49	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₆₇	—	2	4	no	—
51	C₂³×C₄×C₆₇	—	2	2	no	—
52	C₂³×C₄×C₆₇	—	2	4	no	—
53	C₂³×C₄×C₆₇	—	2	2	no	—
54	C₂³×C₄×C₆₇	D₄	1	2	yes	—
55	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
56	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
58	C₂×SD₁₆×C₆₇	—	2	2	no	—
59	C₂×SD₁₆×C₆₇	—	2	4	no	—
60	C₂×M₁₆×C₆₇	—	2	32	no	—
61	C₂×M₁₆×C₆₇	—	2	4	no	—
62	C₂×M₁₆×C₆₇	—	2	4	no	—
63	C₂×M₁₆×C₆₇	—	2	4	no	—
64	C₂×M₁₆×C₆₇	—	2	4	no	—
65	C₂×M₁₆×C₆₇	—	2	4	no	—
66	C₂×M₁₆×C₆₇	D₄	1	8	yes	complemented
67	C₂²×C₈×C₆₇	—	2	2	no	—
68	C₂²×C₈×C₆₇	—	2	4	no	—
69	C₂²×C₈×C₆₇	—	2	4	no	—
70	C₂²×C₈×C₆₇	D₄	1	4	yes	complemented
71	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
72	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₂²⋊Q₈×C₆₇	—	2	2	no	—
75	C₂²⋊Q₈×C₆₇	—	2	4	no	—
76	C₂²⋊Q₈×C₆₇	—	2	2	no	—
77	C₂²⋊Q₈×C₆₇	—	2	4	no	—
78	#121×C₆₇	—	2	2	no	—
79	#121×C₆₇	—	2	4	no	—
80	#121×C₆₇	—	2	2	no	—
81	#121×C₆₇	—	2	4	no	—
82	#120×C₆₇	—	4	4	no	—
83	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
84	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
85	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
89	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
90	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
91	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
93	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
95	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
96	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
97	#107×C₆₇	—	2	4	no	—
98	#106×C₆₇	—	2	2	no	—
99	#103×C₆₇	—	2	2	no	—
100	#103×C₆₇	—	2	4	no	—
101	#102×C₆₇	—	2	2	no	—
102	#102×C₆₇	—	2	4	no	—
103	#95×C₆₇	—	2	1	no	—
104	#95×C₆₇	—	2	2	no	—
105	#95×C₆₇	—	2	1	no	—
106	#95×C₆₇	—	2	2	no	—
107	C₂⁴×C₆₇	—	2	4	no	—
108	C₂⁴×C₆₇	—	2	2	no	—
109	C₂⁴×C₆₇	—	2	2	no	—
110	C₂⁴×C₆₇	—	2	2	no	—
111	C₂⁴×C₆₇	—	2	1	no	—
112	C₂⁴×C₆₇	—	4	2	no	—
113	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
114	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂×Q₈×C₆₇	—	2	1	no	—
116	C₂×Q₈×C₆₇	—	2	2	no	—
117	C₂×Q₈×C₆₇	—	2	1	no	—
118	C₂×Q₈×C₆₇	—	2	2	no	—
119	C₂×D₄×C₆₇	—	2	1	no	—
120	C₂×D₄×C₆₇	—	2	2	no	—
121	C₂×D₄×C₆₇	—	2	1	no	—
122	C₂×D₄×C₆₇	—	2	2	no	—
123	C₂×D₄×C₆₇	—	4	4	no	—
124	C₂×D₄×C₆₇	—	4	4	no	—
125	C₂×D₄×C₆₇	—	4	4	no	—
126	C₂×D₄×C₆₇	—	4	2	no	—
127	C₂×D₄×C₆₇	—	4	2	no	—
128	C₂×D₄×C₆₇	—	4	2	no	—
129	C₂²×C₄×C₆₇	—	2	8	no	—
130	C₂²×C₄×C₆₇	—	2	8	no	—
131	C₂²×C₄×C₆₇	—	2	16	no	—
132	C₂²×C₄×C₆₇	—	2	4	no	—
133	C₂²×C₄×C₆₇	—	2	2	no	—
134	C₂²×C₄×C₆₇	—	2	4	no	—
135	C₂²×C₄×C₆₇	—	2	4	no	—
136	C₂²×C₄×C₆₇	—	2	4	no	—
137	C₂²×C₄×C₆₇	—	2	2	no	—
138	C₂²×C₄×C₆₇	—	2	2	no	—
139	C₂²×C₄×C₆₇	—	2	2	no	—
140	C₂²×C₄×C₆₇	—	2	2	no	—
141	C₂²×C₄×C₆₇	—	2	2	no	—
142	C₂²×C₄×C₆₇	—	2	2	no	—
143	C₂²×C₄×C₆₇	—	2	1	no	—
144	C₂²×C₄×C₆₇	—	2	2	no	—
145	C₂²×C₄×C₆₇	—	2	1	no	—
146	C₂²×C₄×C₆₇	—	2	1	no	—
147	C₂²×C₄×C₆₇	—	2	2	no	—
148	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
150	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
151	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
152	C₂²×C₄×C₆₇	#41	1	4	yes	—
153	C₂²×C₄×C₆₇	#41	1	2	yes	—
154	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
155	SD₁₆×C₆₇	—	4	4	no	—
156	M₁₆×C₆₇	—	2	16	no	—
157	M₁₆×C₆₇	—	2	16	no	—
158	C₂×C₈×C₆₇	—	2	8	no	—
159	C₂×C₈×C₆₇	—	2	2	no	—
160	C₂×C₈×C₆₇	—	2	2	no	—
161	C₂×C₈×C₆₇	—	2	2	no	—
162	C₂×C₈×C₆₇	—	2	8	no	—
163	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
164	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
165	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
166	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
167	C₄⋊C₄×C₆₇	—	2	1	no	—
168	C₄⋊C₄×C₆₇	—	2	2	no	—
169	C₄⋊C₄×C₆₇	—	2	1	no	—
170	C₄⋊C₄×C₆₇	—	2	2	no	—
171	C₄⋊C₄×C₆₇	—	2	1	no	—
172	C₄⋊C₄×C₆₇	—	2	2	no	—
173	C₄⋊C₄×C₆₇	—	2	1	no	—
174	C₄⋊C₄×C₆₇	—	2	2	no	—
175	C₄⋊C₄×C₆₇	—	4	2	no	—
176	C₄⋊C₄×C₆₇	—	4	4	no	—
177	C₂²⋊C₄×C₆₇	—	4	4	no	—
178	C₂²⋊C₄×C₆₇	—	4	4	no	—
179	C₂²⋊C₄×C₆₇	—	4	4	no	—
180	C₂²⋊C₄×C₆₇	—	4	2	no	—
181	C₂²⋊C₄×C₆₇	—	4	2	no	—
182	C₂²⋊C₄×C₆₇	—	4	2	no	—
183	C₂²⋊C₄×C₆₇	—	4	2	no	—
184	C₂²⋊C₄×C₆₇	—	4	4	no	—
185	C₂³×C₆₇	—	2	2	no	—
186	C₂³×C₆₇	—	2	2	no	—
187	C₂³×C₆₇	—	2	8	no	—
188	C₂³×C₆₇	—	2	2	no	—
189	C₂³×C₆₇	—	2	2	no	—
190	C₂³×C₆₇	—	2	2	no	—
191	C₂³×C₆₇	—	2	2	no	—
192	C₂³×C₆₇	—	2	2	no	—
193	C₂³×C₆₇	—	2	2	no	—
194	C₂³×C₆₇	—	4	8	no	—
195	C₂³×C₆₇	—	4	8	no	—
196	C₂³×C₆₇	—	4	4	no	—
197	C₂³×C₆₇	—	4	4	no	—
198	C₂³×C₆₇	—	4	1	no	—
199	C₂³×C₆₇	—	4	2	no	—
200	C₂³×C₆₇	—	4	1	no	—
201	C₂³×C₆₇	—	4	1	no	—
202	C₂³×C₆₇	—	4	2	no	—
203	C₂³×C₆₇	#121	1	4	yes	—
204	C₂³×C₆₇	#121	1	2	yes	—
205	C₂³×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
206	C₂³×C₆₇	#137	1	2	yes	—
207	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
208	Q₈×C₆₇	—	4	1	no	—
209	Q₈×C₆₇	—	4	1	no	—
210	Q₈×C₆₇	—	4	2	no	—
211	D₄×C₆₇	—	4	1	no	—
212	D₄×C₆₇	—	4	1	no	—
213	D₄×C₆₇	—	4	2	no	—
214	D₄×C₆₇	—	8	8	no	—
215	D₄×C₆₇	—	8	4	no	—
216	C₂×C₄×C₆₇	—	2	8	no	—
217	C₂×C₄×C₆₇	—	2	16	no	—
218	C₂×C₄×C₆₇	—	2	2	no	—
219	C₂×C₄×C₆₇	—	2	2	no	—
220	C₂×C₄×C₆₇	—	2	2	no	—
221	C₂×C₄×C₆₇	—	2	2	no	—
222	C₂×C₄×C₆₇	—	2	4	no	—
223	C₂×C₄×C₆₇	—	2	2	no	—
224	C₂×C₄×C₆₇	—	2	4	no	—
225	C₂×C₄×C₆₇	—	2	4	no	—
226	C₂×C₄×C₆₇	—	2	4	no	—
227	C₂×C₄×C₆₇	—	2	4	no	—
228	C₂×C₄×C₆₇	—	4	1	no	—
229	C₂×C₄×C₆₇	—	4	2	no	—
230	C₂×C₄×C₆₇	—	4	1	no	—
231	C₂×C₄×C₆₇	—	4	2	no	—
232	C₂×C₄×C₆₇	—	4	1	no	—
233	C₂×C₄×C₆₇	—	4	2	no	—
234	C₂×C₄×C₆₇	—	4	1	no	—
235	C₂×C₄×C₆₇	—	4	2	no	—
236	C₂×C₄×C₆₇	—	4	2	no	—
237	C₂×C₄×C₆₇	—	4	1	no	—
238	C₂×C₄×C₆₇	—	4	2	no	—
239	C₂×C₄×C₆₇	—	4	2	no	—
240	C₂×C₄×C₆₇	—	4	2	no	—
241	C₂×C₄×C₆₇	#121	1	2	yes	—
242	C₂×C₄×C₆₇	#121	1	4	yes	—
243	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
244	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
245	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
246	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
247	C₈×C₆₇	—	2	4	no	—
248	C₈×C₆₇	—	2	2	no	—
249	C₈×C₆₇	#120	1	4	yes	complemented
250	C₂²×C₆₇	—	2	2	no	—
251	C₂²×C₆₇	—	2	2	no	—
252	C₂²×C₆₇	—	2	4	no	—
253	C₂²×C₆₇	—	2	4	no	—
254	C₂²×C₆₇	—	2	4	no	—
255	C₂²×C₆₇	—	2	4	no	—
256	C₂²×C₆₇	—	2	4	no	—
257	C₂²×C₆₇	—	4	16	no	—
258	C₂²×C₆₇	—	4	2	no	—
259	C₂²×C₆₇	—	4	2	no	—
260	C₂²×C₆₇	—	4	2	no	—
261	C₂²×C₆₇	—	4	2	no	—
262	C₂²×C₆₇	—	8	1	no	—
263	C₂²×C₆₇	—	8	1	no	—
264	C₂²×C₆₇	—	8	1	no	—
265	C₂²×C₆₇	—	8	2	no	—
266	C₂²×C₆₇	—	8	2	no	—
267	C₂²×C₆₇	#468	1	4	yes	complemented
268	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
269	C₂²×C₆₇	C₂×#121	1	2	yes	—
270	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
271	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
272	C₂²×C₆₇	#575	1	2	yes	—
273	C₄×C₆₇	—	2	2	no	—
274	C₄×C₆₇	—	2	4	no	—
275	C₄×C₆₇	—	8	1	no	—
276	C₄×C₆₇	—	8	1	no	—
277	C₄×C₆₇	—	8	1	no	—
278	C₄×C₆₇	—	8	2	no	—
279	C₄×C₆₇	—	8	2	no	—
280	C₄×C₆₇	C₂×#120	1	1	yes	characteristic
281	C₄×C₆₇	#525	1	1	yes	characteristic
282	C₄×C₆₇	#525	1	2	yes	—
283	#32935	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
284	C₂×C₆₇	—	2	4	no	—
285	C₂×C₆₇	—	4	4	no	—
286	C₂×C₆₇	—	16	1	no	—
287	C₂×C₆₇	#2847	1	1	yes	characteristic
288	C₂×C₆₇	C₂×#468	1	1	yes	characteristic
289	C₂×C₆₇	#3054	1	1	yes	characteristic
290	C₂×C₆₇	#3055	1	2	yes	—
291	C₂×C₆₇	#3056	1	2	yes	—
292	C₂³×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
293	C₂×#468	C₂×C₆₇	1	2	yes	complemented
294	C₂×#465	C₂×C₆₇	1	1	yes	characteristic, complemented
295	#2848	C₂×C₆₇	1	1	yes	characteristic, complemented
296	#2847	C₂×C₆₇	1	1	yes	characteristic
297	#1895	C₂×C₆₇	1	2	yes	complemented
298	#1894	C₂×C₆₇	1	2	yes	complemented
299	#1893	C₂×C₆₇	1	1	yes	characteristic, complemented
300	#1892	C₂×C₆₇	1	1	yes	characteristic, complemented
301	#1809	C₂×C₆₇	1	2	yes	complemented
302	#1808	C₂×C₆₇	1	1	yes	characteristic, complemented
303	C₆₇	#32935	1	1	yes	characteristic, direct product factor, Sylow
304	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
305	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
306	C₂²×M₁₆	—	2	8	no	—
307	C₂²×M₁₆	—	2	4	no	—
308	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
309	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
310	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
311	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
312	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
313	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
314	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
315	C₂×#121	C₂²×C₆₇	1	2	yes	—
316	C₂×#120	—	2	1	no	—
317	#468	—	2	8	no	—
318	#465	—	2	4	no	—
319	C₂×#107	C₂²×C₆₇	1	2	yes	complemented
320	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic, complemented
321	C₂×#103	C₂²×C₆₇	1	2	yes	complemented
322	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic, complemented
323	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
324	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic, complemented
325	#386	—	2	2	no	—
326	#386	—	2	1	no	—
327	#385	—	2	2	no	—
328	#385	—	2	1	no	—
329	#380	C₂²×C₆₇	1	1	yes	characteristic
330	#380	C₂²×C₆₇	1	1	yes	characteristic
331	#380	C₂²×C₆₇	1	2	yes	—
332	#380	C₂²×C₆₇	1	2	yes	—
333	#379	—	2	1	no	—
334	#340	C₂²×C₆₇	1	2	yes	complemented
335	#340	C₂²×C₆₇	1	2	yes	complemented
336	#340	C₂²×C₆₇	1	2	yes	complemented
337	C₂⁵	D₄×C₆₇	1	1	yes	characteristic
338	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
339	C₂²×D₄	—	2	2	no	—
340	C₂²×D₄	—	2	1	no	—
341	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
342	C₂³×C₄	—	2	4	no	—
343	C₂³×C₄	—	2	2	no	—
344	C₂³×C₄	—	2	4	no	—
345	C₂³×C₄	—	2	2	no	—
346	C₂³×C₄	D₄×C₆₇	1	2	yes	—
347	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
348	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
349	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
350	C₂×SD₁₆	—	2	2	no	—
351	C₂×SD₁₆	—	2	4	no	—
352	C₂×M₁₆	—	2	32	no	—
353	C₂×M₁₆	—	2	4	no	—
354	C₂×M₁₆	—	2	4	no	—
355	C₂×M₁₆	—	2	4	no	—
356	C₂×M₁₆	—	2	4	no	—
357	C₂×M₁₆	—	2	4	no	—
358	C₂×M₁₆	D₄×C₆₇	1	8	yes	complemented
359	C₂²×C₈	—	2	2	no	—
360	C₂²×C₈	—	2	4	no	—
361	C₂²×C₈	—	2	4	no	—
362	C₂²×C₈	D₄×C₆₇	1	4	yes	complemented
363	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
364	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
365	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
366	C₂²⋊Q₈	—	2	2	no	—
367	C₂²⋊Q₈	—	2	4	no	—
368	C₂²⋊Q₈	—	2	2	no	—
369	C₂²⋊Q₈	—	2	4	no	—
370	#121	—	2	2	no	—
371	#121	—	2	4	no	—
372	#121	—	2	2	no	—
373	#121	—	2	4	no	—
374	#120	—	4	4	no	—
375	C₂×C₄⋊C₄	—	2	1	no	—
376	C₂×C₄⋊C₄	—	2	2	no	—
377	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
378	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
379	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
380	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
381	C₂×C₂²⋊C₄	—	2	2	no	—
382	C₂×C₂²⋊C₄	—	2	1	no	—
383	C₂×C₂²⋊C₄	—	2	2	no	—
384	C₂×C₂²⋊C₄	—	2	1	no	—
385	C₂×C₂²⋊C₄	—	2	2	no	—
386	C₂×C₂²⋊C₄	—	2	1	no	—
387	C₂×C₂²⋊C₄	—	2	1	no	—
388	C₂×C₂²⋊C₄	—	2	2	no	—
389	#107	—	2	4	no	—
390	#106	—	2	2	no	—
391	#103	—	2	2	no	—
392	#103	—	2	4	no	—
393	#102	—	2	2	no	—
394	#102	—	2	4	no	—
395	#95	—	2	1	no	—
396	#95	—	2	2	no	—
397	#95	—	2	1	no	—
398	#95	—	2	2	no	—
399	C₂⁴	—	2	4	no	—
400	C₂⁴	—	2	2	no	—
401	C₂⁴	—	2	2	no	—
402	C₂⁴	—	2	2	no	—
403	C₂⁴	—	2	1	no	—
404	C₂⁴	—	4	2	no	—
405	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
406	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
407	C₂×Q₈	—	2	1	no	—
408	C₂×Q₈	—	2	2	no	—
409	C₂×Q₈	—	2	1	no	—
410	C₂×Q₈	—	2	2	no	—
411	C₂×D₄	—	2	1	no	—
412	C₂×D₄	—	2	2	no	—
413	C₂×D₄	—	2	1	no	—
414	C₂×D₄	—	2	2	no	—
415	C₂×D₄	—	4	4	no	—
416	C₂×D₄	—	4	4	no	—
417	C₂×D₄	—	4	4	no	—
418	C₂×D₄	—	4	2	no	—
419	C₂×D₄	—	4	2	no	—
420	C₂×D₄	—	4	2	no	—
421	C₂²×C₄	—	2	8	no	—
422	C₂²×C₄	—	2	8	no	—
423	C₂²×C₄	—	2	16	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	2	2	no	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	4	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	2	2	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	2	no	—
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	2	2	no	—
435	C₂²×C₄	—	2	2	no	—
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
441	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
442	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
443	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
444	C₂²×C₄	#41×C₆₇	1	4	yes	—
445	C₂²×C₄	#41×C₆₇	1	2	yes	—
446	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
447	SD₁₆	—	4	4	no	—
448	M₁₆	—	2	16	no	—
449	M₁₆	—	2	16	no	—
450	C₂×C₈	—	2	8	no	—
451	C₂×C₈	—	2	2	no	—
452	C₂×C₈	—	2	2	no	—
453	C₂×C₈	—	2	2	no	—
454	C₂×C₈	—	2	8	no	—
455	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
456	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
457	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
458	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	2	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	2	2	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	2	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	2	no	—
467	C₄⋊C₄	—	4	2	no	—
468	C₄⋊C₄	—	4	4	no	—
469	C₂²⋊C₄	—	4	4	no	—
470	C₂²⋊C₄	—	4	4	no	—
471	C₂²⋊C₄	—	4	4	no	—
472	C₂²⋊C₄	—	4	2	no	—
473	C₂²⋊C₄	—	4	2	no	—
474	C₂²⋊C₄	—	4	2	no	—
475	C₂²⋊C₄	—	4	2	no	—
476	C₂²⋊C₄	—	4	4	no	—
477	C₂³	—	2	2	no	—
478	C₂³	—	2	2	no	—
479	C₂³	—	2	8	no	—
480	C₂³	—	2	2	no	—
481	C₂³	—	2	2	no	—
482	C₂³	—	2	2	no	—
483	C₂³	—	2	2	no	—
484	C₂³	—	2	2	no	—
485	C₂³	—	2	2	no	—
486	C₂³	—	4	8	no	—
487	C₂³	—	4	8	no	—
488	C₂³	—	4	4	no	—
489	C₂³	—	4	4	no	—
490	C₂³	—	4	1	no	—
491	C₂³	—	4	2	no	—
492	C₂³	—	4	1	no	—
493	C₂³	—	4	1	no	—
494	C₂³	—	4	2	no	—
495	C₂³	#121×C₆₇	1	4	yes	—
496	C₂³	#121×C₆₇	1	2	yes	—
497	C₂³	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
498	C₂³	#137×C₆₇	1	2	yes	—
499	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
500	Q₈	—	4	1	no	—
501	Q₈	—	4	1	no	—
502	Q₈	—	4	2	no	—
503	D₄	—	4	1	no	—
504	D₄	—	4	1	no	—
505	D₄	—	4	2	no	—
506	D₄	—	8	8	no	—
507	D₄	—	8	4	no	—
508	C₂×C₄	—	2	8	no	—
509	C₂×C₄	—	2	16	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	4	2	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	4	2	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	—	4	2	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	2	no	—
528	C₂×C₄	—	4	2	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	4	2	no	—
531	C₂×C₄	—	4	2	no	—
532	C₂×C₄	—	4	2	no	—
533	C₂×C₄	#121×C₆₇	1	2	yes	—
534	C₂×C₄	#121×C₆₇	1	4	yes	—
535	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
536	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
537	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
538	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
539	C₈	—	2	4	no	—
540	C₈	—	2	2	no	—
541	C₈	#120×C₆₇	1	4	yes	complemented
542	C₂²	—	2	2	no	—
543	C₂²	—	2	4	no	—
544	C₂²	—	2	4	no	—
545	C₂²	—	2	2	no	—
546	C₂²	—	2	4	no	—
547	C₂²	—	2	4	no	—
548	C₂²	—	2	4	no	—
549	C₂²	—	4	16	no	—
550	C₂²	—	4	2	no	—
551	C₂²	—	4	2	no	—
552	C₂²	—	4	2	no	—
553	C₂²	—	4	2	no	—
554	C₂²	—	8	1	no	—
555	C₂²	—	8	1	no	—
556	C₂²	—	8	2	no	—
557	C₂²	—	8	2	no	—
558	C₂²	—	8	1	no	—
559	C₂²	#468×C₆₇	1	4	yes	complemented
560	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
561	C₂²	C₂×#121×C₆₇	1	2	yes	—
562	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
563	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
564	C₂²	#575×C₆₇	1	2	yes	—
565	C₄	—	2	2	no	—
566	C₄	—	2	4	no	—
567	C₄	—	8	1	no	—
568	C₄	—	8	1	no	—
569	C₄	—	8	2	no	—
570	C₄	—	8	2	no	—
571	C₄	—	8	1	no	—
572	C₄	C₂×#120×C₆₇	1	1	yes	characteristic
573	C₄	#525×C₆₇	1	1	yes	characteristic
574	C₄	#525×C₆₇	1	2	yes	—
575	C₂	—	2	4	no	—
576	C₂	—	4	4	no	—
577	C₂	—	16	1	no	—
578	C₂	C₆₇×#2847	1	1	yes	characteristic
579	C₂	C₂×#468×C₆₇	1	1	yes	characteristic
580	C₂	C₆₇×#3054	1	1	yes	characteristic
581	C₂	C₆₇×#3055	1	2	yes	—
582	C₂	C₆₇×#3056	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#2847	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂⁴×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×M₁₆×C₆₇	C₂	semidirect
2. C₂×#468×C₆₇	C₂	semidirect
3. C₂×#465×C₆₇	C₂	semidirect
4. C₆₇×#2848	C₂	semidirect
6. C₆₇×#1895	C₂	semidirect
7. C₆₇×#1894	C₂	semidirect
8. C₆₇×#1893	C₂	semidirect
9. C₆₇×#1892	C₂	semidirect
10. C₆₇×#1809	C₂	semidirect
11. C₆₇×#1808	C₂	semidirect
13. C₂²×SD₁₆×C₆₇	C₂²	semidirect
27. C₂×#107×C₆₇	C₂²	semidirect
28. C₂×#106×C₆₇	C₂²	semidirect
29. C₂×#103×C₆₇	C₂²	semidirect
30. C₂×#103×C₆₇	C₂²	semidirect
31. C₂×#102×C₆₇	C₂²	semidirect
32. C₂×#102×C₆₇	C₂²	semidirect
42. #340×C₆₇	C₂²	semidirect
43. #340×C₆₇	C₂²	semidirect
44. #340×C₆₇	C₂²	semidirect
66. C₂×M₁₆×C₆₇	D₄	semidirect
70. C₂²×C₈×C₆₇	D₄	semidirect
249. C₈×C₆₇	#120	semidirect
267. C₂²×C₆₇	#468	semidirect
283. #32935	C₆₇	direct (N × H)
292. C₂³×M₁₆	C₂×C₆₇	semidirect
293. C₂×#468	C₂×C₆₇	semidirect
294. C₂×#465	C₂×C₆₇	semidirect
295. #2848	C₂×C₆₇	semidirect
297. #1895	C₂×C₆₇	semidirect
298. #1894	C₂×C₆₇	semidirect
299. #1893	C₂×C₆₇	semidirect
300. #1892	C₂×C₆₇	semidirect
301. #1809	C₂×C₆₇	semidirect
302. #1808	C₂×C₆₇	semidirect
303. C₆₇	#32935	direct (N × H)
305. C₂²×SD₁₆	C₂²×C₆₇	semidirect
319. C₂×#107	C₂²×C₆₇	semidirect
320. C₂×#106	C₂²×C₆₇	semidirect
321. C₂×#103	C₂²×C₆₇	semidirect
322. C₂×#103	C₂²×C₆₇	semidirect
323. C₂×#102	C₂²×C₆₇	semidirect
324. C₂×#102	C₂²×C₆₇	semidirect
334. #340	C₂²×C₆₇	semidirect
335. #340	C₂²×C₆₇	semidirect
336. #340	C₂²×C₆₇	semidirect
358. C₂×M₁₆	D₄×C₆₇	semidirect
362. C₂²×C₈	D₄×C₆₇	semidirect
541. C₈	#120×C₆₇	semidirect
559. C₂²	#468×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₆₇	2
C₂×#468×C₆₇	2
C₂×#465×C₆₇	2
C₆₇×#2848	2
C₆₇×#2847	2
C₆₇×#1895	2
C₆₇×#1894	2
C₆₇×#1893	2
C₆₇×#1892	2
C₆₇×#1809	2
C₆₇×#1808	2
#32935	67

Subgroups — C67×#32935 #30712186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32935 #30712186