Subgroups — C₆₇×#32880 #30712131

Order 17152.

4004 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#449×C₆₇	C₂	1	4	yes	maximal, complemented
3	C₂×#448×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#447×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2847	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1788	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1787	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1786	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1779	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₆₇	—	2	16	no	—
11	C₂²×#41×C₆₇	—	2	4	no	—
12	C₂²×#41×C₆₇	—	2	4	no	—
13	C₂²×#41×C₆₇	—	2	2	no	—
14	C₂²×#41×C₆₇	C₂²	1	4	yes	—
15	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂³×D₄×C₆₇	—	2	1	no	—
17	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄×C₆₇	—	2	1	no	—
19	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	complemented
21	C₂²×D₈×C₆₇	C₂²	1	2	yes	complemented
22	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#121×C₆₇	C₂²	1	2	yes	complemented
24	C₂×#120×C₆₇	—	2	1	no	—
25	#449×C₆₇	—	2	16	no	—
26	#448×C₆₇	—	2	4	no	—
27	#447×C₆₇	—	2	4	no	—
28	C₂×#103×C₆₇	C₂²	1	4	yes	complemented
29	C₂×#103×C₆₇	C₂²	1	2	yes	complemented
30	C₂×#102×C₆₇	C₂²	1	4	yes	complemented
31	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
32	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	complemented
35	#386×C₆₇	—	2	1	no	—
36	#386×C₆₇	—	2	2	no	—
37	#380×C₆₇	C₂²	1	1	yes	characteristic
38	#380×C₆₇	C₂²	1	2	yes	complemented
39	C₂⁵×C₆₇	—	2	1	no	—
40	C₂⁵×C₆₇	D₄	1	1	yes	characteristic
41	C₂×#41×C₆₇	—	2	64	no	—
42	C₂×#41×C₆₇	—	2	64	no	—
43	C₂×#41×C₆₇	—	2	16	no	—
44	C₂×#41×C₆₇	—	2	16	no	—
45	C₂×#41×C₆₇	—	2	16	no	—
46	C₂×#41×C₆₇	—	2	8	no	—
47	C₂×#41×C₆₇	—	2	8	no	—
48	C₂×#41×C₆₇	—	2	8	no	—
49	C₂×#41×C₆₇	—	2	8	no	—
50	C₂×#41×C₆₇	—	2	8	no	—
51	C₂×#41×C₆₇	D₄	1	16	yes	complemented
52	C₂²×Q₈×C₆₇	—	2	8	no	—
53	C₂²×Q₈×C₆₇	—	2	2	no	—
54	C₂²×Q₈×C₆₇	—	2	1	no	—
55	C₂²×Q₈×C₆₇	D₄	1	4	yes	complemented
56	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
57	C₂²×D₄×C₆₇	—	2	16	no	—
58	C₂²×D₄×C₆₇	—	2	8	no	—
59	C₂²×D₄×C₆₇	—	2	4	no	—
60	C₂²×D₄×C₆₇	—	2	8	no	—
61	C₂²×D₄×C₆₇	—	2	1	no	—
62	C₂²×D₄×C₆₇	—	2	2	no	—
63	C₂²×D₄×C₆₇	—	2	1	no	—
64	C₂²×D₄×C₆₇	—	2	2	no	—
65	C₂²×D₄×C₆₇	D₄	1	4	yes	complemented
66	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
68	C₂³×C₄×C₆₇	—	2	16	no	—
69	C₂³×C₄×C₆₇	—	2	8	no	—
70	C₂³×C₄×C₆₇	—	2	1	no	—
71	C₂³×C₄×C₆₇	—	2	4	no	—
72	C₂³×C₄×C₆₇	—	2	2	no	—
73	C₂³×C₄×C₆₇	—	2	2	no	—
74	C₂³×C₄×C₆₇	—	2	8	no	—
75	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
76	C₂³×C₄×C₆₇	D₄	1	2	yes	—
77	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
79	C₂×SD₁₆×C₆₇	—	2	4	no	—
80	C₂×SD₁₆×C₆₇	—	2	8	no	—
81	C₂×D₈×C₆₇	—	2	8	no	—
82	C₂×D₈×C₆₇	—	2	4	no	—
83	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
84	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
85	#121×C₆₇	—	2	2	no	—
86	#121×C₆₇	—	2	8	no	—
87	#121×C₆₇	—	2	2	no	—
88	#120×C₆₇	—	4	4	no	—
89	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
91	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
92	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
93	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
94	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
95	#103×C₆₇	—	2	4	no	—
96	#103×C₆₇	—	2	8	no	—
97	#102×C₆₇	—	2	8	no	—
98	#102×C₆₇	—	2	4	no	—
99	C₂²⋊C₈×C₆₇	—	2	2	no	—
100	C₂²⋊C₈×C₆₇	—	2	8	no	—
101	C₂²⋊C₈×C₆₇	—	2	2	no	—
102	#95×C₆₇	—	2	1	no	—
103	#95×C₆₇	—	2	2	no	—
104	C₂⁴×C₆₇	—	2	8	no	—
105	C₂⁴×C₆₇	—	2	4	no	—
106	C₂⁴×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	—	4	4	no	—
114	C₂⁴×C₆₇	—	4	2	no	—
115	C₂⁴×C₆₇	—	4	2	no	—
116	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
117	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
118	#41×C₆₇	—	2	128	no	—
119	#41×C₆₇	—	2	32	no	—
120	#41×C₆₇	—	2	32	no	—
121	#41×C₆₇	—	2	32	no	—
122	#41×C₆₇	—	2	32	no	—
123	C₂×Q₈×C₆₇	—	2	16	no	—
124	C₂×Q₈×C₆₇	—	2	8	no	—
125	C₂×Q₈×C₆₇	—	2	16	no	—
126	C₂×Q₈×C₆₇	—	2	4	no	—
127	C₂×Q₈×C₆₇	—	2	4	no	—
128	C₂×Q₈×C₆₇	—	2	2	no	—
129	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	—
130	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
131	C₂×D₄×C₆₇	—	2	64	no	—
132	C₂×D₄×C₆₇	—	2	16	no	—
133	C₂×D₄×C₆₇	—	2	16	no	—
134	C₂×D₄×C₆₇	—	2	8	no	—
135	C₂×D₄×C₆₇	—	2	16	no	—
136	C₂×D₄×C₆₇	—	2	16	no	—
137	C₂×D₄×C₆₇	—	2	16	no	—
138	C₂×D₄×C₆₇	—	2	4	no	—
139	C₂×D₄×C₆₇	—	2	2	no	—
140	C₂×D₄×C₆₇	—	2	4	no	—
141	C₂×D₄×C₆₇	—	2	1	no	—
142	C₂×D₄×C₆₇	—	2	4	no	—
143	C₂×D₄×C₆₇	—	2	1	no	—
144	C₂×D₄×C₆₇	—	4	4	no	—
145	C₂×D₄×C₆₇	—	4	4	no	—
146	C₂×D₄×C₆₇	—	4	4	no	—
147	C₂×D₄×C₆₇	—	4	2	no	—
148	C₂×D₄×C₆₇	—	4	2	no	—
149	C₂×D₄×C₆₇	—	4	2	no	—
150	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
151	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
152	C₂²×C₄×C₆₇	—	2	8	no	—
153	C₂²×C₄×C₆₇	—	2	64	no	—
154	C₂²×C₄×C₆₇	—	2	16	no	—
155	C₂²×C₄×C₆₇	—	2	16	no	—
156	C₂²×C₄×C₆₇	—	2	16	no	—
157	C₂²×C₄×C₆₇	—	2	16	no	—
158	C₂²×C₄×C₆₇	—	2	2	no	—
159	C₂²×C₄×C₆₇	—	2	4	no	—
160	C₂²×C₄×C₆₇	—	2	4	no	—
161	C₂²×C₄×C₆₇	—	2	2	no	—
162	C₂²×C₄×C₆₇	—	2	4	no	—
163	C₂²×C₄×C₆₇	—	2	16	no	—
164	C₂²×C₄×C₆₇	—	2	4	no	—
165	C₂²×C₄×C₆₇	—	2	4	no	—
166	C₂²×C₄×C₆₇	—	2	4	no	—
167	C₂²×C₄×C₆₇	—	2	2	no	—
168	C₂²×C₄×C₆₇	—	2	4	no	—
169	C₂²×C₄×C₆₇	—	2	2	no	—
170	C₂²×C₄×C₆₇	—	2	1	no	—
171	C₂²×C₄×C₆₇	—	2	2	no	—
172	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
173	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
176	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
177	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
178	SD₁₆×C₆₇	—	4	8	no	—
179	D₈×C₆₇	—	4	8	no	—
180	C₂×C₈×C₆₇	—	2	2	no	—
181	C₂×C₈×C₆₇	—	2	4	no	—
182	C₂×C₈×C₆₇	—	2	4	no	—
183	C₂×C₈×C₆₇	—	2	2	no	—
184	C₄⋊C₄×C₆₇	—	2	1	no	—
185	C₄⋊C₄×C₆₇	—	2	4	no	—
186	C₄⋊C₄×C₆₇	—	2	1	no	—
187	C₂²⋊C₄×C₆₇	—	4	4	no	—
188	C₂²⋊C₄×C₆₇	—	4	2	no	—
189	C₂²⋊C₄×C₆₇	—	4	4	no	—
190	C₂²⋊C₄×C₆₇	—	4	2	no	—
191	C₂³×C₆₇	—	2	2	no	—
192	C₂³×C₆₇	—	2	2	no	—
193	C₂³×C₆₇	—	2	8	no	—
194	C₂³×C₆₇	—	2	16	no	—
195	C₂³×C₆₇	—	2	2	no	—
196	C₂³×C₆₇	—	2	2	no	—
197	C₂³×C₆₇	—	2	4	no	—
198	C₂³×C₆₇	—	2	8	no	—
199	C₂³×C₆₇	—	2	2	no	—
200	C₂³×C₆₇	—	2	2	no	—
201	C₂³×C₆₇	—	2	2	no	—
202	C₂³×C₆₇	—	2	2	no	—
203	C₂³×C₆₇	—	4	8	no	—
204	C₂³×C₆₇	—	4	8	no	—
205	C₂³×C₆₇	—	4	16	no	—
206	C₂³×C₆₇	—	4	4	no	—
207	C₂³×C₆₇	—	4	8	no	—
208	C₂³×C₆₇	—	4	16	no	—
209	C₂³×C₆₇	—	4	2	no	—
210	C₂³×C₆₇	—	4	2	no	—
211	C₂³×C₆₇	—	4	4	no	—
212	C₂³×C₆₇	—	4	4	no	—
213	C₂³×C₆₇	—	4	8	no	—
214	C₂³×C₆₇	—	4	1	no	—
215	C₂³×C₆₇	—	4	2	no	—
216	C₂³×C₆₇	—	4	1	no	—
217	C₂³×C₆₇	—	4	1	no	—
218	C₂³×C₆₇	—	4	2	no	—
219	C₂³×C₆₇	#120	1	4	yes	—
220	C₂³×C₆₇	#120	1	2	yes	—
221	C₂³×C₆₇	#135	1	2	yes	—
222	C₂³×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
223	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
224	Q₈×C₆₇	—	2	16	no	—
225	Q₈×C₆₇	—	2	4	no	—
226	Q₈×C₆₇	—	2	8	no	—
227	Q₈×C₆₇	#120	1	8	yes	complemented
228	D₄×C₆₇	—	2	64	no	—
229	D₄×C₆₇	—	2	16	no	—
230	D₄×C₆₇	—	2	8	no	—
231	D₄×C₆₇	—	2	4	no	—
232	D₄×C₆₇	—	4	2	no	—
233	D₄×C₆₇	—	4	2	no	—
234	D₄×C₆₇	—	8	8	no	—
235	D₄×C₆₇	—	8	4	no	—
236	D₄×C₆₇	#120	1	8	yes	complemented
237	C₂×C₄×C₆₇	—	2	8	no	—
238	C₂×C₄×C₆₇	—	2	16	no	—
239	C₂×C₄×C₆₇	—	2	64	no	—
240	C₂×C₄×C₆₇	—	2	16	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	4	no	—
244	C₂×C₄×C₆₇	—	2	8	no	—
245	C₂×C₄×C₆₇	—	2	2	no	—
246	C₂×C₄×C₆₇	—	2	4	no	—
247	C₂×C₄×C₆₇	—	2	4	no	—
248	C₂×C₄×C₆₇	—	2	8	no	—
249	C₂×C₄×C₆₇	—	2	8	no	—
250	C₂×C₄×C₆₇	—	4	1	no	—
251	C₂×C₄×C₆₇	—	4	2	no	—
252	C₂×C₄×C₆₇	—	4	1	no	—
253	C₂×C₄×C₆₇	—	4	1	no	—
254	C₂×C₄×C₆₇	—	4	2	no	—
255	C₂×C₄×C₆₇	—	4	2	no	—
256	C₂×C₄×C₆₇	#120	1	4	yes	—
257	C₂×C₄×C₆₇	#120	1	2	yes	—
258	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
259	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
260	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
261	C₈×C₆₇	—	4	2	no	—
262	C₈×C₆₇	—	4	2	no	—
263	C₂²×C₆₇	—	2	2	no	—
264	C₂²×C₆₇	—	2	4	no	—
265	C₂²×C₆₇	—	2	2	no	—
266	C₂²×C₆₇	—	2	4	no	—
267	C₂²×C₆₇	—	2	4	no	—
268	C₂²×C₆₇	—	2	8	no	—
269	C₂²×C₆₇	—	2	4	no	—
270	C₂²×C₆₇	—	2	4	no	—
271	C₂²×C₆₇	—	4	16	no	—
272	C₂²×C₆₇	—	4	32	no	—
273	C₂²×C₆₇	—	4	2	no	—
274	C₂²×C₆₇	—	4	4	no	—
275	C₂²×C₆₇	—	4	2	no	—
276	C₂²×C₆₇	—	4	8	no	—
277	C₂²×C₆₇	—	4	2	no	—
278	C₂²×C₆₇	—	4	4	no	—
279	C₂²×C₆₇	—	4	2	no	—
280	C₂²×C₆₇	—	4	8	no	—
281	C₂²×C₆₇	—	8	1	no	—
282	C₂²×C₆₇	—	8	1	no	—
283	C₂²×C₆₇	—	8	2	no	—
284	C₂²×C₆₇	—	8	1	no	—
285	C₂²×C₆₇	—	8	2	no	—
286	C₂²×C₆₇	#449	1	4	yes	complemented
287	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
288	C₂²×C₆₇	C₂×#120	1	2	yes	—
289	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
290	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
291	C₂²×C₆₇	#575	1	2	yes	—
292	C₄×C₆₇	—	2	4	no	—
293	C₄×C₆₇	—	2	8	no	—
294	C₄×C₆₇	—	2	2	no	—
295	C₄×C₆₇	—	8	1	no	—
296	C₄×C₆₇	—	8	2	no	—
297	C₄×C₆₇	C₂×#120	1	2	yes	—
298	C₄×C₆₇	C₂×#120	1	2	yes	—
299	#32880	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
300	C₂×C₆₇	—	2	4	no	—
301	C₂×C₆₇	—	4	4	no	—
302	C₂×C₆₇	—	4	8	no	—
303	C₂×C₆₇	—	16	1	no	—
304	C₂×C₆₇	#2846	1	1	yes	characteristic
305	C₂×C₆₇	C₂×#449	1	1	yes	characteristic
306	C₂×C₆₇	#3002	1	2	yes	—
307	C₂×C₆₇	#3005	1	1	yes	characteristic
308	C₂×C₆₇	#3006	1	2	yes	—
309	C₂³×#41	C₂×C₆₇	1	1	yes	characteristic, complemented
310	C₂×#449	C₂×C₆₇	1	4	yes	complemented
311	C₂×#448	C₂×C₆₇	1	1	yes	characteristic, complemented
312	C₂×#447	C₂×C₆₇	1	1	yes	characteristic, complemented
313	#2847	C₂×C₆₇	1	1	yes	characteristic, complemented
314	#1788	C₂×C₆₇	1	4	yes	complemented
315	#1787	C₂×C₆₇	1	1	yes	characteristic, complemented
316	#1786	C₂×C₆₇	1	1	yes	characteristic, complemented
317	#1779	C₂×C₆₇	1	1	yes	characteristic, complemented
318	C₆₇	#32880	1	1	yes	characteristic, direct product factor, Sylow
319	C₂²×#41	—	2	16	no	—
320	C₂²×#41	—	2	4	no	—
321	C₂²×#41	—	2	4	no	—
322	C₂²×#41	—	2	2	no	—
323	C₂²×#41	C₂²×C₆₇	1	4	yes	—
324	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
325	C₂³×D₄	—	2	1	no	—
326	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
327	C₂⁴×C₄	—	2	1	no	—
328	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
329	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	complemented
330	C₂²×D₈	C₂²×C₆₇	1	2	yes	complemented
331	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
332	C₂×#121	C₂²×C₆₇	1	2	yes	complemented
333	C₂×#120	—	2	1	no	—
334	#449	—	2	16	no	—
335	#448	—	2	4	no	—
336	#447	—	2	4	no	—
337	C₂×#103	C₂²×C₆₇	1	2	yes	complemented
338	C₂×#103	C₂²×C₆₇	1	4	yes	complemented
339	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
340	C₂×#102	C₂²×C₆₇	1	4	yes	complemented
341	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
342	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
343	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	complemented
344	#386	—	2	2	no	—
345	#386	—	2	1	no	—
346	#380	C₂²×C₆₇	1	1	yes	characteristic
347	#380	C₂²×C₆₇	1	2	yes	complemented
348	C₂⁵	—	2	1	no	—
349	C₂⁵	D₄×C₆₇	1	1	yes	characteristic
350	C₂×#41	—	2	64	no	—
351	C₂×#41	—	2	64	no	—
352	C₂×#41	—	2	16	no	—
353	C₂×#41	—	2	16	no	—
354	C₂×#41	—	2	16	no	—
355	C₂×#41	—	2	8	no	—
356	C₂×#41	—	2	8	no	—
357	C₂×#41	—	2	8	no	—
358	C₂×#41	—	2	8	no	—
359	C₂×#41	—	2	8	no	—
360	C₂×#41	D₄×C₆₇	1	16	yes	complemented
361	C₂²×Q₈	—	2	8	no	—
362	C₂²×Q₈	—	2	2	no	—
363	C₂²×Q₈	—	2	1	no	—
364	C₂²×Q₈	D₄×C₆₇	1	4	yes	complemented
365	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
366	C₂²×D₄	—	2	16	no	—
367	C₂²×D₄	—	2	8	no	—
368	C₂²×D₄	—	2	4	no	—
369	C₂²×D₄	—	2	8	no	—
370	C₂²×D₄	—	2	1	no	—
371	C₂²×D₄	—	2	2	no	—
372	C₂²×D₄	—	2	1	no	—
373	C₂²×D₄	—	2	2	no	—
374	C₂²×D₄	D₄×C₆₇	1	4	yes	complemented
375	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
376	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
377	C₂³×C₄	—	2	16	no	—
378	C₂³×C₄	—	2	8	no	—
379	C₂³×C₄	—	2	1	no	—
380	C₂³×C₄	—	2	4	no	—
381	C₂³×C₄	—	2	2	no	—
382	C₂³×C₄	—	2	2	no	—
383	C₂³×C₄	—	2	8	no	—
384	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
385	C₂³×C₄	D₄×C₆₇	1	2	yes	—
386	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
387	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
388	C₂×SD₁₆	—	2	4	no	—
389	C₂×SD₁₆	—	2	8	no	—
390	C₂×D₈	—	2	4	no	—
391	C₂×D₈	—	2	8	no	—
392	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
393	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
394	#121	—	2	2	no	—
395	#121	—	2	2	no	—
396	#121	—	2	8	no	—
397	#120	—	4	4	no	—
398	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
399	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
400	C₂×C₂²⋊C₄	—	2	1	no	—
401	C₂×C₂²⋊C₄	—	2	2	no	—
402	C₂×C₂²⋊C₄	—	2	1	no	—
403	C₂×C₂²⋊C₄	—	2	2	no	—
404	#103	—	2	4	no	—
405	#103	—	2	8	no	—
406	#102	—	2	4	no	—
407	#102	—	2	8	no	—
408	C₂²⋊C₈	—	2	2	no	—
409	C₂²⋊C₈	—	2	2	no	—
410	C₂²⋊C₈	—	2	8	no	—
411	#95	—	2	1	no	—
412	#95	—	2	2	no	—
413	C₂⁴	—	2	8	no	—
414	C₂⁴	—	2	4	no	—
415	C₂⁴	—	2	2	no	—
416	C₂⁴	—	2	4	no	—
417	C₂⁴	—	2	2	no	—
418	C₂⁴	—	2	2	no	—
419	C₂⁴	—	2	2	no	—
420	C₂⁴	—	2	1	no	—
421	C₂⁴	—	4	2	no	—
422	C₂⁴	—	4	4	no	—
423	C₂⁴	—	4	2	no	—
424	C₂⁴	—	4	2	no	—
425	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
426	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
427	#41	—	2	128	no	—
428	#41	—	2	32	no	—
429	#41	—	2	32	no	—
430	#41	—	2	32	no	—
431	#41	—	2	32	no	—
432	C₂×Q₈	—	2	16	no	—
433	C₂×Q₈	—	2	8	no	—
434	C₂×Q₈	—	2	16	no	—
435	C₂×Q₈	—	2	4	no	—
436	C₂×Q₈	—	2	4	no	—
437	C₂×Q₈	—	2	2	no	—
438	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	—
439	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
440	C₂×D₄	—	2	64	no	—
441	C₂×D₄	—	2	16	no	—
442	C₂×D₄	—	2	16	no	—
443	C₂×D₄	—	2	8	no	—
444	C₂×D₄	—	2	16	no	—
445	C₂×D₄	—	2	16	no	—
446	C₂×D₄	—	2	16	no	—
447	C₂×D₄	—	2	4	no	—
448	C₂×D₄	—	2	2	no	—
449	C₂×D₄	—	2	4	no	—
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	—	2	4	no	—
452	C₂×D₄	—	2	1	no	—
453	C₂×D₄	—	4	4	no	—
454	C₂×D₄	—	4	4	no	—
455	C₂×D₄	—	4	4	no	—
456	C₂×D₄	—	4	2	no	—
457	C₂×D₄	—	4	2	no	—
458	C₂×D₄	—	4	2	no	—
459	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
460	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
461	C₂²×C₄	—	2	8	no	—
462	C₂²×C₄	—	2	64	no	—
463	C₂²×C₄	—	2	16	no	—
464	C₂²×C₄	—	2	16	no	—
465	C₂²×C₄	—	2	16	no	—
466	C₂²×C₄	—	2	16	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	4	no	—
469	C₂²×C₄	—	2	4	no	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	—	2	4	no	—
472	C₂²×C₄	—	2	16	no	—
473	C₂²×C₄	—	2	4	no	—
474	C₂²×C₄	—	2	4	no	—
475	C₂²×C₄	—	2	4	no	—
476	C₂²×C₄	—	2	2	no	—
477	C₂²×C₄	—	2	4	no	—
478	C₂²×C₄	—	2	2	no	—
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	2	no	—
481	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
482	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
483	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
484	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
485	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
486	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
487	SD₁₆	—	4	8	no	—
488	D₈	—	4	8	no	—
489	C₂×C₈	—	2	2	no	—
490	C₂×C₈	—	2	4	no	—
491	C₂×C₈	—	2	4	no	—
492	C₂×C₈	—	2	2	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	4	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₂²⋊C₄	—	4	4	no	—
497	C₂²⋊C₄	—	4	2	no	—
498	C₂²⋊C₄	—	4	4	no	—
499	C₂²⋊C₄	—	4	2	no	—
500	C₂³	—	2	2	no	—
501	C₂³	—	2	2	no	—
502	C₂³	—	2	8	no	—
503	C₂³	—	2	16	no	—
504	C₂³	—	2	2	no	—
505	C₂³	—	2	2	no	—
506	C₂³	—	2	4	no	—
507	C₂³	—	2	8	no	—
508	C₂³	—	2	2	no	—
509	C₂³	—	2	2	no	—
510	C₂³	—	2	2	no	—
511	C₂³	—	2	2	no	—
512	C₂³	—	4	8	no	—
513	C₂³	—	4	8	no	—
514	C₂³	—	4	16	no	—
515	C₂³	—	4	4	no	—
516	C₂³	—	4	8	no	—
517	C₂³	—	4	16	no	—
518	C₂³	—	4	4	no	—
519	C₂³	—	4	8	no	—
520	C₂³	—	4	2	no	—
521	C₂³	—	4	2	no	—
522	C₂³	—	4	4	no	—
523	C₂³	—	4	1	no	—
524	C₂³	—	4	2	no	—
525	C₂³	—	4	1	no	—
526	C₂³	—	4	1	no	—
527	C₂³	—	4	2	no	—
528	C₂³	#120×C₆₇	1	4	yes	—
529	C₂³	#120×C₆₇	1	2	yes	—
530	C₂³	#135×C₆₇	1	2	yes	—
531	C₂³	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
532	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
533	Q₈	—	2	16	no	—
534	Q₈	—	2	4	no	—
535	Q₈	—	2	8	no	—
536	Q₈	#120×C₆₇	1	8	yes	complemented
537	D₄	—	2	64	no	—
538	D₄	—	2	16	no	—
539	D₄	—	2	8	no	—
540	D₄	—	2	4	no	—
541	D₄	—	4	2	no	—
542	D₄	—	4	2	no	—
543	D₄	—	8	8	no	—
544	D₄	—	8	4	no	—
545	D₄	#120×C₆₇	1	8	yes	complemented
546	C₂×C₄	—	2	8	no	—
547	C₂×C₄	—	2	16	no	—
548	C₂×C₄	—	2	64	no	—
549	C₂×C₄	—	2	16	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	8	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	2	4	no	—
557	C₂×C₄	—	2	8	no	—
558	C₂×C₄	—	2	8	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	2	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	1	no	—
563	C₂×C₄	—	4	2	no	—
564	C₂×C₄	—	4	2	no	—
565	C₂×C₄	#120×C₆₇	1	4	yes	—
566	C₂×C₄	#120×C₆₇	1	2	yes	—
567	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
568	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
569	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
570	C₈	—	4	2	no	—
571	C₈	—	4	2	no	—
572	C₂²	—	2	2	no	—
573	C₂²	—	2	4	no	—
574	C₂²	—	2	2	no	—
575	C₂²	—	2	4	no	—
576	C₂²	—	2	4	no	—
577	C₂²	—	2	8	no	—
578	C₂²	—	2	4	no	—
579	C₂²	—	2	4	no	—
580	C₂²	—	4	16	no	—
581	C₂²	—	4	32	no	—
582	C₂²	—	4	2	no	—
583	C₂²	—	4	4	no	—
584	C₂²	—	4	2	no	—
585	C₂²	—	4	8	no	—
586	C₂²	—	4	2	no	—
587	C₂²	—	4	4	no	—
588	C₂²	—	4	2	no	—
589	C₂²	—	4	8	no	—
590	C₂²	—	8	1	no	—
591	C₂²	—	8	1	no	—
592	C₂²	—	8	2	no	—
593	C₂²	—	8	1	no	—
594	C₂²	—	8	2	no	—
595	C₂²	#449×C₆₇	1	4	yes	complemented
596	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
597	C₂²	C₂×#120×C₆₇	1	2	yes	—
598	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
599	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
600	C₂²	#575×C₆₇	1	2	yes	—
601	C₄	—	2	4	no	—
602	C₄	—	2	8	no	—
603	C₄	—	2	2	no	—
604	C₄	—	8	1	no	—
605	C₄	—	8	2	no	—
606	C₄	C₂×#120×C₆₇	1	2	yes	—
607	C₄	C₂×#120×C₆₇	1	2	yes	—
608	C₂	—	2	4	no	—
609	C₂	—	4	4	no	—
610	C₂	—	4	8	no	—
611	C₂	—	16	1	no	—
612	C₂	C₆₇×#2846	1	1	yes	characteristic
613	C₂	C₂×#449×C₆₇	1	1	yes	characteristic
614	C₂	C₆₇×#3002	1	2	yes	—
615	C₂	C₆₇×#3005	1	1	yes	characteristic
616	C₂	C₆₇×#3006	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 ◁ C₂⁴×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×#41×C₆₇	C₂	semidirect
2. C₂×#449×C₆₇	C₂	semidirect
3. C₂×#448×C₆₇	C₂	semidirect
4. C₂×#447×C₆₇	C₂	semidirect
5. C₆₇×#2847	C₂	semidirect
6. C₆₇×#1788	C₂	semidirect
7. C₆₇×#1787	C₂	semidirect
8. C₆₇×#1786	C₂	semidirect
9. C₆₇×#1779	C₂	semidirect
20. C₂²×SD₁₆×C₆₇	C₂²	semidirect
21. C₂²×D₈×C₆₇	C₂²	semidirect
23. C₂×#121×C₆₇	C₂²	semidirect
28. C₂×#103×C₆₇	C₂²	semidirect
29. C₂×#103×C₆₇	C₂²	semidirect
30. C₂×#102×C₆₇	C₂²	semidirect
31. C₂×#102×C₆₇	C₂²	semidirect
34. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
38. #380×C₆₇	C₂²	semidirect
51. C₂×#41×C₆₇	D₄	semidirect
55. C₂²×Q₈×C₆₇	D₄	semidirect
65. C₂²×D₄×C₆₇	D₄	semidirect
227. Q₈×C₆₇	#120	semidirect
236. D₄×C₆₇	#120	semidirect
286. C₂²×C₆₇	#449	semidirect
299. #32880	C₆₇	direct (N × H)
309. C₂³×#41	C₂×C₆₇	semidirect
310. C₂×#449	C₂×C₆₇	semidirect
311. C₂×#448	C₂×C₆₇	semidirect
312. C₂×#447	C₂×C₆₇	semidirect
313. #2847	C₂×C₆₇	semidirect
314. #1788	C₂×C₆₇	semidirect
315. #1787	C₂×C₆₇	semidirect
316. #1786	C₂×C₆₇	semidirect
317. #1779	C₂×C₆₇	semidirect
318. C₆₇	#32880	direct (N × H)
329. C₂²×SD₁₆	C₂²×C₆₇	semidirect
330. C₂²×D₈	C₂²×C₆₇	semidirect
332. C₂×#121	C₂²×C₆₇	semidirect
337. C₂×#103	C₂²×C₆₇	semidirect
338. C₂×#103	C₂²×C₆₇	semidirect
339. C₂×#102	C₂²×C₆₇	semidirect
340. C₂×#102	C₂²×C₆₇	semidirect
343. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
347. #380	C₂²×C₆₇	semidirect
360. C₂×#41	D₄×C₆₇	semidirect
364. C₂²×Q₈	D₄×C₆₇	semidirect
374. C₂²×D₄	D₄×C₆₇	semidirect
536. Q₈	#120×C₆₇	semidirect
545. D₄	#120×C₆₇	semidirect
595. C₂²	#449×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×#41×C₆₇	2
C₂×#449×C₆₇	2
C₂×#448×C₆₇	2
C₂×#447×C₆₇	2
C₆₇×#2847	2
C₆₇×#1788	2
C₆₇×#1787	2
C₆₇×#1786	2
C₆₇×#1779	2
#32880	67

Subgroups — C67×#32880 #30712131

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32880 #30712131