Subgroups — C₆₇×#2491 #22556922

Order 8576.

632 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#124×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#121×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#394×C₆₇	C₂	1	2	yes	maximal, complemented
5	#393×C₆₇	C₂	1	2	yes	maximal, complemented
6	#390×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#390×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#388×C₆₇	C₂	1	2	yes	maximal, complemented
9	#380×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₆₇	C₂²	1	2	yes	—
13	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₆₇	C₂²	1	2	yes	—
15	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₆₇	—	2	4	no	—
17	#124×C₆₇	—	2	8	no	—
18	#121×C₆₇	—	2	4	no	—
19	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₆₇	C₂²	1	2	yes	—
32	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
34	#95×C₆₇	C₂²	1	1	yes	characteristic
35	#95×C₆₇	C₂²	1	1	yes	characteristic
36	#95×C₆₇	C₂²	1	2	yes	—
37	#95×C₆₇	C₂²	1	2	yes	—
38	C₂⁴×C₆₇	C₂³	1	2	yes	—
39	C₂⁴×C₆₇	C₂³	1	1	yes	characteristic
40	C₂×Q₈×C₆₇	—	2	4	no	—
41	C₂×Q₈×C₆₇	—	2	4	no	—
42	C₂×Q₈×C₆₇	—	4	2	no	—
43	C₂×D₄×C₆₇	—	2	4	no	—
44	C₂×D₄×C₆₇	—	2	4	no	—
45	C₂×D₄×C₆₇	—	2	2	no	—
46	C₂×D₄×C₆₇	—	2	2	no	—
47	C₂×D₄×C₆₇	—	2	2	no	—
48	C₂×D₄×C₆₇	—	4	2	no	—
49	C₂²×C₄×C₆₇	—	2	2	no	—
50	C₂²×C₄×C₆₇	—	2	2	no	—
51	C₂²×C₄×C₆₇	—	2	2	no	—
52	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
55	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
59	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₆₇	—	2	4	no	—
62	C₄⋊C₄×C₆₇	—	2	2	no	—
63	C₄⋊C₄×C₆₇	—	2	2	no	—
64	C₄⋊C₄×C₆₇	D₄	1	4	yes	—
65	C₂²⋊C₄×C₆₇	—	2	4	no	—
66	C₂²⋊C₄×C₆₇	—	2	4	no	—
67	C₂²⋊C₄×C₆₇	—	2	4	no	—
68	C₂²⋊C₄×C₆₇	—	2	4	no	—
69	C₂²⋊C₄×C₆₇	—	2	2	no	—
70	C₂²⋊C₄×C₆₇	—	2	4	no	—
71	C₂²⋊C₄×C₆₇	—	4	2	no	—
72	C₂²⋊C₄×C₆₇	D₄	1	4	yes	complemented
73	C₄²×C₆₇	—	2	4	no	—
74	C₄²×C₆₇	—	2	2	no	—
75	C₄²×C₆₇	—	2	2	no	—
76	C₂³×C₆₇	—	2	1	no	—
77	C₂³×C₆₇	—	2	1	no	—
78	C₂³×C₆₇	—	2	1	no	—
79	C₂³×C₆₇	—	2	1	no	—
80	C₂³×C₆₇	—	2	1	no	—
81	C₂³×C₆₇	—	2	1	no	—
82	C₂³×C₆₇	—	2	2	no	—
83	C₂³×C₆₇	—	2	2	no	—
84	C₂³×C₆₇	—	2	2	no	—
85	C₂³×C₆₇	—	2	2	no	—
86	C₂³×C₆₇	—	2	2	no	—
87	C₂³×C₆₇	—	2	2	no	—
88	C₂³×C₆₇	—	2	2	no	—
89	C₂³×C₆₇	C₂×D₄	1	2	yes	—
90	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
91	Q₈×C₆₇	—	4	8	no	—
92	D₄×C₆₇	—	4	8	no	—
93	D₄×C₆₇	—	4	4	no	—
94	C₂×C₄×C₆₇	—	2	1	no	—
95	C₂×C₄×C₆₇	—	2	2	no	—
96	C₂×C₄×C₆₇	—	2	1	no	—
97	C₂×C₄×C₆₇	—	2	1	no	—
98	C₂×C₄×C₆₇	—	2	2	no	—
99	C₂×C₄×C₆₇	—	2	1	no	—
100	C₂×C₄×C₆₇	—	2	1	no	—
101	C₂×C₄×C₆₇	—	2	1	no	—
102	C₂×C₄×C₆₇	—	2	2	no	—
103	C₂×C₄×C₆₇	—	2	1	no	—
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	2	1	no	—
109	C₂×C₄×C₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	2	1	no	—
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	—	2	1	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	4	4	no	—
115	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
117	C₂×C₄×C₆₇	C₂×D₄	1	4	yes	—
118	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
119	C₂×C₄×C₆₇	#41	1	2	yes	—
120	C₂×C₄×C₆₇	#41	1	2	yes	—
121	C₂×C₄×C₆₇	#41	1	2	yes	—
122	C₂²×C₆₇	—	2	2	no	—
123	C₂²×C₆₇	—	2	2	no	—
124	C₂²×C₆₇	—	2	2	no	—
125	C₂²×C₆₇	—	4	1	no	—
126	C₂²×C₆₇	—	4	1	no	—
127	C₂²×C₆₇	—	4	1	no	—
128	C₂²×C₆₇	—	4	1	no	—
129	C₂²×C₆₇	—	4	2	no	—
130	C₂²×C₆₇	—	4	2	no	—
131	C₂²×C₆₇	—	4	2	no	—
132	C₂²×C₆₇	—	4	2	no	—
133	C₂²×C₆₇	—	4	2	no	—
134	C₂²×C₆₇	—	4	2	no	—
135	C₂²×C₆₇	—	4	2	no	—
136	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
137	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
138	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
139	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
140	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
141	C₂²×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
142	C₂²×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
143	C₄×C₆₇	—	2	4	no	—
144	C₄×C₆₇	—	2	2	no	—
145	C₄×C₆₇	—	2	2	no	—
146	C₄×C₆₇	—	2	2	no	—
147	C₄×C₆₇	—	4	1	no	—
148	C₄×C₆₇	—	4	2	no	—
149	C₄×C₆₇	—	4	2	no	—
150	C₄×C₆₇	—	4	1	no	—
151	C₄×C₆₇	#124	1	4	yes	complemented
152	C₂×C₆₇	—	4	2	no	—
153	C₂×C₆₇	—	8	2	no	—
154	C₂×C₆₇	C₂×#124	1	1	yes	characteristic
155	C₂×C₆₇	#532	1	1	yes	characteristic
156	C₂×C₆₇	#542	1	1	yes	characteristic
157	C₂×C₆₇	D₄²	1	1	yes	characteristic
158	C₂×C₆₇	#547	1	1	yes	characteristic
159	C₂×C₆₇	#548	1	1	yes	characteristic
160	C₂×C₆₇	#555	1	1	yes	characteristic
161	#2491	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₆₇	#2491	1	1	yes	characteristic, direct product factor, Sylow
163	C₂×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
164	C₂×#124	C₂×C₆₇	1	2	yes	complemented
165	C₂×#121	C₂×C₆₇	1	1	yes	characteristic
166	#394	C₂×C₆₇	1	2	yes	complemented
167	#393	C₂×C₆₇	1	2	yes	complemented
168	#390	C₂×C₆₇	1	1	yes	characteristic, complemented
169	#390	C₂×C₆₇	1	1	yes	characteristic, complemented
170	#388	C₂×C₆₇	1	2	yes	complemented
171	#380	C₂×C₆₇	1	1	yes	characteristic, complemented
172	C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
173	C₄×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
174	C₂²×Q₈	C₂²×C₆₇	1	2	yes	—
175	C₂²×D₄	C₂²×C₆₇	1	1	yes	characteristic
176	C₂²×D₄	C₂²×C₆₇	1	2	yes	—
177	C₂³×C₄	C₂²×C₆₇	1	1	yes	characteristic
178	C₄⋊Q₈	—	2	4	no	—
179	#124	—	2	8	no	—
180	#121	—	2	4	no	—
181	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
182	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
183	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
185	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
186	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
187	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
188	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
189	C₂×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
190	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
191	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
192	C₂×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
193	C₂×C₄²	C₂²×C₆₇	1	2	yes	—
194	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
195	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
196	#95	C₂²×C₆₇	1	1	yes	characteristic
197	#95	C₂²×C₆₇	1	1	yes	characteristic
198	#95	C₂²×C₆₇	1	2	yes	—
199	#95	C₂²×C₆₇	1	2	yes	—
200	C₂⁴	C₂³×C₆₇	1	2	yes	—
201	C₂⁴	C₂³×C₆₇	1	1	yes	characteristic
202	C₂×Q₈	—	2	4	no	—
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	—	4	2	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	2	4	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
215	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
216	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
217	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
218	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
219	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
220	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
221	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
222	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	D₄×C₆₇	1	4	yes	—
227	C₂²⋊C₄	—	2	4	no	—
228	C₂²⋊C₄	—	2	4	no	—
229	C₂²⋊C₄	—	2	4	no	—
230	C₂²⋊C₄	—	2	4	no	—
231	C₂²⋊C₄	—	2	2	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	—	4	2	no	—
234	C₂²⋊C₄	D₄×C₆₇	1	4	yes	complemented
235	C₄²	—	2	4	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	2	2	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	2	1	no	—
241	C₂³	—	2	1	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	C₂×D₄×C₆₇	1	2	yes	—
252	C₂³	C₂⁴×C₆₇	1	1	yes	characteristic
253	Q₈	—	4	8	no	—
254	D₄	—	4	8	no	—
255	D₄	—	4	4	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
278	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
279	C₂×C₄	C₂×D₄×C₆₇	1	4	yes	—
280	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
281	C₂×C₄	#41×C₆₇	1	2	yes	—
282	C₂×C₄	#41×C₆₇	1	2	yes	—
283	C₂×C₄	#41×C₆₇	1	2	yes	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	2	no	—
298	C₂²	C₂²×D₄×C₆₇	1	1	yes	characteristic
299	C₂²	C₂²×D₄×C₆₇	1	1	yes	characteristic
300	C₂²	C₂×#41×C₆₇	1	1	yes	characteristic
301	C₂²	C₂×#41×C₆₇	1	1	yes	characteristic
302	C₂²	C₂×#41×C₆₇	1	1	yes	characteristic
303	C₂²	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
304	C₂²	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
305	C₄	—	2	4	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	2	no	—
312	C₄	—	4	1	no	—
313	C₄	#124×C₆₇	1	4	yes	complemented
314	C₂	—	4	2	no	—
315	C₂	—	8	2	no	—
316	C₂	C₂×#124×C₆₇	1	1	yes	characteristic
317	C₂	#532×C₆₇	1	1	yes	characteristic
318	C₂	#542×C₆₇	1	1	yes	characteristic
319	C₂	D₄²×C₆₇	1	1	yes	characteristic
320	C₂	#547×C₆₇	1	1	yes	characteristic
321	C₂	#548×C₆₇	1	1	yes	characteristic
322	C₂	#555×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₆₇
prime radical	C₂×#121×C₆₇	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊Q₈×C₆₇	C₂	semidirect
2. C₂×#124×C₆₇	C₂	semidirect
4. #394×C₆₇	C₂	semidirect
5. #393×C₆₇	C₂	semidirect
6. #390×C₆₇	C₂	semidirect
7. #390×C₆₇	C₂	semidirect
8. #388×C₆₇	C₂	semidirect
9. #380×C₆₇	C₂	semidirect
10. C₄×C₄⋊C₄×C₆₇	C₂	semidirect
11. C₄×C₂²⋊C₄×C₆₇	C₂	semidirect
72. C₂²⋊C₄×C₆₇	D₄	semidirect
151. C₄×C₆₇	#124	semidirect
161. #2491	C₆₇	direct (N × H)
162. C₆₇	#2491	direct (N × H)
163. C₂×C₄⋊Q₈	C₂×C₆₇	semidirect
164. C₂×#124	C₂×C₆₇	semidirect
166. #394	C₂×C₆₇	semidirect
167. #393	C₂×C₆₇	semidirect
168. #390	C₂×C₆₇	semidirect
169. #390	C₂×C₆₇	semidirect
170. #388	C₂×C₆₇	semidirect
171. #380	C₂×C₆₇	semidirect
172. C₄×C₄⋊C₄	C₂×C₆₇	semidirect
173. C₄×C₂²⋊C₄	C₂×C₆₇	semidirect
234. C₂²⋊C₄	D₄×C₆₇	semidirect
313. C₄	#124×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Q₈×C₆₇	2
C₂×#124×C₆₇	2
C₂×#121×C₆₇	2
#394×C₆₇	2
#393×C₆₇	2
#390×C₆₇	2
#388×C₆₇	2
#380×C₆₇	2
C₄×C₄⋊C₄×C₆₇	2
C₄×C₂²⋊C₄×C₆₇	2
#2491	67

Subgroups — C67×#2491 #22556922

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#2491 #22556922