Subgroups — C₆₇×#2408 #22556839

Order 8576.

1232 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#121×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#120×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#394×C₆₇	C₂	1	2	yes	maximal, complemented
6	#392×C₆₇	C₂	1	2	yes	maximal, complemented
7	#390×C₆₇	C₂	1	2	yes	maximal, complemented
8	#380×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#379×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂⁵×C₆₇	C₂²	1	2	yes	—
11	C₂²×D₄×C₆₇	—	2	8	no	—
12	C₂²×D₄×C₆₇	—	2	4	no	—
13	C₂²×D₄×C₆₇	C₂²	1	2	yes	—
14	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×D₄×C₆₇	C₂²	1	2	yes	—
16	C₂²×D₄×C₆₇	C₂²	1	4	yes	complemented
17	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
19	#121×C₆₇	—	2	8	no	—
20	#120×C₆₇	—	2	8	no	—
21	C₄×D₄×C₆₇	—	2	4	no	—
22	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
24	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
27	C₂×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	complemented
28	C₂×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	complemented
29	C₂×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	complemented
30	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic, complemented
31	#95×C₆₇	C₂²	1	2	yes	complemented
32	C₂⁴×C₆₇	—	2	8	no	—
33	C₂⁴×C₆₇	—	2	4	no	—
34	C₂⁴×C₆₇	—	2	4	no	—
35	C₂⁴×C₆₇	—	2	4	no	—
36	C₂⁴×C₆₇	—	4	4	no	—
37	C₂⁴×C₆₇	C₂³	1	4	yes	—
38	C₂⁴×C₆₇	C₂³	1	1	yes	characteristic
39	C₂⁴×C₆₇	C₂³	1	1	yes	characteristic
40	C₂×D₄×C₆₇	—	2	32	no	—
41	C₂×D₄×C₆₇	—	2	2	no	—
42	C₂×D₄×C₆₇	—	2	8	no	—
43	C₂×D₄×C₆₇	—	2	2	no	—
44	C₂×D₄×C₆₇	—	2	8	no	—
45	C₂×D₄×C₆₇	—	2	8	no	—
46	C₂×D₄×C₆₇	—	2	4	no	—
47	C₂×D₄×C₆₇	—	2	4	no	—
48	C₂×D₄×C₆₇	—	2	4	no	—
49	C₂×D₄×C₆₇	—	4	4	no	—
50	C₂×D₄×C₆₇	D₄	1	8	yes	complemented
51	C₂×D₄×C₆₇	D₄	1	4	yes	complemented
52	C₂²×C₄×C₆₇	—	2	4	no	—
53	C₂²×C₄×C₆₇	—	2	2	no	—
54	C₂²×C₄×C₆₇	—	2	4	no	—
55	C₂²×C₄×C₆₇	D₄	1	4	yes	complemented
56	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
59	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₆₇	—	2	2	no	—
63	C₄⋊C₄×C₆₇	—	2	4	no	—
64	C₂²⋊C₄×C₆₇	—	2	2	no	—
65	C₂²⋊C₄×C₆₇	—	2	4	no	—
66	C₂²⋊C₄×C₆₇	—	2	8	no	—
67	C₂²⋊C₄×C₆₇	—	2	2	no	—
68	C₂²⋊C₄×C₆₇	—	2	8	no	—
69	C₂²⋊C₄×C₆₇	—	4	4	no	—
70	C₄²×C₆₇	—	2	2	no	—
71	C₂³×C₆₇	—	2	2	no	—
72	C₂³×C₆₇	—	2	2	no	—
73	C₂³×C₆₇	—	2	16	no	—
74	C₂³×C₆₇	—	2	1	no	—
75	C₂³×C₆₇	—	2	4	no	—
76	C₂³×C₆₇	—	2	2	no	—
77	C₂³×C₆₇	—	2	4	no	—
78	C₂³×C₆₇	—	2	4	no	—
79	C₂³×C₆₇	—	2	1	no	—
80	C₂³×C₆₇	—	2	4	no	—
81	C₂³×C₆₇	—	2	4	no	—
82	C₂³×C₆₇	—	2	2	no	—
83	C₂³×C₆₇	—	2	4	no	—
84	C₂³×C₆₇	—	4	16	no	—
85	C₂³×C₆₇	—	4	16	no	—
86	C₂³×C₆₇	—	4	8	no	—
87	C₂³×C₆₇	—	4	8	no	—
88	C₂³×C₆₇	C₂×D₄	1	8	yes	—
89	C₂³×C₆₇	C₂×D₄	1	2	yes	—
90	C₂³×C₆₇	C₂×D₄	1	2	yes	—
91	C₂³×C₆₇	C₂×D₄	1	2	yes	complemented
92	C₂³×C₆₇	#41	1	2	yes	—
93	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
94	D₄×C₆₇	—	2	16	no	—
95	D₄×C₆₇	—	2	8	no	—
96	D₄×C₆₇	—	2	16	no	—
97	D₄×C₆₇	—	4	16	no	—
98	C₂×C₄×C₆₇	—	2	1	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	2	1	no	—
101	C₂×C₄×C₆₇	—	2	8	no	—
102	C₂×C₄×C₆₇	—	2	1	no	—
103	C₂×C₄×C₆₇	—	2	2	no	—
104	C₂×C₄×C₆₇	—	2	4	no	—
105	C₂×C₄×C₆₇	—	2	1	no	—
106	C₂×C₄×C₆₇	—	2	4	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	2	4	no	—
109	C₂×C₄×C₆₇	—	4	4	no	—
110	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
112	C₂×C₄×C₆₇	C₂×D₄	1	4	yes	—
113	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	complemented
114	C₂²×C₆₇	—	2	2	no	—
115	C₂²×C₆₇	—	2	2	no	—
116	C₂²×C₆₇	—	2	4	no	—
117	C₂²×C₆₇	—	2	4	no	—
118	C₂²×C₆₇	—	2	8	no	—
119	C₂²×C₆₇	—	2	2	no	—
120	C₂²×C₆₇	—	2	4	no	—
121	C₂²×C₆₇	—	2	8	no	—
122	C₂²×C₆₇	—	2	8	no	—
123	C₂²×C₆₇	—	4	32	no	—
124	C₂²×C₆₇	—	4	1	no	—
125	C₂²×C₆₇	—	4	4	no	—
126	C₂²×C₆₇	—	4	1	no	—
127	C₂²×C₆₇	—	4	4	no	—
128	C₂²×C₆₇	—	4	4	no	—
129	C₂²×C₆₇	—	4	2	no	—
130	C₂²×C₆₇	—	4	4	no	—
131	C₂²×C₆₇	#121	1	4	yes	complemented
132	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
133	C₂²×C₆₇	C₂²×D₄	1	2	yes	—
134	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
135	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
136	C₂²×C₆₇	H(5,𝔽₂)	1	2	yes	—
137	C₄×C₆₇	—	2	2	no	—
138	C₄×C₆₇	—	2	2	no	—
139	C₄×C₆₇	—	4	1	no	—
140	C₄×C₆₇	—	4	1	no	—
141	C₄×C₆₇	—	4	4	no	—
142	C₄×C₆₇	#120	1	4	yes	complemented
143	C₂×C₆₇	—	2	4	no	—
144	C₂×C₆₇	—	4	2	no	—
145	C₂×C₆₇	—	4	8	no	—
146	C₂×C₆₇	C₂×#120	1	1	yes	characteristic
147	C₂×C₆₇	C₂×#121	1	1	yes	characteristic
148	C₂×C₆₇	#535	1	1	yes	characteristic
149	C₂×C₆₇	D₄²	1	2	yes	—
150	C₂×C₆₇	#546	1	2	yes	—
151	#2408	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₆₇	#2408	1	1	yes	characteristic, direct product factor, Sylow
153	C₂³×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
154	C₂×#121	C₂×C₆₇	1	2	yes	complemented
155	C₂×#120	C₂×C₆₇	1	2	yes	complemented
156	C₂×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
157	#394	C₂×C₆₇	1	2	yes	complemented
158	#392	C₂×C₆₇	1	2	yes	complemented
159	#390	C₂×C₆₇	1	2	yes	complemented
160	#380	C₂×C₆₇	1	1	yes	characteristic, complemented
161	#379	C₂×C₆₇	1	2	yes	complemented
162	C₂⁵	C₂²×C₆₇	1	2	yes	—
163	C₂²×D₄	—	2	8	no	—
164	C₂²×D₄	—	2	4	no	—
165	C₂²×D₄	C₂²×C₆₇	1	2	yes	—
166	C₂²×D₄	C₂²×C₆₇	1	1	yes	characteristic
167	C₂²×D₄	C₂²×C₆₇	1	2	yes	—
168	C₂²×D₄	C₂²×C₆₇	1	4	yes	complemented
169	C₂³×C₄	C₂²×C₆₇	1	1	yes	characteristic
170	C₂³×C₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
171	#121	—	2	8	no	—
172	#120	—	2	8	no	—
173	C₄×D₄	—	2	4	no	—
174	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
175	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
176	C₂×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
177	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
178	C₂×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
179	C₂×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	complemented
180	C₂×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	complemented
181	C₂×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	complemented
182	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic, complemented
183	#95	C₂²×C₆₇	1	2	yes	complemented
184	C₂⁴	—	2	8	no	—
185	C₂⁴	—	2	4	no	—
186	C₂⁴	—	2	4	no	—
187	C₂⁴	—	2	4	no	—
188	C₂⁴	—	4	4	no	—
189	C₂⁴	C₂³×C₆₇	1	4	yes	—
190	C₂⁴	C₂³×C₆₇	1	1	yes	characteristic
191	C₂⁴	C₂³×C₆₇	1	1	yes	characteristic
192	C₂×D₄	—	2	32	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	2	8	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	2	8	no	—
197	C₂×D₄	—	2	8	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	—	2	4	no	—
200	C₂×D₄	—	2	4	no	—
201	C₂×D₄	—	4	4	no	—
202	C₂×D₄	D₄×C₆₇	1	8	yes	complemented
203	C₂×D₄	D₄×C₆₇	1	4	yes	complemented
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	D₄×C₆₇	1	4	yes	complemented
208	C₂²×C₄	C₂³×C₆₇	1	4	yes	—
209	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
210	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
211	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
212	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
213	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₂²⋊C₄	—	2	8	no	—
219	C₂²⋊C₄	—	2	2	no	—
220	C₂²⋊C₄	—	2	8	no	—
221	C₂²⋊C₄	—	4	4	no	—
222	C₄²	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	16	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	4	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	4	no	—
230	C₂³	—	2	4	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	2	4	no	—
233	C₂³	—	2	4	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	4	no	—
236	C₂³	—	4	16	no	—
237	C₂³	—	4	16	no	—
238	C₂³	—	4	8	no	—
239	C₂³	—	4	8	no	—
240	C₂³	C₂×D₄×C₆₇	1	8	yes	—
241	C₂³	C₂×D₄×C₆₇	1	2	yes	—
242	C₂³	C₂×D₄×C₆₇	1	2	yes	—
243	C₂³	C₂×D₄×C₆₇	1	2	yes	complemented
244	C₂³	#41×C₆₇	1	2	yes	—
245	C₂³	C₂⁴×C₆₇	1	1	yes	characteristic
246	D₄	—	2	16	no	—
247	D₄	—	2	8	no	—
248	D₄	—	2	16	no	—
249	D₄	—	4	16	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
263	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
264	C₂×C₄	C₂×D₄×C₆₇	1	4	yes	—
265	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	complemented
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	8	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	8	no	—
274	C₂²	—	2	8	no	—
275	C₂²	—	4	32	no	—
276	C₂²	—	4	1	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	4	4	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	4	no	—
281	C₂²	—	4	4	no	—
282	C₂²	—	4	2	no	—
283	C₂²	#121×C₆₇	1	4	yes	complemented
284	C₂²	C₂²×D₄×C₆₇	1	1	yes	characteristic
285	C₂²	C₂²×D₄×C₆₇	1	2	yes	—
286	C₂²	C₂²×D₄×C₆₇	1	1	yes	characteristic
287	C₂²	C₂×#41×C₆₇	1	1	yes	characteristic
288	C₂²	H(5,𝔽₂)×C₆₇	1	2	yes	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	4	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	4	4	no	—
294	C₄	#120×C₆₇	1	4	yes	complemented
295	C₂	—	2	4	no	—
296	C₂	—	4	8	no	—
297	C₂	—	4	2	no	—
298	C₂	C₂×#120×C₆₇	1	1	yes	characteristic
299	C₂	C₂×#121×C₆₇	1	1	yes	characteristic
300	C₂	#535×C₆₇	1	1	yes	characteristic
301	C₂	D₄²×C₆₇	1	2	yes	—
302	C₂	#546×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₆₇
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. C₂³×D₄×C₆₇	C₂	semidirect
2. C₂×#121×C₆₇	C₂	semidirect
3. C₂×#120×C₆₇	C₂	semidirect
4. C₂×C₄×D₄×C₆₇	C₂	semidirect
5. #394×C₆₇	C₂	semidirect
6. #392×C₆₇	C₂	semidirect
7. #390×C₆₇	C₂	semidirect
8. #380×C₆₇	C₂	semidirect
9. #379×C₆₇	C₂	semidirect
16. C₂²×D₄×C₆₇	C₂²	semidirect
18. C₂³×C₄×C₆₇	C₂²	semidirect
22. C₂×C₄⋊C₄×C₆₇	C₂²	semidirect
23. C₂×C₄⋊C₄×C₆₇	C₂²	semidirect
26. C₂×C₂²⋊C₄×C₆₇	C₂²	semidirect
27. C₂×C₂²⋊C₄×C₆₇	C₂²	semidirect
28. C₂×C₂²⋊C₄×C₆₇	C₂²	semidirect
29. C₂×C₂²⋊C₄×C₆₇	C₂²	semidirect
30. C₂×C₄²×C₆₇	C₂²	semidirect
31. #95×C₆₇	C₂²	semidirect
50. C₂×D₄×C₆₇	D₄	semidirect
51. C₂×D₄×C₆₇	D₄	semidirect
55. C₂²×C₄×C₆₇	D₄	semidirect
91. C₂³×C₆₇	C₂×D₄	semidirect
113. C₂×C₄×C₆₇	C₂×D₄	semidirect
131. C₂²×C₆₇	#121	semidirect
142. C₄×C₆₇	#120	semidirect
151. #2408	C₆₇	direct (N × H)
152. C₆₇	#2408	direct (N × H)
153. C₂³×D₄	C₂×C₆₇	semidirect
154. C₂×#121	C₂×C₆₇	semidirect
155. C₂×#120	C₂×C₆₇	semidirect
156. C₂×C₄×D₄	C₂×C₆₇	semidirect
157. #394	C₂×C₆₇	semidirect
158. #392	C₂×C₆₇	semidirect
159. #390	C₂×C₆₇	semidirect
160. #380	C₂×C₆₇	semidirect
161. #379	C₂×C₆₇	semidirect
168. C₂²×D₄	C₂²×C₆₇	semidirect
170. C₂³×C₄	C₂²×C₆₇	semidirect
174. C₂×C₄⋊C₄	C₂²×C₆₇	semidirect
175. C₂×C₄⋊C₄	C₂²×C₆₇	semidirect
178. C₂×C₂²⋊C₄	C₂²×C₆₇	semidirect
179. C₂×C₂²⋊C₄	C₂²×C₆₇	semidirect
180. C₂×C₂²⋊C₄	C₂²×C₆₇	semidirect
181. C₂×C₂²⋊C₄	C₂²×C₆₇	semidirect
182. C₂×C₄²	C₂²×C₆₇	semidirect
183. #95	C₂²×C₆₇	semidirect
202. C₂×D₄	D₄×C₆₇	semidirect
203. C₂×D₄	D₄×C₆₇	semidirect
207. C₂²×C₄	D₄×C₆₇	semidirect
243. C₂³	C₂×D₄×C₆₇	semidirect
265. C₂×C₄	C₂×D₄×C₆₇	semidirect
283. C₂²	#121×C₆₇	semidirect
294. C₄	#120×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₄×C₆₇	2
C₂×#121×C₆₇	2
C₂×#120×C₆₇	2
C₂×C₄×D₄×C₆₇	2
#394×C₆₇	2
#392×C₆₇	2
#390×C₆₇	2
#380×C₆₇	2
#379×C₆₇	2
#2408	67

Subgroups — C67×#2408 #22556839

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#2408 #22556839