Subgroups — #37554422

Order 23488.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₉₃₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#26044235	C₂	1	2	yes	maximal, complemented
3	#26044195	C₂	1	1	yes	maximal, characteristic, complemented
4	#26044183	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₄₆₈	C₂²	1	2	yes	—
7	C₂×C₂²⋊D₃₆₇	—	2	2	no	—
8	D₂₉₃₆	—	2	2	no	—
9	C₄⋊Dic₃₆₇	C₂²	1	2	yes	—
10	(C₂×D₃₆₇)⋊C₄	—	2	2	no	—
11	C₂²×C₄×C₃₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₃₆₇	—	2	2	no	—
13	C₂×C₈×C₃₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₃₆₇	C₂²	1	1	yes	characteristic
15	C₂²×D₃₆₇	—	2	2	no	—
16	D₁₄₆₈	—	2	2	no	—
17	D₁₄₆₈	—	2	2	no	—
18	C₂²⋊D₃₆₇	—	4	4	no	—
19	C₂×Dic₃₆₇	—	2	2	no	—
20	C₂³×C₃₆₇	D₄	1	1	yes	characteristic
21	C₂×C₄×C₃₆₇	—	2	1	no	—
22	C₂×C₄×C₃₆₇	—	2	1	no	—
23	C₂×C₄×C₃₆₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
25	C₈×C₃₆₇	—	2	1	no	—
26	C₈×C₃₆₇	D₄	1	2	yes	complemented
27	C₂²×C₃₆₇	—	2	1	no	—
28	C₂²×C₃₆₇	—	2	1	no	—
29	C₂²×C₃₆₇	D₈	1	2	yes	complemented
30	C₂²×C₃₆₇	C₂×D₄	1	1	yes	characteristic
31	C₂×D₃₆₇	—	4	2	no	—
32	C₂×D₃₆₇	—	4	2	no	—
33	C₂×D₃₆₇	—	4	2	no	—
34	C₄×C₃₆₇	—	2	1	no	—
35	C₄×C₃₆₇	C₂×D₄	1	1	yes	characteristic
36	C₄×C₃₆₇	#41	1	1	yes	characteristic
37	Dic₃₆₇	—	4	2	no	—
38	C₂×C₃₆₇	—	2	2	no	—
39	C₂×C₃₆₇	#121	1	1	yes	characteristic
40	C₂×C₃₆₇	C₂×D₈	1	1	yes	characteristic
41	C₂×C₃₆₇	#135	1	1	yes	characteristic
42	D₃₆₇	—	8	2	no	—
43	C₃₆₇	#466	1	1	yes	characteristic, complemented, Sylow
44	#466	—	367	1	no	maximal, Sylow
45	C₂×D₈	—	367	1	no	—
46	C₂²×C₈	D₃₆₇	1	1	yes	characteristic, complemented
47	#121	—	367	2	no	—
48	#107	—	367	1	no	—
49	#102	—	367	2	no	—
50	C₂×D₄	—	367	2	no	—
51	C₂×D₄	—	734	2	no	—
52	C₂²×C₄	C₂×D₃₆₇	1	1	yes	characteristic
53	D₈	—	734	2	no	—
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	C₂×D₃₆₇	1	1	yes	characteristic
56	C₂×C₈	C₂×D₃₆₇	1	1	yes	characteristic
57	C₄⋊C₄	—	367	2	no	—
58	C₂²⋊C₄	—	734	2	no	—
59	C₂³	—	734	2	no	—
60	C₂³	D₁₄₆₈	1	1	yes	characteristic
61	D₄	—	1468	4	no	—
62	D₄	—	734	2	no	—
63	D₄	—	734	2	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	734	2	no	—
67	C₂×C₄	D₁₄₆₈	1	1	yes	characteristic
68	C₂×C₄	C₂²×D₃₆₇	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	C₂²⋊D₃₆₇	1	2	yes	complemented
71	C₂²	—	2	1	no	—
72	C₂²	—	2	1	no	—
73	C₂²	—	1468	2	no	—
74	C₂²	—	1468	2	no	—
75	C₂²	—	1468	2	no	—
76	C₂²	D₂₉₃₆	1	2	yes	complemented
77	C₂²	C₂×D₁₄₆₈	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	1468	2	no	—
80	C₄	C₂×C₂²⋊D₃₆₇	1	1	yes	characteristic
81	C₄	#19154577	1	1	yes	characteristic
82	C₂	—	2	2	no	—
83	C₂	—	2936	2	no	—
84	C₂	#26044235	1	1	yes	characteristic
85	C₂	C₂×D₂₉₃₆	1	1	yes	characteristic
86	C₂	#26044283	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₄₆₈
commutator, cosocle	C₂×C₄×C₃₆₇	C₂³
Frattini	C₂×C₄	C₂²×D₃₆₇
Fitting	C₂²×C₈×C₃₆₇	C₂
socle	C₂²×C₃₆₇	C₂×D₄
hypercenter	C₂²×C₈	D₃₆₇
hypocenter	C₃₆₇	#466
LCS	G ▷ C₂×C₄×C₃₆₇ ▷ C₂×C₃₆₇ ▷ C₃₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×D₂₉₃₆	C₂	semidirect
2. #26044235	C₂	semidirect
3. #26044195	C₂	semidirect
4. #26044183	C₂	semidirect
5. C₂²×C₈×C₃₆₇	C₂	semidirect
26. C₈×C₃₆₇	D₄	semidirect
29. C₂²×C₃₆₇	D₈	semidirect
43. C₃₆₇	#466	semidirect
46. C₂²×C₈	D₃₆₇	semidirect
70. C₈	C₂²⋊D₃₆₇	semidirect
76. C₂²	D₂₉₃₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₉₃₆	2
#26044235	2
#26044195	2
#26044183	2
C₂²×C₈×C₃₆₇	2
#466	367