Subgroups — C₄×#19154560 #37554136

Order 23488.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19154560	C₂	1	2	yes	maximal
2	C₄²×D₃₆₇	C₂	1	1	yes	maximal, characteristic
3	#26044171	C₂	1	2	yes	maximal, complemented
4	C₄×C₃₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₈×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₃₆₇	C₄	1	2	yes	—
7	C₂×C₄×D₃₆₇	C₂²	1	1	yes	characteristic
8	#19154560	C₄	1	8	yes	direct product factor
9	C₂×C₃₆₇⋊C₈	C₂²	1	2	yes	—
10	C₄×Dic₃₆₇	C₄	1	2	yes	—
11	C₄×Dic₃₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₃₆₇	C₂²	1	2	yes	—
13	C₄²×C₃₆₇	C₂²	1	1	yes	characteristic
14	C₂²×D₃₆₇	C₂×C₄	1	1	yes	characteristic
15	C₂×Dic₃₆₇	C₂×C₄	1	2	yes	—
16	C₂×Dic₃₆₇	C₂×C₄	1	1	yes	characteristic
17	C₄×D₃₆₇	—	2	4	no	—
18	C₄×D₃₆₇	C₂×C₄	1	4	yes	—
19	C₃₆₇⋊C₈	C₂×C₄	1	4	yes	complemented
20	C₂×C₄×C₃₆₇	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
22	C₈×C₃₆₇	C₂×C₄	1	4	yes	complemented
23	C₂²×C₃₆₇	C₂²×C₄	1	1	yes	characteristic
24	C₂×D₃₆₇	—	2	2	no	—
25	C₂×D₃₆₇	C₄²	1	2	yes	—
26	C₄×C₃₆₇	M₁₆	1	4	yes	complemented
27	C₄×C₃₆₇	C₂²×C₄	1	2	yes	—
28	Dic₃₆₇	—	2	2	no	—
29	Dic₃₆₇	C₄²	1	2	yes	—
30	C₂×C₃₆₇	C₂×C₄²	1	1	yes	characteristic
31	C₂×C₃₆₇	C₂×M₁₆	1	2	yes	—
32	D₃₆₇	—	2	2	no	—
33	C₃₆₇	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₄×M₁₆	—	367	1	no	maximal, Sylow
35	C₂×M₁₆	—	367	2	no	—
36	C₂×C₄²	—	367	1	no	—
37	#97	—	367	2	no	—
38	C₄×C₈	—	367	1	no	—
39	C₄×C₈	D₃₆₇	1	1	yes	characteristic, complemented
40	C₂²×C₄	—	367	1	no	—
41	C₂²×C₄	—	367	2	no	—
42	M₁₆	—	367	8	no	—
43	C₂×C₈	—	367	2	no	—
44	C₂×C₈	C₂×D₃₆₇	1	2	yes	—
45	C₄²	—	367	2	no	—
46	C₄²	—	367	1	no	—
47	C₄²	C₂×D₃₆₇	1	1	yes	characteristic
48	C₂³	—	367	1	no	—
49	C₂×C₄	—	367	1	no	—
50	C₂×C₄	—	367	4	no	—
51	C₂×C₄	—	367	2	no	—
52	C₂×C₄	—	734	4	no	—
53	C₂×C₄	C₄×D₃₆₇	1	2	yes	—
54	C₂×C₄	C₂²×D₃₆₇	1	1	yes	characteristic
55	C₈	—	367	4	no	—
56	C₈	C₄×D₃₆₇	1	4	yes	complemented
57	C₂²	—	367	2	no	—
58	C₂²	—	734	2	no	—
59	C₂²	C₂×C₄×D₃₆₇	1	1	yes	characteristic
60	C₄	—	367	2	no	—
61	C₄	—	734	2	no	—
62	C₄	#19154560	1	4	yes	direct product factor
63	C₄	C₂×C₄×D₃₆₇	1	2	yes	—
64	C₂	—	734	2	no	—
65	C₂	C₄²×D₃₆₇	1	1	yes	characteristic
66	C₂	C₂×#19154560	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×D₃₆₇
commutator	C₂×C₃₆₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×D₃₆₇
Fitting	C₄×C₈×C₃₆₇	C₂
socle	C₂²×C₃₆₇	C₂²×C₄
prime radical	C₂²×D₃₆₇	C₂×C₄
cosocle	C₂×C₄×C₃₆₇	C₂³
hypercenter	C₄×C₈	D₃₆₇
hypocenter	C₃₆₇	C₄×M₁₆
LCS	G ▷ C₂×C₃₆₇ ▷ C₃₆₇
UCS	C₁ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #26044171	C₂	semidirect
4. C₄×C₃₆₇⋊C₈	C₂	semidirect
5. C₄×C₈×C₃₆₇	C₂	semidirect
8. #19154560	C₄	direct (N × H)
19. C₃₆₇⋊C₈	C₂×C₄	semidirect
22. C₈×C₃₆₇	C₂×C₄	semidirect
26. C₄×C₃₆₇	M₁₆	semidirect
33. C₃₆₇	C₄×M₁₆	semidirect
39. C₄×C₈	D₃₆₇	semidirect
56. C₈	C₄×D₃₆₇	semidirect
62. C₄	#19154560	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19154560	2
C₄²×D₃₆₇	2
#26044171	2
C₄×C₃₆₇⋊C₈	2
C₄×C₈×C₃₆₇	2
C₄×M₁₆	367

Subgroups — C4×#19154560 #37554136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19154560 #37554136