Subgroups — #38762961

Order 25408.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27163115	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×D₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₃₁₇₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#27163042	C₂	1	1	yes	maximal, characteristic, complemented
5	#27163040	C₂	1	1	yes	maximal, characteristic, complemented
6	#27163035	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₃₂×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#20263730	—	2	1	no	—
9	Q₈×D₃₉₇	—	2	1	no	—
10	Dic₁₅₈₈	C₂²	1	1	yes	characteristic
11	Dic₁₅₈₈	C₂²	1	1	yes	characteristic
12	#20263721	—	2	1	no	—
13	#20263719	—	2	1	no	—
14	C₈×D₃₉₇	C₂²	1	1	yes	characteristic
15	#20263706	C₂²	1	1	yes	characteristic
16	Q₁₆×C₃₉₇	C₂²	1	1	yes	characteristic
17	D₈×C₃₉₇	C₂²	1	1	yes	characteristic
18	C₁₆×C₃₉₇	C₂²	1	1	yes	characteristic
19	Dic₇₉₄	—	2	1	no	—
20	Dic₇₉₄	—	2	1	no	—
21	Dic₇₉₄	—	4	1	no	—
22	C₂²⋊D₃₉₇	—	4	1	no	—
23	C₂×Dic₃₉₇	—	4	1	no	—
24	C₄×D₃₉₇	—	4	1	no	—
25	C₄×D₃₉₇	D₄	1	1	yes	characteristic
26	#15691787	D₄	1	1	yes	characteristic
27	Q₈×C₃₉₇	—	2	1	no	—
28	D₄×C₃₉₇	—	2	1	no	—
29	C₈×C₃₉₇	C₂³	1	1	yes	characteristic
30	C₂²×C₃₉₇	—	4	1	no	—
31	C₂×D₃₉₇	D₈	1	1	yes	characteristic
32	C₄×C₃₉₇	—	4	1	no	—
33	C₄×C₃₉₇	C₂×D₄	1	1	yes	characteristic
34	Dic₃₉₇	—	4	1	no	—
35	Dic₃₉₇	—	4	1	no	—
36	Dic₃₉₇	D₈	1	1	yes	characteristic
37	C₂×C₃₉₇	—	8	1	no	—
38	C₂×C₃₉₇	C₂×D₈	1	1	yes	characteristic
39	D₃₉₇	—	2	1	no	—
40	C₃₉₇	#510	1	1	yes	characteristic, complemented, Sylow
41	#510	—	397	1	no	maximal, Sylow
42	#135	—	397	1	no	—
43	C₂×Q₁₆	—	397	1	no	—
44	Q₃₂	—	397	1	no	—
45	Q₃₂	—	397	1	no	—
46	SD₃₂	—	397	1	no	—
47	SD₃₂	D₃₉₇	1	1	yes	characteristic, complemented
48	M₃₂	—	397	1	no	—
49	#41	—	794	1	no	—
50	C₂×Q₈	—	794	1	no	—
51	Q₁₆	—	794	1	no	—
52	Q₁₆	—	397	1	no	—
53	Q₁₆	—	397	1	no	—
54	Q₁₆	C₂×D₃₉₇	1	1	yes	characteristic
55	SD₁₆	—	794	1	no	—
56	D₈	C₂×D₃₉₇	1	1	yes	characteristic
57	C₂×C₈	—	397	1	no	—
58	C₁₆	—	397	1	no	—
59	C₁₆	C₂×D₃₉₇	1	1	yes	characteristic
60	Q₈	—	2	1	no	—
61	Q₈	—	794	1	no	—
62	Q₈	—	794	1	no	—
63	Q₈	—	1588	1	no	—
64	D₄	—	2	1	no	—
65	D₄	—	1588	1	no	—
66	C₂×C₄	—	1588	1	no	—
67	C₂×C₄	—	1588	1	no	—
68	C₂×C₄	—	397	1	no	—
69	C₈	—	397	1	no	—
70	C₈	C₂²×D₃₉₇	1	1	yes	characteristic
71	C₂²	—	4	1	no	—
72	C₂²	—	397	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	1588	1	no	—
75	C₄	—	1588	1	no	—
76	C₄	—	397	1	no	—
77	C₄	D₄×D₃₉₇	1	1	yes	characteristic
78	C₂	—	8	1	no	—
79	C₂	—	794	1	no	—
80	C₂	D₈×D₃₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₃₉₇
commutator, cosocle	C₈×C₃₉₇	C₂³
Frattini	C₈	C₂²×D₃₉₇
Fitting	SD₃₂×C₃₉₇	C₂
socle	C₂×C₃₉₇	C₂×D₈
prime radical	#27163115	C₂
hypercenter	SD₃₂	D₃₉₇
hypocenter	C₃₉₇	#510
LCS	G ▷ C₈×C₃₉₇ ▷ C₄×C₃₉₇ ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

Complements

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

N	H ≅ G/N	type
2. Q₁₆×D₃₉₇	C₂	semidirect
3. Dic₃₁₇₆	C₂	semidirect
4. #27163042	C₂	semidirect
5. #27163040	C₂	semidirect
6. #27163035	C₂	semidirect
7. SD₃₂×C₃₉₇	C₂	semidirect
40. C₃₉₇	#510	semidirect
47. SD₃₂	D₃₉₇	semidirect

Maximal subgroups

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

subgroup	index
#27163115	2
Q₁₆×D₃₉₇	2
Dic₃₁₇₆	2
#27163042	2
#27163040	2
#27163035	2
SD₃₂×C₃₉₇	2
#510	397