Subgroups — #38762797

Order 25408.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27163126	C₂	1	2	yes	maximal
2	C₂×Dic₁₅₈₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#27163032	C₂	1	1	yes	maximal, characteristic, complemented
4	#27163014	C₂	1	2	yes	maximal, complemented
5	#131×C₃₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₉₄	C₂²	1	2	yes	—
7	Dic₁₅₈₈	—	2	1	no	—
8	Dic₁₅₈₈	—	2	1	no	—
9	#20263721	—	2	2	no	—
10	#20263719	—	2	2	no	—
11	#20263715	C₂²	1	2	yes	—
12	#41×C₃₉₇	C₂²	1	1	yes	characteristic
13	M₁₆×C₃₉₇	—	2	1	no	—
14	M₁₆×C₃₉₇	C₂²	1	1	yes	characteristic
15	C₂×C₈×C₃₉₇	—	2	1	no	—
16	C₂×C₈×C₃₉₇	C₂²	1	1	yes	characteristic
17	Dic₇₉₄	—	2	2	no	—
18	Dic₇₉₄	—	2	2	no	—
19	C₂×Dic₃₉₇	—	2	2	no	—
20	#15691787	—	2	2	no	—
21	Q₈×C₃₉₇	D₄	1	1	yes	characteristic
22	D₄×C₃₉₇	—	2	1	no	—
23	D₄×C₃₉₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₃₉₇	—	2	1	no	—
25	C₂×C₄×C₃₉₇	C₂³	1	1	yes	characteristic
26	C₈×C₃₉₇	—	2	1	no	—
27	C₈×C₃₉₇	D₄	1	1	yes	characteristic
28	C₈×C₃₉₇	D₄	1	1	yes	characteristic
29	C₂²×C₃₉₇	—	2	1	no	—
30	C₂²×C₃₉₇	#41	1	1	yes	characteristic
31	C₄×C₃₉₇	—	2	1	no	—
32	C₄×C₃₉₇	C₂×D₄	1	1	yes	characteristic
33	C₄×C₃₉₇	C₂×D₄	1	1	yes	characteristic
34	Dic₃₉₇	—	4	2	no	—
35	C₂×C₃₉₇	—	2	1	no	—
36	C₂×C₃₉₇	—	4	1	no	—
37	C₂×C₃₉₇	#121	1	1	yes	characteristic
38	C₃₉₇	#473	1	1	yes	characteristic, complemented, Sylow
39	#473	—	397	1	no	maximal, Sylow
40	#137	—	397	2	no	—
41	C₂×Q₁₆	—	397	1	no	—
42	#131	D₃₉₇	1	1	yes	characteristic
43	#108	—	397	1	no	—
44	#101	—	397	2	no	—
45	#41	C₂×D₃₉₇	1	1	yes	characteristic
46	C₂×Q₈	—	397	2	no	—
47	Q₁₆	—	794	2	no	—
48	Q₁₆	—	794	1	no	—
49	Q₁₆	—	794	1	no	—
50	SD₁₆	—	794	2	no	—
51	M₁₆	—	2	1	no	—
52	M₁₆	—	397	2	no	—
53	M₁₆	C₂×D₃₉₇	1	1	yes	characteristic
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	C₂×D₃₉₇	1	1	yes	characteristic
56	Q₈	—	794	2	no	—
57	Q₈	—	794	2	no	—
58	Q₈	D₁₅₈₈	1	1	yes	characteristic
59	D₄	—	2	1	no	—
60	D₄	D₁₅₈₈	1	1	yes	characteristic
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	794	2	no	—
63	C₂×C₄	C₂²×D₃₉₇	1	1	yes	characteristic
64	C₈	—	2	1	no	—
65	C₈	—	794	2	no	—
66	C₈	C₂²⋊D₃₉₇	1	1	yes	characteristic
67	C₈	C₂²⋊D₃₉₇	1	1	yes	characteristic
68	C₂²	—	2	1	no	—
69	C₂²	#20263731	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	1588	2	no	—
72	C₄	C₂×C₂²⋊D₃₉₇	1	1	yes	characteristic
73	C₄	C₂×D₁₅₈₈	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	4	1	no	—
76	C₂	#27163070	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27163070
commutator, cosocle	C₂×C₄×C₃₉₇	C₂³
Frattini	C₂×C₄	C₂²×D₃₉₇
Fitting	#131×C₃₉₇	C₂
socle	C₂×C₃₉₇	#121
prime radical	#41×C₃₉₇	C₂²
hypercenter	#131	D₃₉₇
hypocenter	C₃₉₇	#473
LCS	G ▷ C₂×C₄×C₃₉₇ ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₅₈₈	C₂	semidirect
3. #27163032	C₂	semidirect
4. #27163014	C₂	semidirect
38. C₃₉₇	#473	semidirect

Maximal subgroups

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

subgroup	index
#27163126	2
C₂×Dic₁₅₈₈	2
#27163032	2
#27163014	2
#131×C₃₉₇	2
#473	397