Subgroups — #27163068

Order 12704.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊D₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊D₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×Dic₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263710	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊Dic₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#20263708	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₄×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₃₉₇	C₂²	1	1	yes	characteristic
9	C₂²⋊D₃₉₇	—	2	2	no	—
10	C₂²⋊D₃₉₇	—	2	2	no	—
11	C₂×Dic₃₉₇	—	2	2	no	—
12	C₂×Dic₃₉₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₃₉₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₉₇	C₂²	1	1	yes	characteristic
15	C₂³×C₃₉₇	C₂²	1	1	yes	characteristic
16	C₂³×C₃₉₇	C₂²	1	1	yes	characteristic
17	D₄×C₃₉₇	—	2	2	no	—
18	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
19	C₂²×C₃₉₇	—	2	1	no	—
20	C₂²×C₃₉₇	—	2	1	no	—
21	C₂²×C₃₉₇	—	2	1	no	—
22	C₂²×C₃₉₇	—	2	1	no	—
23	C₂²×C₃₉₇	—	2	1	no	—
24	C₂²×C₃₉₇	D₄	1	2	yes	complemented
25	C₂²×C₃₉₇	C₂³	1	1	yes	characteristic
26	C₂×D₃₉₇	—	2	1	no	—
27	C₂×D₃₉₇	—	2	1	no	—
28	C₂×D₃₉₇	—	2	1	no	—
29	C₄×C₃₉₇	—	2	1	no	—
30	Dic₃₉₇	—	2	1	no	—
31	Dic₃₉₇	—	2	1	no	—
32	Dic₃₉₇	D₄	1	2	yes	complemented
33	C₂×C₃₉₇	—	2	2	no	—
34	C₂×C₃₉₇	—	4	1	no	—
35	C₂×C₃₉₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₃₉₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₃₉₇	#41	1	1	yes	characteristic
38	D₃₉₇	—	4	1	no	—
39	C₃₉₇	#121	1	1	yes	characteristic, complemented, Sylow
40	#121	—	397	1	no	maximal, Sylow
41	C₂×D₄	—	397	1	no	—
42	C₂×D₄	—	397	1	no	—
43	C₂×D₄	D₃₉₇	1	1	yes	characteristic, complemented
44	C₂²×C₄	—	397	1	no	—
45	C₄⋊C₄	—	397	1	no	—
46	C₂²⋊C₄	—	397	1	no	—
47	C₂²⋊C₄	—	397	1	no	—
48	C₂³	—	397	1	no	—
49	C₂³	C₂×D₃₉₇	1	1	yes	characteristic
50	C₂³	C₂×D₃₉₇	1	1	yes	characteristic
51	D₄	—	2	2	no	—
52	D₄	—	794	2	no	—
53	D₄	—	794	2	no	—
54	C₂×C₄	—	794	2	no	—
55	C₂×C₄	—	397	1	no	—
56	C₂×C₄	—	397	1	no	—
57	C₂×C₄	—	397	1	no	—
58	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic
59	C₂²	—	2	1	no	—
60	C₂²	—	2	1	no	—
61	C₂²	—	2	1	no	—
62	C₂²	—	2	1	no	—
63	C₂²	—	2	1	no	—
64	C₂²	—	794	1	no	—
65	C₂²	—	794	1	no	—
66	C₂²	—	794	1	no	—
67	C₂²	C₂²⋊D₃₉₇	1	2	yes	complemented
68	C₂²	C₂²×D₃₉₇	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	794	1	no	—
71	C₄	—	794	1	no	—
72	C₄	—	397	2	no	—
73	C₂	—	2	2	no	—
74	C₂	—	4	1	no	—
75	C₂	—	1588	1	no	—
76	C₂	D₄×D₃₉₇	1	1	yes	characteristic
77	C₂	C₂×C₂²⋊D₃₉₇	1	1	yes	characteristic
78	C₂	#20263730	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃₉₇
commutator, socle, cosocle	C₂²×C₃₉₇	C₂³
Fitting	C₂×D₄×C₃₉₇	C₂
hypercenter	C₂×D₄	D₃₉₇
hypocenter	C₃₉₇	#121
LCS	G ▷ C₂²×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊D₃₉₇	C₂	semidirect
2. C₂×C₂²⋊D₃₉₇	C₂	semidirect
3. C₂²×Dic₃₉₇	C₂	semidirect
4. #20263710	C₂	semidirect
5. C₂²⋊Dic₃₉₇	C₂	semidirect
6. #20263708	C₂	semidirect
7. C₂×D₄×C₃₉₇	C₂	semidirect
24. C₂²×C₃₉₇	D₄	semidirect
32. Dic₃₉₇	D₄	semidirect
39. C₃₉₇	#121	semidirect
43. C₂×D₄	D₃₉₇	semidirect
67. C₂²	C₂²⋊D₃₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊D₃₉₇	2
C₂²×Dic₃₉₇	2
#20263710	2
C₂²⋊Dic₃₉₇	2
#20263708	2
C₂×D₄×C₃₉₇	2
#121	397