Subgroups — #27161751

Order 12608.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691359	C₂	1	2	yes	maximal, complemented
2	#20263323	C₂	1	3	yes	maximal, complemented
3	#20263301	C₂	1	6	yes	maximal, complemented
4	C₄×D₇₈₈	C₂	1	3	yes	maximal, complemented
5	C₄×Q₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#15691359	—	2	8	no	—
7	C₂×D₇₈₈	C₂²	1	3	yes	—
8	C₂×D₇₈₈	C₂²	1	6	yes	—
9	C₂×D₇₈₈	C₂²	1	6	yes	—
10	C₂×C₄×D₁₉₇	C₂²	1	6	yes	—
11	C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
12	#15691335	C₂²	1	6	yes	—
13	C₂×Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₁₉₇	C₂²	1	3	yes	—
15	C₄²×C₁₉₇	C₂²	1	3	yes	—
16	C₂²×D₁₉₇	C₂³	1	6	yes	—
17	D₇₈₈	—	2	12	no	—
18	D₇₈₈	—	2	6	no	—
19	D₇₈₈	—	4	6	no	—
20	C₂×Dic₁₉₇	C₂³	1	2	yes	—
21	C₄×D₁₉₇	—	2	12	no	—
22	Q₈×C₁₉₇	D₄	1	4	yes	complemented
23	C₂×C₄×C₁₉₇	C₂³	1	3	yes	—
24	C₂×C₄×C₁₉₇	C₂³	1	3	yes	—
25	C₂×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
26	C₂²×C₁₉₇	C₂⁴	1	1	yes	characteristic
27	C₂×D₁₉₇	—	2	6	no	—
28	C₂×D₁₉₇	—	2	6	no	—
29	C₂×D₁₉₇	—	2	6	no	—
30	C₄×C₁₉₇	—	2	3	no	—
31	C₄×C₁₉₇	C₂×D₄	1	6	yes	—
32	C₄×C₁₉₇	#41	1	2	yes	complemented
33	Dic₁₉₇	—	2	2	no	—
34	C₂×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
35	C₂×C₁₉₇	C₂×#41	1	1	yes	characteristic
36	C₂×C₁₉₇	H(5,𝔽₂)	1	1	yes	characteristic
37	D₁₉₇	—	4	6	no	—
38	C₁₉₇	#550	1	1	yes	characteristic, complemented, Sylow
39	#550	—	197	1	no	maximal, Sylow
40	C₂×#41	—	197	2	no	—
41	#127	—	197	3	no	—
42	#121	—	197	6	no	—
43	C₄×Q₈	D₁₉₇	1	1	yes	characteristic, complemented
44	C₄×D₄	—	197	3	no	—
45	#41	—	394	8	no	—
46	C₂×Q₈	C₂×D₁₉₇	1	1	yes	characteristic
47	C₂×D₄	—	197	6	no	—
48	C₂×D₄	—	197	3	no	—
49	C₂×D₄	—	197	6	no	—
50	C₂²×C₄	—	197	6	no	—
51	C₄⋊C₄	—	197	1	no	—
52	C₄⋊C₄	C₂×D₁₉₇	1	3	yes	—
53	C₂²⋊C₄	—	197	6	no	—
54	C₄²	C₂×D₁₉₇	1	3	yes	—
55	C₂³	—	197	6	no	—
56	Q₈	D₇₈₈	1	4	yes	complemented
57	D₄	—	788	6	no	—
58	D₄	—	394	12	no	—
59	D₄	—	394	6	no	—
60	C₂×C₄	—	394	12	no	—
61	C₂×C₄	—	197	2	no	—
62	C₂×C₄	C₂²×D₁₉₇	1	3	yes	—
63	C₂×C₄	C₂²×D₁₉₇	1	3	yes	—
64	C₂×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
65	C₂²	—	394	6	no	—
66	C₂²	—	394	6	no	—
67	C₂²	—	394	6	no	—
68	C₂²	C₂³×D₁₉₇	1	1	yes	characteristic
69	C₄	—	2	3	no	—
70	C₄	—	394	2	no	—
71	C₄	C₂×D₇₈₈	1	6	yes	—
72	C₄	#15691359	1	2	yes	complemented
73	C₂	—	788	6	no	—
74	C₂	C₂²×D₇₈₈	1	1	yes	characteristic
75	C₂	C₂×#15691359	1	1	yes	characteristic
76	C₂	#20263372	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₉₇
commutator, socle, cosocle	C₂²×C₁₉₇	C₂⁴
Fitting	C₄×Q₈×C₁₉₇	C₂
hypercenter	C₄×Q₈	D₁₉₇
hypocenter	C₁₉₇	#550
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#15691359	C₂	semidirect
2. #20263323	C₂	semidirect
3. #20263301	C₂	semidirect
4. C₄×D₇₈₈	C₂	semidirect
5. C₄×Q₈×C₁₉₇	C₂	semidirect
22. Q₈×C₁₉₇	D₄	semidirect
32. C₄×C₁₉₇	#41	semidirect
38. C₁₉₇	#550	semidirect
43. C₄×Q₈	D₁₉₇	semidirect
56. Q₈	D₇₈₈	semidirect
72. C₄	#15691359	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15691359	2
#20263323	2
#20263301	2
C₄×D₇₈₈	2
C₄×Q₈×C₁₉₇	2
#550	197