Subgroups — #27161417

Order 12608.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263285	C₂	1	1	yes	maximal, characteristic
2	#20263259	C₂	1	1	yes	maximal, characteristic, complemented
3	#20263257	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263250	C₂	1	1	yes	maximal, characteristic, complemented
5	#20263250	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²⋊#13096802	C₂	1	1	yes	maximal, characteristic
7	C₂²⋊Q₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#13096802	C₂²	1	1	yes	characteristic
9	C₂×#13096802	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
12	C₂²⋊Dic₁₉₇	—	2	1	no	—
13	C₄×Dic₁₉₇	—	2	1	no	—
14	C₂×Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₁₉₇	—	2	1	no	—
17	C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
18	C₂²⋊C₄×C₁₉₇	—	2	1	no	—
19	C₂×Dic₁₉₇	—	2	1	no	—
20	C₂×Dic₁₉₇	—	2	1	no	—
21	#13096802	—	2	1	no	—
22	#13096802	—	2	1	no	—
23	C₂³×C₁₉₇	D₄	1	1	yes	characteristic
24	Q₈×C₁₉₇	—	2	1	no	—
25	Q₈×C₁₉₇	—	2	1	no	—
26	C₂×C₄×C₁₉₇	—	2	1	no	—
27	C₂×C₄×C₁₉₇	—	2	1	no	—
28	C₂×C₄×C₁₉₇	—	2	1	no	—
29	C₂×C₄×C₁₉₇	—	2	1	no	—
30	C₂×C₄×C₁₉₇	D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
32	C₂²×C₁₉₇	—	2	1	no	—
33	C₂²×C₁₉₇	—	2	1	no	—
34	C₂²×C₁₉₇	—	2	1	no	—
35	C₂²×C₁₉₇	C₂×D₄	1	1	yes	characteristic
36	C₄×C₁₉₇	—	2	1	no	—
37	C₄×C₁₉₇	—	4	1	no	—
38	C₄×C₁₉₇	—	4	1	no	—
39	C₄×C₁₉₇	#41	1	1	yes	characteristic
40	C₄×C₁₉₇	#41	1	1	yes	characteristic
41	Dic₁₉₇	—	4	1	no	—
42	Dic₁₉₇	—	4	1	no	—
43	C₂×C₁₉₇	—	4	1	no	—
44	C₂×C₁₉₇	#123	1	1	yes	characteristic
45	C₂×C₁₉₇	#135	1	1	yes	characteristic
46	C₂×C₁₉₇	#137	1	1	yes	characteristic
47	C₁₉₇	#485	1	1	yes	characteristic, complemented, Sylow
48	#485	—	197	1	no	maximal, Sylow
49	C₂²⋊Q₈	D₁₉₇	1	1	yes	characteristic
50	#117	—	197	1	no	—
51	#107	—	197	1	no	—
52	#106	—	197	1	no	—
53	#103	—	197	1	no	—
54	#103	—	197	1	no	—
55	C₂²⋊C₈	—	197	1	no	—
56	C₂×Q₈	C₂×D₁₉₇	1	1	yes	characteristic
57	C₂²×C₄	C₂×D₁₉₇	1	1	yes	characteristic
58	C₂×C₈	—	197	1	no	—
59	C₂×C₈	—	197	1	no	—
60	C₄⋊C₄	—	2	1	no	—
61	C₄⋊C₄	—	197	1	no	—
62	C₄⋊C₄	—	197	1	no	—
63	C₄⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
64	C₂²⋊C₄	—	2	1	no	—
65	C₂²⋊C₄	—	394	1	no	—
66	C₄²	—	394	1	no	—
67	C₂³	C₂²⋊D₁₉₇	1	1	yes	characteristic
68	Q₈	—	2	1	no	—
69	Q₈	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	394	1	no	—
75	C₂×C₄	—	394	1	no	—
76	C₂×C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
77	C₂×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
78	C₈	—	394	1	no	—
79	C₈	—	394	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	4	1	no	—
87	C₄	—	788	1	no	—
88	C₄	—	788	1	no	—
89	C₄	#15691357	1	1	yes	characteristic
90	C₄	#15691357	1	1	yes	characteristic
91	C₂	—	4	1	no	—
92	C₂	#20263307	1	1	yes	characteristic
93	C₂	#20263346	1	1	yes	characteristic
94	C₂	#20263357	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₇
commutator, cosocle	C₂×C₄×C₁₉₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₇
Fitting	C₂²⋊Q₈×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂×D₄
prime radical	C₂³×C₁₉₇	D₄
hypercenter	C₂²⋊Q₈	D₁₉₇
hypocenter	C₁₉₇	#485
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #20263259	C₂	semidirect
3. #20263257	C₂	semidirect
4. #20263250	C₂	semidirect
5. #20263250	C₂	semidirect
47. C₁₉₇	#485	semidirect

Maximal subgroups

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

subgroup	index
#20263285	2
#20263259	2
#20263257	2
#20263250	2
C₂²⋊#13096802	2
C₂²⋊Q₈×C₁₉₇	2
#485	197