Subgroups — #38740679

Order 25216.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27160855	C₂	1	1	yes	maximal, characteristic, complemented
2	#27160830	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#15691333	—	2	1	no	—
5	C₄⋊#13096802	—	2	1	no	—
6	C₂×D₈×C₁₉₇	C₄	1	1	yes	characteristic
7	C₄×D₄×C₁₉₇	—	2	1	no	—
8	#107×C₁₉₇	C₄	1	1	yes	characteristic
9	#102×C₁₉₇	—	2	1	no	—
10	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
11	C₂×#13096802	—	4	1	no	—
12	#15691333	—	2	2	no	—
13	C₂×D₄×C₁₉₇	—	2	1	no	—
14	C₂²×C₄×C₁₉₇	—	4	1	no	—
15	D₈×C₁₉₇	—	2	1	no	—
16	D₈×C₁₉₇	C₈	1	2	yes	complemented
17	C₂×C₈×C₁₉₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
19	C₄⋊C₄×C₁₉₇	—	2	1	no	—
20	C₂²⋊C₄×C₁₉₇	—	4	1	no	—
21	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
22	#13096802	—	8	1	no	—
23	C₂³×C₁₉₇	—	4	1	no	—
24	D₄×C₁₉₇	—	2	2	no	—
25	D₄×C₁₉₇	—	4	1	no	—
26	C₂×C₄×C₁₉₇	—	4	1	no	—
27	C₂×C₄×C₁₉₇	—	8	2	no	—
28	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₉₇	D₈	1	1	yes	characteristic
30	C₂×C₄×C₁₉₇	SD₁₆	1	1	yes	characteristic
31	C₈×C₁₉₇	—	2	1	no	—
32	C₈×C₁₉₇	C₂×C₈	1	1	yes	characteristic
33	C₈×C₁₉₇	M₁₆	1	1	yes	characteristic
34	C₂²×C₁₉₇	—	4	2	no	—
35	C₂²×C₁₉₇	—	8	1	no	—
36	C₂²×C₁₉₇	—	8	1	no	—
37	C₂²×C₁₉₇	#102	1	1	yes	characteristic
38	C₄×C₁₉₇	—	2	1	no	—
39	C₄×C₁₉₇	—	2	1	no	—
40	C₄×C₁₉₇	—	8	1	no	—
41	C₄×C₁₉₇	C₂²⋊C₈	1	1	yes	characteristic
42	C₄×C₁₉₇	#104	1	1	yes	characteristic
43	C₂×C₁₉₇	—	8	2	no	—
44	C₂×C₁₉₇	D₄⋊C₈	1	1	yes	characteristic
45	C₂×C₁₉₇	#360	1	1	yes	characteristic
46	C₂×C₁₉₇	#361	1	1	yes	characteristic
47	C₁₉₇	#1333	1	1	yes	characteristic, complemented, Sylow
48	#1333	—	197	1	no	maximal, Sylow
49	C₄×D₈	D₁₉₇	1	1	yes	characteristic
50	#346	—	197	1	no	—
51	#334	—	197	1	no	—
52	C₂×D₈	Dic₁₉₇	1	1	yes	characteristic
53	C₄×D₄	—	2	1	no	—
54	C₂×C₁₆	—	394	1	no	—
55	#107	Dic₁₉₇	1	1	yes	characteristic
56	C₄⋊C₈	—	394	1	no	—
57	#102	—	2	1	no	—
58	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
59	C₂×D₄	—	2	1	no	—
60	C₂²×C₄	—	4	1	no	—
61	D₈	—	2	1	no	—
62	D₈	#13096802	1	2	yes	complemented
63	C₂×C₈	—	788	1	no	—
64	C₂×C₈	C₂×Dic₁₉₇	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₁₉₇	1	1	yes	characteristic
66	C₄⋊C₄	—	2	1	no	—
67	C₂²⋊C₄	—	4	1	no	—
68	C₄²	C₂²⋊D₁₉₇	1	1	yes	characteristic
69	C₁₆	—	394	2	no	—
70	C₂³	—	4	1	no	—
71	D₄	—	2	2	no	—
72	D₄	—	4	1	no	—
73	C₂×C₄	—	4	1	no	—
74	C₂×C₄	—	8	2	no	—
75	C₂×C₄	C₂²⋊Dic₁₉₇	1	1	yes	characteristic
76	C₂×C₄	#15691343	1	1	yes	characteristic
77	C₂×C₄	#15691346	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	1576	1	no	—
80	C₈	C₂×#13096802	1	1	yes	characteristic
81	C₈	#15691342	1	1	yes	characteristic
82	C₂²	—	4	2	no	—
83	C₂²	—	8	1	no	—
84	C₂²	—	8	1	no	—
85	C₂²	#20263247	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	2	1	no	—
88	C₄	—	8	1	no	—
89	C₄	C₂²⋊#13096802	1	1	yes	characteristic
90	C₄	#20263253	1	1	yes	characteristic
91	C₂	—	8	2	no	—
92	C₂	D₄⋊#13096802	1	1	yes	characteristic
93	C₂	#27160889	1	1	yes	characteristic
94	C₂	#27160892	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#20263247
commutator	C₈×C₁₉₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₉₇
Fitting	C₄×D₈×C₁₉₇	C₂
socle	C₂²×C₁₉₇	#102
prime radical	C₂×D₈×C₁₉₇	C₄
cosocle	C₄×C₈×C₁₉₇	C₂²
hypercenter	C₄×D₈	D₁₉₇
hypocenter	C₁₉₇	#1333
LCS	G ▷ C₈×C₁₉₇ ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #27160855	C₂	semidirect
2. #27160830	C₂	semidirect
16. D₈×C₁₉₇	C₈	semidirect
47. C₁₉₇	#1333	semidirect
62. D₈	#13096802	semidirect

Maximal subgroups

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

subgroup	index
#27160855	2
#27160830	2
C₄×D₈×C₁₉₇	2
#1333	197