Subgroups — #38740872

Order 25216.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27160876	C₂	1	1	yes	maximal, characteristic
2	#27160874	C₂	1	1	yes	maximal, characteristic
3	#497×C₁₉₇	C₂	1	1	yes	maximal, characteristic
4	#20263245	—	2	1	no	—
5	#20263241	—	2	1	no	—
6	C₂×Q₁₆×C₁₉₇	—	2	1	no	—
7	C₂×SD₁₆×C₁₉₇	—	2	1	no	—
8	C₄⋊Q₈×C₁₉₇	C₄	1	1	yes	characteristic, complemented
9	#124×C₁₉₇	C₂²	1	1	yes	characteristic
10	#97×C₁₉₇	C₄	1	1	yes	characteristic, complemented
11	#15691342	—	4	1	no	—
12	C₂²⋊Dic₁₉₇	—	4	1	no	—
13	C₂×Q₈×C₁₉₇	—	2	1	no	—
14	C₂×Q₈×C₁₉₇	D₄	1	1	yes	characteristic
15	C₂×D₄×C₁₉₇	D₄	1	1	yes	characteristic
16	Q₁₆×C₁₉₇	—	4	2	no	—
17	SD₁₆×C₁₉₇	—	4	2	no	—
18	C₂×C₈×C₁₉₇	—	2	1	no	—
19	C₄⋊C₄×C₁₉₇	—	4	1	no	—
20	C₂²⋊C₄×C₁₉₇	—	4	1	no	—
21	C₄²×C₁₉₇	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₁₉₇	—	8	1	no	—
23	#13096802	—	8	1	no	—
24	C₂³×C₁₉₇	—	2	1	no	—
25	Q₈×C₁₉₇	—	4	1	no	—
26	Q₈×C₁₉₇	—	4	1	no	—
27	Q₈×C₁₉₇	—	4	1	no	—
28	D₄×C₁₉₇	—	4	1	no	—
29	C₂×C₄×C₁₉₇	—	2	1	no	—
30	C₂×C₄×C₁₉₇	—	2	1	no	—
31	C₂×C₄×C₁₉₇	—	4	1	no	—
32	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₈×C₁₉₇	—	2	2	no	—
34	C₂²×C₁₉₇	—	4	1	no	—
35	C₂²×C₁₉₇	—	8	1	no	—
36	C₂²×C₁₉₇	#99	1	1	yes	characteristic
37	C₄×C₁₉₇	—	2	1	no	—
38	C₄×C₁₉₇	—	4	1	no	—
39	C₄×C₁₉₇	—	4	1	no	—
40	C₄×C₁₉₇	—	8	1	no	—
41	Dic₁₉₇	—	16	1	no	—
42	C₂×C₁₉₇	—	2	1	no	—
43	C₂×C₁₉₇	—	8	1	no	—
44	C₂×C₁₉₇	#353	1	1	yes	characteristic
45	C₁₉₇	#1407	1	1	yes	characteristic, complemented, Sylow
46	#1407	—	197	1	no	maximal, Sylow
47	#497	D₁₉₇	1	1	yes	characteristic
48	#355	—	197	1	no	—
49	#354	—	197	1	no	—
50	C₂×Q₁₆	—	2	1	no	—
51	C₂×SD₁₆	—	2	1	no	—
52	C₄⋊Q₈	Dic₁₉₇	1	1	yes	characteristic, complemented
53	#124	C₂×D₁₉₇	1	1	yes	characteristic
54	#101	—	394	1	no	—
55	#99	—	394	1	no	—
56	#97	Dic₁₉₇	1	1	yes	characteristic, complemented
57	C₂×Q₈	—	2	1	no	—
58	C₂×Q₈	C₂²⋊D₁₉₇	1	1	yes	characteristic
59	C₂×D₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
60	Q₁₆	—	4	2	no	—
61	SD₁₆	—	4	2	no	—
62	M₁₆	—	788	1	no	—
63	C₂×C₈	—	2	1	no	—
64	C₄⋊C₄	—	4	1	no	—
65	C₂²⋊C₄	—	4	1	no	—
66	C₂²⋊C₄	—	788	1	no	—
67	C₄²	C₂×Dic₁₉₇	1	1	yes	characteristic
68	C₂³	—	2	1	no	—
69	Q₈	—	4	1	no	—
70	Q₈	—	4	1	no	—
71	Q₈	—	4	1	no	—
72	D₄	—	4	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	1576	1	no	—
77	C₂×C₄	C₂²⋊Dic₁₉₇	1	1	yes	characteristic
78	C₈	—	2	2	no	—
79	C₈	—	1576	1	no	—
80	C₂²	—	4	1	no	—
81	C₂²	—	8	1	no	—
82	C₂²	#20263241	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	8	1	no	—
87	C₄	—	3152	1	no	—
88	C₂	—	2	1	no	—
89	C₂	—	8	1	no	—
90	C₂	#27160872	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27160872
commutator	C₄²×C₁₉₇	C₂×C₄
Frattini	#124	C₂×D₁₉₇
Fitting	#497×C₁₉₇	C₂
socle	C₂×C₁₉₇	#353
prime radical	C₂×D₄×C₁₉₇	D₄
cosocle	#124×C₁₉₇	C₂²
hypercenter	#497	D₁₉₇
hypocenter	C₁₉₇	#1407
LCS	G ▷ C₄²×C₁₉₇ ▷ C₂×C₄×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #497

Complements

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

N	H ≅ G/N	type
8. C₄⋊Q₈×C₁₉₇	C₄	semidirect
10. #97×C₁₉₇	C₄	semidirect
45. C₁₉₇	#1407	semidirect
52. C₄⋊Q₈	Dic₁₉₇	semidirect
56. #97	Dic₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
#27160876	2
#27160874	2
#497×C₁₉₇	2
#1407	197