Subgroups — #27143517

Order 12416.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₆×D₉₇	C₂	1	1	yes	maximal, characteristic
2	#20260772	C₂	1	2	yes	maximal
3	#20260562	C₂	1	2	yes	maximal, complemented
4	#20260561	C₂	1	1	yes	maximal, characteristic, complemented
5	#366×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
7	#15690768	—	2	2	no	—
8	C₂×#13095486	C₂²	1	1	yes	characteristic
9	C₁₆×D₉₇	C₄	1	4	yes	complemented
10	#15690744	C₂²	1	2	yes	—
11	#15690741	—	2	2	no	—
12	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
13	#107×C₉₇	C₂²	1	2	yes	—
14	C₂×C₄×D₉₇	D₄	1	1	yes	characteristic
15	C₂×#11715895	D₄	1	1	yes	characteristic
16	C₈×D₉₇	C₂×C₄	1	2	yes	—
17	C₈×D₉₇	Q₈	1	2	yes	—
18	C₄⋊Dic₉₇	—	2	2	no	—
19	#13095488	—	4	2	no	—
20	C₄×Dic₉₇	—	4	2	no	—
21	#13095486	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₉₇	—	2	2	no	—
24	C₁₆×C₉₇	C₂×C₄	1	2	yes	—
25	C₂²×D₉₇	D₈	1	1	yes	characteristic
26	C₂×C₄×C₉₇	—	4	2	no	—
27	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₉₇	—	4	2	no	—
29	C₂×Dic₉₇	D₈	1	1	yes	characteristic
30	C₄×D₉₇	C₄⋊C₄	1	2	yes	—
31	C₄×D₉₇	Q₁₆	1	2	yes	—
32	C₈×C₉₇	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₉₇	C₂×Q₈	1	1	yes	characteristic
34	#11715895	C₄⋊C₄	1	2	yes	—
35	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
36	C₂×D₉₇	—	2	1	no	—
37	C₂×D₉₇	—	2	1	no	—
38	C₂×D₉₇	#107	1	2	yes	—
39	C₄×C₉₇	—	8	2	no	—
40	C₄×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₄×C₉₇	C₂×Q₁₆	1	1	yes	characteristic
42	Dic₉₇	—	8	2	no	—
43	Dic₉₇	#107	1	2	yes	—
44	C₂×C₉₇	C₂×#107	1	1	yes	characteristic
45	C₂×C₉₇	#508	1	1	yes	characteristic
46	C₂×C₉₇	#508	1	1	yes	characteristic
47	D₉₇	—	2	2	no	—
48	#2158	—	97	1	no	maximal, Sylow
49	C₉₇	#2158	1	1	yes	characteristic, complemented, Sylow
50	C₂²×C₁₆	—	97	1	no	—
51	#427	—	97	2	no	—
52	#367	—	97	2	no	—
53	#366	—	97	1	no	—
54	#366	D₉₇	1	1	yes	characteristic, complemented
55	C₂²×C₈	—	97	1	no	—
56	#117	—	194	2	no	—
57	C₂×C₁₆	—	97	1	no	—
58	C₂×C₁₆	—	97	4	no	—
59	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
60	#107	—	97	2	no	—
61	#107	C₂×D₉₇	1	2	yes	—
62	#106	—	194	2	no	—
63	C₂²×C₄	—	97	1	no	—
64	C₂×C₈	—	97	1	no	—
65	C₂×C₈	—	97	2	no	—
66	C₂×C₈	—	97	2	no	—
67	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	194	2	no	—
70	C₂²⋊C₄	—	388	2	no	—
71	C₄²	—	388	2	no	—
72	C₁₆	—	97	2	no	—
73	C₁₆	C₄×D₉₇	1	2	yes	—
74	C₂³	—	97	1	no	—
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	—	97	1	no	—
77	C₂×C₄	—	97	2	no	—
78	C₂×C₄	—	97	2	no	—
79	C₂×C₄	—	388	2	no	—
80	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
81	C₈	—	97	2	no	—
82	C₈	C₂×C₄×D₉₇	1	1	yes	characteristic
83	C₈	Q₈×D₉₇	1	1	yes	characteristic
84	C₂²	—	97	2	no	—
85	C₂²	—	194	1	no	—
86	C₂²	—	194	1	no	—
87	C₂²	D₈×D₉₇	1	1	yes	characteristic
88	C₄	—	8	2	no	—
89	C₄	—	776	2	no	—
90	C₄	—	97	2	no	—
91	C₄	C₄⋊C₄×D₉₇	1	1	yes	characteristic
92	C₄	Q₁₆×D₉₇	1	1	yes	characteristic
93	C₂	—	194	2	no	—
94	C₂	#107×D₉₇	1	1	yes	characteristic
95	C₂	#20261164	1	1	yes	characteristic
96	C₂	#20261168	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₉₇
commutator	C₈×C₉₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₉₇
Fitting	#366×C₉₇	C₂
socle	C₂²×C₉₇	C₂×D₈
prime radical	C₂²×D₉₇	D₈
cosocle	C₂×C₈×C₉₇	C₂³
hypercenter	#366	D₉₇
hypocenter	C₉₇	#2158
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
3. #20260562	C₂	semidirect
4. #20260561	C₂	semidirect
5. #366×C₉₇	C₂	semidirect
9. C₁₆×D₉₇	C₄	semidirect
49. C₉₇	#2158	semidirect
54. #366	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₆×D₉₇	2
#20260772	2
#20260562	2
#20260561	2
#366×C₉₇	2
#2158	97