Subgroups — #27139680

Order 12416.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260461	C₂	1	2	yes	maximal
2	#382×C₉₇	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#11715895	—	2	2	no	—
4	C₂×C₄⋊C₄×C₉₇	C₄	1	1	yes	characteristic
5	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₉₇	C₄	1	1	yes	characteristic
7	#95×C₉₇	—	2	2	no	—
8	C₂×#11715895	—	4	2	no	—
9	C₂²×C₄×C₉₇	—	2	1	no	—
10	C₂²×C₄×C₉₇	—	2	1	no	—
11	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₉₇	D₄	1	2	yes	—
13	C₄⋊C₄×C₉₇	—	4	1	no	—
14	C₄⋊C₄×C₉₇	—	4	1	no	—
15	C₄²×C₉₇	—	2	1	no	—
16	C₄²×C₉₇	C₈	1	2	yes	complemented
17	C₂³×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
18	C₂×C₄×C₉₇	—	2	1	no	—
19	C₂×C₄×C₉₇	—	2	1	no	—
20	C₂×C₄×C₉₇	—	2	2	no	—
21	C₂×C₄×C₉₇	—	2	2	no	—
22	C₂×C₄×C₉₇	—	4	1	no	—
23	C₂×C₄×C₉₇	—	4	1	no	—
24	C₂×C₄×C₉₇	—	4	2	no	—
25	C₂×C₄×C₉₇	—	4	2	no	—
26	C₂×C₄×C₉₇	—	4	1	no	—
27	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₉₇	M₁₆	1	1	yes	characteristic
29	#11715895	—	8	2	no	—
30	C₂²×C₉₇	—	2	1	no	—
31	C₂²×C₉₇	—	2	1	no	—
32	C₂²×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
33	C₂²×C₉₇	#99	1	1	yes	characteristic
34	C₂²×C₉₇	#101	1	1	yes	characteristic
35	C₄×C₉₇	—	2	1	no	—
36	C₄×C₉₇	—	2	1	no	—
37	C₄×C₉₇	—	4	2	no	—
38	C₄×C₉₇	—	4	2	no	—
39	C₄×C₉₇	—	8	1	no	—
40	C₂×C₉₇	—	2	1	no	—
41	C₂×C₉₇	—	2	1	no	—
42	C₂×C₉₇	#324	1	1	yes	characteristic
43	C₂×C₉₇	#354	1	1	yes	characteristic
44	C₂×C₉₇	#355	1	1	yes	characteristic
45	#1324	—	97	1	no	maximal, Sylow
46	C₉₇	#1324	1	1	yes	characteristic, complemented, Sylow
47	#382	D₉₇	1	1	yes	characteristic
48	#324	—	97	2	no	—
49	C₂×C₄⋊C₄	Dic₉₇	1	1	yes	characteristic
50	C₂×C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
51	C₂×C₄²	Dic₉₇	1	1	yes	characteristic
52	C₂²⋊C₈	—	194	2	no	—
53	#95	—	2	2	no	—
54	C₂²×C₄	—	2	1	no	—
55	C₂²×C₄	—	2	1	no	—
56	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
57	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
58	C₂×C₈	—	388	2	no	—
59	C₄⋊C₄	—	4	1	no	—
60	C₄⋊C₄	—	4	1	no	—
61	C₄²	—	2	1	no	—
62	C₄²	#11715895	1	2	yes	complemented
63	C₂³	C₂²⋊Dic₉₇	1	1	yes	characteristic
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	4	1	no	—
69	C₂×C₄	—	4	1	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	4	2	no	—
72	C₂×C₄	—	4	1	no	—
73	C₂×C₄	C₂×#11715895	1	1	yes	characteristic
74	C₂×C₄	#13095495	1	1	yes	characteristic
75	C₈	—	776	2	no	—
76	C₂²	—	2	1	no	—
77	C₂²	—	2	1	no	—
78	C₂²	C₂²⋊#11715895	1	1	yes	characteristic
79	C₂²	#15690725	1	1	yes	characteristic
80	C₂²	#15690729	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	—	2	1	no	—
83	C₄	—	4	2	no	—
84	C₄	—	4	2	no	—
85	C₄	—	8	1	no	—
86	C₂	—	2	1	no	—
87	C₂	—	2	1	no	—
88	C₂	#20260461	1	1	yes	characteristic
89	C₂	#20260530	1	1	yes	characteristic
90	C₂	#20260532	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690725
commutator	C₂×C₄×C₉₇	C₂×C₈
Frattini	C₂×C₄⋊C₄	C₂×D₉₇
Fitting	#382×C₉₇	C₂
socle	C₂²×C₉₇	#99
prime radical	C₂³×C₉₇	C₂²⋊C₄
cosocle	C₂×C₄⋊C₄×C₉₇	C₂²
hypercenter	#382	D₉₇
hypocenter	C₉₇	#1324
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #382

Complements

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

N	H ≅ G/N	type
16. C₄²×C₉₇	C₈	semidirect
46. C₉₇	#1324	semidirect
62. C₄²	#11715895	semidirect

Maximal subgroups

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

subgroup	index
#20260461	2
#382×C₉₇	2
#1324	97