Subgroups — #27158776

Order 12416.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261834	C₂	1	2	yes	maximal
2	#20261739	C₂	1	1	yes	maximal, characteristic
3	#15690906	—	2	2	no	—
4	C₂×#13095521	C₄	1	1	yes	characteristic
5	C₂×#13095521	C₂²	1	1	yes	characteristic
6	C₄⋊#11715897	—	2	2	no	—
7	C₄²×D₉₇	C₄	1	1	yes	characteristic
8	#13095523	—	4	2	no	—
9	#13095521	—	4	1	no	—
10	#13095521	—	4	1	no	—
11	C₂×#11715897	—	2	1	no	—
12	C₂×#11715897	D₄	1	2	yes	—
13	C₂×C₄×D₉₇	—	2	1	no	—
14	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
15	C₄×Dic₉₇	—	2	1	no	—
16	C₄×Dic₉₇	C₈	1	1	yes	characteristic
17	C₄²×C₉₇	C₈	1	1	yes	characteristic
18	C₂²×D₉₇	M₁₆	1	1	yes	characteristic
19	C₂×C₄×C₉₇	—	2	1	no	—
20	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
21	C₂×Dic₉₇	—	2	1	no	—
22	C₂×Dic₉₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×D₉₇	—	2	1	no	—
24	C₄×D₉₇	—	2	1	no	—
25	C₄×D₉₇	—	4	2	no	—
26	#11715897	—	2	2	no	—
27	#11715897	—	4	1	no	—
28	C₂²×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
29	C₂×D₉₇	—	2	1	no	—
30	C₂×D₉₇	—	4	1	no	—
31	C₄×C₉₇	—	2	2	no	—
32	C₄×C₉₇	—	2	1	no	—
33	Dic₉₇	—	4	1	no	—
34	Dic₉₇	#99	1	1	yes	characteristic
35	Dic₉₇	#101	1	1	yes	characteristic
36	C₂×C₉₇	—	2	1	no	—
37	C₂×C₉₇	#324	1	1	yes	characteristic
38	D₉₇	—	4	1	no	—
39	#1327	—	97	1	no	maximal, Sylow
40	C₉₇	#1327	1	1	yes	characteristic, complemented, Sylow
41	#423	—	97	1	no	—
42	#349	—	97	2	no	—
43	C₂×M₁₆	—	97	1	no	—
44	C₂×M₁₆	—	97	1	no	—
45	C₂×C₄²	—	97	1	no	—
46	M₃₂	—	194	2	no	—
47	C₄⋊C₈	—	194	2	no	—
48	C₂²×C₄	—	97	1	no	—
49	C₂²×C₄	—	194	1	no	—
50	M₁₆	—	388	1	no	—
51	M₁₆	—	388	1	no	—
52	C₂×C₈	—	97	2	no	—
53	C₂×C₈	—	194	1	no	—
54	C₄²	—	97	1	no	—
55	C₄²	—	194	1	no	—
56	C₄²	#11715906	1	1	yes	characteristic
57	C₁₆	—	388	2	no	—
58	C₂³	—	97	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	97	1	no	—
61	C₂×C₄	—	388	2	no	—
62	C₂×C₄	—	194	1	no	—
63	C₂×C₄	—	194	1	no	—
64	C₂×C₄	—	194	1	no	—
65	C₂×C₄	C₂×#11715906	1	1	yes	characteristic
66	C₈	—	388	1	no	—
67	C₈	—	194	2	no	—
68	C₂²	—	388	1	no	—
69	C₂²	—	194	1	no	—
70	C₂²	C₂²⋊#11715906	1	1	yes	characteristic
71	C₄	—	2	2	no	—
72	C₄	—	2	1	no	—
73	C₄	—	97	1	no	—
74	C₄	—	97	1	no	—
75	C₄	—	388	1	no	—
76	C₂	—	2	1	no	—
77	C₂	—	388	1	no	—
78	C₂	#20261821	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20261821
commutator	C₂×C₄×C₉₇	C₂×C₈
Frattini	C₂×C₄	C₂×#11715906
Fitting	C₄²×C₉₇	C₈
socle	C₂×C₉₇	#324
prime radical	C₂²×D₉₇	M₁₆
cosocle	C₂×#13095521	C₂²
hypercenter	C₄²	#11715906
hypocenter	C₉₇	#1327
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
40. C₉₇	#1327	semidirect

Maximal subgroups

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

subgroup	index
#20261834	2
#20261739	2
#1327	97