Subgroups — #27139677

Order 12416.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260761	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊#13095486	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊C₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095495	C₄	1	1	yes	characteristic
5	#15690769	C₄	1	1	yes	characteristic
6	C₂×#13095486	—	2	1	no	—
7	#15690739	—	2	1	no	—
8	#15690739	—	2	1	no	—
9	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×C₉₇	—	2	1	no	—
11	#13095495	—	4	1	no	—
12	C₂×#11715895	—	2	1	no	—
13	C₄⋊Dic₉₇	C₈	1	1	yes	characteristic
14	C₄⋊Dic₉₇	C₈	1	1	yes	characteristic
15	C₂²⋊Dic₉₇	—	2	1	no	—
16	C₄×Dic₉₇	—	2	1	no	—
17	#13095486	—	4	1	no	—
18	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₉₇	—	4	1	no	—
20	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
21	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
22	C₁₆×C₉₇	—	4	1	no	—
23	C₂³×C₉₇	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₉₇	—	2	1	no	—
25	C₂×C₄×C₉₇	—	2	1	no	—
26	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
28	C₂×Dic₉₇	—	2	1	no	—
29	C₂×Dic₉₇	—	2	1	no	—
30	C₈×C₉₇	—	2	1	no	—
31	C₈×C₉₇	—	2	1	no	—
32	#11715895	—	4	1	no	—
33	C₂²×C₉₇	—	2	1	no	—
34	C₂²×C₉₇	—	2	1	no	—
35	C₂²×C₉₇	—	2	1	no	—
36	C₂²×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
37	C₄×C₉₇	—	2	1	no	—
38	C₄×C₉₇	#99	1	1	yes	characteristic
39	C₄×C₉₇	#101	1	1	yes	characteristic
40	Dic₉₇	—	4	1	no	—
41	Dic₉₇	—	4	1	no	—
42	C₂×C₉₇	—	4	1	no	—
43	C₂×C₉₇	#324	1	1	yes	characteristic
44	C₂×C₉₇	#350	1	1	yes	characteristic
45	C₂×C₉₇	#350	1	1	yes	characteristic
46	#1323	—	97	1	no	maximal, Sylow
47	C₉₇	#1323	1	1	yes	characteristic, complemented, Sylow
48	#424	—	97	1	no	—
49	C₂²⋊C₁₆	—	97	1	no	—
50	C₂²⋊C₁₆	D₉₇	1	1	yes	characteristic
51	C₂×M₁₆	—	97	1	no	—
52	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
53	#117	—	97	1	no	—
54	C₂×C₁₆	—	2	1	no	—
55	C₂×C₁₆	—	194	1	no	—
56	C₄⋊C₈	—	194	1	no	—
57	C₄⋊C₈	—	194	1	no	—
58	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
59	M₁₆	—	388	1	no	—
60	C₂×C₈	—	4	1	no	—
61	C₂×C₈	—	194	1	no	—
62	C₂×C₈	D₃₈₈	1	1	yes	characteristic
63	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
64	C₄⋊C₄	—	97	1	no	—
65	C₄⋊C₄	—	97	1	no	—
66	C₂²⋊C₄	—	194	1	no	—
67	C₄²	—	194	1	no	—
68	C₁₆	—	4	1	no	—
69	C₁₆	—	388	1	no	—
70	C₂³	#13095494	1	1	yes	characteristic
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	194	1	no	—
74	C₂×C₄	—	194	1	no	—
75	C₂×C₄	#13095488	1	1	yes	characteristic
76	C₂×C₄	C₈×D₉₇	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	2	1	no	—
79	C₈	—	388	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	#15690722	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	388	1	no	—
86	C₄	—	388	1	no	—
87	C₄	#15690724	1	1	yes	characteristic
88	C₄	#15690728	1	1	yes	characteristic
89	C₂	—	4	1	no	—
90	C₂	#20260460	1	1	yes	characteristic
91	C₂	#20260518	1	1	yes	characteristic
92	C₂	#20260520	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13095488
commutator	C₂×C₄×C₉₇	C₂×C₈
Frattini	C₂²×C₈	C₂×D₉₇
Fitting	C₂²⋊C₁₆×C₉₇	C₂
socle	C₂²×C₉₇	C₂²⋊C₈
prime radical	C₂³×C₉₇	M₁₆
cosocle	C₂²×C₈×C₉₇	C₂²
hypercenter	C₂²⋊C₁₆	D₉₇
hypocenter	C₉₇	#1323
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
47. C₉₇	#1323	semidirect

Maximal subgroups

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

subgroup	index
#20260761	2
C₂²⋊#13095486	2
C₂²⋊C₁₆×C₉₇	2
#1323	97