Subgroups — #27156672

Order 12416.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊#13095514	C₂	1	2	yes	maximal
2	C₂×C₄⋊#11715895	C₂	1	1	yes	maximal, characteristic
3	C₂×#13095514	—	2	2	no	—
4	C₂²×#11715895	C₄	1	1	yes	characteristic
5	C₂²×#11715895	C₂²	1	1	yes	characteristic
6	C₄⋊#11715895	—	2	2	no	—
7	C₂×C₄²×C₉₇	C₄	1	1	yes	characteristic
8	#13095514	—	4	2	no	—
9	C₂×#11715895	—	2	1	no	—
10	C₂×#11715895	—	4	1	no	—
11	C₂×#11715895	—	4	1	no	—
12	C₂×#11715895	D₄	1	2	yes	—
13	C₂²×C₄×C₉₇	C₈	1	1	yes	characteristic
14	C₂²×C₄×C₉₇	C₈	1	1	yes	characteristic
15	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
16	C₄²×C₉₇	—	2	1	no	—
17	C₄²×C₉₇	—	2	1	no	—
18	C₂³×C₉₇	C₂×C₈	1	1	yes	characteristic
19	C₂×C₄×C₉₇	—	2	1	no	—
20	C₂×C₄×C₉₇	—	2	1	no	—
21	C₂×C₄×C₉₇	—	2	1	no	—
22	C₂×C₄×C₉₇	—	2	1	no	—
23	C₂×C₄×C₉₇	—	2	1	no	—
24	C₂×C₄×C₉₇	—	4	1	no	—
25	C₂×C₄×C₉₇	C₁₆	1	2	yes	complemented
26	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₉₇	M₁₆	1	1	yes	characteristic
28	#11715895	—	2	2	no	—
29	#11715895	—	4	1	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	C₂×C₁₆	1	1	yes	characteristic
34	C₂²×C₉₇	M₃₂	1	1	yes	characteristic
35	C₄×C₉₇	—	2	1	no	—
36	C₄×C₉₇	—	2	2	no	—
37	C₄×C₉₇	—	4	1	no	—
38	C₄×C₉₇	#99	1	1	yes	characteristic
39	C₄×C₉₇	#101	1	1	yes	characteristic
40	C₂×C₉₇	—	2	1	no	—
41	C₂×C₉₇	—	2	1	no	—
42	C₂×C₉₇	#324	1	1	yes	characteristic
43	C₂×C₉₇	C₂²⋊C₁₆	1	1	yes	characteristic
44	C₂×C₉₇	#349	1	1	yes	characteristic
45	#1322	—	97	1	no	maximal, Sylow
46	C₉₇	#1322	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₄⋊C₈	—	97	1	no	—
48	C₂²⋊C₁₆	—	97	2	no	—
49	C₂²×C₈	—	97	1	no	—
50	C₂²×C₈	—	97	1	no	—
51	C₂×C₄²	#87604	1	1	yes	characteristic
52	C₂×C₁₆	—	194	2	no	—
53	C₄⋊C₈	—	194	2	no	—
54	C₂²×C₄	#11715897	1	1	yes	characteristic
55	C₂²×C₄	#11715897	1	1	yes	characteristic
56	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
57	C₂×C₈	—	97	2	no	—
58	C₂×C₈	—	388	1	no	—
59	C₂×C₈	—	388	1	no	—
60	C₂×C₈	—	194	1	no	—
61	C₄²	—	2	1	no	—
62	C₄²	—	2	1	no	—
63	C₁₆	—	388	2	no	—
64	C₂³	C₂×#11715897	1	1	yes	characteristic
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	4	1	no	—
71	C₂×C₄	#13095514	1	2	yes	complemented
72	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
73	C₂×C₄	#13095520	1	1	yes	characteristic
74	C₈	—	388	1	no	—
75	C₈	—	194	2	no	—
76	C₂²	—	2	1	no	—
77	C₂²	—	2	1	no	—
78	C₂²	C₂²⋊#11715897	1	1	yes	characteristic
79	C₂²	C₂×#13095514	1	1	yes	characteristic
80	C₂²	#15690884	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	—	2	2	no	—
83	C₄	—	4	1	no	—
84	C₄	#15690869	1	1	yes	characteristic
85	C₄	#15690871	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2	1	no	—
88	C₂	#20261580	1	1	yes	characteristic
89	C₂	C₂²⋊#13095514	1	1	yes	characteristic
90	C₂	#20261631	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#87604
commutator	C₂²×C₉₇	C₂×C₁₆
Frattini	C₂²×C₄	C₂×#87604
Fitting	C₂×C₄²×C₉₇	C₄
socle	C₂²×C₉₇	C₂²⋊C₈
prime radical	C₂³×C₉₇	C₂×C₈
cosocle	C₂²×#11715895	C₂²
hypercenter	C₂×C₄²	#87604
hypocenter	C₉₇	#1322
LCS	G ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
25. C₂×C₄×C₉₇	C₁₆	semidirect
46. C₉₇	#1322	semidirect
71. C₂×C₄	#13095514	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊#13095514	2
C₂×C₄⋊#11715895	2
#1322	97