Subgroups — #27156674

Order 12416.

94 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	#20260762	C₂	1	1	yes	maximal, characteristic
3	C₂×#13095514	—	2	2	no	—
4	C₂×#13095495	C₄	1	1	yes	characteristic
5	C₂²×#11715895	C₂²	1	1	yes	characteristic
6	C₄⋊#11715895	—	2	2	no	—
7	#117×C₉₇	C₄	1	1	yes	characteristic
8	#13095514	—	4	2	no	—
9	#13095495	—	4	1	no	—
10	C₂×#11715895	—	2	1	no	—
11	C₂×#11715895	—	4	1	no	—
12	C₂×#11715895	D₄	1	2	yes	—
13	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
14	C₄⋊C₄×C₉₇	C₈	1	2	yes	—
15	C₂²⋊C₄×C₉₇	—	2	1	no	—
16	C₄²×C₉₇	—	2	1	no	—
17	C₂³×C₉₇	M₁₆	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	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
23	#11715895	—	2	2	no	—
24	#11715895	—	4	1	no	—
25	C₂²×C₉₇	—	2	1	no	—
26	C₂²×C₉₇	—	2	2	no	—
27	C₂²×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
28	C₄×C₉₇	—	2	1	no	—
29	C₄×C₉₇	—	4	2	no	—
30	C₄×C₉₇	#99	1	1	yes	characteristic
31	C₄×C₉₇	#101	1	1	yes	characteristic
32	C₂×C₉₇	—	4	1	no	—
33	C₂×C₉₇	#324	1	1	yes	characteristic
34	C₂×C₉₇	#350	1	2	yes	—
35	#1323	—	97	1	no	maximal, Sylow
36	C₉₇	#1323	1	1	yes	characteristic, complemented, Sylow
37	#424	—	97	1	no	—
38	C₂²⋊C₁₆	—	97	2	no	—
39	C₂×M₁₆	—	97	1	no	—
40	C₂²×C₈	—	97	1	no	—
41	#117	#87604	1	1	yes	characteristic
42	C₂×C₁₆	—	194	2	no	—
43	C₄⋊C₈	—	194	2	no	—
44	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
45	M₁₆	—	388	1	no	—
46	C₂×C₈	—	97	2	no	—
47	C₂×C₈	—	388	1	no	—
48	C₂×C₈	—	194	1	no	—
49	C₄⋊C₄	#11715897	1	2	yes	—
50	C₂²⋊C₄	—	2	1	no	—
51	C₄²	—	2	1	no	—
52	C₁₆	—	388	2	no	—
53	C₂³	#13095520	1	1	yes	characteristic
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
58	C₂×C₄	C₂×#11715897	1	1	yes	characteristic
59	C₈	—	388	1	no	—
60	C₈	—	194	2	no	—
61	C₂²	—	2	2	no	—
62	C₂²	—	2	1	no	—
63	C₂²	C₂²⋊#11715897	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	4	2	no	—
66	C₄	#15690869	1	1	yes	characteristic
67	C₄	#15690871	1	1	yes	characteristic
68	C₂	—	4	1	no	—
69	C₂	#20261580	1	1	yes	characteristic
70	C₂	#20261633	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊#87604
commutator	C₂×C₄×C₉₇	C₂×C₈
Frattini	C₂²×C₄	C₂×#87604
Fitting	#117×C₉₇	C₄
socle	C₂²×C₉₇	C₂²⋊C₈
prime radical	C₂³×C₉₇	M₁₆
cosocle	C₂²×#11715895	C₂²
hypercenter	#117	#87604
hypocenter	C₉₇	#1323
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
36. C₉₇	#1323	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
#20260762	2
#1323	97