Subgroups — #27158970

Order 12416.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261844	C₂	1	2	yes	maximal, complemented
2	#20261843	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261842	C₂	1	2	yes	maximal, complemented
4	C₂×#15690900	C₂	1	1	yes	maximal, characteristic, complemented
5	#20261794	C₂	1	1	yes	maximal, characteristic
6	#15690900	C₂²	1	1	yes	characteristic
7	#15690900	C₂²	1	1	yes	characteristic
8	#15690900	C₂²	1	2	yes	—
9	#15690885	C₄	1	2	yes	—
10	#15690884	C₄	1	1	yes	characteristic
11	#15690883	C₂²	1	2	yes	—
12	C₂×#13095514	C₂²	1	1	yes	characteristic
13	#15690818	C₄	1	1	yes	characteristic
14	#13095514	C₂×C₄	1	1	yes	characteristic
15	#13095514	C₂×C₄	1	2	yes	—
16	#13095514	C₂³	1	1	yes	characteristic
17	#13095511	C₈	1	1	yes	characteristic
18	#13095494	C₈	1	2	yes	—
19	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
20	C₈×D₉₇	C₂×C₄	1	2	yes	—
21	M₁₆×C₉₇	C₈	1	1	yes	characteristic
22	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
23	C₂²⋊D₉₇	C₁₆	1	2	yes	—
24	D₃₈₈	C₁₆	1	1	yes	characteristic
25	C₄×D₉₇	C₂×C₈	1	2	yes	—
26	Dic₁₉₄	C₁₆	1	1	yes	characteristic
27	C₈×C₉₇	C₂×C₈	1	2	yes	—
28	#11715895	C₂×C₈	1	1	yes	characteristic
29	#11715895	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₉₇	C₂×C₁₆	1	1	yes	characteristic
31	C₂×D₉₇	C₂×C₁₆	1	2	yes	—
32	C₄×C₉₇	C₂×C₁₆	1	1	yes	characteristic
33	C₄×C₉₇	C₂²×C₈	1	1	yes	characteristic
34	Dic₉₇	C₂×C₁₆	1	2	yes	—
35	C₂×C₉₇	—	2	1	no	—
36	C₂×C₉₇	C₂²×C₁₆	1	1	yes	characteristic
37	D₉₇	—	2	2	no	—
38	#2258	—	97	1	no	maximal, Sylow
39	C₉₇	#2258	1	1	yes	characteristic, complemented, Sylow
40	#504	—	97	1	no	—
41	M₆₄	—	97	1	no	—
42	M₆₄	—	97	2	no	—
43	C₂×C₃₂	—	97	1	no	—
44	C₂×C₃₂	—	97	2	no	—
45	#131	—	97	1	no	—
46	M₃₂	—	97	1	no	—
47	M₃₂	—	97	2	no	—
48	C₂×C₁₆	—	97	1	no	—
49	C₂×C₁₆	—	97	2	no	—
50	C₃₂	—	97	1	no	—
51	C₃₂	—	97	1	no	—
52	C₃₂	—	97	2	no	—
53	#41	—	97	1	no	—
54	M₁₆	—	97	2	no	—
55	M₁₆	#11715906	1	1	yes	characteristic
56	C₂×C₈	—	97	1	no	—
57	C₂×C₈	—	97	2	no	—
58	C₁₆	—	97	1	no	—
59	C₁₆	—	97	2	no	—
60	C₁₆	—	97	1	no	—
61	Q₈	—	97	1	no	—
62	D₄	—	97	2	no	—
63	D₄	—	97	1	no	—
64	C₂×C₄	—	97	2	no	—
65	C₂×C₄	C₂×#11715906	1	1	yes	characteristic
66	C₈	—	97	1	no	—
67	C₈	—	97	1	no	—
68	C₈	C₂×#11715906	1	2	yes	—
69	C₂²	—	97	2	no	—
70	C₂²	#15690905	1	1	yes	characteristic
71	C₄	—	97	2	no	—
72	C₄	#15690905	1	1	yes	characteristic
73	C₄	C₂²×#11715906	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	194	2	no	—
76	C₂	C₂×#15690905	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×#11715906
commutator, socle	C₂×C₉₇	C₂²×C₁₆
Fitting	M₁₆×C₉₇	C₈
prime radical	#13095511	C₈
cosocle	#13095514	C₂³
hypercenter	M₁₆	#11715906
hypocenter	C₉₇	#2258
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
1. #20261844	C₂	semidirect
2. #20261843	C₂	semidirect
3. #20261842	C₂	semidirect
4. C₂×#15690900	C₂	semidirect
39. C₉₇	#2258	semidirect

Maximal subgroups

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

subgroup	index
#20261844	2
#20261843	2
#20261842	2
C₂×#15690900	2
#20261794	2
#2258	97