Subgroups — #27143315

Order 12416.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260763	C₂	1	1	yes	maximal, characteristic
2	#20260506	C₂	1	2	yes	maximal
3	#20260499	C₂	1	2	yes	maximal
4	#20260493	C₂	1	1	yes	maximal, characteristic
5	#430×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13095495	C₂²	1	2	yes	—
7	C₂²×#11715895	C₂²	1	1	yes	characteristic
8	#15690769	C₂²	1	1	yes	characteristic
9	#15690739	—	2	2	no	—
10	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
11	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
12	#108×C₉₇	—	2	2	no	—
13	#13095495	—	2	2	no	—
14	#13095495	—	2	2	no	—
15	C₂×#11715895	—	2	1	no	—
16	C₂×#11715895	—	2	2	no	—
17	C₂×#11715895	—	2	2	no	—
18	C₄⋊Dic₉₇	—	2	1	no	—
19	C₂²⋊Dic₉₇	Q₈	1	2	yes	—
20	C₄×Dic₉₇	—	2	1	no	—
21	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
22	M₁₆×C₉₇	—	2	2	no	—
23	M₁₆×C₉₇	—	2	2	no	—
24	M₁₆×C₉₇	—	2	2	no	—
25	C₂×C₈×C₉₇	—	2	2	no	—
26	C₂×C₈×C₉₇	D₄	1	2	yes	—
27	C₂³×C₉₇	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₉₇	#41	1	2	yes	—
30	C₂×C₄×C₉₇	#41	1	2	yes	—
31	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
32	C₂×Dic₉₇	—	2	2	no	—
33	C₈×C₉₇	—	2	2	no	—
34	C₈×C₉₇	—	4	2	no	—
35	#11715895	—	4	1	no	—
36	#11715895	—	4	2	no	—
37	C₂²×C₉₇	—	4	1	no	—
38	C₂²×C₉₇	C₂²⋊Q₈	1	1	yes	characteristic
39	C₂²×C₉₇	#125	1	2	yes	—
40	C₄×C₉₇	#123	1	1	yes	characteristic
41	C₄×C₉₇	#124	1	1	yes	characteristic
42	C₄×C₉₇	#126	1	2	yes	—
43	Dic₉₇	—	4	2	no	—
44	C₂×C₉₇	—	2	1	no	—
45	C₂×C₉₇	—	2	2	no	—
46	C₂×C₉₇	#400	1	1	yes	characteristic
47	#2103	—	97	1	no	maximal, Sylow
48	C₉₇	#2103	1	1	yes	characteristic, complemented, Sylow
49	#430	D₉₇	1	1	yes	characteristic
50	#424	—	97	1	no	—
51	#344	—	97	2	no	—
52	#341	—	97	2	no	—
53	#338	—	97	1	no	—
54	C₂×M₁₆	—	97	2	no	—
55	C₂×M₁₆	C₂×D₉₇	1	2	yes	—
56	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
57	C₂²×C₈	—	97	1	no	—
58	#117	—	97	1	no	—
59	#108	—	2	2	no	—
60	C₄⋊C₈	—	194	2	no	—
61	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
62	M₁₆	—	2	2	no	—
63	M₁₆	—	2	2	no	—
64	M₁₆	—	2	2	no	—
65	M₁₆	—	194	2	no	—
66	M₁₆	—	194	2	no	—
67	C₂×C₈	—	2	2	no	—
68	C₂×C₈	—	194	1	no	—
69	C₂×C₈	—	194	2	no	—
70	C₂×C₈	—	194	2	no	—
71	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
72	C₄⋊C₄	—	194	1	no	—
73	C₂²⋊C₄	—	97	2	no	—
74	C₄²	—	194	1	no	—
75	C₂³	Q₈×D₉₇	1	1	yes	characteristic
76	C₂×C₄	—	194	2	no	—
77	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
78	C₂×C₄	#13095510	1	2	yes	—
79	C₂×C₄	#13095511	1	2	yes	—
80	C₂×C₄	#13095512	1	1	yes	characteristic
81	C₈	—	2	2	no	—
82	C₈	—	4	2	no	—
83	C₈	—	388	1	no	—
84	C₈	—	388	2	no	—
85	C₂²	—	4	1	no	—
86	C₂²	#15690786	1	1	yes	characteristic
87	C₂²	#15690800	1	2	yes	—
88	C₄	—	388	2	no	—
89	C₄	#15690793	1	1	yes	characteristic
90	C₄	#15690797	1	1	yes	characteristic
91	C₄	#15690805	1	2	yes	—
92	C₂	—	2	1	no	—
93	C₂	—	2	2	no	—
94	C₂	#20260673	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15690797
commutator, cosocle	C₂²×C₄×C₉₇	C₂³
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#430×C₉₇	C₂
socle	C₂×C₉₇	#400
prime radical	C₂³×C₉₇	C₂×Q₈
hypercenter	#430	D₉₇
hypocenter	C₉₇	#2103
LCS	G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

Complements

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

N	H ≅ G/N	type
48. C₉₇	#2103	semidirect

Maximal subgroups

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

subgroup	index
#20260763	2
#20260506	2
#20260499	2
#20260493	2
#430×C₉₇	2
#2103	97