Subgroups — #27141407

Order 12416.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261140	C₂	1	2	yes	maximal, complemented
2	#20261115	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261108	C₂	1	1	yes	maximal, characteristic
4	#20261082	C₂	1	2	yes	maximal
5	#322×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃₈₈	—	2	2	no	—
7	C₂×#13095500	—	2	2	no	—
8	#15690810	C₂²	1	1	yes	characteristic
9	#15690810	C₂²	1	2	yes	—
10	#15690807	C₂²	1	1	yes	characteristic
11	#15690744	—	2	4	no	—
12	#15690732	—	2	4	no	—
13	C₄×C₈×C₉₇	C₂²	1	2	yes	—
14	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₉₄	—	2	1	no	—
16	C₂×Dic₁₉₄	—	2	2	no	—
17	C₂×D₃₈₈	—	2	1	no	—
18	C₂×D₃₈₈	—	2	2	no	—
19	Dic₃₈₈	—	4	4	no	—
20	#13095500	—	4	4	no	—
21	C₄⋊Dic₉₇	—	2	2	no	—
22	C₄⋊Dic₉₇	—	2	2	no	—
23	C₄⋊Dic₉₇	—	2	2	no	—
24	C₂×C₈×C₉₇	D₄	1	4	yes	—
25	C₂×C₈×C₉₇	D₄	1	2	yes	—
26	C₄²×C₉₇	C₂³	1	1	yes	characteristic
27	C₂²×D₉₇	—	4	1	no	—
28	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
30	C₂×Dic₉₇	—	4	2	no	—
31	C₂×Dic₉₇	—	4	1	no	—
32	D₃₈₈	—	4	2	no	—
33	D₃₈₈	—	4	4	no	—
34	Dic₁₉₄	—	4	2	no	—
35	Dic₁₉₄	—	4	4	no	—
36	C₈×C₉₇	—	2	4	no	—
37	C₈×C₉₇	D₈	1	4	yes	—
38	C₂²×C₉₇	#127	1	1	yes	characteristic
39	C₂×D₉₇	—	8	1	no	—
40	C₂×D₉₇	—	8	2	no	—
41	C₄×C₉₇	C₂×D₈	1	2	yes	—
42	C₄×C₉₇	#137	1	4	yes	—
43	Dic₉₇	—	8	1	no	—
44	Dic₉₇	—	8	2	no	—
45	C₂×C₉₇	#493	1	1	yes	characteristic
46	C₂×C₉₇	#497	1	2	yes	—
47	D₉₇	—	16	1	no	—
48	#1722	—	97	1	no	maximal, Sylow
49	C₉₇	#1722	1	1	yes	characteristic, complemented, Sylow
50	#500	—	97	2	no	—
51	#494	—	97	1	no	—
52	#492	—	97	1	no	—
53	#486	—	97	2	no	—
54	#322	D₉₇	1	1	yes	characteristic, complemented
55	C₂×Q₁₆	—	194	2	no	—
56	C₂×SD₁₆	—	194	2	no	—
57	C₄⋊Q₈	—	97	2	no	—
58	C₄⋊Q₈	—	97	1	no	—
59	#127	—	97	1	no	—
60	#107	—	194	4	no	—
61	#102	—	194	4	no	—
62	C₄×C₈	C₂×D₉₇	1	2	yes	—
63	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
64	C₂×Q₈	—	194	1	no	—
65	C₂×Q₈	—	194	2	no	—
66	C₂×D₄	—	194	2	no	—
67	C₂×D₄	—	194	1	no	—
68	Q₁₆	—	388	4	no	—
69	SD₁₆	—	388	4	no	—
70	C₂×C₈	D₃₈₈	1	4	yes	—
71	C₂×C₈	D₃₈₈	1	2	yes	—
72	C₄⋊C₄	—	194	2	no	—
73	C₄⋊C₄	—	194	2	no	—
74	C₄⋊C₄	—	194	2	no	—
75	C₄²	C₂²×D₉₇	1	1	yes	characteristic
76	C₂³	—	388	1	no	—
77	Q₈	—	388	2	no	—
78	Q₈	—	388	4	no	—
79	D₄	—	388	2	no	—
80	D₄	—	388	4	no	—
81	C₂×C₄	—	388	2	no	—
82	C₂×C₄	—	388	1	no	—
83	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
84	C₂×C₄	C₂×D₃₈₈	1	2	yes	—
85	C₈	—	2	4	no	—
86	C₈	D₇₇₆	1	4	yes	—
87	C₂²	—	776	2	no	—
88	C₂²	—	776	1	no	—
89	C₂²	#15690807	1	1	yes	characteristic
90	C₄	—	776	1	no	—
91	C₄	—	776	2	no	—
92	C₄	C₂×D₇₇₆	1	2	yes	—
93	C₄	#15690843	1	4	yes	—
94	C₂	—	1552	1	no	—
95	C₂	#20261112	1	1	yes	characteristic
96	C₂	#20261129	1	2	yes	—

	subgroup	G/H
center	C₂²	#15690807
commutator, cosocle	C₄²×C₉₇	C₂³
Frattini	C₄²	C₂²×D₉₇
Fitting	#322×C₉₇	C₂
socle	C₂²×C₉₇	#127
prime radical	#15690807	C₂²
hypercenter	#322	D₉₇
hypocenter	C₉₇	#1722
LCS	G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
1. #20261140	C₂	semidirect
2. #20261115	C₂	semidirect
5. #322×C₉₇	C₂	semidirect
49. C₉₇	#1722	semidirect
54. #322	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261140	2
#20261115	2
#20261108	2
#20261082	2
#322×C₉₇	2
#1722	97