Subgroups — #27141369

Order 12416.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261133	C₂	1	1	yes	maximal, characteristic
2	#20261133	C₂	1	2	yes	maximal
3	#20261115	C₂	1	1	yes	maximal, characteristic
4	#20261086	C₂	1	2	yes	maximal
5	#322×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₈₈	—	2	2	no	—
7	#15690810	C₂²	1	2	yes	—
8	#15690810	C₂²	1	1	yes	characteristic
9	#15690810	C₂²	1	1	yes	characteristic
10	#15690742	—	2	4	no	—
11	#15690742	—	2	2	no	—
12	#15690735	—	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	1	no	—
17	C₂×Dic₁₉₄	—	2	2	no	—
18	Dic₃₈₈	—	4	4	no	—
19	C₄⋊Dic₉₇	—	2	2	no	—
20	C₄⋊Dic₉₇	—	2	2	no	—
21	C₄⋊Dic₉₇	—	2	2	no	—
22	C₄⋊Dic₉₇	—	2	2	no	—
23	C₂×C₈×C₉₇	D₄	1	4	yes	—
24	C₂×C₈×C₉₇	D₄	1	2	yes	—
25	C₄²×C₉₇	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
28	C₂×Dic₉₇	—	4	1	no	—
29	C₂×Dic₉₇	—	4	1	no	—
30	C₂×Dic₉₇	—	4	2	no	—
31	Dic₁₉₄	—	4	2	no	—
32	Dic₁₉₄	—	4	2	no	—
33	Dic₁₉₄	—	4	4	no	—
34	C₈×C₉₇	—	2	4	no	—
35	C₈×C₉₇	SD₁₆	1	4	yes	—
36	C₂²×C₉₇	#127	1	1	yes	characteristic
37	C₄×C₉₇	C₂×SD₁₆	1	2	yes	—
38	C₄×C₉₇	#137	1	4	yes	—
39	Dic₉₇	—	8	1	no	—
40	Dic₉₇	—	8	1	no	—
41	Dic₉₇	—	8	2	no	—
42	C₂×C₉₇	#492	1	1	yes	characteristic
43	C₂×C₉₇	#497	1	2	yes	—
44	#1716	—	97	1	no	maximal, Sylow
45	C₉₇	#1716	1	1	yes	characteristic, complemented, Sylow
46	#498	—	97	2	no	—
47	#498	—	97	1	no	—
48	#494	—	97	1	no	—
49	#487	—	97	2	no	—
50	#322	D₉₇	1	1	yes	characteristic
51	C₂×Q₁₆	—	194	2	no	—
52	C₄⋊Q₈	—	97	1	no	—
53	C₄⋊Q₈	—	97	2	no	—
54	C₄⋊Q₈	—	97	1	no	—
55	#106	—	194	4	no	—
56	#106	—	194	2	no	—
57	#103	—	194	4	no	—
58	C₄×C₈	C₂×D₉₇	1	2	yes	—
59	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
60	C₂×Q₈	—	194	2	no	—
61	C₂×Q₈	—	194	1	no	—
62	C₂×Q₈	—	194	1	no	—
63	Q₁₆	—	388	4	no	—
64	C₂×C₈	D₃₈₈	1	4	yes	—
65	C₂×C₈	D₃₈₈	1	2	yes	—
66	C₄⋊C₄	—	194	2	no	—
67	C₄⋊C₄	—	194	2	no	—
68	C₄⋊C₄	—	194	2	no	—
69	C₄⋊C₄	—	194	2	no	—
70	C₄²	C₂²×D₉₇	1	1	yes	characteristic
71	Q₈	—	388	2	no	—
72	Q₈	—	388	2	no	—
73	Q₈	—	388	4	no	—
74	C₂×C₄	—	388	2	no	—
75	C₂×C₄	—	388	1	no	—
76	C₂×C₄	—	388	1	no	—
77	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
78	C₂×C₄	C₂×D₃₈₈	1	2	yes	—
79	C₈	—	2	4	no	—
80	C₈	#13095500	1	4	yes	—
81	C₂²	#15690807	1	1	yes	characteristic
82	C₄	—	776	2	no	—
83	C₄	—	776	1	no	—
84	C₄	—	776	1	no	—
85	C₄	C₂×#13095500	1	2	yes	—
86	C₄	#15690843	1	4	yes	—
87	C₂	#20261108	1	1	yes	characteristic
88	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, prime radical	C₂²×C₉₇	#127
hypercenter	#322	D₉₇
hypocenter	C₉₇	#1716
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
45. C₉₇	#1716	semidirect

Maximal subgroups

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

subgroup	index
#20261133	2
#20261115	2
#20261086	2
#322×C₉₇	2
#1716	97