Subgroups — #27139623

Order 12416.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#11715895	C₂	1	2	yes	maximal
2	#407×C₉₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11715895	—	2	2	no	—
4	C₂²⋊#11715895	—	2	2	no	—
5	C₂²⋊#11715895	C₄	1	4	yes	—
6	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₉₇	C₄	1	2	yes	—
8	C₂²⋊C₈×C₉₇	—	2	2	no	—
9	C₂×#11715895	—	2	4	no	—
10	C₂×#11715895	—	4	4	no	—
11	C₂⁴×C₉₇	Q₈	1	1	yes	characteristic
12	C₂²×C₄×C₉₇	—	4	2	no	—
13	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
14	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₉₇	D₄	1	2	yes	—
17	M₁₆×C₉₇	—	4	2	no	—
18	C₂×C₈×C₉₇	—	2	2	no	—
19	C₂³×C₉₇	—	2	4	no	—
20	C₂³×C₉₇	—	4	1	no	—
21	C₂³×C₉₇	C₄⋊C₄	1	2	yes	—
22	C₂³×C₉₇	C₄⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₉₇	—	2	2	no	—
24	C₂×C₄×C₉₇	—	2	4	no	—
25	C₂×C₄×C₉₇	—	4	2	no	—
26	C₂×C₄×C₉₇	—	4	1	no	—
27	C₂×C₄×C₉₇	C₄²	1	1	yes	characteristic
28	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
29	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₈×C₉₇	—	4	2	no	—
31	#11715895	—	4	4	no	—
32	C₂²×C₉₇	—	2	2	no	—
33	C₂²×C₉₇	—	2	2	no	—
34	C₂²×C₉₇	—	2	2	no	—
35	C₂²×C₉₇	—	2	1	no	—
36	C₂²×C₉₇	—	4	4	no	—
37	C₂²×C₉₇	#95	1	1	yes	characteristic
38	C₂²×C₉₇	#108	1	4	yes	—
39	C₄×C₉₇	—	2	2	no	—
40	C₄×C₉₇	—	2	1	no	—
41	C₄×C₉₇	#99	1	2	yes	—
42	C₂×C₉₇	—	2	4	no	—
43	C₂×C₉₇	—	4	1	no	—
44	C₂×C₉₇	#341	1	2	yes	—
45	C₂×C₉₇	#342	1	1	yes	characteristic
46	#1298	—	97	1	no	maximal, Sylow
47	C₉₇	#1298	1	1	yes	characteristic, complemented, Sylow
48	#407	D₉₇	1	1	yes	characteristic
49	C₂×C₂²⋊C₈	—	97	2	no	—
50	C₂³×C₄	C₂×D₉₇	1	1	yes	characteristic
51	C₂×M₁₆	Dic₉₇	1	2	yes	—
52	C₂²×C₈	—	194	2	no	—
53	C₂²⋊C₈	—	2	2	no	—
54	C₂²⋊C₈	—	97	4	no	—
55	C₂²⋊C₈	—	194	2	no	—
56	C₂⁴	Dic₁₉₄	1	1	yes	characteristic
57	C₂²×C₄	—	4	2	no	—
58	C₂²×C₄	C₄×D₉₇	1	2	yes	—
59	C₂²×C₄	D₃₈₈	1	1	yes	characteristic
60	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
61	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
62	M₁₆	—	4	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	388	4	no	—
65	C₂×C₈	—	194	4	no	—
66	C₂³	—	2	4	no	—
67	C₂³	—	4	1	no	—
68	C₂³	#13095490	1	2	yes	—
69	C₂³	C₄⋊Dic₉₇	1	1	yes	characteristic
70	C₂×C₄	—	2	4	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	4	1	no	—
74	C₂×C₄	C₄×Dic₉₇	1	1	yes	characteristic
75	C₂×C₄	#13095488	1	2	yes	—
76	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
77	C₈	—	4	2	no	—
78	C₈	—	388	4	no	—
79	C₂²	—	2	2	no	—
80	C₂²	—	2	2	no	—
81	C₂²	—	2	2	no	—
82	C₂²	—	2	1	no	—
83	C₂²	—	4	4	no	—
84	C₂²	#15690717	1	1	yes	characteristic
85	C₂²	#15690745	1	4	yes	—
86	C₄	—	2	2	no	—
87	C₄	—	2	1	no	—
88	C₄	#15690725	1	2	yes	—
89	C₂	—	2	4	no	—
90	C₂	—	4	1	no	—
91	C₂	#20260499	1	2	yes	—
92	C₂	#20260502	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₉₇
commutator	C₂×C₄×C₉₇	C₄²
Frattini	C₂³×C₄	C₂×D₉₇
Fitting	#407×C₉₇	C₂
socle	C₂²×C₉₇	#95
prime radical	C₂⁴×C₉₇	Q₈
cosocle	C₂³×C₄×C₉₇	C₂²
hypercenter	#407	D₉₇
hypocenter	C₉₇	#1298
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

Complements

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

N	H ≅ G/N	type
47. C₉₇	#1298	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊#11715895	2
#407×C₉₇	2
#1298	97