Subgroups — #27139611

Order 12416.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260759	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#13095495	—	2	2	no	—
4	C₄⋊#11715895	—	2	2	no	—
5	C₄⋊#11715895	C₄	1	4	yes	—
6	C₂²×C₈×C₉₇	C₄	1	2	yes	—
7	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
8	C₄⋊C₈×C₉₇	—	2	2	no	—
9	#13095495	—	4	2	no	—
10	#13095495	—	4	2	no	—
11	C₂×#11715895	—	2	4	no	—
12	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₉₇	Q₈	1	1	yes	characteristic
15	C₂×C₈×C₉₇	—	2	2	no	—
16	C₂×C₈×C₉₇	—	4	2	no	—
17	C₄²×C₉₇	C₂×C₄	1	2	yes	—
18	C₄²×C₉₇	D₄	1	2	yes	—
19	C₂³×C₉₇	C₄⋊C₄	1	1	yes	characteristic
20	C₂×C₄×C₉₇	—	2	1	no	—
21	C₂×C₄×C₉₇	—	2	1	no	—
22	C₂×C₄×C₉₇	—	4	1	no	—
23	C₂×C₄×C₉₇	C₄²	1	1	yes	characteristic
24	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
25	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
27	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
28	C₂×C₄×C₉₇	SD₁₆	1	2	yes	—
29	C₂×C₄×C₉₇	Q₁₆	1	1	yes	characteristic
30	C₈×C₉₇	—	4	2	no	—
31	#11715895	—	4	4	no	—
32	C₂²×C₉₇	—	2	1	no	—
33	C₂²×C₉₇	—	2	1	no	—
34	C₂²×C₉₇	#95	1	1	yes	characteristic
35	C₂²×C₉₇	#106	1	1	yes	characteristic
36	C₂²×C₉₇	#107	1	1	yes	characteristic
37	C₄×C₉₇	—	2	2	no	—
38	C₄×C₉₇	—	2	1	no	—
39	C₄×C₉₇	#100	1	1	yes	characteristic
40	C₄×C₉₇	#101	1	1	yes	characteristic
41	C₄×C₉₇	#102	1	2	yes	—
42	C₄×C₉₇	#103	1	2	yes	—
43	C₂×C₉₇	—	2	1	no	—
44	C₂×C₉₇	—	2	1	no	—
45	C₂×C₉₇	#338	1	1	yes	characteristic
46	C₂×C₉₇	#340	1	1	yes	characteristic
47	C₂×C₉₇	#343	1	1	yes	characteristic
48	#1292	—	97	1	no	maximal, Sylow
49	C₉₇	#1292	1	1	yes	characteristic, complemented, Sylow
50	#423	—	97	2	no	—
51	C₂×C₄⋊C₈	D₉₇	1	1	yes	characteristic
52	C₂×M₁₆	—	194	2	no	—
53	C₂²×C₈	Dic₉₇	1	2	yes	—
54	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
55	C₄⋊C₈	—	2	2	no	—
56	C₄⋊C₈	—	97	4	no	—
57	C₄⋊C₈	—	194	2	no	—
58	C₂²×C₄	Dic₁₉₄	1	1	yes	characteristic
59	C₂²×C₄	D₃₈₈	1	1	yes	characteristic
60	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
61	M₁₆	—	388	2	no	—
62	M₁₆	—	388	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	4	2	no	—
65	C₂×C₈	—	194	4	no	—
66	C₄²	C₄×D₉₇	1	2	yes	—
67	C₄²	C₂²⋊D₉₇	1	2	yes	—
68	C₂³	C₄⋊Dic₉₇	1	1	yes	characteristic
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	4	1	no	—
72	C₂×C₄	C₄×Dic₉₇	1	1	yes	characteristic
73	C₂×C₄	#13095488	1	2	yes	—
74	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
75	C₂×C₄	#13095490	1	2	yes	—
76	C₂×C₄	D₇₇₆	1	1	yes	characteristic
77	C₂×C₄	#13095500	1	2	yes	—
78	C₂×C₄	Dic₃₈₈	1	1	yes	characteristic
79	C₈	—	4	2	no	—
80	C₈	—	388	4	no	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	#15690717	1	1	yes	characteristic
84	C₂²	#15690742	1	1	yes	characteristic
85	C₂²	#15690744	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	2	1	no	—
88	C₄	#15690727	1	1	yes	characteristic
89	C₄	#15690729	1	1	yes	characteristic
90	C₄	#15690732	1	2	yes	—
91	C₄	#15690735	1	2	yes	—
92	C₂	—	2	1	no	—
93	C₂	—	2	1	no	—
94	C₂	#20260493	1	1	yes	characteristic
95	C₂	#20260498	1	1	yes	characteristic
96	C₂	#20260504	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690717
commutator	C₂×C₄×C₉₇	C₄²
Frattini	C₂×C₄²	C₂×D₉₇
Fitting	C₂×C₄⋊C₈×C₉₇	C₂
socle	C₂²×C₉₇	#95
prime radical	C₂³×C₉₇	C₄⋊C₄
cosocle	C₂×C₄²×C₉₇	C₂²
hypercenter	C₂×C₄⋊C₈	D₉₇
hypocenter	C₉₇	#1292
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
49. C₉₇	#1292	semidirect

Maximal subgroups

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

subgroup	index
#20260759	2
C₂×C₄⋊C₈×C₉₇	2
#1292	97