Subgroups — #27140019

Order 12416.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260788	C₂	1	2	yes	maximal
2	#20260456	C₂	1	2	yes	maximal, complemented
3	C₈×#11715895	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#11715895	C₄	1	4	yes	—
6	C₂²⋊#11715895	C₄	1	4	yes	—
7	C₄×#11715895	C₂²	1	4	yes	—
8	C₂²×C₈×C₉₇	C₄	1	2	yes	—
9	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
10	C₄×C₈×C₉₇	C₄	1	2	yes	—
11	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
13	C₂×#11715895	C₂×C₄	1	8	yes	—
14	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
15	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₉₇	C₈	1	8	yes	complemented
17	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
19	C₄²×C₉₇	C₂×C₄	1	2	yes	—
20	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₉₇	C₂³	1	1	yes	characteristic
22	C₂³×C₉₇	C₄²	1	1	yes	characteristic
23	C₂×C₄×C₉₇	—	2	4	no	—
24	C₂×C₄×C₉₇	C₄²	1	2	yes	—
25	C₂×C₄×C₉₇	C₄²	1	1	yes	characteristic
26	C₂×C₄×C₉₇	C₂×C₈	1	4	yes	—
27	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
28	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
29	C₈×C₉₇	C₂×C₈	1	4	yes	—
30	C₈×C₉₇	C₂×C₈	1	4	yes	—
31	#11715895	—	2	8	no	—
32	C₂²×C₉₇	—	2	2	no	—
33	C₂²×C₉₇	C₄×C₈	1	2	yes	—
34	C₂²×C₉₇	C₂×C₄²	1	1	yes	characteristic
35	C₄×C₉₇	—	2	2	no	—
36	C₄×C₉₇	C₄×C₈	1	2	yes	—
37	C₄×C₉₇	C₂²×C₈	1	2	yes	—
38	C₄×C₉₇	#131	1	4	yes	—
39	C₂×C₉₇	—	2	2	no	—
40	C₂×C₉₇	C₂×C₄×C₈	1	1	yes	characteristic
41	C₂×C₉₇	#405	1	2	yes	—
42	#1450	—	97	1	no	maximal, Sylow
43	C₉₇	#1450	1	1	yes	characteristic, complemented, Sylow
44	#431	—	97	2	no	—
45	C₂×C₄×C₈	D₉₇	1	1	yes	characteristic
46	#322	—	97	2	no	—
47	C₈²	—	97	2	no	—
48	C₂²×C₈	Dic₉₇	1	2	yes	—
49	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
50	C₄⋊C₈	—	97	4	no	—
51	C₂²⋊C₈	—	97	4	no	—
52	C₄×C₈	—	97	4	no	—
53	C₄×C₈	Dic₉₇	1	2	yes	—
54	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
55	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
56	C₂²×C₄	C₄×D₉₇	1	2	yes	—
57	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
58	C₂×C₈	—	97	8	no	—
59	C₂×C₈	#11715895	1	8	yes	complemented
60	C₂×C₈	C₂×Dic₉₇	1	2	yes	—
61	C₂×C₈	C₂×Dic₉₇	1	2	yes	—
62	C₄²	C₄×D₉₇	1	2	yes	—
63	C₄²	C₂×Dic₉₇	1	1	yes	characteristic
64	C₄²	C₂²×D₉₇	1	1	yes	characteristic
65	C₂³	C₄×Dic₉₇	1	1	yes	characteristic
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	C₄×Dic₉₇	1	2	yes	—
68	C₂×C₄	C₄×Dic₉₇	1	1	yes	characteristic
69	C₂×C₄	C₂×#11715895	1	4	yes	—
70	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
71	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
72	C₈	—	194	8	no	—
73	C₈	C₂×#11715895	1	4	yes	—
74	C₈	C₂×#11715895	1	4	yes	—
75	C₂²	—	2	2	no	—
76	C₂²	C₄×#11715895	1	2	yes	—
77	C₂²	C₂×C₄×Dic₉₇	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	C₄×#11715895	1	2	yes	—
80	C₄	C₂²×#11715895	1	2	yes	—
81	C₄	#15690816	1	4	yes	—
82	C₂	—	2	2	no	—
83	C₂	C₂×C₄×#11715895	1	1	yes	characteristic
84	C₂	#20260690	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂×D₉₇
commutator	C₂×C₉₇	C₂×C₄×C₈
Frattini	C₄²	C₂²×D₉₇
Fitting	C₂×C₄×C₈×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₄²
prime radical	C₂³×C₉₇	C₄²
cosocle	C₄²×C₉₇	C₂³
hypercenter	C₂×C₄×C₈	D₉₇
hypocenter	C₉₇	#1450
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20260456	C₂	semidirect
3. C₈×#11715895	C₂	semidirect
16. C₂×C₈×C₉₇	C₈	semidirect
43. C₉₇	#1450	semidirect
59. C₂×C₈	#11715895	semidirect

Maximal subgroups

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

subgroup	index
#20260788	2
#20260456	2
C₈×#11715895	2
C₂×C₄×C₈×C₉₇	2
#1450	97