Subgroups — #27141850

Order 12416.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690729	C₂	1	2	yes	maximal
2	#20260505	C₂	1	4	yes	maximal
3	#519×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095495	C₂²	1	4	yes	—
5	#15690729	—	2	4	no	—
6	C₂²×Q₈×C₉₇	C₂²	1	1	yes	characteristic
7	C₄×Q₈×C₉₇	—	2	4	no	—
8	#117×C₉₇	C₄	1	4	yes	—
9	#117×C₉₇	C₂²	1	2	yes	—
10	#13095495	—	2	4	no	—
11	#13095495	—	2	4	no	—
12	C₂×#11715895	—	2	4	no	—
13	C₂×Q₈×C₉₇	—	2	4	no	—
14	C₂×Q₈×C₉₇	D₄	1	4	yes	—
15	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
16	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
17	C₄⋊C₄×C₉₇	—	2	4	no	—
18	C₄⋊C₄×C₉₇	—	2	2	no	—
19	C₂²⋊C₄×C₉₇	C₂×C₄	1	4	yes	—
20	C₄²×C₉₇	—	2	4	no	—
21	C₄²×C₉₇	—	2	2	no	—
22	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
23	Q₈×C₉₇	—	2	8	no	—
24	C₂×C₄×C₉₇	—	2	4	no	—
25	C₂×C₄×C₉₇	—	2	4	no	—
26	C₂×C₄×C₉₇	C₂²⋊C₄	1	4	yes	—
27	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
28	C₂×C₄×C₉₇	#41	1	4	yes	—
29	#11715895	—	4	4	no	—
30	C₂²×C₉₇	—	4	1	no	—
31	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₉₇	#117	1	2	yes	—
33	C₄×C₉₇	—	2	4	no	—
34	C₄×C₉₇	—	4	4	no	—
35	C₄×C₉₇	#123	1	4	yes	—
36	C₂×C₉₇	—	2	1	no	—
37	C₂×C₉₇	—	2	2	no	—
38	C₂×C₉₇	#381	1	1	yes	characteristic
39	#1830	—	97	1	no	maximal, Sylow
40	C₉₇	#1830	1	1	yes	characteristic, complemented, Sylow
41	#519	D₉₇	1	1	yes	characteristic
42	C₂×#101	—	97	2	no	—
43	#344	—	97	4	no	—
44	C₂²×Q₈	C₂×D₉₇	1	1	yes	characteristic
45	C₂×M₁₆	—	97	4	no	—
46	C₄×Q₈	—	2	4	no	—
47	#117	Dic₉₇	1	4	yes	—
48	#117	C₂×D₉₇	1	2	yes	—
49	#101	—	194	4	no	—
50	C₂×Q₈	—	2	4	no	—
51	C₂×Q₈	C₂²⋊D₉₇	1	4	yes	—
52	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
53	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
54	M₁₆	—	194	4	no	—
55	M₁₆	—	194	4	no	—
56	C₂×C₈	—	194	4	no	—
57	C₄⋊C₄	—	2	4	no	—
58	C₄⋊C₄	—	2	2	no	—
59	C₂²⋊C₄	C₂×Dic₉₇	1	4	yes	—
60	C₄²	—	2	4	no	—
61	C₄²	—	2	2	no	—
62	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
63	Q₈	—	2	8	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	C₂²⋊Dic₉₇	1	4	yes	—
67	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
68	C₂×C₄	#13095511	1	4	yes	—
69	C₈	—	388	4	no	—
70	C₂²	—	4	1	no	—
71	C₂²	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
72	C₂²	#15690769	1	2	yes	—
73	C₄	—	2	4	no	—
74	C₄	—	4	4	no	—
75	C₄	#15690793	1	4	yes	—
76	C₂	—	2	1	no	—
77	C₂	—	2	2	no	—
78	C₂	#20260598	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20260598
commutator, prime radical	C₂³×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#519×C₉₇	C₂
socle	C₂×C₉₇	#381
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#519	D₉₇
hypocenter	C₉₇	#1830
LCS	G ▷ C₂³×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #519

Complements

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

N	H ≅ G/N	type
40. C₉₇	#1830	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690729	2
#20260505	2
#519×C₉₇	2
#1830	97