Subgroups — #27139649

Order 12416.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260759	C₂	1	2	yes	maximal
2	#423×C₉₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#13095495	C₄	1	4	yes	—
4	C₄⋊#11715895	—	2	2	no	—
5	C₄⋊#11715895	—	2	2	no	—
6	C₂×M₁₆×C₉₇	C₄	1	2	yes	—
7	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
8	C₄⋊C₈×C₉₇	—	2	2	no	—
9	#13095495	—	4	4	no	—
10	C₂×#11715895	—	2	4	no	—
11	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
12	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
13	M₁₆×C₉₇	—	4	2	no	—
14	C₂×C₈×C₉₇	—	2	2	no	—
15	C₄²×C₉₇	D₄	1	1	yes	characteristic
16	C₄²×C₉₇	D₄	1	2	yes	—
17	C₄²×C₉₇	Q₈	1	1	yes	characteristic
18	C₂³×C₉₇	C₄²	1	1	yes	characteristic
19	C₂×C₄×C₉₇	—	2	1	no	—
20	C₂×C₄×C₉₇	—	2	2	no	—
21	C₂×C₄×C₉₇	—	2	2	no	—
22	C₂×C₄×C₉₇	—	2	1	no	—
23	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
24	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
27	C₈×C₉₇	—	4	2	no	—
28	#11715895	—	4	4	no	—
29	C₂²×C₉₇	—	2	2	no	—
30	C₂²×C₉₇	—	2	1	no	—
31	C₂²×C₉₇	#95	1	1	yes	characteristic
32	C₄×C₉₇	—	2	2	no	—
33	C₄×C₉₇	—	2	1	no	—
34	C₄×C₉₇	#100	1	2	yes	—
35	C₄×C₉₇	#100	1	1	yes	characteristic
36	C₄×C₉₇	#101	1	2	yes	—
37	C₄×C₉₇	#101	1	1	yes	characteristic
38	C₂×C₉₇	—	4	1	no	—
39	C₂×C₉₇	#343	1	2	yes	—
40	C₂×C₉₇	#343	1	1	yes	characteristic
41	#1309	—	97	1	no	maximal, Sylow
42	C₉₇	#1309	1	1	yes	characteristic, complemented, Sylow
43	#423	—	97	2	no	—
44	#423	D₉₇	1	1	yes	characteristic
45	C₂×M₁₆	—	97	4	no	—
46	C₂×M₁₆	Dic₉₇	1	2	yes	—
47	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
48	C₄⋊C₈	—	2	2	no	—
49	C₄⋊C₈	—	194	2	no	—
50	C₄⋊C₈	—	194	2	no	—
51	C₂²×C₄	C₄×D₉₇	1	2	yes	—
52	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
53	M₁₆	—	4	2	no	—
54	M₁₆	—	388	4	no	—
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	194	4	no	—
57	C₄²	Dic₁₉₄	1	1	yes	characteristic
58	C₄²	D₃₈₈	1	1	yes	characteristic
59	C₄²	C₂²⋊D₉₇	1	2	yes	—
60	C₂³	C₄×Dic₉₇	1	1	yes	characteristic
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	#13095488	1	2	yes	—
66	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
67	C₂×C₄	#13095490	1	2	yes	—
68	C₂×C₄	C₄⋊Dic₉₇	1	1	yes	characteristic
69	C₈	—	4	2	no	—
70	C₈	—	388	4	no	—
71	C₂²	—	2	2	no	—
72	C₂²	—	2	1	no	—
73	C₂²	#15690717	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	2	1	no	—
76	C₄	#15690726	1	2	yes	—
77	C₄	#15690727	1	1	yes	characteristic
78	C₄	#15690728	1	2	yes	—
79	C₄	#15690729	1	1	yes	characteristic
80	C₂	—	4	1	no	—
81	C₂	#20260503	1	2	yes	—
82	C₂	#20260504	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690717
commutator, prime radical	C₂³×C₉₇	C₄²
Frattini	C₂×C₄²	C₂×D₉₇
Fitting	#423×C₉₇	C₂
socle	C₂²×C₉₇	#95
cosocle	C₂×C₄²×C₉₇	C₂²
hypercenter	#423	D₉₇
hypocenter	C₉₇	#1309
LCS	G ▷ C₂³×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
42. C₉₇	#1309	semidirect

Maximal subgroups

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

subgroup	index
#20260759	2
#423×C₉₇	2
#1309	97