Subgroups — #27139647

Order 12416.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260792	C₂	1	2	yes	maximal
2	#423×C₉₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11715895	—	2	2	no	—
4	C₂²⋊#11715895	C₄	1	4	yes	—
5	#15690721	—	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	C₂×#11715895	—	2	4	no	—
10	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
11	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
12	M₁₆×C₉₇	—	4	2	no	—
13	C₂×C₈×C₉₇	—	2	2	no	—
14	C₄²×C₉₇	D₄	1	1	yes	characteristic
15	C₄²×C₉₇	D₄	1	2	yes	—
16	C₄²×C₉₇	Q₈	1	1	yes	characteristic
17	C₂³×C₉₇	C₄²	1	1	yes	characteristic
18	C₂×C₄×C₉₇	—	2	1	no	—
19	C₂×C₄×C₉₇	—	2	1	no	—
20	C₂×C₄×C₉₇	—	4	2	no	—
21	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
22	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
24	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
25	C₈×C₉₇	—	4	2	no	—
26	#11715895	—	4	4	no	—
27	C₂²×C₉₇	—	2	1	no	—
28	C₂²×C₉₇	—	2	2	no	—
29	C₂²×C₉₇	#95	1	1	yes	characteristic
30	C₄×C₉₇	—	2	1	no	—
31	C₄×C₉₇	—	2	2	no	—
32	C₄×C₉₇	—	2	2	no	—
33	C₄×C₉₇	#100	1	1	yes	characteristic
34	C₄×C₉₇	#101	1	1	yes	characteristic
35	C₂×C₉₇	—	4	1	no	—
36	C₂×C₉₇	#343	1	1	yes	characteristic
37	C₂×C₉₇	#344	1	2	yes	—
38	#1307	—	97	1	no	maximal, Sylow
39	C₉₇	#1307	1	1	yes	characteristic, complemented, Sylow
40	#432	—	97	2	no	—
41	#423	D₉₇	1	1	yes	characteristic
42	C₂×M₁₆	Dic₉₇	1	2	yes	—
43	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
44	C₄⋊C₈	—	2	2	no	—
45	C₄⋊C₈	—	194	2	no	—
46	C₂²⋊C₈	—	97	4	no	—
47	#97	—	194	2	no	—
48	C₂²×C₄	C₄×D₉₇	1	2	yes	—
49	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
50	M₁₆	—	4	2	no	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₈	—	194	4	no	—
53	C₄²	Dic₁₉₄	1	1	yes	characteristic
54	C₄²	D₃₈₈	1	1	yes	characteristic
55	C₄²	C₂²⋊D₉₇	1	2	yes	—
56	C₂³	C₄×Dic₉₇	1	1	yes	characteristic
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	4	2	no	—
60	C₂×C₄	#13095488	1	2	yes	—
61	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
62	C₂×C₄	#13095490	1	2	yes	—
63	C₂×C₄	C₄⋊Dic₉₇	1	1	yes	characteristic
64	C₈	—	4	2	no	—
65	C₈	—	388	4	no	—
66	C₂²	—	2	1	no	—
67	C₂²	—	2	2	no	—
68	C₂²	#15690717	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	2	2	no	—
71	C₄	—	2	2	no	—
72	C₄	#15690727	1	1	yes	characteristic
73	C₄	#15690729	1	1	yes	characteristic
74	C₂	—	4	1	no	—
75	C₂	#20260504	1	1	yes	characteristic
76	C₂	#20260507	1	2	yes	—

	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	#1307
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
39. C₉₇	#1307	semidirect

Maximal subgroups

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

subgroup	index
#20260792	2
#423×C₉₇	2
#1307	97