Subgroups — #27141423

Order 12416.

328 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690721	C₂	1	3	yes	maximal
2	C₂×C₄×#11715895	C₂	1	3	yes	maximal
3	#376×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11715895	C₂²	1	1	yes	characteristic
5	C₂²×#11715895	C₂²	1	3	yes	—
6	#15690721	C₄	1	12	yes	complemented
7	C₄×#11715895	C₄	1	12	yes	complemented
8	C₂×C₄²×C₉₇	C₂²	1	3	yes	—
9	#95×C₉₇	C₄	1	4	yes	—
10	C₂×#11715895	C₂×C₄	1	6	yes	—
11	C₂×#11715895	C₂×C₄	1	6	yes	—
12	C₂×#11715895	C₂×C₄	1	12	yes	—
13	C₂²×C₄×C₉₇	C₂×C₄	1	6	yes	—
14	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
15	C₄²×C₉₇	C₂×C₄	1	6	yes	—
16	C₄²×C₉₇	C₂×C₄	1	6	yes	—
17	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₄×C₉₇	—	2	12	no	—
19	C₂×C₄×C₉₇	C₄²	1	12	yes	—
20	C₂×C₄×C₉₇	C₂²×C₄	1	3	yes	—
21	C₂×C₄×C₉₇	C₂²×C₄	1	3	yes	—
22	#11715895	C₄²	1	4	yes	—
23	#11715895	C₄²	1	12	yes	—
24	C₂²×C₉₇	C₂×C₄²	1	3	yes	—
25	C₂²×C₉₇	#131	1	4	yes	—
26	C₄×C₉₇	—	2	12	no	—
27	C₄×C₉₇	C₂×C₄²	1	1	yes	characteristic
28	C₄×C₉₇	C₂×C₄²	1	3	yes	—
29	C₂×C₉₇	C₄³	1	1	yes	characteristic
30	C₂×C₉₇	#405	1	6	yes	—
31	#1726	—	97	1	no	maximal, Sylow
32	C₉₇	#1726	1	1	yes	characteristic, complemented, Sylow
33	C₂×#97	—	97	3	no	—
34	C₂×C₄×C₈	—	97	3	no	—
35	#376	D₉₇	1	1	yes	characteristic
36	C₂²×C₈	—	97	1	no	—
37	C₂²×C₈	—	97	3	no	—
38	C₂×C₄²	C₂×D₉₇	1	3	yes	—
39	#97	—	97	12	no	—
40	C₄×C₈	—	97	12	no	—
41	#95	Dic₉₇	1	4	yes	—
42	C₂²×C₄	C₂×Dic₉₇	1	6	yes	—
43	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
44	C₂×C₈	—	97	6	no	—
45	C₂×C₈	—	97	6	no	—
46	C₂×C₈	—	97	12	no	—
47	C₄²	C₄×D₉₇	1	6	yes	—
48	C₄²	C₄×D₉₇	1	6	yes	—
49	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
50	C₂×C₄	—	2	12	no	—
51	C₂×C₄	C₄×Dic₉₇	1	12	yes	—
52	C₂×C₄	C₂×C₄×D₉₇	1	3	yes	—
53	C₂×C₄	C₂×C₄×D₉₇	1	3	yes	—
54	C₈	—	97	4	no	—
55	C₈	—	97	12	no	—
56	C₂²	C₂×C₄×Dic₉₇	1	3	yes	—
57	C₂²	#15690817	1	4	yes	—
58	C₄	—	2	12	no	—
59	C₄	C₄²×D₉₇	1	1	yes	characteristic
60	C₄	C₄²×D₉₇	1	3	yes	—
61	C₂	C₄²×Dic₉₇	1	1	yes	characteristic
62	C₂	#20260691	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₉₇
commutator	C₂×C₉₇	C₄³
Fitting	#376×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#376	D₉₇
hypocenter	C₉₇	#1726
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
6. #15690721	C₄	semidirect
7. C₄×#11715895	C₄	semidirect
32. C₉₇	#1726	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690721	2
C₂×C₄×#11715895	2
#376×C₉₇	2
#1726	97