Subgroups — #27141890

Order 12416.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11715895	C₂	1	6	yes	maximal
2	#376×C₉₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11715895	C₂²	1	4	yes	—
4	C₄⋊#11715895	—	2	12	no	—
5	C₂×C₄²×C₉₇	C₂²	1	3	yes	—
6	#95×C₉₇	C₄	1	4	yes	—
7	C₂×#11715895	—	2	12	no	—
8	C₂²×C₄×C₉₇	C₂×C₄	1	6	yes	—
9	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
10	C₄²×C₉₇	D₄	1	6	yes	—
11	C₄²×C₉₇	Q₈	1	6	yes	—
12	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
13	C₂×C₄×C₉₇	—	2	12	no	—
14	C₂×C₄×C₉₇	C₄⋊C₄	1	12	yes	—
15	C₂×C₄×C₉₇	C₂×D₄	1	3	yes	—
16	C₂×C₄×C₉₇	C₂×Q₈	1	3	yes	—
17	#11715895	—	4	4	no	—
18	C₂²×C₉₇	C₂×C₄⋊C₄	1	3	yes	—
19	C₂²×C₉₇	#131	1	4	yes	—
20	C₄×C₉₇	—	2	12	no	—
21	C₄×C₉₇	#127	1	1	yes	characteristic
22	C₄×C₉₇	C₄⋊Q₈	1	3	yes	—
23	C₂×C₉₇	#384	1	1	yes	characteristic
24	C₂×C₉₇	#424	1	6	yes	—
25	#1843	—	97	1	no	maximal, Sylow
26	C₉₇	#1843	1	1	yes	characteristic, complemented, Sylow
27	C₂×C₄⋊C₈	—	97	6	no	—
28	#376	D₉₇	1	1	yes	characteristic
29	C₂²×C₈	—	97	4	no	—
30	C₂×C₄²	C₂×D₉₇	1	3	yes	—
31	C₄⋊C₈	—	194	12	no	—
32	#95	Dic₉₇	1	4	yes	—
33	C₂²×C₄	C₂×Dic₉₇	1	6	yes	—
34	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
35	C₂×C₈	—	194	12	no	—
36	C₄²	Dic₁₉₄	1	6	yes	—
37	C₄²	D₃₈₈	1	6	yes	—
38	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
39	C₂×C₄	—	2	12	no	—
40	C₂×C₄	C₄⋊Dic₉₇	1	12	yes	—
41	C₂×C₄	C₂×D₃₈₈	1	3	yes	—
42	C₂×C₄	C₂×Dic₁₉₄	1	3	yes	—
43	C₈	—	388	4	no	—
44	C₂²	C₂×C₄⋊Dic₉₇	1	3	yes	—
45	C₂²	#15690817	1	4	yes	—
46	C₄	—	2	12	no	—
47	C₄	#15690807	1	1	yes	characteristic
48	C₄	#15690810	1	3	yes	—
49	C₂	#20260605	1	1	yes	characteristic
50	C₂	#20260762	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₉₇
commutator, socle, prime radical	C₂³×C₉₇	C₂²×C₄
Fitting	#376×C₉₇	C₂
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#376	D₉₇
hypocenter	C₉₇	#1843
LCS	G ▷ C₂³×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
26. C₉₇	#1843	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11715895	2
#376×C₉₇	2
#1843	97