Subgroups — #27156651

Order 12416.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13095514	C₂	1	2	yes	maximal
2	C₈×#11715895	C₂	1	1	yes	maximal, characteristic
3	C₂×#13095514	C₄	1	4	yes	—
4	C₄×#11715895	C₄	1	1	yes	characteristic
5	C₄×#11715895	C₂²	1	1	yes	characteristic
6	C₄×C₈×C₉₇	C₄	1	1	yes	characteristic
7	#13095514	—	2	4	no	—
8	C₂×#11715895	C₈	1	2	yes	—
9	C₂×#11715895	C₂×C₄	1	2	yes	—
10	C₂×C₈×C₉₇	C₈	1	2	yes	—
11	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₉₇	C₄²	1	1	yes	characteristic
13	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
14	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
15	C₈×C₉₇	—	2	2	no	—
16	#11715895	C₁₆	1	4	yes	complemented
17	#11715895	M₁₆	1	4	yes	—
18	C₂²×C₉₇	C₄×C₈	1	1	yes	characteristic
19	C₄×C₉₇	#97	1	2	yes	—
20	C₄×C₉₇	C₂×C₁₆	1	2	yes	—
21	C₄×C₉₇	M₃₂	1	2	yes	—
22	C₂×C₉₇	#322	1	1	yes	characteristic
23	C₂×C₉₇	C₄×C₁₆	1	1	yes	characteristic
24	C₂×C₉₇	#346	1	1	yes	characteristic
25	#1312	—	97	1	no	maximal, Sylow
26	C₉₇	#1312	1	1	yes	characteristic, complemented, Sylow
27	C₄×C₁₆	—	97	2	no	—
28	C₈²	—	97	1	no	—
29	C₂×C₁₆	—	97	4	no	—
30	C₄×C₈	—	97	1	no	—
31	C₄×C₈	—	97	1	no	—
32	C₄×C₈	#87604	1	1	yes	characteristic
33	C₂×C₈	—	97	2	no	—
34	C₂×C₈	—	97	2	no	—
35	C₂×C₈	#11715897	1	2	yes	—
36	C₄²	C₂×#87604	1	1	yes	characteristic
37	C₁₆	—	194	4	no	—
38	C₂×C₄	C₄×#87604	1	1	yes	characteristic
39	C₂×C₄	C₂×#11715897	1	1	yes	characteristic
40	C₂×C₄	#13095519	1	1	yes	characteristic
41	C₈	—	2	2	no	—
42	C₈	—	97	4	no	—
43	C₈	—	97	4	no	—
44	C₂²	C₄×#11715897	1	1	yes	characteristic
45	C₄	#15690864	1	2	yes	—
46	C₄	#15690883	1	2	yes	—
47	C₄	#15690884	1	2	yes	—
48	C₂	#20261576	1	1	yes	characteristic
49	C₂	#20261621	1	1	yes	characteristic
50	C₂	#20261623	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂×#87604
commutator	C₂×C₉₇	C₄×C₁₆
Fitting	C₄×C₈×C₉₇	C₄
socle, prime radical	C₂²×C₉₇	C₄×C₈
cosocle	C₄×#11715895	C₂²
hypercenter	C₄×C₈	#87604
hypocenter	C₉₇	#1312
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
16. #11715895	C₁₆	semidirect
26. C₉₇	#1312	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13095514	2
C₈×#11715895	2
#1312	97