Subgroups — #27156650

Order 12416.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261621	C₂	1	1	yes	maximal, characteristic
2	C₈×#11715897	C₂	1	1	yes	maximal, characteristic
3	C₁₆×Dic₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095514	C₄	1	2	yes	—
5	C₄×#11715897	C₄	1	1	yes	characteristic
6	C₄×#11715897	C₄	1	1	yes	characteristic
7	C₂×#13095486	C₄	1	1	yes	characteristic
8	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₉₇	C₄	1	1	yes	characteristic
10	C₂×#11715897	C₈	1	2	yes	—
11	C₂×#11715897	C₈	1	2	yes	—
12	#13095514	—	2	2	no	—
13	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
14	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
15	#13095486	—	2	1	no	—
16	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₉₇	—	2	1	no	—
18	C₂×C₄×C₉₇	C₄²	1	1	yes	characteristic
19	C₂×Dic₉₇	C₂×C₈	1	1	yes	characteristic
20	C₂×Dic₉₇	C₂×C₈	1	1	yes	characteristic
21	#11715897	—	2	2	no	—
22	#11715897	C₁₆	1	4	yes	complemented
23	C₈×C₉₇	M₁₆	1	1	yes	characteristic
24	C₈×C₉₇	M₁₆	1	1	yes	characteristic
25	#11715895	M₁₆	1	2	yes	—
26	C₂²×C₉₇	C₄×C₈	1	1	yes	characteristic
27	C₄×C₉₇	#97	1	1	yes	characteristic
28	C₄×C₉₇	#97	1	1	yes	characteristic
29	Dic₉₇	C₂×C₁₆	1	2	yes	—
30	Dic₉₇	M₃₂	1	2	yes	—
31	C₂×C₉₇	#322	1	1	yes	characteristic
32	C₂×C₉₇	C₄×C₁₆	1	1	yes	characteristic
33	C₂×C₉₇	#346	1	1	yes	characteristic
34	#1312	—	97	1	no	maximal, Sylow
35	C₉₇	#1312	1	1	yes	characteristic, complemented, Sylow
36	C₄×C₁₆	—	97	1	no	—
37	C₄×C₁₆	—	97	1	no	—
38	C₈²	—	97	1	no	—
39	C₂×C₁₆	—	97	2	no	—
40	C₂×C₁₆	—	97	1	no	—
41	C₂×C₁₆	#87604	1	1	yes	characteristic
42	C₄×C₈	—	97	1	no	—
43	C₄×C₈	—	97	1	no	—
44	C₄×C₈	—	97	1	no	—
45	C₂×C₈	—	97	2	no	—
46	C₂×C₈	—	97	1	no	—
47	C₂×C₈	—	97	2	no	—
48	C₂×C₈	C₂×#87604	1	1	yes	characteristic
49	C₄²	—	97	1	no	—
50	C₁₆	—	2	1	no	—
51	C₁₆	—	194	2	no	—
52	C₁₆	—	194	1	no	—
53	C₂×C₄	—	97	1	no	—
54	C₂×C₄	—	97	1	no	—
55	C₂×C₄	C₄×#87604	1	1	yes	characteristic
56	C₈	—	97	4	no	—
57	C₈	—	97	2	no	—
58	C₈	—	194	2	no	—
59	C₈	#13095520	1	1	yes	characteristic
60	C₈	#13095521	1	1	yes	characteristic
61	C₂²	C₈×#87604	1	1	yes	characteristic
62	C₄	—	97	2	no	—
63	C₄	—	97	2	no	—
64	C₄	#15690864	1	1	yes	characteristic
65	C₄	#15690865	1	1	yes	characteristic
66	C₂	#20261577	1	1	yes	characteristic
67	C₂	C₁₆×#87604	1	1	yes	characteristic
68	C₂	#20261622	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×#87604
commutator	C₂×C₉₇	C₄×C₁₆
Fitting	C₂×C₁₆×C₉₇	C₄
socle, prime radical	C₂²×C₉₇	C₄×C₈
cosocle	C₈×Dic₉₇	C₂²
hypercenter	C₂×C₁₆	#87604
hypocenter	C₉₇	#1312
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
22. #11715897	C₁₆	semidirect
35. C₉₇	#1312	semidirect

Maximal subgroups

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

subgroup	index
#20261621	2
C₈×#11715897	2
C₁₆×Dic₉₇	2
#1312	97