Subgroups — #27141485

Order 12416.

272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690721	C₂	1	2	yes	maximal
2	#20260490	C₂	1	4	yes	maximal
3	C₄³×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11715895	C₂²	1	4	yes	—
5	#15690721	C₄	1	8	yes	complemented
6	C₂×C₄²×C₉₇	C₄	1	4	yes	—
7	C₂×C₄²×C₉₇	C₂²	1	2	yes	—
8	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
9	C₂×#11715895	—	2	8	no	—
10	C₂×#11715895	C₂×C₄	1	8	yes	—
11	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
12	C₂²×C₄×C₉₇	C₂×C₄	1	4	yes	—
13	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
14	C₄²×C₉₇	—	2	8	no	—
15	C₄²×C₉₇	—	2	4	no	—
16	C₄²×C₉₇	C₂×C₄	1	4	yes	—
17	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₄×C₉₇	—	2	4	no	—
19	C₂×C₄×C₉₇	—	2	4	no	—
20	C₂×C₄×C₉₇	—	2	4	no	—
21	C₂×C₄×C₉₇	C₄²	1	4	yes	—
22	C₂×C₄×C₉₇	M₁₆	1	8	yes	—
23	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
24	C₂×C₄×C₉₇	#41	1	4	yes	—
25	#11715895	—	2	8	no	—
26	C₂²×C₉₇	C₂×C₄²	1	1	yes	characteristic
27	C₂²×C₉₇	#117	1	2	yes	—
28	C₂²×C₉₇	C₂×M₁₆	1	4	yes	—
29	C₄×C₉₇	—	2	4	no	—
30	C₄×C₉₇	—	2	8	no	—
31	C₄×C₉₇	#117	1	4	yes	—
32	C₂×C₉₇	#376	1	1	yes	characteristic
33	C₂×C₉₇	C₄×M₁₆	1	2	yes	—
34	C₂×C₉₇	#432	1	4	yes	—
35	#1745	—	97	1	no	maximal, Sylow
36	C₉₇	#1745	1	1	yes	characteristic, complemented, Sylow
37	C₂×#97	—	97	2	no	—
38	C₄³	D₉₇	1	1	yes	characteristic
39	#336	—	97	4	no	—
40	C₂²×C₈	—	97	4	no	—
41	C₂×C₄²	Dic₉₇	1	4	yes	—
42	C₂×C₄²	C₂×D₉₇	1	2	yes	—
43	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
44	#97	—	97	8	no	—
45	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
46	C₂²×C₄	C₂×Dic₉₇	1	4	yes	—
47	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
48	C₂×C₈	—	97	8	no	—
49	C₂×C₈	—	194	8	no	—
50	C₄²	—	2	8	no	—
51	C₄²	—	2	4	no	—
52	C₄²	C₄×D₉₇	1	4	yes	—
53	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
54	C₂×C₄	—	2	4	no	—
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	2	4	no	—
57	C₂×C₄	C₄×Dic₉₇	1	4	yes	—
58	C₂×C₄	#13095495	1	8	yes	—
59	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
60	C₂×C₄	#13095511	1	4	yes	—
61	C₈	—	194	8	no	—
62	C₂²	C₂×C₄×Dic₉₇	1	1	yes	characteristic
63	C₂²	#15690769	1	2	yes	—
64	C₂²	C₂×#13095495	1	4	yes	—
65	C₄	—	2	4	no	—
66	C₄	—	2	8	no	—
67	C₄	#15690767	1	4	yes	—
68	C₂	#20260581	1	1	yes	characteristic
69	C₂	C₄×#13095495	1	2	yes	—
70	C₂	#20260792	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₉₇
commutator	C₂²×C₉₇	C₂×C₄²
Fitting	C₄³×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₄³	D₉₇
hypocenter	C₉₇	#1745
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
5. #15690721	C₄	semidirect
36. C₉₇	#1745	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
#20260490	2
C₄³×C₉₇	2
#1745	97