Subgroups — #27158879

Order 12416.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	(C₂×#11715906)⋊C₄	C₂	1	6	yes	maximal
2	C₄²×#87604	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11715906	C₂²	1	4	yes	—
4	C₂×C₄×#87604	C₄	1	3	yes	—
5	C₂×C₄×#87604	C₂²	1	3	yes	—
6	C₄²×D₉₇	C₄	1	1	yes	characteristic
7	C₂×#11715906	—	2	12	no	—
8	C₂²×#87604	C₂×C₄	1	3	yes	—
9	C₂²×#87604	C₂³	1	1	yes	characteristic
10	C₄×#87604	—	2	6	no	—
11	C₄×#87604	—	2	6	no	—
12	C₂×C₄×D₉₇	C₂×C₄	1	3	yes	—
13	C₄×Dic₉₇	C₈	1	3	yes	complemented
14	C₄²×C₉₇	C₈	1	1	yes	characteristic, complemented
15	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
16	C₂×#87604	—	2	3	no	—
17	C₂×#87604	—	2	3	no	—
18	C₂×#87604	M₁₆	1	6	yes	—
19	C₂×#87604	#41	1	6	yes	—
20	#11715906	—	4	4	no	—
21	C₂×C₄×C₉₇	C₂×C₈	1	3	yes	—
22	C₂×Dic₉₇	C₂×C₈	1	3	yes	—
23	C₄×D₉₇	—	2	3	no	—
24	C₄×D₉₇	—	2	3	no	—
25	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
26	C₂×D₉₇	#117	1	3	yes	—
27	C₂×D₉₇	C₂×M₁₆	1	3	yes	—
28	#87604	—	2	6	no	—
29	#87604	#126	1	4	yes	—
30	C₄×C₉₇	—	2	3	no	—
31	Dic₉₇	—	2	3	no	—
32	C₂×C₉₇	#431	1	3	yes	—
33	D₉₇	#383	1	1	yes	characteristic
34	D₉₇	#432	1	3	yes	—
35	#1839	—	97	1	no	maximal, Sylow
36	C₉₇	#1839	1	1	yes	characteristic, complemented, Sylow
37	C₄³	—	97	1	no	—
38	#336	—	97	6	no	—
39	C₂²×C₈	—	97	4	no	—
40	C₂×C₄²	—	97	1	no	—
41	C₂×C₄²	—	97	3	no	—
42	C₂×C₄²	—	97	3	no	—
43	C₂²×C₄	—	97	1	no	—
44	C₂²×C₄	—	97	3	no	—
45	C₂²×C₄	—	97	3	no	—
46	C₂×C₈	—	194	12	no	—
47	C₄²	—	97	3	no	—
48	C₄²	—	194	6	no	—
49	C₄²	—	194	6	no	—
50	C₄²	#11715906	1	1	yes	characteristic, complemented
51	C₂³	—	97	1	no	—
52	C₂×C₄	—	97	6	no	—
53	C₂×C₄	—	97	6	no	—
54	C₂×C₄	—	97	3	no	—
55	C₂×C₄	—	194	3	no	—
56	C₂×C₄	—	194	3	no	—
57	C₂×C₄	—	194	3	no	—
58	C₂×C₄	—	194	3	no	—
59	C₂×C₄	C₂×#11715906	1	3	yes	—
60	C₈	—	388	4	no	—
61	C₂²	—	97	3	no	—
62	C₂²	—	97	3	no	—
63	C₂²	C₂²×#11715906	1	1	yes	characteristic
64	C₄	—	2	3	no	—
65	C₄	—	97	4	no	—
66	C₄	—	194	6	no	—
67	C₄	—	194	3	no	—
68	C₂	—	97	1	no	—
69	C₂	—	97	3	no	—
70	C₂	#20261848	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11715906
commutator, socle	C₂²×C₉₇	C₂²×C₈
Fitting	C₄²×C₉₇	C₈
prime radical	C₂²×D₉₇	C₂²×C₄
cosocle	C₂²×#87604	C₂³
hypercenter	C₄²	#11715906
hypocenter	C₉₇	#1839
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
13. C₄×Dic₉₇	C₈	semidirect
14. C₄²×C₉₇	C₈	semidirect
36. C₉₇	#1839	semidirect
50. C₄²	#11715906	semidirect

Maximal subgroups

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

subgroup	index
(C₂×#11715906)⋊C₄	2
C₄²×#87604	2
#1839	97