Subgroups — #27141489

Order 12416.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11715895	C₂	1	1	yes	maximal, characteristic
2	#20260581	C₂	1	1	yes	maximal, characteristic
3	#20260489	C₂	1	4	yes	maximal
4	C₂×#97×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11715895	C₂²	1	2	yes	—
6	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
7	C₄×#11715895	C₄	1	4	yes	complemented
8	#15690717	C₄	1	4	yes	—
9	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
10	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
11	#97×C₉₇	C₄	1	4	yes	complemented
12	C₂²×Dic₉₇	C₂×C₄	1	4	yes	—
13	C₂×#11715895	—	2	4	no	—
14	C₂×#11715895	C₂×C₄	1	4	yes	—
15	C₄×Dic₉₇	—	2	4	no	—
16	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
18	C₂×C₈×C₉₇	—	2	4	no	—
19	C₂×C₈×C₉₇	C₂×C₄	1	4	yes	—
20	C₄²×C₉₇	C₂×C₄	1	2	yes	—
21	C₄²×C₉₇	C₂×C₄	1	2	yes	—
22	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
23	C₂×C₄×C₉₇	—	2	4	no	—
24	C₂×C₄×C₉₇	C₄²	1	4	yes	—
25	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₉₇	#41	1	4	yes	—
28	C₂×Dic₉₇	—	2	4	no	—
29	C₂×Dic₉₇	—	2	8	no	—
30	C₈×C₉₇	—	2	4	no	—
31	#11715895	—	2	4	no	—
32	C₂²×C₉₇	C₂×C₄²	1	1	yes	characteristic
33	C₂²×C₉₇	#117	1	2	yes	—
34	C₂²×C₉₇	#131	1	4	yes	—
35	C₄×C₉₇	—	2	4	no	—
36	C₄×C₉₇	#117	1	2	yes	—
37	C₄×C₉₇	#117	1	2	yes	—
38	Dic₉₇	—	4	4	no	—
39	C₂×C₉₇	#376	1	1	yes	characteristic
40	C₂×C₉₇	#405	1	2	yes	—
41	C₂×C₉₇	#433	1	4	yes	—
42	#1746	—	97	1	no	maximal, Sylow
43	C₉₇	#1746	1	1	yes	characteristic, complemented, Sylow
44	C₂×#97	D₉₇	1	1	yes	characteristic
45	C₂×C₄×C₈	—	97	1	no	—
46	#376	—	97	1	no	—
47	#336	—	97	4	no	—
48	C₂²×C₈	—	97	2	no	—
49	C₂²×C₈	C₂×D₉₇	1	2	yes	—
50	C₂×C₄²	—	97	2	no	—
51	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
52	#97	Dic₉₇	1	4	yes	complemented
53	C₄×C₈	—	97	4	no	—
54	#95	—	97	4	no	—
55	C₂²×C₄	—	97	4	no	—
56	C₂²×C₄	C₄×D₉₇	1	2	yes	—
57	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
58	C₂×C₈	—	2	4	no	—
59	C₂×C₈	—	97	4	no	—
60	C₂×C₈	—	194	4	no	—
61	C₂×C₈	C₂×Dic₉₇	1	4	yes	—
62	C₄²	—	194	4	no	—
63	C₄²	C₄×D₉₇	1	2	yes	—
64	C₄²	C₂×Dic₉₇	1	2	yes	—
65	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	194	4	no	—
68	C₂×C₄	—	194	8	no	—
69	C₂×C₄	C₄×Dic₉₇	1	4	yes	—
70	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
71	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
72	C₂×C₄	#13095511	1	4	yes	—
73	C₈	—	2	4	no	—
74	C₈	—	194	4	no	—
75	C₂²	C₂×C₄×Dic₉₇	1	1	yes	characteristic
76	C₂²	#15690767	1	2	yes	—
77	C₂²	#15690818	1	4	yes	—
78	C₄	—	2	4	no	—
79	C₄	—	388	4	no	—
80	C₄	#15690767	1	2	yes	—
81	C₄	#15690769	1	2	yes	—
82	C₂	#20260581	1	1	yes	characteristic
83	C₂	#20260693	1	2	yes	—
84	C₂	#20260797	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₉₇
commutator	C₂²×C₉₇	C₂×C₄²
Fitting	C₂×#97×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×#97	D₉₇
hypocenter	C₉₇	#1746
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
7. C₄×#11715895	C₄	semidirect
11. #97×C₉₇	C₄	semidirect
43. C₉₇	#1746	semidirect
52. #97	Dic₉₇	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
#20260581	2
#20260489	2
C₂×#97×C₉₇	2
#1746	97