Subgroups — #27141484

Order 12416.

272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690721	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₉₇	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	C₄	1	4	yes	—
7	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
8	#15690721	C₄	1	4	yes	complemented
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	C₄×Dic₉₇	—	2	8	no	—
17	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
19	C₂×C₈×C₉₇	—	2	4	no	—
20	C₂×C₈×C₉₇	C₂×C₄	1	4	yes	—
21	C₄²×C₉₇	C₂×C₄	1	2	yes	—
22	C₄²×C₉₇	C₂×C₄	1	2	yes	—
23	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
24	C₂×C₄×C₉₇	—	2	4	no	—
25	C₂×C₄×C₉₇	C₄²	1	4	yes	—
26	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₉₇	#41	1	4	yes	—
29	C₂×Dic₉₇	—	2	4	no	—
30	C₂×Dic₉₇	—	2	4	no	—
31	C₂×Dic₉₇	M₁₆	1	8	yes	—
32	C₈×C₉₇	—	2	4	no	—
33	#11715895	—	2	4	no	—
34	C₂²×C₉₇	C₂×C₄²	1	1	yes	characteristic
35	C₂²×C₉₇	#117	1	2	yes	—
36	C₂²×C₉₇	C₂×M₁₆	1	4	yes	—
37	C₄×C₉₇	—	2	4	no	—
38	C₄×C₉₇	#117	1	2	yes	—
39	C₄×C₉₇	#117	1	2	yes	—
40	Dic₉₇	—	2	8	no	—
41	C₂×C₉₇	#376	1	1	yes	characteristic
42	C₂×C₉₇	C₄×M₁₆	1	2	yes	—
43	C₂×C₉₇	#432	1	4	yes	—
44	#1745	—	97	1	no	maximal, Sylow
45	C₉₇	#1745	1	1	yes	characteristic, complemented, Sylow
46	C₂×#97	—	97	1	no	—
47	C₂×#97	D₉₇	1	1	yes	characteristic
48	C₄³	—	97	1	no	—
49	#336	—	97	4	no	—
50	C₂²×C₈	—	97	2	no	—
51	C₂²×C₈	C₂×D₉₇	1	2	yes	—
52	C₂×C₄²	—	97	4	no	—
53	C₂×C₄²	—	97	2	no	—
54	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
55	#97	—	97	4	no	—
56	#97	Dic₉₇	1	4	yes	complemented
57	C₂²×C₄	—	97	4	no	—
58	C₂²×C₄	C₄×D₉₇	1	2	yes	—
59	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
60	C₂×C₈	—	2	4	no	—
61	C₂×C₈	—	97	4	no	—
62	C₂×C₈	—	194	4	no	—
63	C₂×C₈	C₂×Dic₉₇	1	4	yes	—
64	C₄²	—	194	8	no	—
65	C₄²	—	194	4	no	—
66	C₄²	C₄×D₉₇	1	2	yes	—
67	C₄²	C₂×Dic₉₇	1	2	yes	—
68	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
69	C₂×C₄	—	2	4	no	—
70	C₂×C₄	—	97	8	no	—
71	C₂×C₄	—	194	4	no	—
72	C₂×C₄	—	194	4	no	—
73	C₂×C₄	C₄×Dic₉₇	1	4	yes	—
74	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
75	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
76	C₂×C₄	#13095511	1	4	yes	—
77	C₈	—	2	4	no	—
78	C₈	—	194	4	no	—
79	C₂²	C₂×C₄×Dic₉₇	1	1	yes	characteristic
80	C₂²	#15690767	1	2	yes	—
81	C₂²	M₁₆×D₉₇	1	4	yes	—
82	C₄	—	2	4	no	—
83	C₄	—	194	8	no	—
84	C₄	#15690767	1	2	yes	—
85	C₄	#15690769	1	2	yes	—
86	C₂	#20260581	1	1	yes	characteristic
87	C₂	M₁₆×Dic₉₇	1	2	yes	—
88	C₂	#20260790	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	#1745
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
8. #15690721	C₄	semidirect
11. #97×C₉₇	C₄	semidirect
45. C₉₇	#1745	semidirect
56. #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×#15690721	2
C₄²×Dic₉₇	2
#20260489	2
C₂×#97×C₉₇	2
#1745	97