Subgroups — #27150123

Order 12416.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260633	C₂	1	4	yes	maximal
2	#20260619	C₂	1	8	yes	maximal
3	#20260601	C₂	1	2	yes	maximal
4	C₄³×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₉₇	C₂²	1	4	yes	—
6	C₂×#13095490	C₂²	1	8	yes	—
7	#15690717	C₂²	1	8	yes	—
8	#15690717	C₂²	1	8	yes	—
9	C₂×C₄²×C₉₇	C₂²	1	4	yes	—
10	C₂×C₄²×C₉₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
12	C₂²×Dic₉₇	C₂³	1	8	yes	—
13	C₄⋊Dic₉₇	—	2	8	no	—
14	#13095490	—	2	16	no	—
15	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
16	C₂²×C₄×C₉₇	C₂³	1	4	yes	—
17	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
18	C₄²×C₉₇	—	2	8	no	—
19	C₄²×C₉₇	—	2	4	no	—
20	C₄²×C₉₇	Q₈	1	4	yes	—
21	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
22	C₂×C₄×C₉₇	—	2	4	no	—
23	C₂×C₄×C₉₇	—	2	4	no	—
24	C₂×C₄×C₉₇	—	2	4	no	—
25	C₂×C₄×C₉₇	C₂×Q₈	1	4	yes	—
26	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
27	C₂×C₄×C₉₇	#41	1	8	yes	—
28	C₂×C₄×C₉₇	#41	1	4	yes	—
29	C₂×Dic₉₇	—	2	16	no	—
30	C₂×Dic₉₇	—	2	8	no	—
31	C₂²×C₉₇	C₂²×Q₈	1	1	yes	characteristic
32	C₂²×C₉₇	C₂×#41	1	2	yes	—
33	C₂²×C₉₇	C₂×#41	1	4	yes	—
34	C₄×C₉₇	—	2	8	no	—
35	C₄×C₉₇	—	2	4	no	—
36	C₄×C₉₇	#125	1	4	yes	—
37	Dic₉₇	—	4	8	no	—
38	C₂×C₉₇	C₂×#125	1	1	yes	characteristic
39	C₂×C₉₇	#529	1	4	yes	—
40	C₂×C₉₇	#533	1	2	yes	—
41	#2864	—	97	1	no	maximal, Sylow
42	C₉₇	#2864	1	1	yes	characteristic, complemented, Sylow
43	#389	—	97	4	no	—
44	#387	—	97	8	no	—
45	#382	—	97	2	no	—
46	C₄³	D₉₇	1	1	yes	characteristic
47	C₂×C₄⋊C₄	—	97	8	no	—
48	C₂×C₄⋊C₄	—	97	4	no	—
49	C₂×C₄²	C₂×D₉₇	1	4	yes	—
50	C₂×C₄²	C₂×D₉₇	1	2	yes	—
51	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
52	#95	—	97	8	no	—
53	#95	—	97	8	no	—
54	C₂²×C₄	—	97	8	no	—
55	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
56	C₂²×C₄	C₂²×D₉₇	1	4	yes	—
57	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
58	C₄⋊C₄	—	194	16	no	—
59	C₄⋊C₄	—	194	8	no	—
60	C₄²	—	2	8	no	—
61	C₄²	—	2	4	no	—
62	C₄²	Dic₁₉₄	1	4	yes	—
63	C₂³	C₂³×D₉₇	1	1	yes	characteristic
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	194	8	no	—
68	C₂×C₄	—	194	16	no	—
69	C₂×C₄	C₂×Dic₁₉₄	1	4	yes	—
70	C₂×C₄	C₂×Dic₁₉₄	1	2	yes	—
71	C₂×C₄	#13095511	1	8	yes	—
72	C₂×C₄	#13095511	1	4	yes	—
73	C₂²	C₂²×Dic₁₉₄	1	1	yes	characteristic
74	C₂²	C₂×#13095511	1	2	yes	—
75	C₂²	C₂×#13095511	1	4	yes	—
76	C₄	—	2	8	no	—
77	C₄	—	2	4	no	—
78	C₄	—	388	8	no	—
79	C₄	#15690802	1	4	yes	—
80	C₂	C₂×#15690802	1	1	yes	characteristic
81	C₂	#20261269	1	4	yes	—
82	C₂	#20261292	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₉₇
commutator, socle, prime radical, cosocle	C₂³×C₉₇	C₂⁴
Fitting	C₄³×C₉₇	C₂
hypercenter	C₄³	D₉₇
hypocenter	C₉₇	#2864
LCS	G ▷ C₂³×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₄³

Complements

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

N	H ≅ G/N	type
42. C₉₇	#2864	semidirect

Maximal subgroups

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

subgroup	index
#20260633	2
#20260619	2
#20260601	2
C₄³×C₉₇	2
#2864	97