Subgroups — #27140007

Order 12416.

736 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690717	C₂	1	6	yes	maximal
2	C₂³×Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic
3	C₂³×Dic₉₇	C₂²	1	4	yes	—
4	#15690717	—	2	12	no	—
5	C₂²×Q₈×C₉₇	—	2	12	no	—
6	C₂²×Q₈×C₉₇	C₄	1	4	yes	complemented
7	C₂³×C₄×C₉₇	C₂²	1	3	yes	—
8	C₂²×Dic₉₇	—	2	12	no	—
9	C₂²×Dic₉₇	—	2	12	no	—
10	C₂²×Dic₉₇	Q₈	1	8	yes	complemented
11	C₂⁴×C₉₇	C₂³	1	1	yes	characteristic
12	C₂×Q₈×C₉₇	—	2	32	no	—
13	C₂×Q₈×C₉₇	—	2	24	no	—
14	C₂²×C₄×C₉₇	—	2	12	no	—
15	C₂²×C₄×C₉₇	C₂×C₄	1	6	yes	—
16	C₂²×C₄×C₉₇	D₄	1	12	yes	—
17	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
18	C₂³×C₉₇	C₂×D₄	1	6	yes	—
19	C₂³×C₉₇	C₂×Q₈	1	4	yes	—
20	C₂³×C₉₇	#41	1	4	yes	—
21	Q₈×C₉₇	—	2	32	no	—
22	C₂×C₄×C₉₇	—	2	12	no	—
23	C₂×C₄×C₉₇	—	2	24	no	—
24	C₂×C₄×C₉₇	C₂²⋊C₄	1	12	yes	—
25	C₂×Dic₉₇	—	2	24	no	—
26	C₂×Dic₉₇	—	4	4	no	—
27	C₂×Dic₉₇	—	4	12	no	—
28	C₂²×C₉₇	C₂×C₂²⋊C₄	1	3	yes	—
29	C₂²×C₉₇	C₄×Q₈	1	4	yes	—
30	C₂²×C₉₇	#120	1	4	yes	—
31	C₂²×C₉₇	C₂²⋊Q₈	1	12	yes	—
32	C₂²×C₉₇	#124	1	6	yes	—
33	C₂²×C₉₇	C₄⋊Q₈	1	6	yes	—
34	C₄×C₉₇	—	2	12	no	—
35	Dic₉₇	—	4	8	no	—
36	C₂×C₉₇	#379	1	1	yes	characteristic
37	C₂×C₉₇	#391	1	6	yes	—
38	C₂×C₉₇	#393	1	4	yes	—
39	C₂×C₉₇	#395	1	4	yes	—
40	#1446	—	97	1	no	maximal, Sylow
41	C₉₇	#1446	1	1	yes	characteristic, complemented, Sylow
42	C₂³×Q₈	D₉₇	1	1	yes	characteristic
43	C₂×#95	—	97	6	no	—
44	C₂²×Q₈	—	2	12	no	—
45	C₂²×Q₈	Dic₉₇	1	4	yes	complemented
46	C₂³×C₄	—	97	4	no	—
47	C₂³×C₄	C₂×D₉₇	1	3	yes	—
48	#95	—	194	12	no	—
49	C₂⁴	C₂²×D₉₇	1	1	yes	characteristic
50	C₂×Q₈	—	2	32	no	—
51	C₂×Q₈	—	2	24	no	—
52	C₂²×C₄	—	2	12	no	—
53	C₂²×C₄	—	97	8	no	—
54	C₂²×C₄	—	194	12	no	—
55	C₂²×C₄	—	194	12	no	—
56	C₂²×C₄	C₂×Dic₉₇	1	6	yes	—
57	C₂²×C₄	C₂²⋊D₉₇	1	12	yes	—
58	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
59	C₂³	C₂×C₂²⋊D₉₇	1	6	yes	—
60	C₂³	Q₈×D₉₇	1	4	yes	—
61	C₂³	#13095512	1	4	yes	—
62	Q₈	—	2	32	no	—
63	C₂×C₄	—	2	12	no	—
64	C₂×C₄	—	2	24	no	—
65	C₂×C₄	—	388	4	no	—
66	C₂×C₄	—	388	12	no	—
67	C₂×C₄	—	194	24	no	—
68	C₂×C₄	C₂²⋊Dic₉₇	1	12	yes	—
69	C₂²	C₂×C₂²⋊Dic₉₇	1	3	yes	—
70	C₂²	Q₈×Dic₉₇	1	4	yes	—
71	C₂²	#15690779	1	4	yes	—
72	C₂²	#15690786	1	12	yes	—
73	C₂²	#15690799	1	6	yes	—
74	C₂²	#15690808	1	6	yes	—
75	C₄	—	2	12	no	—
76	C₄	—	388	8	no	—
77	C₂	#20260591	1	1	yes	characteristic
78	C₂	#20260642	1	6	yes	—
79	C₂	#20260646	1	4	yes	—
80	C₂	#20260654	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₉₇
commutator	C₂³×C₉₇	C₂²×C₄
Fitting	C₂³×Q₈×C₉₇	C₂
socle, prime radical, cosocle	C₂⁴×C₉₇	C₂³
hypercenter	C₂³×Q₈	D₉₇
hypocenter	C₉₇	#1446
LCS	G ▷ C₂³×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
6. C₂²×Q₈×C₉₇	C₄	semidirect
10. C₂²×Dic₉₇	Q₈	semidirect
41. C₉₇	#1446	semidirect
45. C₂²×Q₈	Dic₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690717	2
C₂³×Q₈×C₉₇	2
#1446	97