Subgroups — #27144917

Order 12416.

424 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260642	C₂	1	4	yes	maximal
2	#20260621	C₂	1	8	yes	maximal
3	#20260603	C₂	1	2	yes	maximal
4	C₂×C₄⋊Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₉₇	C₂²	1	4	yes	—
6	C₂×C₄×Dic₉₇	C₂²	1	4	yes	—
7	#15690717	C₂²	1	8	yes	—
8	#15690717	C₂²	1	8	yes	—
9	#15690717	C₂²	1	4	yes	—
10	C₂²×Q₈×C₉₇	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₉₇	C₄	1	8	yes	complemented
12	C₂×C₄⋊C₄×C₉₇	C₂²	1	4	yes	—
13	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₉₇	C₂³	1	8	yes	—
15	C₄⋊Dic₉₇	—	4	4	no	—
16	C₄×Dic₉₇	—	4	4	no	—
17	C₂×Q₈×C₉₇	—	4	2	no	—
18	C₂×Q₈×C₉₇	—	4	2	no	—
19	C₂×Q₈×C₉₇	C₂×C₄	1	8	yes	—
20	C₂²×C₄×C₉₇	C₂³	1	4	yes	—
21	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
22	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₉₇	C₂×C₄	1	16	yes	—
24	C₄²×C₉₇	C₂×C₄	1	4	yes	—
25	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
26	Q₈×C₉₇	—	4	8	no	—
27	C₂×C₄×C₉₇	—	2	2	no	—
28	C₂×C₄×C₉₇	—	2	4	no	—
29	C₂×C₄×C₉₇	—	2	4	no	—
30	C₂×C₄×C₉₇	—	2	4	no	—
31	C₂×C₄×C₉₇	C₂²×C₄	1	4	yes	—
32	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
33	C₂×C₄×C₉₇	C₂²×C₄	1	8	yes	—
34	C₂×Dic₉₇	—	2	8	no	—
35	C₂×Dic₉₇	—	2	16	no	—
36	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
37	C₂²×C₉₇	H(5,𝔽₂)	1	2	yes	—
38	C₂²×C₉₇	2₋¹⁺⁴	1	4	yes	—
39	C₄×C₉₇	—	2	8	no	—
40	C₄×C₉₇	—	2	2	no	—
41	C₄×C₉₇	—	2	4	no	—
42	Dic₉₇	—	4	8	no	—
43	C₂×C₉₇	#518	1	1	yes	characteristic
44	C₂×C₉₇	#519	1	2	yes	—
45	C₂×C₉₇	#563	1	4	yes	—
46	#2381	—	97	1	no	maximal, Sylow
47	C₉₇	#2381	1	1	yes	characteristic, complemented, Sylow
48	C₂×C₄⋊Q₈	D₉₇	1	1	yes	characteristic
49	#391	—	97	4	no	—
50	#387	—	97	8	no	—
51	#383	—	97	2	no	—
52	C₂²×Q₈	C₂×D₉₇	1	2	yes	—
53	C₄⋊Q₈	Dic₉₇	1	8	yes	complemented
54	C₂×C₄⋊C₄	—	97	4	no	—
55	C₂×C₄⋊C₄	C₂×D₉₇	1	4	yes	—
56	C₂×C₄²	—	97	4	no	—
57	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
58	#95	—	97	8	no	—
59	#95	—	97	4	no	—
60	#95	—	97	8	no	—
61	C₂×Q₈	—	4	2	no	—
62	C₂×Q₈	—	4	2	no	—
63	C₂×Q₈	C₂×Dic₉₇	1	8	yes	—
64	C₂²×C₄	—	97	8	no	—
65	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
66	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
67	C₂²×C₄	C₂²×D₉₇	1	4	yes	—
68	C₄⋊C₄	—	388	4	no	—
69	C₄⋊C₄	C₂×Dic₉₇	1	16	yes	—
70	C₄²	—	388	4	no	—
71	C₄²	C₂×Dic₉₇	1	4	yes	—
72	C₂³	C₂³×D₉₇	1	1	yes	characteristic
73	Q₈	—	4	8	no	—
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	2	4	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	4	no	—
78	C₂×C₄	—	194	8	no	—
79	C₂×C₄	—	194	16	no	—
80	C₂×C₄	C₂²×Dic₉₇	1	4	yes	—
81	C₂×C₄	C₂²×Dic₉₇	1	2	yes	—
82	C₂×C₄	C₂²×Dic₉₇	1	8	yes	—
83	C₂²	C₂³×Dic₉₇	1	1	yes	characteristic
84	C₂²	#15690855	1	2	yes	—
85	C₂²	#15690858	1	4	yes	—
86	C₄	—	2	2	no	—
87	C₄	—	2	8	no	—
88	C₄	—	2	4	no	—
89	C₄	—	388	8	no	—
90	C₂	#20261210	1	1	yes	characteristic
91	C₂	#20261213	1	2	yes	—
92	C₂	#20261484	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₉₇
commutator	C₂²×C₉₇	C₂³×C₄
Fitting	C₂×C₄⋊Q₈×C₉₇	C₂
socle, prime radical, cosocle	C₂³×C₉₇	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₉₇
hypocenter	C₉₇	#2381
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
11. C₄⋊Q₈×C₉₇	C₄	semidirect
47. C₉₇	#2381	semidirect
53. C₄⋊Q₈	Dic₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20260642	2
#20260621	2
#20260603	2
C₂×C₄⋊Q₈×C₉₇	2
#2381	97