Subgroups — #27141893

Order 12416.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690742	C₂	1	4	yes	maximal
2	#20260605	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Dic₉₇	—	2	4	no	—
5	C₂×C₄⋊Dic₉₇	C₂²	1	4	yes	—
6	#15690742	—	2	8	no	—
7	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
8	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₉₇	C₄	1	4	yes	complemented
10	C₂²×Dic₉₇	—	2	4	no	—
11	C₄⋊Dic₉₇	—	2	8	no	—
12	C₄⋊Dic₉₇	—	4	8	no	—
13	C₂²×C₄×C₉₇	D₄	1	2	yes	—
14	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
15	C₂×C₈×C₉₇	C₂×C₄	1	4	yes	—
16	C₂×C₈×C₉₇	D₄	1	4	yes	—
17	C₂×C₈×C₉₇	Q₈	1	4	yes	—
18	C₄²×C₉₇	C₂×C₄	1	2	yes	—
19	C₄²×C₉₇	Q₈	1	2	yes	—
20	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
21	C₂×C₄×C₉₇	C₄⋊C₄	1	4	yes	—
22	C₂×C₄×C₉₇	SD₁₆	1	8	yes	—
23	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
24	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
25	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
26	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
27	C₂×Dic₉₇	—	4	4	no	—
28	C₂×Dic₉₇	—	4	8	no	—
29	C₈×C₉₇	C₄⋊C₄	1	8	yes	complemented
30	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₉₇	#127	1	1	yes	characteristic
32	C₂²×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
33	C₂²×C₉₇	C₂×SD₁₆	1	4	yes	—
34	C₄×C₉₇	#106	1	8	yes	complemented
35	C₄×C₉₇	C₂×C₄⋊C₄	1	2	yes	—
36	C₄×C₉₇	C₄⋊Q₈	1	2	yes	—
37	Dic₉₇	—	8	4	no	—
38	C₂×C₉₇	#384	1	1	yes	characteristic
39	C₂×C₉₇	C₂×#106	1	2	yes	—
40	C₂×C₉₇	#492	1	2	yes	—
41	C₂×C₉₇	#498	1	2	yes	—
42	#1844	—	97	1	no	maximal, Sylow
43	C₉₇	#1844	1	1	yes	characteristic, complemented, Sylow
44	C₂×#106	—	97	4	no	—
45	C₂×C₄×C₈	D₉₇	1	1	yes	characteristic
46	#384	—	97	2	no	—
47	C₂²×C₈	C₂×D₉₇	1	2	yes	—
48	C₂×C₄⋊C₄	—	97	4	no	—
49	C₂×C₄⋊C₄	—	194	4	no	—
50	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
51	#106	—	194	8	no	—
52	C₄×C₈	Dic₉₇	1	4	yes	complemented
53	C₂²×C₄	—	194	4	no	—
54	C₂²×C₄	D₃₈₈	1	2	yes	—
55	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
56	C₂×C₈	Dic₁₉₄	1	4	yes	—
57	C₂×C₈	D₃₈₈	1	4	yes	—
58	C₂×C₈	C₂×Dic₉₇	1	4	yes	—
59	C₄⋊C₄	—	388	8	no	—
60	C₄⋊C₄	—	194	8	no	—
61	C₄²	Dic₁₉₄	1	2	yes	—
62	C₄²	C₂×Dic₉₇	1	2	yes	—
63	C₂³	C₂×D₃₈₈	1	1	yes	characteristic
64	C₂×C₄	—	388	4	no	—
65	C₂×C₄	—	388	8	no	—
66	C₂×C₄	C₄⋊Dic₉₇	1	4	yes	—
67	C₂×C₄	#13095500	1	8	yes	—
68	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
69	C₂×C₄	C₂×D₃₈₈	1	2	yes	—
70	C₂×C₄	C₂×Dic₁₉₄	1	2	yes	—
71	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
72	C₈	C₄⋊Dic₉₇	1	8	yes	complemented
73	C₂²	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
74	C₂²	#15690807	1	1	yes	characteristic
75	C₂²	#15690810	1	1	yes	characteristic
76	C₂²	C₂×#13095500	1	4	yes	—
77	C₄	—	776	4	no	—
78	C₄	#15690742	1	8	yes	complemented
79	C₄	C₂×C₄⋊Dic₉₇	1	2	yes	—
80	C₄	#15690810	1	2	yes	—
81	C₂	#20260605	1	1	yes	characteristic
82	C₂	C₂×#15690742	1	2	yes	—
83	C₂	#20261108	1	2	yes	—
84	C₂	#20261133	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₃₈₈
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂×C₄×C₈×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×C₄×C₈	D₉₇
hypocenter	C₉₇	#1844
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
9. C₄×C₈×C₉₇	C₄	semidirect
29. C₈×C₉₇	C₄⋊C₄	semidirect
34. C₄×C₉₇	#106	semidirect
43. C₉₇	#1844	semidirect
52. C₄×C₈	Dic₉₇	semidirect
72. C₈	C₄⋊Dic₉₇	semidirect
78. C₄	#15690742	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690742	2
#20260605	2
C₂×C₄×C₈×C₉₇	2
#1844	97