Subgroups — #27156804

Order 12416.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261745	C₂	1	2	yes	maximal
2	C₂×#15690749	C₂	1	1	yes	maximal, characteristic
3	#15690889	—	2	2	no	—
4	#15690879	—	2	2	no	—
5	#15690878	C₄	1	4	yes	complemented
6	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
7	#15690749	—	2	2	no	—
8	C₂×#13095486	C₄	1	1	yes	characteristic, complemented
9	C₂×C₁₆×C₉₇	C₄	1	1	yes	characteristic, complemented
10	C₄⋊#87604	—	2	4	no	—
11	C₂²⋊#87604	—	4	2	no	—
12	C₄×#87604	—	4	2	no	—
13	C₂×C₄×D₉₇	D₄	1	1	yes	characteristic
14	C₂×#11715895	Q₈	1	1	yes	characteristic
15	C₈×D₉₇	C₂×C₄	1	2	yes	—
16	C₈×D₉₇	D₄	1	2	yes	—
17	#13095486	—	2	1	no	—
18	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
19	C₁₆×C₉₇	—	2	1	no	—
20	C₂²×D₉₇	D₈	1	1	yes	characteristic
21	C₂×#87604	—	4	4	no	—
22	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
23	C₂×Dic₉₇	Q₁₆	1	1	yes	characteristic
24	C₄×D₉₇	C₂²⋊C₄	1	2	yes	—
25	C₄×D₉₇	SD₁₆	1	2	yes	—
26	C₈×C₉₇	C₄²	1	1	yes	characteristic
27	C₈×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
28	#11715895	C₄⋊C₄	1	2	yes	—
29	C₂²×C₉₇	#107	1	1	yes	characteristic
30	C₂×D₉₇	—	4	1	no	—
31	C₂×D₉₇	#102	1	2	yes	—
32	#87604	—	8	4	no	—
33	C₄×C₉₇	#95	1	1	yes	characteristic
34	C₄×C₉₇	#106	1	1	yes	characteristic
35	Dic₉₇	#103	1	2	yes	—
36	C₂×C₉₇	—	2	1	no	—
37	C₂×C₉₇	#340	1	1	yes	characteristic
38	D₉₇	—	2	2	no	—
39	#1384	—	97	1	no	maximal, Sylow
40	C₉₇	#1384	1	1	yes	characteristic, complemented, Sylow
41	C₂×M₃₂	—	97	1	no	—
42	#427	—	97	2	no	—
43	C₂²×C₈	—	97	1	no	—
44	#117	—	194	2	no	—
45	M₃₂	—	194	2	no	—
46	C₂×C₁₆	—	97	1	no	—
47	C₂×C₁₆	#87604	1	1	yes	characteristic, complemented
48	#107	—	194	2	no	—
49	#106	—	97	4	no	—
50	C₂²×C₄	—	97	1	no	—
51	C₂×C₈	—	97	1	no	—
52	C₂×C₈	—	97	2	no	—
53	C₂×C₈	—	97	2	no	—
54	C₂×C₈	C₂×#87604	1	1	yes	characteristic
55	C₄⋊C₄	—	194	4	no	—
56	C₂²⋊C₄	—	388	2	no	—
57	C₄²	—	388	2	no	—
58	C₁₆	—	2	1	no	—
59	C₁₆	—	194	1	no	—
60	C₂³	—	97	1	no	—
61	C₂×C₄	—	97	1	no	—
62	C₂×C₄	—	97	2	no	—
63	C₂×C₄	—	97	2	no	—
64	C₂×C₄	—	388	4	no	—
65	C₂×C₄	C₄⋊#87604	1	1	yes	characteristic
66	C₈	—	97	2	no	—
67	C₈	C₄×#87604	1	1	yes	characteristic
68	C₈	C₂²⋊#87604	1	1	yes	characteristic
69	C₂²	—	97	2	no	—
70	C₂²	—	388	1	no	—
71	C₂²	#15690879	1	1	yes	characteristic
72	C₄	—	776	4	no	—
73	C₄	—	97	2	no	—
74	C₄	(C₂×#87604)⋊C₄	1	1	yes	characteristic
75	C₄	#15690878	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	194	2	no	—
78	C₂	#20261609	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15690878
commutator	C₈×C₉₇	C₄²
Frattini	C₂×C₈	C₂×#87604
Fitting	C₂×C₁₆×C₉₇	C₄
socle	C₂×C₉₇	#340
prime radical	C₂²×D₉₇	D₈
cosocle	C₂×C₈×D₉₇	C₂²
hypercenter	C₂×C₁₆	#87604
hypocenter	C₉₇	#1384
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
5. #15690878	C₄	semidirect
8. C₂×#13095486	C₄	semidirect
9. C₂×C₁₆×C₉₇	C₄	semidirect
40. C₉₇	#1384	semidirect
47. C₂×C₁₆	#87604	semidirect

Maximal subgroups

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

subgroup	index
#20261745	2
C₂×#15690749	2
#1384	97