Subgroups — #27158991

Order 12416.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261857	C₂	1	8	yes	maximal, complemented
2	#20261854	C₂	1	6	yes	maximal, complemented
3	C₂×#15690898	C₂	1	1	yes	maximal, characteristic
4	#15690907	C₂²	1	12	yes	complemented
5	#15690906	C₂²	1	4	yes	—
6	#15690905	C₂²	1	12	yes	complemented
7	#15690898	C₄	1	4	yes	—
8	#15690898	C₂²	1	4	yes	—
9	C₂×#13095521	C₄	1	3	yes	—
10	C₂×#13095519	C₂²	1	3	yes	—
11	C₂×#13095512	C₄	1	1	yes	characteristic
12	#13095523	C₂³	1	8	yes	—
13	#13095521	C₂×C₄	1	12	yes	—
14	#13095519	C₂×C₄	1	6	yes	—
15	#13095519	C₂³	1	6	yes	—
16	C₂×#11715897	C₂×C₄	1	3	yes	—
17	C₂×#11715897	C₂³	1	1	yes	characteristic
18	#13095512	C₈	1	4	yes	—
19	#13095512	C₂×C₄	1	4	yes	—
20	C₂×D₃₈₈	C₈	1	3	yes	—
21	C₂×C₄×D₉₇	C₂×C₄	1	3	yes	—
22	C₂×Q₈×C₉₇	C₈	1	1	yes	characteristic
23	C₂²×D₉₇	C₂×C₈	1	3	yes	—
24	Q₈×C₉₇	C₂×C₈	1	4	yes	—
25	C₂×C₄×C₉₇	C₂×C₈	1	3	yes	—
26	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
27	D₃₈₈	C₂×C₈	1	12	yes	—
28	C₄×D₉₇	C₂×C₈	1	6	yes	—
29	C₄×D₉₇	C₂²×C₄	1	6	yes	—
30	#11715897	C₂²×C₄	1	6	yes	—
31	#11715897	C₂²×C₄	1	1	yes	characteristic
32	#11715897	C₂⁴	1	1	yes	characteristic
33	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
34	C₂×D₉₇	—	4	3	no	—
35	C₂×D₉₇	C₂²×C₈	1	6	yes	—
36	C₄×C₉₇	C₂²×C₈	1	6	yes	—
37	Dic₉₇	C₂²×C₈	1	1	yes	characteristic
38	Dic₉₇	C₂³×C₄	1	1	yes	characteristic
39	C₂×C₉₇	—	2	1	no	—
40	C₂×C₉₇	C₂³×C₈	1	1	yes	characteristic
41	D₉₇	—	2	6	no	—
42	#3407	—	97	1	no	maximal, Sylow
43	C₉₇	#3407	1	1	yes	characteristic, complemented, Sylow
44	C₂×#131	—	97	1	no	—
45	#504	—	97	8	no	—
46	C₂×M₃₂	—	97	6	no	—
47	C₂×#41	—	97	1	no	—
48	#131	—	97	4	no	—
49	#131	—	97	4	no	—
50	C₂×M₁₆	—	97	3	no	—
51	C₂²×C₈	—	97	3	no	—
52	M₃₂	—	97	4	no	—
53	M₃₂	—	97	12	no	—
54	C₂×C₁₆	—	97	12	no	—
55	#41	—	97	4	no	—
56	#41	—	97	4	no	—
57	C₂×Q₈	#11715906	1	1	yes	characteristic
58	C₂×D₄	—	97	3	no	—
59	C₂²×C₄	—	97	3	no	—
60	M₁₆	—	97	12	no	—
61	C₂×C₈	—	97	3	no	—
62	C₂×C₈	—	97	1	no	—
63	C₂×C₈	—	97	6	no	—
64	C₂×C₈	—	97	6	no	—
65	C₁₆	—	97	8	no	—
66	C₂³	—	97	3	no	—
67	Q₈	C₂×#11715906	1	4	yes	—
68	D₄	—	97	12	no	—
69	C₂×C₄	—	97	1	no	—
70	C₂×C₄	—	97	6	no	—
71	C₂×C₄	—	97	6	no	—
72	C₂×C₄	C₂×#11715906	1	3	yes	—
73	C₈	—	97	1	no	—
74	C₈	—	97	1	no	—
75	C₈	—	97	6	no	—
76	C₂²	—	97	6	no	—
77	C₂²	—	388	3	no	—
78	C₂²	C₂²×#11715906	1	1	yes	characteristic
79	C₄	—	97	1	no	—
80	C₄	—	97	1	no	—
81	C₄	C₂²×#11715906	1	6	yes	—
82	C₂	—	2	1	no	—
83	C₂	—	194	6	no	—
84	C₂	C₂³×#11715906	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#11715906
commutator, socle	C₂×C₉₇	C₂³×C₈
Fitting	C₂×Q₈×C₉₇	C₈
prime radical	C₂×#13095512	C₄
cosocle	#11715897	C₂⁴
hypercenter	C₂×Q₈	#11715906
hypocenter	C₉₇	#3407
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. #20261857	C₂	semidirect
2. #20261854	C₂	semidirect
4. #15690907	C₂²	semidirect
6. #15690905	C₂²	semidirect
43. C₉₇	#3407	semidirect

Maximal subgroups

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

subgroup	index
#20261857	2
#20261854	2
C₂×#15690898	2
#3407	97