Subgroups — #27143993

Order 12416.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690753	C₂	1	4	yes	maximal, complemented
2	#20261112	C₂	1	1	yes	maximal, characteristic
3	C₄×#13095486	C₂	1	1	yes	maximal, characteristic, complemented
4	#494×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₇₇₆	—	2	1	no	—
6	C₂×D₇₇₆	C₂²	1	2	yes	—
7	#15690807	—	2	1	no	—
8	#15690753	—	2	8	no	—
9	C₂×#13095486	C₂²	1	2	yes	—
10	C₂×Q₁₆×C₉₇	—	2	1	no	—
11	C₂×Q₁₆×C₉₇	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₉₇	—	2	1	no	—
13	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
14	C₂×D₃₈₈	—	2	2	no	—
15	C₂×D₃₈₈	—	4	2	no	—
16	D₇₇₆	—	2	4	no	—
17	D₇₇₆	—	4	2	no	—
18	#13095486	D₄	1	4	yes	complemented
19	C₂×Q₈×C₉₇	—	2	2	no	—
20	Q₁₆×C₉₇	—	2	4	no	—
21	Q₁₆×C₉₇	—	4	2	no	—
22	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
23	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₉₇	—	4	2	no	—
25	C₄²×C₉₇	D₄	1	1	yes	characteristic
26	C₂²×D₉₇	—	4	2	no	—
27	Q₈×C₉₇	—	4	2	no	—
28	Q₈×C₉₇	—	4	2	no	—
29	C₂×C₄×C₉₇	—	4	2	no	—
30	C₂×C₄×C₉₇	D₈	1	2	yes	—
31	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
32	D₃₈₈	—	4	2	no	—
33	D₃₈₈	—	4	2	no	—
34	D₃₈₈	—	8	4	no	—
35	C₈×C₉₇	D₈	1	2	yes	—
36	C₈×C₉₇	C₂×D₄	1	2	yes	—
37	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
38	C₂×D₉₇	—	8	2	no	—
39	C₂×D₉₇	—	8	4	no	—
40	C₄×C₉₇	—	8	2	no	—
41	C₄×C₉₇	SD₃₂	1	4	yes	complemented
42	C₄×C₉₇	#127	1	1	yes	characteristic
43	C₄×C₉₇	C₂×D₈	1	1	yes	characteristic
44	C₂×C₉₇	#493	1	1	yes	characteristic
45	C₂×C₉₇	C₂×SD₃₂	1	2	yes	—
46	D₉₇	—	16	2	no	—
47	#2248	—	97	1	no	maximal, Sylow
48	C₉₇	#2248	1	1	yes	characteristic, complemented, Sylow
49	C₂×SD₃₂	—	97	4	no	—
50	#494	D₉₇	1	1	yes	characteristic, complemented
51	#493	—	97	1	no	—
52	C₄×C₁₆	—	97	1	no	—
53	C₂×Q₁₆	—	2	1	no	—
54	C₂×Q₁₆	C₂×D₉₇	1	2	yes	—
55	C₂×D₈	—	97	2	no	—
56	C₂×D₈	—	194	1	no	—
57	C₄⋊Q₈	—	2	1	no	—
58	#127	—	194	1	no	—
59	SD₃₂	—	194	8	no	—
60	C₂×C₁₆	—	97	2	no	—
61	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
62	C₂×Q₈	—	2	2	no	—
63	C₂×D₄	—	388	2	no	—
64	C₂×D₄	—	194	2	no	—
65	Q₁₆	—	2	4	no	—
66	Q₁₆	—	4	2	no	—
67	D₈	—	388	2	no	—
68	D₈	—	194	4	no	—
69	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
70	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
71	C₄⋊C₄	—	4	2	no	—
72	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
73	C₁₆	—	97	4	no	—
74	C₂³	—	388	2	no	—
75	Q₈	—	4	2	no	—
76	Q₈	—	4	2	no	—
77	D₄	—	776	4	no	—
78	D₄	—	388	2	no	—
79	D₄	—	388	2	no	—
80	C₂×C₄	—	4	2	no	—
81	C₂×C₄	#13095496	1	2	yes	—
82	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
83	C₈	#13095496	1	2	yes	—
84	C₈	D₄×D₉₇	1	2	yes	—
85	C₂²	—	776	4	no	—
86	C₂²	—	776	2	no	—
87	C₂²	C₂×#13095496	1	1	yes	characteristic
88	C₄	—	8	2	no	—
89	C₄	#15690753	1	4	yes	complemented
90	C₄	#15690806	1	1	yes	characteristic
91	C₄	C₂×#13095496	1	1	yes	characteristic
92	C₂	—	1552	2	no	—
93	C₂	#20261111	1	1	yes	characteristic
94	C₂	C₂×#15690753	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13095496
commutator, cosocle	C₂×C₈×C₉₇	C₂³
Frattini	C₂×C₈	C₂²×D₉₇
Fitting	#494×C₉₇	C₂
socle	C₂²×C₉₇	C₂×D₈
prime radical	#20261112	C₂
hypercenter	#494	D₉₇
hypocenter	C₉₇	#2248
LCS	G ▷ C₂×C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
1. C₂×#15690753	C₂	semidirect
3. C₄×#13095486	C₂	semidirect
4. #494×C₉₇	C₂	semidirect
18. #13095486	D₄	semidirect
41. C₄×C₉₇	SD₃₂	semidirect
48. C₉₇	#2248	semidirect
50. #494	D₉₇	semidirect
89. C₄	#15690753	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690753	2
#20261112	2
C₄×#13095486	2
#494×C₉₇	2
#2248	97