Subgroups — #27139781

Order 12416.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261112	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#13095486	C₂	1	1	yes	maximal, characteristic, complemented
3	#335×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₇₇₆	—	2	1	no	—
5	C₂×D₇₇₆	C₄	1	2	yes	—
6	#15690807	—	2	1	no	—
7	C₂×#13095486	—	2	1	no	—
8	C₄⋊C₈×C₉₇	—	2	1	no	—
9	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
10	C₂×D₃₈₈	—	2	2	no	—
11	C₂×D₃₈₈	—	4	1	no	—
12	C₂×D₃₈₈	—	4	1	no	—
13	D₇₇₆	—	4	2	no	—
14	D₇₇₆	—	4	1	no	—
15	D₇₇₆	—	4	1	no	—
16	#13095486	—	4	1	no	—
17	C₂×C₈×C₉₇	—	4	1	no	—
18	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
20	C₄²×C₉₇	D₄	1	1	yes	characteristic
21	C₂²×D₉₇	—	4	2	no	—
22	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
24	C₂×C₄×C₉₇	SD₁₆	1	1	yes	characteristic
25	D₃₈₈	—	4	2	no	—
26	D₃₈₈	—	4	2	no	—
27	D₃₈₈	—	8	2	no	—
28	D₃₈₈	—	8	1	no	—
29	D₃₈₈	—	8	1	no	—
30	C₈×C₉₇	—	2	1	no	—
31	C₈×C₉₇	—	8	1	no	—
32	C₈×C₉₇	D₈	1	1	yes	characteristic
33	C₈×C₉₇	SD₁₆	1	1	yes	characteristic
34	C₂²×C₉₇	#102	1	1	yes	characteristic
35	C₂×D₉₇	—	8	2	no	—
36	C₂×D₉₇	—	8	2	no	—
37	C₂×D₉₇	—	8	2	no	—
38	C₄×C₉₇	—	2	1	no	—
39	C₄×C₉₇	#100	1	1	yes	characteristic
40	C₄×C₉₇	#102	1	1	yes	characteristic
41	C₄×C₉₇	D₁₆	1	1	yes	characteristic
42	C₄×C₉₇	SD₃₂	1	1	yes	characteristic
43	C₂×C₉₇	#331	1	1	yes	characteristic
44	C₂×C₉₇	#357	1	1	yes	characteristic
45	C₂×C₉₇	#361	1	1	yes	characteristic
46	D₉₇	—	16	2	no	—
47	#1361	—	97	1	no	maximal, Sylow
48	C₉₇	#1361	1	1	yes	characteristic, complemented, Sylow
49	#493	—	97	1	no	—
50	C₄⋊C₁₆	—	97	1	no	—
51	#335	D₉₇	1	1	yes	characteristic, complemented
52	C₂×D₈	—	97	2	no	—
53	C₂×D₈	—	194	1	no	—
54	#127	—	194	1	no	—
55	C₂×C₁₆	—	194	1	no	—
56	C₄⋊C₈	—	2	1	no	—
57	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
58	C₂×D₄	—	388	1	no	—
59	C₂×D₄	—	388	1	no	—
60	C₂×D₄	—	194	2	no	—
61	D₈	—	388	2	no	—
62	D₈	—	388	1	no	—
63	D₈	—	388	1	no	—
64	C₂×C₈	—	4	1	no	—
65	C₂×C₈	C₄×D₉₇	1	1	yes	characteristic
66	C₂×C₈	D₃₈₈	1	1	yes	characteristic
67	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
68	C₁₆	—	388	1	no	—
69	C₂³	—	388	2	no	—
70	D₄	—	776	2	no	—
71	D₄	—	776	1	no	—
72	D₄	—	776	1	no	—
73	D₄	—	388	2	no	—
74	D₄	—	388	2	no	—
75	C₂×C₄	#13095488	1	1	yes	characteristic
76	C₂×C₄	#13095496	1	1	yes	characteristic
77	C₂×C₄	#13095498	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	8	1	no	—
80	C₈	D₇₇₆	1	1	yes	characteristic
81	C₈	#13095500	1	1	yes	characteristic
82	C₂²	—	776	2	no	—
83	C₂²	—	776	2	no	—
84	C₂²	—	776	2	no	—
85	C₂²	#15690730	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	#15690726	1	1	yes	characteristic
88	C₄	#15690732	1	1	yes	characteristic
89	C₄	#15690751	1	1	yes	characteristic
90	C₄	#15690753	1	1	yes	characteristic
91	C₂	—	1552	2	no	—
92	C₂	#20260478	1	1	yes	characteristic
93	C₂	#20260535	1	1	yes	characteristic
94	C₂	#20260547	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690730
commutator	C₂×C₈×C₉₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₉₇
Fitting	#335×C₉₇	C₂
socle	C₂²×C₉₇	#102
prime radical	#20261112	C₂
cosocle	C₄×C₈×C₉₇	C₂²
hypercenter	#335	D₉₇
hypocenter	C₉₇	#1361
LCS	G ▷ C₂×C₈×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

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

N	H ≅ G/N	type
2. C₄⋊#13095486	C₂	semidirect
3. #335×C₉₇	C₂	semidirect
48. C₉₇	#1361	semidirect
51. #335	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261112	2
C₄⋊#13095486	2
#335×C₉₇	2
#1361	97