Subgroups — #27143621

Order 12416.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261167	C₂	1	1	yes	maximal, characteristic
2	#20260851	C₂	1	2	yes	maximal, complemented
3	#20260541	C₂	1	2	yes	maximal, complemented
4	#20260510	C₂	1	1	yes	maximal, characteristic, complemented
5	#368×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#15690832	C₂²	1	2	yes	—
7	#15690818	—	2	2	no	—
8	Dic₇₇₆	C₄	1	1	yes	characteristic, complemented
9	#15690755	C₄	1	2	yes	complemented
10	D₁₅₅₂	C₄	1	1	yes	characteristic, complemented
11	C₂×#13095486	C₂²	1	1	yes	characteristic
12	#15690736	—	2	2	no	—
13	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
15	#108×C₉₇	C₂²	1	2	yes	—
16	#13095511	—	2	2	no	—
17	Dic₃₈₈	C₂×C₄	1	2	yes	—
18	#13095500	—	2	2	no	—
19	D₇₇₆	C₂×C₄	1	2	yes	—
20	#13095494	—	4	2	no	—
21	C₂×#11715895	D₄	1	1	yes	characteristic
22	C₈×D₉₇	—	4	2	no	—
23	C₄×Dic₉₇	D₄	1	1	yes	characteristic
24	#13095486	—	2	1	no	—
25	M₁₆×C₉₇	—	2	2	no	—
26	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
27	C₁₆×C₉₇	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₉₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
30	C₂²⋊D₉₇	—	4	2	no	—
31	C₂×Dic₉₇	—	2	1	no	—
32	D₃₈₈	—	2	2	no	—
33	C₄×D₉₇	—	4	2	no	—
34	Dic₁₉₄	—	2	2	no	—
35	C₈×C₉₇	—	4	2	no	—
36	C₈×C₉₇	C₂²×C₄	1	1	yes	characteristic
37	C₈×C₉₇	#41	1	1	yes	characteristic
38	#11715895	D₈	1	2	yes	—
39	C₂²×C₉₇	#135	1	1	yes	characteristic
40	C₂×D₉₇	—	4	2	no	—
41	C₄×C₉₇	C₄×D₄	1	1	yes	characteristic
42	C₄×C₉₇	C₂×D₈	1	1	yes	characteristic
43	Dic₉₇	—	4	1	no	—
44	Dic₉₇	—	4	2	no	—
45	C₂×C₉₇	—	2	1	no	—
46	C₂×C₉₇	C₄×D₈	1	1	yes	characteristic
47	D₉₇	—	8	2	no	—
48	#2179	—	97	1	no	maximal, Sylow
49	C₉₇	#2179	1	1	yes	characteristic, complemented, Sylow
50	#508	—	97	1	no	—
51	#443	—	97	2	no	—
52	#368	D₉₇	1	1	yes	characteristic, complemented
53	#359	—	97	2	no	—
54	#346	—	97	1	no	—
55	#135	—	97	2	no	—
56	#131	—	194	2	no	—
57	Q₃₂	—	97	1	no	—
58	SD₃₂	—	97	2	no	—
59	D₁₆	—	97	1	no	—
60	C₂×C₁₆	—	97	1	no	—
61	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
62	#108	C₂×D₉₇	1	2	yes	—
63	#104	—	194	2	no	—
64	C₄×C₈	—	97	1	no	—
65	#41	—	194	2	no	—
66	Q₁₆	—	97	2	no	—
67	SD₁₆	—	194	2	no	—
68	D₈	—	97	2	no	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	388	2	no	—
71	C₂×C₈	—	97	1	no	—
72	C₂×C₈	—	388	2	no	—
73	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
74	C₄²	—	97	1	no	—
75	C₁₆	—	194	1	no	—
76	C₁₆	C₄×D₉₇	1	1	yes	characteristic
77	C₁₆	C₄×D₉₇	1	1	yes	characteristic
78	Q₈	—	194	2	no	—
79	D₄	—	388	2	no	—
80	D₄	—	194	2	no	—
81	C₂×C₄	—	388	2	no	—
82	C₂×C₄	—	194	1	no	—
83	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
84	C₈	—	4	2	no	—
85	C₈	—	97	2	no	—
86	C₈	C₂×C₄×D₉₇	1	1	yes	characteristic
87	C₈	#13095512	1	1	yes	characteristic
88	C₂²	—	388	2	no	—
89	C₂²	#15690833	1	1	yes	characteristic
90	C₄	—	388	1	no	—
91	C₄	—	388	2	no	—
92	C₄	#15690772	1	1	yes	characteristic
93	C₄	D₈×D₉₇	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	776	2	no	—
96	C₂	#20260817	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×D₉₇
commutator	C₈×C₉₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₉₇
Fitting	#368×C₉₇	C₂
socle	C₂×C₉₇	C₄×D₈
prime radical	#20261167	C₂
cosocle	C₂×C₈×C₉₇	C₂³
hypercenter	#368	D₉₇
hypocenter	C₉₇	#2179
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

Complements

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

N	H ≅ G/N	type
2. #20260851	C₂	semidirect
3. #20260541	C₂	semidirect
4. #20260510	C₂	semidirect
5. #368×C₉₇	C₂	semidirect
8. Dic₇₇₆	C₄	semidirect
9. #15690755	C₄	semidirect
10. D₁₅₅₂	C₄	semidirect
49. C₉₇	#2179	semidirect
52. #368	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261167	2
#20260851	2
#20260541	2
#20260510	2
#368×C₉₇	2
#2179	97