Group #27140460 — subgroups

Subgroups — #27140460

Order 12416.

228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×D₉₇	C₂	1	1	yes	maximal, characteristic
2	#20260791	C₂	1	2	yes	maximal
3	#20260484	C₂	1	1	yes	maximal, characteristic, complemented
4	#20260483	C₂	1	2	yes	maximal, complemented
5	#333×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×D₉₇	C₄	1	2	yes	—
7	#15690810	C₂²	1	1	yes	characteristic
8	#15690809	—	2	2	no	—
9	#15690808	C₄	1	2	yes	—
10	C₄⋊C₄×D₉₇	—	2	2	no	—
11	C₄²×D₉₇	C₂²	1	1	yes	characteristic
12	C₄⋊#11715895	C₂²	1	2	yes	—
13	#15690722	—	2	2	no	—
14	#15690720	—	2	2	no	—
15	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₉₇	C₂²	1	2	yes	—
17	C₂×Dic₁₉₄	—	2	2	no	—
18	C₂×Dic₁₉₄	C₂×C₄	1	2	yes	—
19	Q₈×D₉₇	—	4	4	no	—
20	C₂×C₄×D₉₇	—	2	2	no	—
21	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₉₇	D₄	1	2	yes	—
23	C₂×#11715895	—	2	2	no	—
24	C₄⋊Dic₉₇	—	2	2	no	—
25	#13095490	—	2	4	no	—
26	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₉₇	D₄	1	2	yes	—
28	C₂×Q₈×C₉₇	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₉₇	—	2	2	no	—
30	C₄⋊C₄×C₉₇	—	2	2	no	—
31	C₄²×C₉₇	C₂³	1	1	yes	characteristic
32	C₂²×D₉₇	C₂²⋊C₄	1	1	yes	characteristic
33	Q₈×C₉₇	—	4	2	no	—
34	C₂×C₄×C₉₇	—	2	2	no	—
35	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
37	C₂×Dic₉₇	—	2	2	no	—
38	C₂×Dic₉₇	C₂²⋊C₄	1	2	yes	—
39	C₂×Dic₉₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₄×D₉₇	—	2	2	no	—
41	C₄×D₉₇	—	2	2	no	—
42	C₄×D₉₇	—	2	2	no	—
43	C₄×D₉₇	—	4	4	no	—
44	Dic₁₉₄	—	4	4	no	—
45	Dic₁₉₄	—	4	2	no	—
46	C₈×C₉₇	—	4	2	no	—
47	#11715895	—	4	2	no	—
48	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂×D₉₇	—	2	2	no	—
50	C₂×D₉₇	#100	1	2	yes	—
51	C₄×C₉₇	—	2	1	no	—
52	C₄×C₉₇	—	4	2	no	—
53	C₄×C₉₇	#137	1	2	yes	—
54	C₄×C₉₇	#137	1	2	yes	—
55	Dic₉₇	—	2	1	no	—
56	Dic₉₇	—	2	2	no	—
57	Dic₉₇	—	4	2	no	—
58	C₂×C₉₇	C₂×#100	1	1	yes	characteristic
59	C₂×C₉₇	#419	1	2	yes	—
60	D₉₇	—	2	2	no	—
61	#1558	—	97	1	no	maximal, Sylow
62	C₉₇	#1558	1	1	yes	characteristic, complemented, Sylow
63	C₂×C₄⋊Q₈	—	97	1	no	—
64	#432	—	97	2	no	—
65	#333	—	97	1	no	—
66	#333	—	97	2	no	—
67	#333	D₉₇	1	1	yes	characteristic, complemented
68	C₂²×Q₈	—	97	2	no	—
69	C₄⋊Q₈	—	97	2	no	—
70	C₄⋊Q₈	—	97	1	no	—
71	C₄⋊Q₈	—	194	2	no	—
72	C₄⋊Q₈	C₂×D₉₇	1	1	yes	characteristic
73	C₂×C₄⋊C₄	—	194	2	no	—
74	C₂×C₄²	—	97	1	no	—
75	C₄⋊C₈	—	97	2	no	—
76	C₄⋊C₈	C₂×D₉₇	1	2	yes	—
77	C₂²⋊C₈	—	194	2	no	—
78	#97	—	194	2	no	—
79	C₂×Q₈	—	97	2	no	—
80	C₂×Q₈	—	388	4	no	—
81	C₂×Q₈	—	194	2	no	—
82	C₂×Q₈	C₄×D₉₇	1	2	yes	—
83	C₂²×C₄	—	97	1	no	—
84	C₂²×C₄	—	97	2	no	—
85	C₂²×C₄	—	194	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	194	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	194	4	no	—
90	C₄⋊C₄	—	194	2	no	—
91	C₄²	—	97	1	no	—
92	C₄²	—	97	2	no	—
93	C₄²	C₂²×D₉₇	1	1	yes	characteristic
94	C₂³	—	97	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	388	2	no	—
97	Q₈	—	388	4	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	97	2	no	—
100	C₂×C₄	—	97	1	no	—
101	C₂×C₄	—	388	4	no	—
102	C₂×C₄	—	194	2	no	—
103	C₂×C₄	—	194	2	no	—
104	C₂×C₄	—	194	2	no	—
105	C₂×C₄	—	194	2	no	—
106	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
107	C₂×C₄	D₄×D₉₇	1	2	yes	—
108	C₈	—	4	2	no	—
109	C₈	—	388	2	no	—
110	C₂²	—	97	2	no	—
111	C₂²	—	194	2	no	—
112	C₂²	C₂²⋊C₄×D₉₇	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	4	2	no	—
115	C₄	—	388	2	no	—
116	C₄	—	194	1	no	—
117	C₄	—	194	2	no	—
118	C₄	#15690839	1	2	yes	—
119	C₄	#15690842	1	2	yes	—
120	C₂	—	194	2	no	—
121	C₂	#100×D₉₇	1	1	yes	characteristic
122	C₂	#20260742	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊Q₈×D₉₇

C₂

yes

maximal, characteristic

#20260791

C₂

yes

maximal

#20260484

C₂

yes

maximal, characteristic, complemented

#20260483

C₂

yes

maximal, complemented

#333×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₉₇

C₄

yes

—

#15690810

C₂²

yes

characteristic

#15690809

—

#15690808

C₄

yes

—

C₄⋊C₄×D₉₇

—

C₄²×D₉₇

C₂²

yes

characteristic

C₄⋊#11715895

C₂²

yes

—

#15690722

—

#15690720

—

C₄⋊Q₈×C₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₉₇

C₂²

yes

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

C₂×C₄

yes

—

Q₈×D₉₇

—

C₂×C₄×D₉₇

—

C₂×C₄×D₉₇

C₂×C₄

yes

characteristic

C₂×C₄×D₉₇

D₄

yes

—

C₂×#11715895

—

C₄⋊Dic₉₇

—

#13095490

—

C₄×Dic₉₇

C₂×C₄

yes

characteristic

C₄×Dic₉₇

D₄

yes

—

C₂×Q₈×C₉₇

C₂×C₄

yes

—

C₂×C₈×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

C₂³

yes

characteristic

C₂²×D₉₇

C₂²⋊C₄

yes

characteristic

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

C₂²⋊C₄

yes

—

C₂×Dic₉₇

C₂²⋊C₄

yes

characteristic

C₄×D₉₇

—

C₄×D₉₇

—

C₄×D₉₇

—

C₄×D₉₇

—

Dic₁₉₄

—

Dic₁₉₄

—

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₉₇

—

C₂×D₉₇

#100

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#137

yes

—

C₄×C₉₇

#137

yes

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

C₂×#100

yes

characteristic

C₂×C₉₇

#419

yes

—

D₉₇

—

#1558

—

maximal, Sylow

C₉₇

#1558

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

—

—

—

D₉₇

yes

characteristic, complemented

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

194

—

C₄⋊Q₈

C₂×D₉₇

yes

characteristic

C₂×C₄⋊C₄

—

194

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₉₇

yes

—

C₂²⋊C₈

—

194

—

#97

—

194

—

C₂×Q₈

—

C₂×Q₈

—

388

—

C₂×Q₈

—

194

—

C₂×Q₈

C₄×D₉₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

194

—

C₂×C₈

—

C₂×C₈

—

194

—

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₄²

—

C₄²

—

C₄²

C₂²×D₉₇

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

388

—

Q₈

—

388

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

388

—

102

C₂×C₄

—

194

—

103

C₂×C₄

—

194

—

104

C₂×C₄

—

194

—

105

C₂×C₄

—

194

—

106

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

107

C₂×C₄

D₄×D₉₇

yes

—

108

C₈

—

109

C₈

—

388

—

110

C₂²

—

111

C₂²

—

194

—

112

C₂²

C₂²⋊C₄×D₉₇

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

388

—

116

C₄

—

194

—

117

C₄

—

194

—

118

C₄

#15690839

yes

—

119

C₄

#15690842

yes

—

120

C₂

—

194

—

121

C₂

#100×D₉₇

yes

characteristic

122

C₂

#20260742

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₄²	C₂²×D₉₇
Fitting	#333×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₉₇	C₂²⋊C₄
cosocle	C₄²×C₉₇	C₂³
hypercenter	#333	D₉₇
hypocenter	C₉₇	#1558
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₉₇

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₄²

C₂²×D₉₇

Fitting

#333×C₉₇

C₂

socle

C₂²×C₉₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₉₇

C₂²⋊C₄

cosocle

C₄²×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
3. #20260484	C₂	semidirect
4. #20260483	C₂	semidirect
5. #333×C₉₇	C₂	semidirect
62. C₉₇	#1558	semidirect
67. #333	D₉₇	semidirect

H ≅ G/N

type

3. #20260484

C₂

semidirect

4. #20260483

C₂

semidirect

5. #333×C₉₇

C₂

semidirect

62. C₉₇

#1558

semidirect

67. #333

D₉₇

semidirect

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

subgroup	index
C₄⋊Q₈×D₉₇	2
#20260791	2
#20260484	2
#20260483	2
#333×C₉₇	2
#1558	97

subgroup

index

C₄⋊Q₈×D₉₇

#333×C₉₇

Subgroups — #27140460

Conjugacy classes of subgroups

Complements

Maximal subgroups