Group #22598849 — subgroups

Subgroups — #22598849

Order 9088.

356 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 145 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	#17986554	C₂	1	2	yes	maximal
2	#17986402	C₂	1	2	yes	maximal, complemented
3	#17986399	C₂	1	4	yes	maximal, complemented
4	#17986397	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986034	C₂	1	4	yes	maximal
6	C₄×C₈×D₇₁	C₂	1	1	yes	maximal, characteristic
7	#498×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₇₁	C₂²	1	2	yes	—
9	#14556128	C₂²	1	2	yes	—
10	#14556119	C₂²	1	4	yes	—
11	#14556108	—	2	2	no	—
12	#14556104	—	2	2	no	—
13	Q₈×Dic₇₁	—	2	2	no	—
14	#14556093	—	2	2	no	—
15	#14556086	C₂²	1	4	yes	—
16	C₄²×D₇₁	C₂²	1	1	yes	characteristic
17	#14556061	C₂²	1	8	yes	—
18	#14556060	C₂²	1	4	yes	—
19	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₈×Dic₇₁	C₂²	1	2	yes	—
21	C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
22	#106×C₇₁	C₂²	1	4	yes	—
23	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₄₂	—	2	2	no	—
25	C₂×C₄×D₇₁	D₄	1	2	yes	—
26	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
27	C₂×C₇₁⋊C₈	C₂³	1	2	yes	—
28	C₈×D₇₁	Q₈	1	8	yes	complemented
29	C₄⋊Dic₇₁	—	2	2	no	—
30	C₄⋊Dic₇₁	—	2	4	no	—
31	C₄⋊Dic₇₁	C₂³	1	4	yes	—
32	Dic₇₁⋊C₄	—	2	4	no	—
33	(C₂×D₇₁)⋊C₄	—	2	4	no	—
34	C₄×Dic₇₁	—	2	4	no	—
35	C₄×Dic₇₁	D₄	1	2	yes	—
36	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
37	C₂×Q₈×C₇₁	—	2	2	no	—
38	C₂×C₈×C₇₁	C₂³	1	2	yes	—
39	C₄⋊C₄×C₇₁	—	2	2	no	—
40	C₄⋊C₄×C₇₁	C₂³	1	4	yes	—
41	C₄²×C₇₁	C₂³	1	1	yes	characteristic
42	C₂²×D₇₁	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₇₁	—	4	4	no	—
44	C₂×C₄×C₇₁	—	2	4	no	—
45	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₇₁	—	2	4	no	—
48	C₂×Dic₇₁	C₂×D₄	1	2	yes	—
49	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
50	C₄×D₇₁	—	2	4	no	—
51	C₄×D₇₁	C₂×Q₈	1	4	yes	—
52	Dic₁₄₂	—	4	4	no	—
53	C₈×C₇₁	C₂×Q₈	1	4	yes	—
54	C₇₁⋊C₈	C₂×Q₈	1	4	yes	—
55	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
56	C₂×D₇₁	—	2	2	no	—
57	C₂×D₇₁	C₄⋊Q₈	1	2	yes	—
58	C₄×C₇₁	—	4	4	no	—
59	C₄×C₇₁	#135	1	4	yes	—
60	C₄×C₇₁	C₂²×Q₈	1	2	yes	—
61	Dic₇₁	—	2	2	no	—
62	Dic₇₁	—	4	4	no	—
63	Dic₇₁	C₄⋊Q₈	1	2	yes	—
64	C₂×C₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₇₁	C₂×#135	1	2	yes	—
66	D₇₁	—	2	2	no	—
67	#3160	—	71	1	no	maximal, Sylow
68	C₇₁	#3160	1	1	yes	characteristic, complemented, Sylow
69	#533	—	71	2	no	—
70	#500	—	71	2	no	—
71	#499	—	71	4	no	—
72	#498	—	71	1	no	—
73	#498	D₇₁	1	1	yes	characteristic, complemented
74	#427	—	71	4	no	—
75	C₂×C₄×C₈	—	71	1	no	—
76	C₂²×C₈	—	71	2	no	—
77	C₄⋊Q₈	—	71	2	no	—
78	C₄⋊Q₈	C₂×D₇₁	1	2	yes	—
79	#125	—	71	4	no	—
80	C₂²⋊Q₈	—	142	2	no	—
81	C₂²⋊Q₈	—	142	2	no	—
82	C₄×Q₈	—	142	2	no	—
83	C₄×Q₈	—	142	2	no	—
84	#117	—	71	4	no	—
85	C₂×C₄²	—	71	1	no	—
86	#107	—	71	8	no	—
87	#106	—	71	4	no	—
88	#106	C₂×D₇₁	1	4	yes	—
89	C₄×C₈	—	71	2	no	—
90	C₄×C₈	—	71	1	no	—
91	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	142	2	no	—
94	C₂²×C₄	—	71	1	no	—
95	C₂²×C₄	—	71	2	no	—
96	C₂×C₈	—	71	2	no	—
97	C₂×C₈	—	71	8	no	—
98	C₂×C₈	C₂²×D₇₁	1	2	yes	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	71	4	no	—
101	C₄⋊C₄	—	142	4	no	—
102	C₄⋊C₄	—	142	4	no	—
103	C₄⋊C₄	—	142	2	no	—
104	C₄⋊C₄	C₂²×D₇₁	1	4	yes	—
105	C₂²⋊C₄	—	142	4	no	—
106	C₄²	—	71	2	no	—
107	C₄²	—	71	1	no	—
108	C₄²	—	142	4	no	—
109	C₄²	C₂²×D₇₁	1	1	yes	characteristic
110	C₂³	—	71	1	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	284	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	71	1	no	—
115	C₂×C₄	—	71	4	no	—
116	C₂×C₄	—	71	2	no	—
117	C₂×C₄	—	142	4	no	—
118	C₂×C₄	—	142	4	no	—
119	C₂×C₄	D₄×D₇₁	1	2	yes	—
120	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
121	C₈	—	71	4	no	—
122	C₈	Q₈×D₇₁	1	4	yes	—
123	C₂²	—	71	2	no	—
124	C₂²	—	142	2	no	—
125	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
126	C₄	—	4	4	no	—
127	C₄	—	284	4	no	—
128	C₄	—	71	2	no	—
129	C₄	—	142	2	no	—
130	C₄	#14556149	1	4	yes	—
131	C₄	C₂×Q₈×D₇₁	1	2	yes	—
132	C₂	—	142	2	no	—
133	C₂	C₄⋊Q₈×D₇₁	1	1	yes	characteristic
134	C₂	C₂×#14556149	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17986554

C₂

yes

maximal

#17986402

C₂

yes

maximal, complemented

#17986399

C₂

yes

maximal, complemented

#17986397

C₂

yes

maximal, characteristic, complemented

#17986034

C₂

yes

maximal

C₄×C₈×D₇₁

C₂

yes

maximal, characteristic

#498×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₇₁

C₂²

yes

—

#14556128

C₂²

yes

—

#14556119

C₂²

yes

—

#14556108

—

#14556104

—

Q₈×Dic₇₁

—

#14556093

—

#14556086

C₂²

yes

—

C₄²×D₇₁

C₂²

yes

characteristic

#14556061

C₂²

yes

—

#14556060

C₂²

yes

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₈×Dic₇₁

C₂²

yes

—

C₄⋊Q₈×C₇₁

C₂²

yes

—

#106×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×C₄×D₇₁

D₄

yes

—

C₂×C₄×D₇₁

C₂³

yes

characteristic

C₂×C₇₁⋊C₈

C₂³

yes

—

C₈×D₇₁

Q₈

yes

complemented

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂³

yes

—

Dic₇₁⋊C₄

—

(C₂×D₇₁)⋊C₄

—

C₄×Dic₇₁

—

C₄×Dic₇₁

D₄

yes

—

C₄×Dic₇₁

C₂³

yes

characteristic

C₂×Q₈×C₇₁

—

C₂×C₈×C₇₁

C₂³

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

C₂³

yes

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂²×D₇₁

C₂×D₄

yes

characteristic

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

C₂×D₄

yes

—

C₂×Dic₇₁

C₂×D₄

yes

characteristic

C₄×D₇₁

—

C₄×D₇₁

C₂×Q₈

yes

—

Dic₁₄₂

—

C₈×C₇₁

C₂×Q₈

yes

—

C₇₁⋊C₈

C₂×Q₈

yes

—

C₂²×C₇₁

C₂²×D₄

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

C₄⋊Q₈

yes

—

C₄×C₇₁

—

C₄×C₇₁

#135

yes

—

C₄×C₇₁

C₂²×Q₈

yes

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

C₄⋊Q₈

yes

—

C₂×C₇₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₇₁

C₂×#135

yes

—

D₇₁

—

#3160

—

maximal, Sylow

C₇₁

#3160

yes

characteristic, complemented, Sylow

—

—

—

—

D₇₁

yes

characteristic, complemented

#427

—

C₂×C₄×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₁

yes

—

#125

—

C₂²⋊Q₈

—

142

—

C₂²⋊Q₈

—

142

—

C₄×Q₈

—

142

—

C₄×Q₈

—

142

—

#117

—

C₂×C₄²

—

#107

—

#106

—

#106

C₂×D₇₁

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

142

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×D₇₁

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

142

—

102

C₄⋊C₄

—

142

—

103

C₄⋊C₄

—

142

—

104

C₄⋊C₄

C₂²×D₇₁

yes

—

105

C₂²⋊C₄

—

142

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

142

—

109

C₄²

C₂²×D₇₁

yes

characteristic

110

C₂³

—

111

Q₈

—

112

Q₈

—

284

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

142

—

118

C₂×C₄

—

142

—

119

C₂×C₄

D₄×D₇₁

yes

—

120

C₂×C₄

C₂³×D₇₁

yes

characteristic

121

C₈

—

122

C₈

Q₈×D₇₁

yes

—

123

C₂²

—

124

C₂²

—

142

—

125

C₂²

C₂×D₄×D₇₁

yes

characteristic

126

C₄

—

127

C₄

—

284

—

128

C₄

—

129

C₄

—

142

—

130

C₄

#14556149

yes

—

131

C₄

C₂×Q₈×D₇₁

yes

—

132

C₂

—

142

—

133

C₂

C₄⋊Q₈×D₇₁

yes

characteristic

134

C₂

C₂×#14556149

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#498×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#498	D₇₁
hypocenter	C₇₁	#3160
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

C₂×D₄×D₇₁

commutator

C₂×C₄×C₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₁

Fitting

#498×C₇₁

C₂

socle

C₂²×C₇₁

C₂²×D₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #17986402	C₂	semidirect
3. #17986399	C₂	semidirect
4. #17986397	C₂	semidirect
7. #498×C₇₁	C₂	semidirect
28. C₈×D₇₁	Q₈	semidirect
68. C₇₁	#3160	semidirect
73. #498	D₇₁	semidirect

H ≅ G/N

type

2. #17986402

C₂

semidirect

3. #17986399

C₂

semidirect

4. #17986397

C₂

semidirect

7. #498×C₇₁

C₂

semidirect

28. C₈×D₇₁

Q₈

semidirect

68. C₇₁

#3160

semidirect

73. #498

D₇₁

semidirect

Subgroups — #22598849

Conjugacy classes of subgroups

Complements