Group #22587065 — subgroups

Subgroups — #22587065

Order 9088.

228 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 77 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	C₂×#14556128	C₂	1	1	yes	maximal, characteristic
2	#17986056	C₂	1	1	yes	maximal, characteristic
3	#17985747	C₂	1	4	yes	maximal, complemented
4	#432×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₄₂	C₄	1	2	yes	—
6	#14556128	—	2	1	no	—
7	#14556128	—	2	1	no	—
8	#14556128	C₄	1	2	yes	—
9	#14556128	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₇₁	—	2	1	no	—
11	C₂×C₄⋊Dic₇₁	—	2	1	no	—
12	C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
13	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
14	#14556039	—	2	1	no	—
15	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
17	C₂²⋊C₈×C₇₁	—	2	1	no	—
18	#97×C₇₁	—	2	1	no	—
19	C₂×Dic₁₄₂	—	2	2	no	—
20	C₂×Dic₁₄₂	—	4	4	no	—
21	C₂×Dic₁₄₂	C₂×C₄	1	4	yes	—
22	C₂²×Dic₇₁	—	2	2	no	—
23	C₂×C₇₁⋊C₈	—	2	2	no	—
24	C₄⋊Dic₇₁	—	2	2	no	—
25	C₄⋊Dic₇₁	—	2	2	no	—
26	C₄⋊Dic₇₁	—	2	2	no	—
27	C₄⋊Dic₇₁	—	2	2	no	—
28	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
31	C₂×C₈×C₇₁	—	2	2	no	—
32	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₇₁	D₄	1	1	yes	characteristic
34	C₄²×C₇₁	D₄	1	1	yes	characteristic
35	C₄²×C₇₁	C₂³	1	1	yes	characteristic
36	C₂³×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₇₁	—	2	2	no	—
38	C₂×C₄×C₇₁	—	2	2	no	—
39	C₂×C₄×C₇₁	—	2	2	no	—
40	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₇₁	—	2	4	no	—
47	C₂×Dic₇₁	—	4	4	no	—
48	Dic₁₄₂	—	4	4	no	—
49	Dic₁₄₂	—	4	4	no	—
50	C₈×C₇₁	—	4	2	no	—
51	C₇₁⋊C₈	—	4	2	no	—
52	C₂²×C₇₁	—	2	1	no	—
53	C₂²×C₇₁	—	2	1	no	—
54	C₂²×C₇₁	#100	1	2	yes	—
55	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₄×C₇₁	—	2	1	no	—
57	C₄×C₇₁	—	2	1	no	—
58	C₄×C₇₁	—	2	1	no	—
59	C₄×C₇₁	—	2	1	no	—
60	C₄×C₇₁	#137	1	2	yes	—
61	C₄×C₇₁	#137	1	2	yes	—
62	Dic₇₁	—	4	4	no	—
63	C₂×C₇₁	—	2	2	no	—
64	C₂×C₇₁	C₂×#100	1	1	yes	characteristic
65	C₂×C₇₁	#419	1	1	yes	characteristic
66	C₂×C₇₁	#419	1	1	yes	characteristic
67	#1558	—	71	1	no	maximal, Sylow
68	C₇₁	#1558	1	1	yes	characteristic, complemented, Sylow
69	C₂×C₄⋊Q₈	—	71	1	no	—
70	#432	—	71	1	no	—
71	#432	D₇₁	1	1	yes	characteristic
72	#333	—	71	4	no	—
73	C₂²×Q₈	—	71	2	no	—
74	C₄⋊Q₈	—	71	2	no	—
75	C₄⋊Q₈	—	71	2	no	—
76	C₄⋊Q₈	—	142	1	no	—
77	C₄⋊Q₈	—	142	1	no	—
78	C₂×C₄⋊C₄	—	142	1	no	—
79	C₂×C₄⋊C₄	—	142	1	no	—
80	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
81	C₄⋊C₈	—	71	2	no	—
82	C₄⋊C₈	C₂×D₇₁	1	2	yes	—
83	C₂²⋊C₈	—	2	1	no	—
84	C₂²⋊C₈	—	142	1	no	—
85	#97	—	2	1	no	—
86	#97	—	142	1	no	—
87	C₂×Q₈	—	284	4	no	—
88	C₂×Q₈	—	71	4	no	—
89	C₂×Q₈	—	142	2	no	—
90	C₂²×C₄	—	142	2	no	—
91	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
92	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
93	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	142	2	no	—
96	C₄⋊C₄	—	142	2	no	—
97	C₄⋊C₄	—	142	2	no	—
98	C₄⋊C₄	—	142	2	no	—
99	C₄⋊C₄	—	142	2	no	—
100	C₄²	C₄×D₇₁	1	1	yes	characteristic
101	C₄²	D₂₈₄	1	1	yes	characteristic
102	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
103	C₄²	C₂²×D₇₁	1	1	yes	characteristic
104	C₂³	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
105	Q₈	—	284	4	no	—
106	Q₈	—	284	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	284	4	no	—
111	C₂×C₄	—	142	4	no	—
112	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
113	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
114	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
116	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
117	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
118	C₈	—	4	2	no	—
119	C₈	—	284	2	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	#14556044	1	2	yes	—
123	C₂²	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	284	4	no	—
129	C₄	#14556158	1	2	yes	—
130	C₄	#14556161	1	2	yes	—
131	C₂	—	2	2	no	—
132	C₂	C₂×#14556044	1	1	yes	characteristic
133	C₂	#17986007	1	1	yes	characteristic
134	C₂	#17986009	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556128

C₂

yes

maximal, characteristic

#17986056

C₂

yes

maximal, characteristic

#17985747

C₂

yes

maximal, complemented

#432×C₇₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₂

C₄

yes

—

#14556128

—

#14556128

—

#14556128

C₄

yes

—

#14556128

C₂²

yes

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

C₂²⋊(C₇₁⋊C₈)

—

#14556039

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

—

C₂²⋊C₈×C₇₁

—

#97×C₇₁

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

C₂×C₄

yes

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#100

yes

—

C₂²×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#137

yes

—

C₄×C₇₁

#137

yes

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#432

—

#432

D₇₁

yes

characteristic

#333

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

142

—

C₄⋊Q₈

—

142

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₁

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

142

—

#97

—

#97

—

142

—

C₂×Q₈

—

284

—

C₂×Q₈

—

C₂×Q₈

—

142

—

C₂²×C₄

—

142

—

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

D₂₈₄

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

142

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

100

C₄²

C₄×D₇₁

yes

characteristic

101

C₄²

D₂₈₄

yes

characteristic

102

C₄²

C₂²⋊D₇₁

yes

characteristic

103

C₄²

C₂²×D₇₁

yes

characteristic

104

C₂³

(C₂×D₇₁)⋊C₄

yes

characteristic

105

Q₈

—

284

—

106

Q₈

—

284

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

284

—

111

C₂×C₄

—

142

—

112

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

113

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

114

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

115

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

116

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

117

C₂×C₄

C₂×D₂₈₄

yes

characteristic

118

C₈

—

119

C₈

—

284

—

120

C₂²

—

121

C₂²

—

122

C₂²

#14556044

yes

—

123

C₂²

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

284

—

129

C₄

#14556158

yes

—

130

C₄

#14556161

yes

—

131

C₂

—

132

C₂

C₂×#14556044

yes

characteristic

133

C₂

#17986007

yes

characteristic

134

C₂

#17986009

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₇₁)⋊C₄
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₄²	C₂²×D₇₁
Fitting	#432×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₄
hypercenter	#432	D₇₁
hypocenter	C₇₁	#1558
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

C₂×(C₂×D₇₁)⋊C₄

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₄²

C₂²×D₇₁

Fitting

#432×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₂²⋊C₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
3. #17985747	C₂	semidirect
68. C₇₁	#1558	semidirect

H ≅ G/N

type

3. #17985747

C₂

semidirect

68. C₇₁

#1558

semidirect

Subgroups — #22587065

Conjugacy classes of subgroups

Complements