Group #22611639 — subgroups

Subgroups — #22611639

Order 9344.

228 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 85 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×#14556613	C₂	1	1	yes	maximal, characteristic
2	#17988625	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988392	C₂	1	2	yes	maximal, complemented
4	#17988389	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11924028	—	2	1	no	—
7	#14556613	—	2	2	no	—
8	#14556613	C₄	1	4	yes	complemented
9	C₂×Dic₂₉₂	C₂²	1	1	yes	characteristic
10	C₂×#11924017	—	2	1	no	—
11	C₂×D₅₈₄	C₂²	1	1	yes	characteristic
12	#14556550	—	2	1	no	—
13	#14556525	C₂²	1	2	yes	—
14	#14556523	—	2	1	no	—
15	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₇₃	—	2	2	no	—
17	C₂×C₁₆×C₇₃	C₂²	1	2	yes	—
18	#11924028	—	2	4	no	—
19	#11924028	—	4	2	no	—
20	C₂×Dic₁₄₆	—	2	1	no	—
21	C₂×D₂₉₂	—	2	1	no	—
22	C₂×C₂²⋊D₇₃	—	4	1	no	—
23	C₂×C₄×D₇₃	—	4	1	no	—
24	Dic₂₉₂	—	2	1	no	—
25	Dic₂₉₂	C₂×C₄	1	2	yes	—
26	#11924017	—	2	4	no	—
27	D₅₈₄	—	2	1	no	—
28	D₅₈₄	C₂×C₄	1	2	yes	—
29	C₄⋊Dic₇₃	—	2	2	no	—
30	C₂²⋊Dic₇₃	—	4	1	no	—
31	C₄×Dic₇₃	—	4	1	no	—
32	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
33	C₂×C₈×C₇₃	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
35	C₂×C₈×C₇₃	D₄	1	2	yes	—
36	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
37	C₁₆×C₇₃	—	2	2	no	—
38	C₂³×C₇₃	D₈	1	1	yes	characteristic
39	C₂²×D₇₃	—	4	1	no	—
40	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₇₃	D₈	1	1	yes	characteristic
42	C₂×C₄×C₇₃	SD₁₆	1	2	yes	—
43	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
44	C₂²⋊D₇₃	—	4	4	no	—
45	C₂×Dic₇₃	—	4	1	no	—
46	C₂×Dic₇₃	—	4	2	no	—
47	D₂₉₂	—	2	2	no	—
48	D₂₉₂	—	4	1	no	—
49	C₄×D₇₃	—	4	4	no	—
50	Dic₁₄₆	—	2	2	no	—
51	Dic₁₄₆	—	4	1	no	—
52	C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
53	C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
54	C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₇₃	—	2	2	no	—
56	C₂²×C₇₃	#102	1	2	yes	—
57	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
58	C₂×D₇₃	—	4	2	no	—
59	C₂×D₇₃	—	8	2	no	—
60	C₄×C₇₃	#102	1	2	yes	—
61	C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
63	Dic₇₃	—	4	2	no	—
64	Dic₇₃	—	8	2	no	—
65	C₂×C₇₃	—	2	2	no	—
66	C₂×C₇₃	C₂×#102	1	1	yes	characteristic
67	C₂×C₇₃	#508	1	2	yes	—
68	D₇₃	—	8	2	no	—
69	#2138	—	73	1	no	maximal, Sylow
70	C₇₃	#2138	1	1	yes	characteristic, complemented, Sylow
71	C₂×#135	—	73	1	no	—
72	C₂²×C₁₆	D₇₃	1	1	yes	characteristic, complemented
73	#427	—	73	1	no	—
74	#358	—	73	2	no	—
75	#357	—	73	2	no	—
76	C₂×#41	—	146	1	no	—
77	#135	—	73	4	no	—
78	#135	—	146	2	no	—
79	C₂×Q₁₆	—	73	1	no	—
80	C₂×SD₁₆	—	146	1	no	—
81	C₂×D₈	—	73	1	no	—
82	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
83	#117	—	146	1	no	—
84	C₂×C₁₆	—	2	2	no	—
85	C₂×C₁₆	C₂×D₇₃	1	2	yes	—
86	#107	—	73	2	no	—
87	#106	—	146	1	no	—
88	#41	—	292	2	no	—
89	#41	—	146	4	no	—
90	C₂×Q₈	—	146	1	no	—
91	C₂×D₄	—	292	1	no	—
92	C₂×D₄	—	146	1	no	—
93	C₂²×C₄	—	292	1	no	—
94	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
95	Q₁₆	—	73	2	no	—
96	Q₁₆	—	146	1	no	—
97	SD₁₆	—	146	4	no	—
98	D₈	—	73	2	no	—
99	D₈	—	146	1	no	—
100	C₂×C₈	C₄×D₇₃	1	2	yes	—
101	C₂×C₈	D₂₉₂	1	1	yes	characteristic
102	C₂×C₈	C₂²⋊D₇₃	1	2	yes	—
103	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
104	C₄⋊C₄	—	146	2	no	—
105	C₂²⋊C₄	—	292	1	no	—
106	C₄²	—	292	1	no	—
107	C₁₆	—	2	2	no	—
108	C₂³	—	292	1	no	—
109	C₂³	D₅₈₄	1	1	yes	characteristic
110	Q₈	—	292	1	no	—
111	Q₈	—	146	2	no	—
112	D₄	—	292	4	no	—
113	D₄	—	292	1	no	—
114	D₄	—	146	2	no	—
115	C₂×C₄	—	292	1	no	—
116	C₂×C₄	—	292	4	no	—
117	C₂×C₄	—	292	2	no	—
118	C₂×C₄	#11924005	1	2	yes	—
119	C₂×C₄	D₅₈₄	1	1	yes	characteristic
120	C₂×C₄	#11924017	1	2	yes	—
121	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
122	C₈	#11924005	1	2	yes	—
123	C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
124	C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
125	C₂²	—	2	2	no	—
126	C₂²	—	292	2	no	—
127	C₂²	—	584	2	no	—
128	C₂²	#14556513	1	2	yes	—
129	C₂²	C₂×D₅₈₄	1	1	yes	characteristic
130	C₄	—	292	2	no	—
131	C₄	—	584	2	no	—
132	C₄	#14556513	1	2	yes	—
133	C₄	C₂×#11924005	1	1	yes	characteristic
134	C₄	C₂×#11924017	1	1	yes	characteristic
135	C₂	—	2	2	no	—
136	C₂	—	584	2	no	—
137	C₂	C₂×#14556513	1	1	yes	characteristic
138	C₂	#17989019	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556613

C₂

yes

maximal, characteristic

#17988625

C₂

yes

maximal, characteristic, complemented

#17988392

C₂

yes

maximal, complemented

#17988389

C₂

yes

maximal, complemented

C₂²×C₁₆×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924028

—

#14556613

—

#14556613

C₄

yes

complemented

C₂×Dic₂₉₂

C₂²

yes

characteristic

C₂×#11924017

—

C₂×D₅₈₄

C₂²

yes

characteristic

#14556550

—

#14556525

C₂²

yes

—

#14556523

—

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

C₂²

yes

—

#11924028

—

#11924028

—

C₂×Dic₁₄₆

—

C₂×D₂₉₂

—

C₂×C₂²⋊D₇₃

—

C₂×C₄×D₇₃

—

Dic₂₉₂

—

Dic₂₉₂

C₂×C₄

yes

—

#11924017

—

D₅₈₄

—

D₅₈₄

C₂×C₄

yes

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

D₄

yes

—

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

D₈

yes

characteristic

C₂²×D₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₇₃

D₈

yes

characteristic

C₂×C₄×C₇₃

SD₁₆

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

C₂²⋊C₄

yes

—

C₈×C₇₃

C₂²×C₄

yes

characteristic

C₈×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

#102

yes

—

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

#102

yes

—

C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₃

C₂×SD₁₆

yes

characteristic

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₂×#102

yes

characteristic

C₂×C₇₃

#508

yes

—

D₇₃

—

#2138

—

maximal, Sylow

C₇₃

#2138

yes

characteristic, complemented, Sylow

C₂×#135

—

C₂²×C₁₆

D₇₃

yes

characteristic, complemented

#427

—

#358

—

#357

—

C₂×#41

—

146

—

#135

—

#135

—

146

—

C₂×Q₁₆

—

C₂×SD₁₆

—

146

—

C₂×D₈

—

C₂²×C₈

C₂×D₇₃

yes

characteristic

#117

—

146

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₃

yes

—

#107

—

#106

—

146

—

#41

—

292

—

#41

—

146

—

C₂×Q₈

—

146

—

C₂×D₄

—

292

—

C₂×D₄

—

146

—

C₂²×C₄

—

292

—

C₂²×C₄

D₂₉₂

yes

characteristic

Q₁₆

—

Q₁₆

—

146

—

SD₁₆

—

146

—

D₈

—

D₈

—

146

—

100

C₂×C₈

C₄×D₇₃

yes

—

101

C₂×C₈

D₂₉₂

yes

characteristic

102

C₂×C₈

C₂²⋊D₇₃

yes

—

103

C₂×C₈

C₂²×D₇₃

yes

characteristic

104

C₄⋊C₄

—

146

—

105

C₂²⋊C₄

—

292

—

106

C₄²

—

292

—

107

C₁₆

—

108

C₂³

—

292

—

109

C₂³

D₅₈₄

yes

characteristic

110

Q₈

—

292

—

111

Q₈

—

146

—

112

D₄

—

292

—

113

D₄

—

292

—

114

D₄

—

146

—

115

C₂×C₄

—

292

—

116

C₂×C₄

—

292

—

117

C₂×C₄

—

292

—

118

C₂×C₄

#11924005

yes

—

119

C₂×C₄

D₅₈₄

yes

characteristic

120

C₂×C₄

#11924017

yes

—

121

C₂×C₄

C₂×D₂₉₂

yes

characteristic

122

C₈

#11924005

yes

—

123

C₈

C₂×C₄×D₇₃

yes

characteristic

124

C₈

C₂×C₂²⋊D₇₃

yes

characteristic

125

C₂²

—

126

C₂²

—

292

—

127

C₂²

—

584

—

128

C₂²

#14556513

yes

—

129

C₂²

C₂×D₅₈₄

yes

characteristic

130

C₄

—

292

—

131

C₄

—

584

—

132

C₄

#14556513

yes

—

133

C₄

C₂×#11924005

yes

characteristic

134

C₄

C₂×#11924017

yes

characteristic

135

C₂

—

136

C₂

—

584

—

137

C₂

C₂×#14556513

yes

characteristic

138

C₂

#17989019

yes

—

	subgroup	G/H
center	C₂×C₄	D₅₈₄
commutator	C₈×C₇₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	C₂²×C₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₂×D₈
hypercenter	C₂²×C₁₆	D₇₃
hypocenter	C₇₃	#2138
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂×C₄

D₅₈₄

commutator

C₈×C₇₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

C₂²×C₁₆×C₇₃

C₂

socle

C₂²×C₇₃

C₂×D₈

hypercenter

C₂²×C₁₆

D₇₃

hypocenter

C₇₃

#2138

LCS

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
2. #17988625	C₂	semidirect
3. #17988392	C₂	semidirect
4. #17988389	C₂	semidirect
5. C₂²×C₁₆×C₇₃	C₂	semidirect
8. #14556613	C₄	semidirect
70. C₇₃	#2138	semidirect
72. C₂²×C₁₆	D₇₃	semidirect

H ≅ G/N

type

2. #17988625

C₂

semidirect

3. #17988392

C₂

semidirect

4. #17988389

C₂

semidirect

5. C₂²×C₁₆×C₇₃

C₂

semidirect

8. #14556613

C₄

semidirect

70. C₇₃

#2138

semidirect

72. C₂²×C₁₆

D₇₃

semidirect

Subgroups — #22611639

Conjugacy classes of subgroups

Complements