Group #22611796 — subgroups

Subgroups — #22611796

Order 9344.

160 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 65 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	Q₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
3	#17988413	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988391	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988388	C₂	1	1	yes	maximal, characteristic
6	#17988362	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×SD₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₇₃	—	2	1	no	—
9	D₄×Dic₇₃	—	2	1	no	—
10	C₂×#11924003	C₂²	1	1	yes	characteristic
11	#14556525	C₂²	1	1	yes	characteristic
12	#14556525	C₂²	1	1	yes	characteristic
13	#14556515	—	2	1	no	—
14	#14556512	—	2	1	no	—
15	C₈×Dic₇₃	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₇₃	C₂²	1	1	yes	characteristic
18	SD₃₂×C₇₃	C₄	1	4	yes	complemented
19	C₂×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
20	C₂²×Dic₇₃	—	4	1	no	—
21	C₂×#10596831	D₄	1	1	yes	characteristic
22	C₄⋊Dic₇₃	—	2	1	no	—
23	C₄⋊Dic₇₃	—	2	1	no	—
24	C₄⋊Dic₇₃	—	4	1	no	—
25	C₂²⋊Dic₇₃	—	4	1	no	—
26	C₄×Dic₇₃	—	4	1	no	—
27	C₄×Dic₇₃	D₄	1	1	yes	characteristic
28	#11924003	—	2	1	no	—
29	C₂×Q₈×C₇₃	—	2	1	no	—
30	C₂×D₄×C₇₃	—	2	1	no	—
31	Q₁₆×C₇₃	—	2	1	no	—
32	Q₁₆×C₇₃	C₂×C₄	1	2	yes	complemented
33	D₈×C₇₃	—	2	1	no	—
34	D₈×C₇₃	C₂×C₄	1	2	yes	—
35	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
36	C₁₆×C₇₃	C₂×C₄	1	2	yes	complemented
37	C₂³×C₇₃	—	4	1	no	—
38	Q₈×C₇₃	—	2	2	no	—
39	Q₈×C₇₃	—	4	1	no	—
40	D₄×C₇₃	—	2	2	no	—
41	D₄×C₇₃	—	4	1	no	—
42	C₂×C₄×C₇₃	—	4	1	no	—
43	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₇₃	—	4	1	no	—
45	C₂×Dic₇₃	—	4	1	no	—
46	C₂×Dic₇₃	—	8	2	no	—
47	C₂×Dic₇₃	D₈	1	1	yes	characteristic
48	C₂×Dic₇₃	D₈	1	1	yes	characteristic
49	C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₈×C₇₃	#41	1	1	yes	characteristic
51	#10596831	—	2	1	no	—
52	C₂²×C₇₃	—	4	2	no	—
53	C₂²×C₇₃	—	8	1	no	—
54	C₂²×C₇₃	—	8	1	no	—
55	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
56	C₄×C₇₃	—	4	2	no	—
57	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
58	C₄×C₇₃	#135	1	1	yes	characteristic
59	Dic₇₃	—	2	1	no	—
60	Dic₇₃	—	2	1	no	—
61	Dic₇₃	—	8	1	no	—
62	Dic₇₃	—	8	1	no	—
63	C₂×C₇₃	—	8	2	no	—
64	C₂×C₇₃	C₄×D₈	1	1	yes	characteristic
65	C₂×C₇₃	#509	1	1	yes	characteristic
66	C₂×C₇₃	#510	1	1	yes	characteristic
67	#2175	—	73	1	no	maximal, Sylow
68	C₇₃	#2175	1	1	yes	characteristic, complemented, Sylow
69	C₂×SD₃₂	D₇₃	1	1	yes	characteristic
70	C₄×Q₁₆	—	73	1	no	—
71	C₄×D₈	—	73	1	no	—
72	#366	—	73	1	no	—
73	#358	—	73	1	no	—
74	#357	—	73	1	no	—
75	#346	—	73	1	no	—
76	C₂×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
77	C₂×D₈	C₂×D₇₃	1	1	yes	characteristic
78	C₄×Q₈	—	146	1	no	—
79	C₄×D₄	—	146	1	no	—
80	SD₃₂	Dic₇₃	1	4	yes	complemented
81	C₂×C₁₆	—	73	1	no	—
82	C₂×C₁₆	C₂×D₇₃	1	1	yes	characteristic
83	#107	—	73	1	no	—
84	#107	—	73	1	no	—
85	#103	—	146	1	no	—
86	#102	—	146	1	no	—
87	C₄×C₈	—	73	1	no	—
88	C₂×Q₈	—	2	1	no	—
89	C₂×D₄	—	2	1	no	—
90	C₂²×C₄	—	292	1	no	—
91	Q₁₆	—	2	1	no	—
92	Q₁₆	C₂×Dic₇₃	1	2	yes	complemented
93	D₈	—	2	1	no	—
94	D₈	C₂×Dic₇₃	1	2	yes	—
95	C₂×C₈	—	73	1	no	—
96	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
97	C₄⋊C₄	—	292	1	no	—
98	C₄⋊C₄	—	146	1	no	—
99	C₄⋊C₄	—	146	1	no	—
100	C₂²⋊C₄	—	292	1	no	—
101	C₄²	—	292	1	no	—
102	C₄²	—	73	1	no	—
103	C₁₆	—	146	1	no	—
104	C₁₆	C₂×Dic₇₃	1	2	yes	complemented
105	C₂³	—	4	1	no	—
106	Q₈	—	2	2	no	—
107	Q₈	—	4	1	no	—
108	D₄	—	2	2	no	—
109	D₄	—	4	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	292	1	no	—
112	C₂×C₄	—	292	1	no	—
113	C₂×C₄	—	584	2	no	—
114	C₂×C₄	—	73	1	no	—
115	C₂×C₄	—	73	1	no	—
116	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
117	C₈	—	146	1	no	—
118	C₈	C₂²×Dic₇₃	1	1	yes	characteristic
119	C₈	#11924027	1	1	yes	characteristic
120	C₂²	—	4	2	no	—
121	C₂²	—	8	1	no	—
122	C₂²	—	8	1	no	—
123	C₂²	D₈×D₇₃	1	1	yes	characteristic
124	C₄	—	4	2	no	—
125	C₄	—	584	1	no	—
126	C₄	—	584	1	no	—
127	C₄	—	146	1	no	—
128	C₄	—	146	1	no	—
129	C₄	D₄×Dic₇₃	1	1	yes	characteristic
130	C₄	#14556610	1	1	yes	characteristic
131	C₂	—	8	2	no	—
132	C₂	D₈×Dic₇₃	1	1	yes	characteristic
133	C₂	#17989024	1	1	yes	characteristic
134	C₂	#17989026	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₁₆×Dic₇₃

C₂

yes

maximal, characteristic, complemented

D₈×Dic₇₃

C₂

yes

maximal, characteristic

#17988413

C₂

yes

maximal, characteristic, complemented

#17988391

C₂

yes

maximal, characteristic, complemented

#17988388

C₂

yes

maximal, characteristic

#17988362

C₂

yes

maximal, characteristic, complemented

C₂×SD₃₂×C₇₃

C₂

yes

maximal, characteristic

Q₈×Dic₇₃

—

D₄×Dic₇₃

—

C₂×#11924003

C₂²

yes

characteristic

#14556525

C₂²

yes

characteristic

#14556525

C₂²

yes

characteristic

#14556515

—

#14556512

—

C₈×Dic₇₃

C₂²

yes

characteristic

C₂×Q₁₆×C₇₃

C₂²

yes

characteristic

C₂×D₈×C₇₃

C₂²

yes

characteristic

SD₃₂×C₇₃

C₄

yes

complemented

C₂×C₁₆×C₇₃

C₂²

yes

characteristic

C₂²×Dic₇₃

—

C₂×#10596831

D₄

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

D₄

yes

characteristic

#11924003

—

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

C₂×C₄

yes

complemented

D₈×C₇₃

—

D₈×C₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₁₆×C₇₃

C₂×C₄

yes

complemented

C₂³×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

D₈

yes

characteristic

C₂×Dic₇₃

D₈

yes

characteristic

C₈×C₇₃

C₂²×C₄

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₄×D₄

yes

characteristic

C₄×C₇₃

#135

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₄×D₈

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×SD₃₂

D₇₃

yes

characteristic

C₄×Q₁₆

—

C₄×D₈

—

—

—

—

—

C₂×Q₁₆

C₂×D₇₃

yes

characteristic

C₂×D₈

C₂×D₇₃

yes

characteristic

C₄×Q₈

—

146

—

C₄×D₄

—

146

—

SD₃₂

Dic₇₃

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₃

yes

—

—

—

146

—

#102

—

146

—

C₄×C₈

—

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

292

—

Q₁₆

—

Q₁₆

C₂×Dic₇₃

yes

complemented

D₈

—

D₈

C₂×Dic₇₃

yes

—

C₂×C₈

—

C₂×C₈

C₂²×D₇₃

yes

characteristic

C₄⋊C₄

—

292

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

100

C₂²⋊C₄

—

292

—

101

C₄²

—

292

—

102

C₄²

—

103

C₁₆

—

146

—

104

C₁₆

C₂×Dic₇₃

yes

complemented

105

C₂³

—

106

Q₈

—

107

Q₈

—

108

D₄

—

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

292

—

112

C₂×C₄

—

292

—

113

C₂×C₄

—

584

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

D₄×D₇₃

yes

characteristic

117

C₈

—

146

—

118

C₈

C₂²×Dic₇₃

yes

characteristic

119

C₈

#11924027

yes

characteristic

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

D₈×D₇₃

yes

characteristic

124

C₄

—

125

C₄

—

584

—

126

C₄

—

584

—

127

C₄

—

146

—

128

C₄

—

146

—

129

C₄

D₄×Dic₇₃

yes

characteristic

130

C₄

#14556610

yes

characteristic

131

C₂

—

132

C₂

D₈×Dic₇₃

yes

characteristic

133

C₂

#17989024

yes

characteristic

134

C₂

#17989026

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₃
commutator	C₈×C₇₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	C₂×SD₃₂×C₇₃	C₂
socle	C₂²×C₇₃	C₂×D₈
hypercenter	C₂×SD₃₂	D₇₃
hypocenter	C₇₃	#2175
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×SD₃₂

subgroup

G/H

center

C₂²

D₈×D₇₃

commutator

C₈×C₇₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

C₂×SD₃₂×C₇₃

C₂

socle

C₂²×C₇₃

C₂×D₈

hypercenter

C₂×SD₃₂

D₇₃

hypocenter

C₇₃

#2175

LCS

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×SD₃₂

N	H ≅ G/N	type
1. Q₁₆×Dic₇₃	C₂	semidirect
3. #17988413	C₂	semidirect
4. #17988391	C₂	semidirect
6. #17988362	C₂	semidirect
18. SD₃₂×C₇₃	C₄	semidirect
32. Q₁₆×C₇₃	C₂×C₄	semidirect
36. C₁₆×C₇₃	C₂×C₄	semidirect
68. C₇₃	#2175	semidirect
80. SD₃₂	Dic₇₃	semidirect
92. Q₁₆	C₂×Dic₇₃	semidirect
104. C₁₆	C₂×Dic₇₃	semidirect

H ≅ G/N

type

1. Q₁₆×Dic₇₃

C₂

semidirect

3. #17988413

C₂

semidirect

4. #17988391

C₂

semidirect

6. #17988362

C₂

semidirect

18. SD₃₂×C₇₃

C₄

semidirect

32. Q₁₆×C₇₃

C₂×C₄

semidirect

36. C₁₆×C₇₃

C₂×C₄

semidirect

68. C₇₃

#2175

semidirect

80. SD₃₂

Dic₇₃

semidirect

92. Q₁₆

C₂×Dic₇₃

semidirect

104. C₁₆

C₂×Dic₇₃

semidirect

Subgroups — #22611796

Conjugacy classes of subgroups

Complements