Group #22610882 — subgroups

Subgroups — #22610882

Order 9344.

292 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 121 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×#14556591	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₇₃	C₂	1	1	yes	maximal, characteristic
3	#17988350	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₄₆	—	2	1	no	—
6	#14556591	—	2	2	no	—
7	#14556591	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₇₃	—	2	1	no	—
9	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
11	C₂×C₄×Dic₇₃	—	2	1	no	—
12	#14556498	—	2	1	no	—
13	C₂²×C₈×C₇₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₇₃	—	2	2	no	—
16	C₂×Dic₁₄₆	—	2	4	no	—
17	C₂×Dic₁₄₆	—	4	2	no	—
18	C₂×Dic₁₄₆	—	4	2	no	—
19	C₂²×Dic₇₃	—	2	2	no	—
20	C₂²×Dic₇₃	—	2	2	no	—
21	C₄⋊Dic₇₃	—	2	4	no	—
22	C₄⋊Dic₇₃	—	2	2	no	—
23	C₄⋊Dic₇₃	—	2	2	no	—
24	C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
25	C₄⋊Dic₇₃	Q₈	1	4	yes	complemented
26	C₄×Dic₇₃	—	4	2	no	—
27	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₇₃	—	2	2	no	—
31	C₂×C₈×C₇₃	—	2	4	no	—
32	C₄²×C₇₃	C₂×C₄	1	2	yes	—
33	C₄²×C₇₃	D₄	1	2	yes	—
34	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₇₃	—	2	2	no	—
36	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₇₃	D₈	1	2	yes	—
39	C₂×C₄×C₇₃	SD₁₆	1	2	yes	—
40	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₇₃	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₇₃	#41	1	2	yes	—
44	C₂×Dic₇₃	—	2	4	no	—
45	C₂×Dic₇₃	—	2	4	no	—
46	C₂×Dic₇₃	—	4	2	no	—
47	C₂×Dic₇₃	—	4	2	no	—
48	C₂×Dic₇₃	—	4	2	no	—
49	C₂×Dic₇₃	—	4	2	no	—
50	Dic₁₄₆	—	4	8	no	—
51	C₈×C₇₃	—	4	2	no	—
52	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₇₃	C₂²⋊Q₈	1	2	yes	—
54	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
55	C₂²×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
56	C₂²×C₇₃	#137	1	2	yes	—
57	C₄×C₇₃	—	2	2	no	—
58	C₄×C₇₃	#102	1	4	yes	—
59	C₄×C₇₃	C₄×Q₈	1	2	yes	—
60	C₄×C₇₃	#124	1	1	yes	characteristic
61	C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
62	Dic₇₃	—	4	4	no	—
63	Dic₇₃	—	4	4	no	—
64	C₂×C₇₃	#391	1	1	yes	characteristic
65	C₂×C₇₃	C₂×#102	1	1	yes	characteristic
66	C₂×C₇₃	#419	1	1	yes	characteristic
67	C₂×C₇₃	#474	1	2	yes	—
68	C₂×C₇₃	#475	1	2	yes	—
69	#1988	—	73	1	no	maximal, Sylow
70	C₇₃	#1988	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄⋊Q₈	—	73	1	no	—
72	C₂×C₄⋊C₈	D₇₃	1	1	yes	characteristic
73	C₄×C₄⋊C₄	—	73	1	no	—
74	#340	—	73	4	no	—
75	C₂²×Q₈	—	146	1	no	—
76	C₂²×C₈	C₂×D₇₃	1	2	yes	—
77	C₄⋊Q₈	—	73	4	no	—
78	C₄⋊Q₈	—	146	2	no	—
79	C₂×C₄⋊C₄	—	73	2	no	—
80	C₂×C₄⋊C₄	—	73	2	no	—
81	C₂×C₄⋊C₄	—	146	1	no	—
82	C₂×C₄²	—	146	1	no	—
83	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
84	C₄⋊C₈	—	2	2	no	—
85	#95	—	146	1	no	—
86	C₂×Q₈	—	292	2	no	—
87	C₂×Q₈	—	292	2	no	—
88	C₂×Q₈	—	146	4	no	—
89	C₂²×C₄	—	146	2	no	—
90	C₂²×C₄	—	146	2	no	—
91	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
92	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	4	no	—
96	C₄⋊C₄	—	73	4	no	—
97	C₄⋊C₄	—	73	4	no	—
98	C₄⋊C₄	—	146	4	no	—
99	C₄⋊C₄	—	146	2	no	—
100	C₄⋊C₄	—	146	2	no	—
101	C₄²	—	292	2	no	—
102	C₄²	C₄×D₇₃	1	2	yes	—
103	C₄²	C₂²⋊D₇₃	1	2	yes	—
104	C₂³	C₂×D₂₉₂	1	1	yes	characteristic
105	Q₈	—	292	8	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	292	2	no	—
108	C₂×C₄	—	292	2	no	—
109	C₂×C₄	—	292	2	no	—
110	C₂×C₄	—	292	2	no	—
111	C₂×C₄	—	146	4	no	—
112	C₂×C₄	—	146	4	no	—
113	C₂×C₄	#11924005	1	2	yes	—
114	C₂×C₄	#11924005	1	2	yes	—
115	C₂×C₄	D₅₈₄	1	2	yes	—
116	C₂×C₄	#11924017	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
119	C₂×C₄	Q₈×D₇₃	1	2	yes	—
120	C₂×C₄	#11924027	1	2	yes	—
121	C₈	—	4	2	no	—
122	C₂²	C₂×#11924005	1	1	yes	characteristic
123	C₂²	#14556571	1	2	yes	—
124	C₂²	C₂×D₅₈₄	1	1	yes	characteristic
125	C₂²	C₂×#11924017	1	1	yes	characteristic
126	C₂²	#14556624	1	2	yes	—
127	C₄	—	2	2	no	—
128	C₄	—	292	4	no	—
129	C₄	—	292	4	no	—
130	C₄	#14556513	1	4	yes	—
131	C₄	#14556556	1	2	yes	—
132	C₄	#14556578	1	1	yes	characteristic
133	C₄	#14556589	1	1	yes	characteristic
134	C₂	#17988492	1	1	yes	characteristic
135	C₂	C₂×#14556513	1	1	yes	characteristic
136	C₂	#17988597	1	1	yes	characteristic
137	C₂	#17988868	1	2	yes	—
138	C₂	#17988873	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556591

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₇₃

C₂

yes

maximal, characteristic

#17988350

C₂

yes

maximal

C₂×C₄⋊C₈×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

—

#14556591

—

#14556591

C₄

yes

complemented

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×C₄×Dic₇₃

—

#14556498

—

C₂²×C₈×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

C₄⋊Dic₇₃

Q₈

yes

complemented

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

D₄

yes

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₇₃

D₈

yes

—

C₂×C₄×C₇₃

SD₁₆

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

—

C₂×C₄×C₇₃

#41

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₃

C₂²⋊Q₈

yes

—

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₂²×C₇₃

C₂×SD₁₆

yes

characteristic

C₂²×C₇₃

#137

yes

—

C₄×C₇₃

—

C₄×C₇₃

#102

yes

—

C₄×C₇₃

C₄×Q₈

yes

—

C₄×C₇₃

#124

yes

characteristic

C₄×C₇₃

C₄⋊Q₈

yes

characteristic

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#475

yes

—

#1988

—

maximal, Sylow

C₇₃

#1988

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

D₇₃

yes

characteristic

C₄×C₄⋊C₄

—

#340

—

C₂²×Q₈

—

146

—

C₂²×C₈

C₂×D₇₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

146

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

—

146

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

C₄⋊C₈

—

#95

—

146

—

C₂×Q₈

—

292

—

C₂×Q₈

—

292

—

C₂×Q₈

—

146

—

C₂²×C₄

—

146

—

C₂²×C₄

—

146

—

C₂²×C₄

D₂₉₂

yes

characteristic

C₂²×C₄

D₂₉₂

yes

characteristic

C₂²×C₄

C₂²×D₇₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

100

C₄⋊C₄

—

146

—

101

C₄²

—

292

—

102

C₄²

C₄×D₇₃

yes

—

103

C₄²

C₂²⋊D₇₃

yes

—

104

C₂³

C₂×D₂₉₂

yes

characteristic

105

Q₈

—

292

—

106

C₂×C₄

—

107

C₂×C₄

—

292

—

108

C₂×C₄

—

292

—

109

C₂×C₄

—

292

—

110

C₂×C₄

—

292

—

111

C₂×C₄

—

146

—

112

C₂×C₄

—

146

—

113

C₂×C₄

#11924005

yes

—

114

C₂×C₄

#11924005

yes

—

115

C₂×C₄

D₅₈₄

yes

—

116

C₂×C₄

#11924017

yes

—

117

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

119

C₂×C₄

Q₈×D₇₃

yes

—

120

C₂×C₄

#11924027

yes

—

121

C₈

—

122

C₂²

C₂×#11924005

yes

characteristic

123

C₂²

#14556571

yes

—

124

C₂²

C₂×D₅₈₄

yes

characteristic

125

C₂²

C₂×#11924017

yes

characteristic

126

C₂²

#14556624

yes

—

127

C₄

—

128

C₄

—

292

—

129

C₄

—

292

—

130

C₄

#14556513

yes

—

131

C₄

#14556556

yes

—

132

C₄

#14556578

yes

characteristic

133

C₄

#14556589

yes

characteristic

134

C₂

#17988492

yes

characteristic

135

C₂

C₂×#14556513

yes

characteristic

136

C₂

#17988597

yes

characteristic

137

C₂

#17988868

yes

—

138

C₂

#17988873

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×C₄⋊C₈×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₇₃
hypocenter	C₇₃	#1988
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₂₉₂

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×C₄⋊C₈×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₇₃

hypocenter

C₇₃

#1988

LCS

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
7. #14556591	C₄	semidirect
25. C₄⋊Dic₇₃	Q₈	semidirect
70. C₇₃	#1988	semidirect

H ≅ G/N

type

7. #14556591

C₄

semidirect

25. C₄⋊Dic₇₃

Q₈

semidirect

70. C₇₃

#1988

semidirect

Subgroups — #22610882

Conjugacy classes of subgroups

Complements