Group #22626050 — subgroups

Subgroups — #22626050

Order 9344.

184 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 66 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	#17989602	C₂	1	2	yes	maximal
2	C₂×#14556661	C₂	1	1	yes	maximal, characteristic
3	#17989550	C₂	1	1	yes	maximal, characteristic
4	#17989470	C₂	1	2	yes	maximal
5	#108×D₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11924038	C₂²	1	2	yes	—
7	D₇₃⋊M₁₆	C₂²	1	1	yes	characteristic
8	#14556670	C₂²	1	1	yes	characteristic
9	#14556662	—	2	2	no	—
10	#14556661	—	2	2	no	—
11	#14556661	—	2	1	no	—
12	#14556661	—	2	1	no	—
13	#14556645	—	2	1	no	—
14	C₈×#64408	—	2	1	no	—
15	M₁₆×D₇₃	C₂²	1	2	yes	—
16	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
17	#14556527	C₄	1	1	yes	characteristic, complemented
18	#14556526	C₄	1	2	yes	complemented
19	#108×C₇₃	C₄	1	1	yes	characteristic, complemented
20	#11924038	—	2	2	no	—
21	#11924038	—	2	2	no	—
22	#11924038	D₄	1	2	yes	—
23	#11924037	—	2	1	no	—
24	#11924036	—	2	1	no	—
25	C₂×#10596833	—	2	2	no	—
26	C₄⋊#64408	—	2	1	no	—
27	C₂²⋊#64408	Q₈	1	2	yes	—
28	C₄×#64408	—	2	1	no	—
29	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
30	#11924012	C₂×C₄	1	2	yes	—
31	#11924011	—	2	2	no	—
32	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
33	C₈×D₇₃	—	2	2	no	—
34	C₈×D₇₃	D₄	1	1	yes	characteristic
35	C₈×D₇₃	D₄	1	1	yes	characteristic
36	C₈×D₇₃	Q₈	1	1	yes	characteristic
37	C₈×D₇₃	Q₈	1	1	yes	characteristic
38	M₁₆×C₇₃	C₂×C₄	1	2	yes	—
39	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×D₇₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×#64408	—	2	2	no	—
42	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₇₃	#41	1	1	yes	characteristic
44	C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
45	C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
46	C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
47	C₄×D₇₃	#41	1	1	yes	characteristic
48	#10596833	—	2	2	no	—
49	#10596833	—	4	2	no	—
50	C₈×C₇₃	—	2	2	no	—
51	C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
52	C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
53	#10596831	—	2	2	no	—
54	#10596831	C₄⋊C₄	1	1	yes	characteristic
55	#10596831	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₇₃	C₄×Q₈	1	1	yes	characteristic
57	C₂×D₇₃	—	4	1	no	—
58	C₂×D₇₃	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂×D₇₃	C₄⋊Q₈	1	1	yes	characteristic
60	#64408	—	4	2	no	—
61	C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
63	Dic₇₃	#121	1	1	yes	characteristic
64	Dic₇₃	#125	1	1	yes	characteristic
65	C₂×C₇₃	—	2	1	no	—
66	C₂×C₇₃	#389	1	1	yes	characteristic
67	D₇₃	—	2	1	no	—
68	D₇₃	—	2	1	no	—
69	#1953	—	73	1	no	maximal, Sylow
70	C₇₃	#1953	1	1	yes	characteristic, complemented, Sylow
71	#430	—	73	2	no	—
72	C₂×#108	—	73	1	no	—
73	C₂×#108	—	73	1	no	—
74	#405	—	73	1	no	—
75	#344	—	73	2	no	—
76	C₂×M₁₆	—	73	1	no	—
77	C₂×M₁₆	—	73	2	no	—
78	C₂×M₁₆	—	73	2	no	—
79	C₂²×C₈	—	73	1	no	—
80	#117	—	73	1	no	—
81	#108	—	73	2	no	—
82	#108	—	73	1	no	—
83	#108	—	146	1	no	—
84	#108	—	146	1	no	—
85	#108	—	146	2	no	—
86	#108	—	146	2	no	—
87	#108	#64408	1	1	yes	characteristic, complemented
88	#97	—	146	1	no	—
89	C₄×C₈	—	146	1	no	—
90	C₂²×C₄	—	73	1	no	—
91	M₁₆	—	73	2	no	—
92	M₁₆	—	73	2	no	—
93	M₁₆	—	146	2	no	—
94	M₁₆	—	146	2	no	—
95	M₁₆	—	146	2	no	—
96	M₁₆	—	146	1	no	—
97	M₁₆	C₂×#64408	1	2	yes	—
98	C₂×C₈	—	73	1	no	—
99	C₂×C₈	—	73	1	no	—
100	C₂×C₈	—	73	1	no	—
101	C₂×C₈	—	73	1	no	—
102	C₂×C₈	—	73	1	no	—
103	C₂×C₈	—	146	2	no	—
104	C₂×C₈	—	146	2	no	—
105	C₂×C₈	—	146	1	no	—
106	C₂×C₈	C₂×#64408	1	1	yes	characteristic
107	C₄⋊C₄	—	146	1	no	—
108	C₂²⋊C₄	—	73	2	no	—
109	C₄²	—	146	1	no	—
110	C₂³	—	73	1	no	—
111	C₂×C₄	—	73	1	no	—
112	C₂×C₄	—	73	1	no	—
113	C₂×C₄	—	73	1	no	—
114	C₂×C₄	—	73	1	no	—
115	C₂×C₄	—	73	1	no	—
116	C₂×C₄	—	146	2	no	—
117	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	—	292	2	no	—
120	C₈	—	73	1	no	—
121	C₈	—	73	1	no	—
122	C₈	—	146	2	no	—
123	C₈	—	146	2	no	—
124	C₈	C₄⋊#64408	1	1	yes	characteristic
125	C₈	C₄⋊#64408	1	1	yes	characteristic
126	C₂²	—	292	1	no	—
127	C₂²	—	73	1	no	—
128	C₂²	—	73	1	no	—
129	C₂²	Q₈×#64408	1	1	yes	characteristic
130	C₄	—	292	2	no	—
131	C₄	—	73	1	no	—
132	C₄	—	73	1	no	—
133	C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
134	C₄	D₄×#64408	1	1	yes	characteristic
135	C₂	—	2	1	no	—
136	C₂	—	146	1	no	—
137	C₂	—	146	1	no	—
138	C₂	#17989538	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989602

C₂

yes

maximal

C₂×#14556661

C₂

yes

maximal, characteristic

#17989550

C₂

yes

maximal, characteristic

#17989470

C₂

yes

maximal

#108×D₇₃

C₂

yes

maximal, characteristic

C₂×#11924038

C₂²

yes

—

D₇₃⋊M₁₆

C₂²

yes

characteristic

#14556670

C₂²

yes

—

—

—

—

—

—

M₁₆×D₇₃

C₂²

yes

—

C₂×C₈×D₇₃

C₂²

yes

characteristic

#14556527

C₄

yes

characteristic, complemented

#14556526

C₄

yes

complemented

#108×C₇₃

C₄

yes

characteristic, complemented

#11924038

—

#11924038

—

#11924038

D₄

yes

—

—

—

—

—

Q₈

yes

—

C₄×#64408

—

C₂×C₄×D₇₃

C₂³

yes

characteristic

#11924012

C₂×C₄

yes

—

#11924011

—

C₂×#10596831

C₂×C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

D₄

yes

characteristic

C₈×D₇₃

D₄

yes

characteristic

C₈×D₇₃

Q₈

yes

characteristic

C₈×D₇₃

Q₈

yes

characteristic

M₁₆×C₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

C₂×Q₈

yes

characteristic

C₂×#64408

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

#41

yes

characteristic

C₄×D₇₃

C₂×D₄

yes

characteristic

C₄×D₇₃

C₂×D₄

yes

characteristic

C₄×D₇₃

C₂×Q₈

yes

C₄×D₇₃

yes

—

—

C₈×C₇₃

—

C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₈×C₇₃

C₄⋊C₄

yes

characteristic

#10596831

—

#10596831

C₄⋊C₄

yes

characteristic

#10596831

C₄⋊C₄

yes

characteristic

C₂²×C₇₃

C₄×Q₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

C₂²⋊Q₈

yes

characteristic

C₂×D₇₃

C₄⋊Q₈

yes

characteristic

#64408

—

C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₃

C₄×D₄

yes

Dic₇₃

yes

Dic₇₃

yes

C₂×C₇₃

—

C₂×C₇₃

#389

yes

characteristic

D₇₃

—

D₇₃

—

#1953

—

maximal, Sylow

C₇₃

#1953

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

—

—

—

—

146

—

#108

—

146

—

#108

—

146

—

#108

—

146

—

#108

#64408

yes

characteristic, complemented

#97

—

146

—

C₄×C₈

—

146

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

—

146

—

M₁₆

—

146

—

M₁₆

—

146

—

M₁₆

—

146

—

M₁₆

C₂×#64408

yes

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

146

—

104

C₂×C₈

—

146

—

105

C₂×C₈

—

146

—

106

C₂×C₈

C₂×#64408

yes

characteristic

107

C₄⋊C₄

—

146

—

108

C₂²⋊C₄

—

109

C₄²

—

146

—

110

C₂³

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

146

—

117

C₂×C₄

C₂²×#64408

yes

characteristic

118

C₈

—

119

C₈

—

292

—

120

C₈

—

121

C₈

—

122

C₈

—

146

—

123

C₈

—

146

—

124

C₈

C₄⋊#64408

yes

characteristic

125

C₈

C₄⋊#64408

yes

characteristic

126

C₂²

—

292

—

127

C₂²

—

128

C₂²

—

129

C₂²

Q₈×#64408

yes

characteristic

130

C₄

—

292

—

131

C₄

—

132

C₄

—

133

C₄

C₂×C₄⋊#64408

yes

characteristic

134

C₄

D₄×#64408

yes

characteristic

135

C₂

—

136

C₂

—

146

—

137

C₂

—

146

—

138

C₂

#17989538

yes

characteristic

	subgroup	G/H
center	C₂	#17989538
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	#108×C₇₃	C₄
socle	C₂×C₇₃	#389
hypercenter	#108	#64408
hypocenter	C₇₃	#1953
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

subgroup

G/H

center

C₂

#17989538

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#64408

Fitting

#108×C₇₃

C₄

C₂×C₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

N	H ≅ G/N	type
17. #14556527	C₄	semidirect
18. #14556526	C₄	semidirect
19. #108×C₇₃	C₄	semidirect
70. C₇₃	#1953	semidirect
87. #108	#64408	semidirect

H ≅ G/N

type

17. #14556527

C₄

semidirect

18. #14556526

C₄

semidirect

19. #108×C₇₃

C₄

semidirect

70. C₇₃

#1953

semidirect

87. #108

#64408

semidirect

Subgroups — #22626050

Conjugacy classes of subgroups

Complements