Group #22610714 — subgroups

Subgroups — #22610714

Order 9344.

196 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 89 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×#14556527	C₂	1	1	yes	maximal, characteristic
2	#17988627	C₂	1	2	yes	maximal
3	#17988542	C₂	1	1	yes	maximal, characteristic
4	#17988356	C₂	1	2	yes	maximal
5	C₂×#107×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11924012	C₂²	1	2	yes	—
7	C₂×#11924012	C₂²	1	1	yes	characteristic
8	#14556550	C₂²	1	1	yes	characteristic
9	#14556527	C₄	1	4	yes	complemented
10	#14556524	—	2	2	no	—
11	#14556522	—	2	2	no	—
12	#14556501	—	2	1	no	—
13	C₈×Dic₇₃	—	2	1	no	—
14	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
16	#107×C₇₃	—	2	1	no	—
17	#107×C₇₃	—	2	1	no	—
18	#11924012	—	2	2	no	—
19	#11924012	—	2	1	no	—
20	#11924012	C₂×C₄	1	4	yes	—
21	#11924012	Q₈	1	2	yes	—
22	C₂×#10596831	—	2	1	no	—
23	C₂×#10596831	—	2	2	no	—
24	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
25	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
26	C₂²⋊Dic₇₃	—	2	1	no	—
27	C₄×Dic₇₃	—	2	1	no	—
28	C₂²×C₄×C₇₃	—	2	2	no	—
29	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₇₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
32	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
33	C₂×C₈×C₇₃	Q₈	1	2	yes	—
34	C₄⋊C₄×C₇₃	—	2	2	no	—
35	C₄⋊C₄×C₇₃	—	2	2	no	—
36	C₂³×C₇₃	#41	1	1	yes	characteristic
37	C₂×C₄×C₇₃	—	4	2	no	—
38	C₂×C₄×C₇₃	—	8	2	no	—
39	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
45	C₂×Dic₇₃	—	2	1	no	—
46	C₂×Dic₇₃	—	2	1	no	—
47	C₈×C₇₃	C₄⋊C₄	1	2	yes	—
48	C₈×C₇₃	C₄⋊C₄	1	2	yes	—
49	#10596831	—	2	4	no	—
50	#10596831	—	2	2	no	—
51	C₂²×C₇₃	—	2	1	no	—
52	C₂²×C₇₃	—	2	1	no	—
53	C₂²×C₇₃	C₄×Q₈	1	1	yes	characteristic
54	C₂²×C₇₃	#121	1	1	yes	characteristic
55	C₂²×C₇₃	#125	1	1	yes	characteristic
56	C₄×C₇₃	—	8	2	no	—
57	C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
59	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
60	C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
61	Dic₇₃	—	4	1	no	—
62	Dic₇₃	—	4	1	no	—
63	C₂×C₇₃	—	2	1	no	—
64	C₂×C₇₃	—	2	1	no	—
65	C₂×C₇₃	#389	1	1	yes	characteristic
66	C₂×C₇₃	#472	1	1	yes	characteristic
67	C₂×C₇₃	#473	1	1	yes	characteristic
68	#1952	—	73	1	no	maximal, Sylow
69	C₇₃	#1952	1	1	yes	characteristic, complemented, Sylow
70	C₂×#108	—	73	1	no	—
71	#428	—	73	2	no	—
72	C₂×#107	D₇₃	1	1	yes	characteristic
73	#405	—	73	1	no	—
74	#343	—	73	2	no	—
75	C₂×M₁₆	—	73	1	no	—
76	C₂×M₁₆	—	73	2	no	—
77	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
78	#117	—	73	1	no	—
79	C₂×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
80	#108	—	73	4	no	—
81	#107	—	2	1	no	—
82	#107	—	2	1	no	—
83	#107	—	146	2	no	—
84	#106	—	146	2	no	—
85	#97	—	146	1	no	—
86	C₄×C₈	—	146	1	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
89	M₁₆	—	73	4	no	—
90	M₁₆	—	73	2	no	—
91	M₁₆	—	146	2	no	—
92	M₁₆	—	146	1	no	—
93	C₂×C₈	—	146	2	no	—
94	C₂×C₈	—	146	1	no	—
95	C₂×C₈	Dic₁₄₆	1	2	yes	—
96	C₂×C₈	C₄×D₇₃	1	2	yes	—
97	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
98	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	73	1	no	—
102	C₄⋊C₄	—	73	1	no	—
103	C₂²⋊C₄	—	146	1	no	—
104	C₄²	—	146	1	no	—
105	C₂³	#11924027	1	1	yes	characteristic
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	8	2	no	—
108	C₂×C₄	—	146	1	no	—
109	C₂×C₄	—	146	1	no	—
110	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
111	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
113	C₂×C₄	Q₈×D₇₃	1	1	yes	characteristic
114	C₂×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
115	C₂×C₄	#11924029	1	1	yes	characteristic
116	C₈	—	146	4	no	—
117	C₈	—	146	2	no	—
118	C₈	#11924007	1	2	yes	—
119	C₈	#11924007	1	2	yes	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	#14556556	1	1	yes	characteristic
123	C₂²	#14556562	1	1	yes	characteristic
124	C₂²	#14556582	1	1	yes	characteristic
125	C₄	—	8	2	no	—
126	C₄	—	292	1	no	—
127	C₄	—	292	1	no	—
128	C₄	C₂×#11924007	1	1	yes	characteristic
129	C₄	#14556553	1	1	yes	characteristic
130	C₄	#14556567	1	1	yes	characteristic
131	C₄	#14556590	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	2	1	no	—
134	C₂	#17988483	1	1	yes	characteristic
135	C₂	#17988854	1	1	yes	characteristic
136	C₂	#17988859	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556527

C₂

yes

maximal, characteristic

#17988627

C₂

yes

maximal

#17988542

C₂

yes

maximal, characteristic

#17988356

C₂

yes

maximal

C₂×#107×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924012

C₂²

yes

—

C₂×#11924012

C₂²

yes

characteristic

#14556550

C₂²

yes

characteristic

#14556527

C₄

yes

—

—

—

C₈×Dic₇₃

—

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₂²

yes

—

#107×C₇₃

—

#107×C₇₃

—

#11924012

—

#11924012

—

#11924012

C₂×C₄

yes

—

#11924012

Q₈

yes

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

D₄

yes

characteristic

C₄⋊Dic₇₃

D₄

yes

characteristic

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

Q₈

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂³×C₇₃

#41

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

C₄⋊C₄

yes

—

C₈×C₇₃

C₄⋊C₄

yes

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₄×Q₈

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₃

C₄×D₄

yes

characteristic

C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₄×C₇₃

C₄⋊Q₈

yes

characteristic

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×#108

—

#428

—

C₂×#107

D₇₃

yes

characteristic

#405

—

#343

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₇₃

yes

characteristic

#117

—

C₂×C₄⋊C₄

C₂×D₇₃

yes

—

—

—

—

—

146

—

#106

—

146

—

#97

—

146

—

C₄×C₈

—

146

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

146

—

M₁₆

—

146

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₂×C₈

Dic₁₄₆

yes

—

C₂×C₈

C₄×D₇₃

yes

—

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

146

—

104

C₄²

—

146

—

105

C₂³

#11924027

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

146

—

109

C₂×C₄

—

146

—

110

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

111

C₂×C₄

D₄×D₇₃

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

113

C₂×C₄

Q₈×D₇₃

yes

characteristic

114

C₂×C₄

C₂×Dic₁₄₆

yes

characteristic

115

C₂×C₄

#11924029

yes

characteristic

116

C₈

—

146

—

117

C₈

—

146

—

118

C₈

#11924007

yes

—

119

C₈

#11924007

yes

—

120

C₂²

—

121

C₂²

—

122

C₂²

#14556556

yes

characteristic

123

C₂²

#14556562

yes

characteristic

124

C₂²

#14556582

yes

characteristic

125

C₄

—

126

C₄

—

292

—

127

C₄

—

292

—

128

C₄

C₂×#11924007

yes

characteristic

129

C₄

#14556553

yes

characteristic

130

C₄

#14556567

yes

characteristic

131

C₄

#14556590

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

#17988483

yes

characteristic

135

C₂

#17988854

yes

characteristic

136

C₂

#17988859

yes

characteristic

	subgroup	G/H
center	C₂²	#14556562
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×#107×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	C₂×#107	D₇₃
hypocenter	C₇₃	#1952
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂²

#14556562

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×#107×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
9. #14556527	C₄	semidirect
69. C₇₃	#1952	semidirect

H ≅ G/N

type

9. #14556527

C₄

semidirect

69. C₇₃

#1952

semidirect

Subgroups — #22610714

Conjugacy classes of subgroups

Complements