Group #22610216 — subgroups

Subgroups — #22610216

Order 9344.

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

normal subgroups only

This group has 260 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	#17989354	C₂	1	1	yes	maximal, characteristic
2	#17988613	C₂	1	2	yes	maximal, complemented
3	#17988561	C₂	1	1	yes	maximal, characteristic
4	#17988359	C₂	1	2	yes	maximal, complemented
5	#408×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#14556598	—	2	4	no	—
7	C₂×#11924012	—	2	2	no	—
8	C₂×#11924012	C₂²	1	2	yes	—
9	#14556550	C₂²	1	2	yes	—
10	#14556520	—	2	4	no	—
11	#14556506	C₄	1	4	yes	—
12	C₂×#41×C₇₃	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₂²⋊C₈×C₇₃	—	2	2	no	—
16	#11924012	—	2	4	no	—
17	#11924012	—	2	2	no	—
18	#11924012	—	2	2	no	—
19	C₂×#10596831	—	2	4	no	—
20	C₂×#10596831	D₄	1	4	yes	—
21	C₄⋊Dic₇₃	—	2	2	no	—
22	C₂²⋊Dic₇₃	C₂×C₄	1	4	yes	—
23	C₄×Dic₇₃	—	2	2	no	—
24	#41×C₇₃	—	2	2	no	—
25	#41×C₇₃	—	2	2	no	—
26	C₂×Q₈×C₇₃	Q₈	1	1	yes	characteristic
27	C₂×D₄×C₇₃	C₂×C₄	1	2	yes	—
28	C₂×D₄×C₇₃	Q₈	1	1	yes	characteristic
29	C₂²×C₄×C₇₃	D₄	1	2	yes	—
30	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
31	M₁₆×C₇₃	—	2	2	no	—
32	C₂×C₈×C₇₃	—	2	1	no	—
33	C₂×C₈×C₇₃	—	2	1	no	—
34	C₂×C₈×C₇₃	—	2	2	no	—
35	C₂³×C₇₃	C₄⋊C₄	1	2	yes	—
36	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₇₃	—	2	2	no	—
38	D₄×C₇₃	—	2	4	no	—
39	D₄×C₇₃	—	2	2	no	—
40	C₂×C₄×C₇₃	—	2	2	no	—
41	C₂×C₄×C₇₃	—	2	2	no	—
42	C₂×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₇₃	#41	1	2	yes	—
47	C₂×Dic₇₃	—	2	4	no	—
48	C₈×C₇₃	—	4	1	no	—
49	C₈×C₇₃	—	4	1	no	—
50	#10596831	—	2	4	no	—
51	C₂²×C₇₃	—	2	2	no	—
52	C₂²×C₇₃	—	4	1	no	—
53	C₂²×C₇₃	—	4	2	no	—
54	C₂²×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₇₃	C₄×D₄	1	2	yes	—
56	C₄×C₇₃	—	2	2	no	—
57	C₄×C₇₃	#120	1	1	yes	characteristic
58	C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₇₃	#123	1	1	yes	characteristic
60	Dic₇₃	—	4	4	no	—
61	C₂×C₇₃	—	2	2	no	—
62	C₂×C₇₃	—	2	1	no	—
63	C₂×C₇₃	—	4	2	no	—
64	C₂×C₇₃	#385	1	1	yes	characteristic
65	#1868	—	73	1	no	maximal, Sylow
66	C₇₃	#1868	1	1	yes	characteristic, complemented, Sylow
67	#568	—	73	1	no	—
68	#424	—	73	2	no	—
69	#410	—	73	1	no	—
70	#408	D₇₃	1	1	yes	characteristic
71	#344	—	73	2	no	—
72	C₂×#41	C₂×D₇₃	1	1	yes	characteristic
73	#131	—	146	4	no	—
74	C₂×M₁₆	—	73	2	no	—
75	C₂×M₁₆	—	146	2	no	—
76	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
77	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
78	#117	—	73	2	no	—
79	C₄⋊C₈	—	146	4	no	—
80	#99	—	73	4	no	—
81	C₂²⋊C₈	—	2	2	no	—
82	#41	—	2	2	no	—
83	#41	—	2	2	no	—
84	C₂×Q₈	Dic₁₄₆	1	1	yes	characteristic
85	C₂×D₄	Dic₁₄₆	1	1	yes	characteristic
86	C₂×D₄	C₄×D₇₃	1	2	yes	—
87	C₂²×C₄	C₂²⋊D₇₃	1	2	yes	—
88	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
89	M₁₆	—	2	2	no	—
90	M₁₆	—	146	4	no	—
91	M₁₆	—	146	2	no	—
92	M₁₆	—	146	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	73	4	no	—
97	C₂×C₈	—	146	4	no	—
98	C₄⋊C₄	—	146	2	no	—
99	C₂²⋊C₄	—	73	4	no	—
100	C₄²	—	146	2	no	—
101	C₂³	#11924007	1	2	yes	—
102	C₂³	C₂×C₄×D₇₃	1	1	yes	characteristic
103	Q₈	—	2	2	no	—
104	D₄	—	2	4	no	—
105	D₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	146	4	no	—
109	C₂×C₄	#11924007	1	2	yes	—
110	C₂×C₄	D₄×D₇₃	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
112	C₂×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
113	C₂×C₄	#11924027	1	2	yes	—
114	C₈	—	4	1	no	—
115	C₈	—	4	1	no	—
116	C₈	—	146	4	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	4	1	no	—
119	C₂²	—	4	2	no	—
120	C₂²	C₂×#11924007	1	1	yes	characteristic
121	C₂²	#14556551	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	292	4	no	—
124	C₄	#14556559	1	1	yes	characteristic
125	C₄	#14556568	1	2	yes	—
126	C₄	#14556572	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	—	2	1	no	—
129	C₂	—	4	2	no	—
130	C₂	#17988459	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989354

C₂

yes

maximal, characteristic

#17988613

C₂

yes

maximal, complemented

#17988561

C₂

yes

maximal, characteristic

#17988359

C₂

yes

maximal, complemented

#408×C₇₃

C₂

yes

maximal, characteristic

#14556598

—

C₂×#11924012

—

C₂×#11924012

C₂²

yes

—

#14556550

C₂²

yes

—

#14556520

—

#14556506

C₄

yes

—

C₂×#41×C₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂²⋊C₈×C₇₃

—

—

—

—

—

D₄

yes

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

C₂×C₄

yes

—

C₄×Dic₇₃

—

#41×C₇₃

—

#41×C₇₃

—

C₂×Q₈×C₇₃

Q₈

yes

characteristic

C₂×D₄×C₇₃

C₂×C₄

yes

—

C₂×D₄×C₇₃

Q₈

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂³×C₇₃

C₄⋊C₄

yes

—

C₂³×C₇₃

C₂²×C₄

yes

characteristic

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₃

#41

yes

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₃

C₄×D₄

yes

—

C₄×C₇₃

—

C₄×C₇₃

#120

yes

characteristic

C₄×C₇₃

C₂²⋊Q₈

yes

—

C₄×C₇₃

#123

yes

characteristic

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

—

—

—

D₇₃

yes

characteristic

#344

—

C₂×#41

C₂×D₇₃

yes

characteristic

#131

—

146

—

C₂×M₁₆

—

C₂×M₁₆

—

146

—

C₂×M₁₆

C₂×D₇₃

yes

characteristic

C₂²×C₈

C₂×D₇₃

yes

characteristic

#117

—

C₄⋊C₈

—

146

—

#99

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

Dic₁₄₆

yes

characteristic

C₂×D₄

Dic₁₄₆

yes

characteristic

C₂×D₄

C₄×D₇₃

yes

—

C₂²×C₄

C₂²⋊D₇₃

yes

—

C₂²×C₄

C₂²×D₇₃

yes

characteristic

M₁₆

—

M₁₆

—

146

—

M₁₆

—

146

—

M₁₆

—

146

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₄⋊C₄

—

146

—

C₂²⋊C₄

—

100

C₄²

—

146

—

101

C₂³

#11924007

yes

—

102

C₂³

C₂×C₄×D₇₃

yes

characteristic

103

Q₈

—

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

146

—

109

C₂×C₄

#11924007

yes

—

110

C₂×C₄

D₄×D₇₃

yes

—

111

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

112

C₂×C₄

C₂×Dic₁₄₆

yes

characteristic

113

C₂×C₄

#11924027

yes

—

114

C₈

—

115

C₈

—

116

C₈

—

146

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×#11924007

yes

characteristic

121

C₂²

#14556551

yes

—

122

C₄

—

123

C₄

—

292

—

124

C₄

#14556559

yes

characteristic

125

C₄

#14556568

yes

—

126

C₄

#14556572

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#17988459

yes

characteristic

	subgroup	G/H
center	C₄	#14556559
commutator	C₂³×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#408×C₇₃	C₂
socle	C₂×C₇₃	#385
hypercenter	#408	D₇₃
hypocenter	C₇₃	#1868
LCS	G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₄

#14556559

commutator

C₂³×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#408×C₇₃

C₂

C₂×C₇₃

D₇₃

C₇₃

G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
2. #17988613	C₂	semidirect
4. #17988359	C₂	semidirect
66. C₇₃	#1868	semidirect

H ≅ G/N

type

2. #17988613

C₂

semidirect

4. #17988359

C₂

semidirect

66. C₇₃

#1868

semidirect

Subgroups — #22610216

Conjugacy classes of subgroups

Complements