Group #34034429 — subgroups

Subgroups — #34034429

Order 18688.

206 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 95 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×#17989696	C₂	1	1	yes	maximal, characteristic
2	C₂×#17989694	C₂	1	1	yes	maximal, characteristic
3	#22627030	C₂	1	2	yes	maximal, complemented
4	C₄⋊#14556681	C₂	1	2	yes	maximal, complemented
5	#22626141	C₂	1	1	yes	maximal, characteristic
6	#17989696	—	2	2	no	—
7	#17989694	—	2	2	no	—
8	C₂×#14556681	C₂²	1	2	yes	—
9	C₂×#14556681	C₂²	1	2	yes	—
10	C₂×#14556666	C₄	1	1	yes	characteristic
11	C₂×#14556664	C₂²	1	1	yes	characteristic
12	#17989476	C₄	1	1	yes	characteristic
13	#17989475	C₄	1	1	yes	characteristic
14	#17989473	C₂²	1	1	yes	characteristic
15	C₄×#11924031	C₂²	1	1	yes	characteristic
16	#17988544	C₄	1	1	yes	characteristic
17	#14556681	—	2	2	no	—
18	#14556681	—	2	2	no	—
19	#14556666	—	2	2	no	—
20	#14556664	D₄	1	2	yes	—
21	#14556664	Q₈	1	2	yes	—
22	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
23	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
24	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
25	C₂×#11924031	C₂³	1	1	yes	characteristic
26	C₂×#11924011	C₈	1	1	yes	characteristic
27	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
28	#14556548	C₈	1	1	yes	characteristic
29	#14556501	C₈	1	1	yes	characteristic
30	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
32	#97×C₇₃	C₈	1	1	yes	characteristic
33	#11924031	—	2	1	no	—
34	#11924031	—	2	1	no	—
35	#11924031	C₄⋊C₄	1	2	yes	—
36	#11924031	C₂×D₄	1	1	yes	characteristic
37	#11924031	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
39	#11924011	—	2	2	no	—
40	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
41	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
42	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
43	C₈×D₇₃	M₁₆	1	2	yes	—
44	#11924007	C₁₆	1	2	yes	—
45	#11924005	C₁₆	1	2	yes	—
46	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
47	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×D₇₃	C₂×C₁₆	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
52	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
53	C₂×Dic₇₃	C₂×C₁₆	1	2	yes	—
54	C₂×Dic₇₃	C₂×C₁₆	1	1	yes	characteristic
55	C₄×D₇₃	C₄⋊C₈	1	2	yes	—
56	C₄×D₇₃	M₃₂	1	2	yes	—
57	C₈×C₇₃	—	2	1	no	—
58	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
59	#10596831	—	2	1	no	—
60	#10596831	C₂×C₄⋊C₄	1	1	yes	characteristic
61	#10596831	C₂×M₁₆	1	1	yes	characteristic
62	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
63	C₂×D₇₃	—	2	2	no	—
64	C₂×D₇₃	C₄⋊C₁₆	1	2	yes	—
65	C₄×C₇₃	—	2	2	no	—
66	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
67	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
68	Dic₇₃	—	2	2	no	—
69	Dic₇₃	C₄⋊C₁₆	1	2	yes	—
70	#13628	—	73	1	no	maximal, Sylow
71	C₂×C₇₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
72	C₂×C₇₃	#2258	1	2	yes	—
73	D₇₃	—	2	2	no	—
74	C₂×M₆₄	—	73	1	no	—
75	C₂²×C₃₂	—	73	1	no	—
76	#2108	—	73	1	no	—
77	C₄⋊C₃₂	—	73	2	no	—
78	C₄⋊C₃₂	—	73	2	no	—
79	C₇₃	#13628	1	1	yes	characteristic, complemented, Sylow
80	C₂×M₃₂	—	73	1	no	—
81	C₂²×C₁₆	—	73	1	no	—
82	#405	—	73	1	no	—
83	M₆₄	—	146	2	no	—
84	C₂×C₃₂	—	73	2	no	—
85	C₂×C₃₂	—	73	2	no	—
86	C₂×C₃₂	—	146	2	no	—
87	#346	—	73	1	no	—
88	#346	—	73	1	no	—
89	C₄×C₁₆	—	73	1	no	—
90	C₄×C₁₆	—	73	1	no	—
91	C₂×M₁₆	—	73	1	no	—
92	C₂²×C₈	—	73	1	no	—
93	#117	—	73	1	no	—
94	M₃₂	—	146	2	no	—
95	C₂×C₁₆	—	73	1	no	—
96	C₂×C₁₆	—	73	1	no	—
97	C₂×C₁₆	—	73	2	no	—
98	C₂×C₁₆	—	73	2	no	—
99	C₂×C₁₆	—	73	1	no	—
100	C₂×C₁₆	—	73	1	no	—
101	#97	—	73	1	no	—
102	#97	#10596842	1	1	yes	characteristic
103	C₄×C₈	—	73	1	no	—
104	C₄×C₈	—	73	1	no	—
105	C₃₂	—	146	2	no	—
106	C₃₂	—	146	2	no	—
107	C₂²×C₄	—	73	1	no	—
108	M₁₆	—	146	2	no	—
109	C₂×C₈	—	73	1	no	—
110	C₂×C₈	—	73	2	no	—
111	C₂×C₈	—	73	2	no	—
112	C₂×C₈	—	73	1	no	—
113	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
114	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
115	C₄⋊C₄	—	73	2	no	—
116	C₂²⋊C₄	—	73	2	no	—
117	C₄²	—	73	1	no	—
118	C₄²	C₂×#10596842	1	1	yes	characteristic
119	C₁₆	—	73	2	no	—
120	C₁₆	—	73	1	no	—
121	C₁₆	—	73	1	no	—
122	C₁₆	—	146	1	no	—
123	C₁₆	—	146	1	no	—
124	C₂³	—	73	1	no	—
125	C₂×C₄	—	73	1	no	—
126	C₂×C₄	—	73	2	no	—
127	C₂×C₄	—	73	2	no	—
128	C₂×C₄	—	73	2	no	—
129	C₂×C₄	#14556686	1	2	yes	—
130	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
131	C₈	—	2	1	no	—
132	C₈	—	73	1	no	—
133	C₈	—	73	1	no	—
134	C₈	—	146	1	no	—
135	C₈	C₄⋊#10596842	1	2	yes	—
136	C₂²	—	73	2	no	—
137	C₂²	—	146	2	no	—
138	C₂²	C₂×#14556686	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	73	2	no	—
141	C₄	—	146	2	no	—
142	C₄	C₂×C₄⋊#10596842	1	1	yes	characteristic
143	C₄	#17989706	1	1	yes	characteristic
144	C₂	—	146	2	no	—
145	C₂	#22627117	1	1	yes	characteristic
146	C₂	#22627165	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989696

C₂

yes

maximal, characteristic

C₂×#17989694

C₂

yes

maximal, characteristic

#22627030

C₂

yes

maximal, complemented

C₄⋊#14556681

C₂

yes

maximal, complemented

#22626141

C₂

yes

maximal, characteristic

#17989696

—

#17989694

—

C₂×#14556681

C₂²

yes

—

C₂×#14556681

C₂²

yes

—

C₂×#14556666

C₄

yes

characteristic

C₂×#14556664

C₂²

yes

characteristic

#17989476

C₄

yes

characteristic

#17989475

C₄

yes

characteristic

#17989473

C₂²

yes

characteristic

C₄×#11924031

C₂²

yes

characteristic

#17988544

C₄

yes

—

—

—

D₄

yes

—

#14556664

Q₈

yes

—

C₂×#11924031

C₂×C₄

yes

characteristic

C₂×#11924031

C₂×C₄

yes

characteristic

C₂×#11924031

C₂×C₄

yes

characteristic

C₂×#11924031

C₂³

yes

characteristic

C₂×#11924011

C₈

yes

characteristic

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

#14556548

C₈

yes

characteristic

#14556501

C₈

yes

characteristic

C₄×#10596831

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

#97×C₇₃

C₈

yes

—

—

C₄⋊C₄

yes

—

#11924031

C₂×D₄

yes

characteristic

#11924031

C₂×Q₈

yes

characteristic

C₂×C₄×D₇₃

C₂×C₈

yes

characteristic

#11924011

—

C₂×#10596831

C₂×C₈

yes

characteristic

C₂×#10596831

C₂²×C₄

yes

characteristic

C₈×D₇₃

C₄⋊C₄

yes

—

C₈×D₇₃

M₁₆

yes

—

#11924007

C₁₆

yes

—

#11924005

C₁₆

yes

—

C₄×Dic₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₂²×D₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×Dic₇₃

C₂×C₁₆

yes

—

C₂×Dic₇₃

C₂×C₁₆

yes

characteristic

C₄×D₇₃

C₄⋊C₈

yes

—

C₄×D₇₃

M₃₂

yes

—

C₈×C₇₃

—

C₈×C₇₃

C₄⋊C₈

yes

—

#10596831

—

#10596831

C₂×C₄⋊C₄

yes

characteristic

#10596831

C₂×M₁₆

yes

characteristic

C₂²×C₇₃

C₂²×C₁₆

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

C₄⋊C₁₆

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₇₃

C₂×M₃₂

yes

characteristic

Dic₇₃

—

Dic₇₃

C₄⋊C₁₆

yes

—

#13628

—

maximal, Sylow

C₂×C₇₃

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₇₃

#2258

yes

—

D₇₃

—

C₂×M₆₄

—

C₂²×C₃₂

—

#2108

—

C₄⋊C₃₂

—

C₄⋊C₃₂

—

C₇₃

#13628

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

M₆₄

—

146

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

146

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

#97

—

102

#97

#10596842

yes

characteristic

103

C₄×C₈

—

104

C₄×C₈

—

105

C₃₂

—

146

—

106

C₃₂

—

146

—

107

C₂²×C₄

—

108

M₁₆

—

146

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₂×#10596842

yes

characteristic

114

C₂×C₈

C₂×#10596842

yes

characteristic

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

C₂×#10596842

yes

characteristic

119

C₁₆

—

120

C₁₆

—

121

C₁₆

—

122

C₁₆

—

146

—

123

C₁₆

—

146

—

124

C₂³

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#14556686

yes

—

130

C₂×C₄

C₂²×#10596842

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

146

—

135

C₈

C₄⋊#10596842

yes

—

136

C₂²

—

137

C₂²

—

146

—

138

C₂²

C₂×#14556686

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

146

—

142

C₄

C₂×C₄⋊#10596842

yes

characteristic

143

C₄

#17989706

yes

characteristic

144

C₂

—

146

—

145

C₂

#22627117

yes

characteristic

146

C₂

#22627165

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10596842
commutator, socle	C₂²×C₇₃	C₂²×C₁₆
Fitting	#97×C₇₃	C₈
prime radical	C₂²×D₇₃	C₂×C₁₆
cosocle	C₂×#11924031	C₂³
hypercenter	#97	#10596842
hypocenter	C₇₃	#13628
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #97

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10596842

commutator, socle

C₂²×C₇₃

C₂²×C₁₆

Fitting

#97×C₇₃

C₈

prime radical

C₂²×D₇₃

C₂×C₁₆

C₂³

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ #97

N	H ≅ G/N	type
3. #22627030	C₂	semidirect
4. C₄⋊#14556681	C₂	semidirect
79. C₇₃	#13628	semidirect

H ≅ G/N

type

3. #22627030

C₂

semidirect

4. C₄⋊#14556681

C₂

semidirect

79. C₇₃

#13628

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#17989696	2
C₂×#17989694	2
#22627030	2
C₄⋊#14556681	2
#22626141	2
#13628	73

subgroup

index

Subgroups — #34034429

Conjugacy classes of subgroups

Complements

Maximal subgroups