Group #34276074 — subgroups

Subgroups — #34276074

Order 20224.

690 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 227 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	#23773986	C₂	1	4	yes	maximal
2	#23773914	C₂	1	2	yes	maximal
3	#23763077	C₂	1	8	yes	maximal, complemented
4	C₂×#322×C₇₉	C₂	1	1	yes	maximal, characteristic
5	#19099120	C₂²	1	4	yes	—
6	#19098968	C₂²	1	4	yes	—
7	#19098968	C₂²	1	4	yes	—
8	#19098965	—	2	4	no	—
9	#19098961	C₂²	1	4	yes	—
10	#19098943	C₂²	1	4	yes	—
11	#19098914	C₂²	1	8	yes	—
12	#19098804	—	2	8	no	—
13	C₄×Dic₃₁₆	—	2	8	no	—
14	#19098601	—	2	4	no	—
15	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	#322×C₇₉	C₂²	1	4	yes	—
18	C₂×Dic₃₁₆	—	2	8	no	—
19	#14579161	C₂³	1	8	yes	—
20	#14579153	—	2	4	no	—
21	#14579140	—	2	8	no	—
22	C₄×Dic₁₅₈	—	2	8	no	—
23	#14579120	—	2	4	no	—
24	#14579095	—	2	8	no	—
25	#14579095	—	2	8	no	—
26	#14579093	—	2	8	no	—
27	#14579086	—	2	16	no	—
28	C₂²×C₈×C₇₉	D₄	1	4	yes	—
29	C₂²×C₈×C₇₉	D₄	1	2	yes	—
30	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₇₉	D₄	1	4	yes	—
32	C₄×C₈×C₇₉	D₄	1	2	yes	—
33	C₄×C₈×C₇₉	C₂³	1	4	yes	—
34	C₄×C₈×C₇₉	C₂³	1	2	yes	—
35	C₂×Dic₁₅₈	—	2	8	no	—
36	Dic₃₁₆	—	4	16	no	—
37	C₄⋊Dic₇₉	—	2	8	no	—
38	C₄⋊Dic₇₉	—	2	8	no	—
39	Dic₇₉⋊C₄	—	4	8	no	—
40	C₂²⋊Dic₇₉	—	4	4	no	—
41	C₄×Dic₇₉	—	4	4	no	—
42	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₇₉	—	4	4	no	—
45	C₂×C₈×C₇₉	Q₁₆	1	8	yes	complemented
46	C₂×C₈×C₇₉	C₂×D₄	1	8	yes	—
47	C₂×C₈×C₇₉	C₂×D₄	1	4	yes	—
48	C₄²×C₇₉	C₂×D₄	1	2	yes	—
49	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
51	C₂³×C₇₉	#127	1	1	yes	characteristic
52	C₂×C₄×C₇₉	—	2	4	no	—
53	C₂×C₄×C₇₉	#127	1	2	yes	—
54	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
55	C₂×C₄×C₇₉	C₂×Q₁₆	1	4	yes	—
56	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
57	C₂×C₄×C₇₉	C₂²×D₄	1	2	yes	—
58	C₂×Dic₇₉	—	4	8	no	—
59	Dic₁₅₈	—	4	16	no	—
60	C₈×C₇₉	—	2	8	no	—
61	C₈×C₇₉	C₂×Q₁₆	1	8	yes	—
62	C₂²×C₇₉	—	2	2	no	—
63	C₂²×C₇₉	#494	1	2	yes	—
64	C₂²×C₇₉	C₂×#127	1	1	yes	characteristic
65	C₄×C₇₉	—	2	2	no	—
66	C₄×C₇₉	#494	1	2	yes	—
67	C₄×C₇₉	C₂²×Q₁₆	1	2	yes	—
68	C₄×C₇₉	#575	1	4	yes	—
69	Dic₇₉	—	8	8	no	—
70	#19618	—	79	1	no	maximal, Sylow
71	C₂×C₇₉	—	2	2	no	—
72	C₂×C₇₉	C₂×#494	1	1	yes	characteristic
73	C₂×C₇₉	#3150	1	2	yes	—
74	#3170	—	79	4	no	—
75	#3160	—	79	2	no	—
76	#1723	—	79	8	no	—
77	C₂×#322	D₇₉	1	1	yes	characteristic
78	C₇₉	#19618	1	1	yes	characteristic, complemented, Sylow
79	#533	—	79	4	no	—
80	#500	—	79	4	no	—
81	#500	—	79	4	no	—
82	#499	—	158	4	no	—
83	#498	—	79	4	no	—
84	#494	—	79	4	no	—
85	#487	—	79	8	no	—
86	#467	—	158	8	no	—
87	C₄×Q₁₆	—	158	8	no	—
88	#427	—	158	4	no	—
89	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
90	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
91	#322	C₂×D₇₉	1	4	yes	—
92	C₂×Q₁₆	—	158	8	no	—
93	C₂²×C₈	D₃₁₆	1	4	yes	—
94	C₂²×C₈	D₃₁₆	1	2	yes	—
95	C₄⋊Q₈	—	79	8	no	—
96	#125	—	158	4	no	—
97	C₂²⋊Q₈	—	158	8	no	—
98	C₄×Q₈	—	158	8	no	—
99	#117	—	158	4	no	—
100	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
101	#107	—	158	8	no	—
102	#107	—	158	8	no	—
103	#106	—	158	8	no	—
104	#103	—	158	16	no	—
105	C₄×C₈	D₃₁₆	1	4	yes	—
106	C₄×C₈	D₃₁₆	1	2	yes	—
107	C₄×C₈	C₂²×D₇₉	1	4	yes	—
108	C₄×C₈	C₂²×D₇₉	1	2	yes	—
109	C₂×Q₈	—	158	8	no	—
110	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
111	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
112	Q₁₆	—	316	16	no	—
113	C₂×C₈	—	4	4	no	—
114	C₂×C₈	Dic₃₁₆	1	8	yes	complemented
115	C₂×C₈	C₂×D₃₁₆	1	8	yes	—
116	C₂×C₈	C₂×D₃₁₆	1	4	yes	—
117	C₄⋊C₄	—	316	8	no	—
118	C₄⋊C₄	—	158	8	no	—
119	C₄⋊C₄	—	158	8	no	—
120	C₂²⋊C₄	—	316	4	no	—
121	C₄²	—	316	4	no	—
122	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
123	C₄²	C₂×D₃₁₆	1	2	yes	—
124	C₄²	C₂³×D₇₉	1	1	yes	characteristic
125	C₂³	#14579158	1	1	yes	characteristic
126	Q₈	—	316	16	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	316	8	no	—
129	C₂×C₄	#14579158	1	2	yes	—
130	C₂×C₄	#14579158	1	1	yes	characteristic
131	C₂×C₄	C₂×Dic₃₁₆	1	4	yes	—
132	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
133	C₂×C₄	C₂²×D₃₁₆	1	2	yes	—
134	C₈	—	2	8	no	—
135	C₈	C₂×Dic₃₁₆	1	8	yes	—
136	C₂²	—	2	2	no	—
137	C₂²	#19098943	1	2	yes	—
138	C₂²	C₂×#14579158	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	632	8	no	—
141	C₄	#19098943	1	2	yes	—
142	C₄	C₂²×Dic₃₁₆	1	2	yes	—
143	C₄	#19099363	1	4	yes	—
144	C₂	—	2	2	no	—
145	C₂	C₂×#19098943	1	1	yes	characteristic
146	C₂	#23773846	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23773986

C₂

yes

maximal

#23773914

C₂

yes

maximal

#23763077

C₂

yes

maximal, complemented

C₂×#322×C₇₉

C₂

yes

maximal, characteristic

#19099120

C₂²

yes

—

#19098968

C₂²

yes

—

#19098968

C₂²

yes

—

#19098965

—

#19098961

C₂²

yes

—

#19098943

C₂²

yes

—

#19098914

C₂²

yes

—

#19098804

—

C₄×Dic₃₁₆

—

#19098601

—

C₂×C₄×C₈×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#322×C₇₉

C₂²

yes

—

C₂×Dic₃₁₆

—

#14579161

C₂³

yes

—

#14579153

—

#14579140

—

C₄×Dic₁₅₈

—

—

—

—

—

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

C₂³

yes

—

C₄×C₈×C₇₉

C₂³

yes

—

C₂×Dic₁₅₈

—

Dic₃₁₆

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

Q₁₆

yes

complemented

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₄²×C₇₉

C₂×D₄

yes

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂⁴

yes

characteristic

C₂³×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#127

yes

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₁₆

yes

—

C₂×C₄×C₇₉

C₂²×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂²×D₄

yes

—

C₂×Dic₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

C₂×Q₁₆

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

#494

yes

—

C₂²×C₇₉

C₂×#127

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

#494

yes

—

C₄×C₇₉

C₂²×Q₁₆

yes

—

C₄×C₇₉

#575

yes

—

Dic₇₉

—

#19618

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

C₂×#494

yes

characteristic

C₂×C₇₉

#3150

yes

—

—

—

—

D₇₉

yes

characteristic

C₇₉

#19618

yes

characteristic, complemented, Sylow

—

—

—

—

158

—

—

—

—

—

158

—

C₄×Q₁₆

—

158

—

#427

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

C₂×C₄×C₈

C₂×D₇₉

yes

—

#322

C₂×D₇₉

yes

—

C₂×Q₁₆

—

158

—

C₂²×C₈

D₃₁₆

yes

—

C₂²×C₈

D₃₁₆

yes

—

C₄⋊Q₈

—

#125

—

158

—

C₂²⋊Q₈

—

158

—

C₄×Q₈

—

158

—

#117

—

158

—

100

C₂×C₄²

C₂²×D₇₉

yes

characteristic

101

#107

—

158

—

102

#107

—

158

—

103

#106

—

158

—

104

#103

—

158

—

105

C₄×C₈

D₃₁₆

yes

—

106

C₄×C₈

D₃₁₆

yes

—

107

C₄×C₈

C₂²×D₇₉

yes

—

108

C₄×C₈

C₂²×D₇₉

yes

—

109

C₂×Q₈

—

158

—

110

C₂²×C₄

C₂×D₃₁₆

yes

—

111

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

112

Q₁₆

—

316

—

113

C₂×C₈

—

114

C₂×C₈

Dic₃₁₆

yes

complemented

115

C₂×C₈

C₂×D₃₁₆

yes

—

116

C₂×C₈

C₂×D₃₁₆

yes

—

117

C₄⋊C₄

—

316

—

118

C₄⋊C₄

—

158

—

119

C₄⋊C₄

—

158

—

120

C₂²⋊C₄

—

316

—

121

C₄²

—

316

—

122

C₄²

C₂×D₃₁₆

yes

characteristic

123

C₄²

C₂×D₃₁₆

yes

—

124

C₄²

C₂³×D₇₉

yes

characteristic

125

C₂³

#14579158

yes

characteristic

126

Q₈

—

316

—

127

C₂×C₄

—

128

C₂×C₄

—

316

—

129

C₂×C₄

#14579158

yes

—

130

C₂×C₄

#14579158

yes

characteristic

131

C₂×C₄

C₂×Dic₃₁₆

yes

—

132

C₂×C₄

C₂²×D₃₁₆

yes

characteristic

133

C₂×C₄

C₂²×D₃₁₆

yes

—

134

C₈

—

135

C₈

C₂×Dic₃₁₆

yes

—

136

C₂²

—

137

C₂²

#19098943

yes

—

138

C₂²

C₂×#14579158

yes

characteristic

139

C₄

—

140

C₄

—

632

—

141

C₄

#19098943

yes

—

142

C₄

C₂²×Dic₃₁₆

yes

—

143

C₄

#19099363

yes

—

144

C₂

—

145

C₂

C₂×#19098943

yes

characteristic

146

C₂

#23773846

yes

—

	subgroup	G/H
center	C₂×C₄	#14579158
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₂×#322×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#127
prime radical	C₂³×C₇₉	#127
hypercenter	C₂×#322	D₇₉
hypocenter	C₇₉	#19618
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#14579158

commutator, cosocle

C₄²×C₇₉

C₂⁴

Frattini

C₄²

C₂³×D₇₉

Fitting

C₂×#322×C₇₉

C₂

C₂²×C₇₉

C₂³×C₇₉

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
3. #23763077	C₂	semidirect
45. C₂×C₈×C₇₉	Q₁₆	semidirect
78. C₇₉	#19618	semidirect
114. C₂×C₈	Dic₃₁₆	semidirect

H ≅ G/N

type

3. #23763077

C₂

semidirect

45. C₂×C₈×C₇₉

Q₁₆

semidirect

78. C₇₉

#19618

semidirect

114. C₂×C₈

Dic₃₁₆

semidirect

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

subgroup	index
#23773986	2
#23773914	2
#23763077	2
C₂×#322×C₇₉	2
#19618	79

subgroup

index

C₂×#322×C₇₉

Subgroups — #34276074

Conjugacy classes of subgroups

Complements

Maximal subgroups