Group #34209231 — subgroups

Subgroups — #34209231

Order 20224.

160 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 55 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	#23765195	C₂	1	1	yes	maximal, characteristic
2	#23761605	C₂	1	2	yes	maximal, complemented
3	#23761593	C₂	1	2	yes	maximal, complemented
4	#23761540	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#2219	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579097	—	2	1	no	—
7	#19098397	C₂²	1	2	yes	—
8	#19098393	—	2	1	no	—
9	#19098393	C₂²	1	2	yes	—
10	C₂×Q₃₂×C₇₉	C₂²	1	1	yes	characteristic
11	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
12	#467×C₇₉	—	2	1	no	—
13	#367×C₇₉	C₂²	1	1	yes	characteristic
14	#358×C₇₉	—	2	1	no	—
15	C₂×#11929608	—	4	1	no	—
16	#14579097	—	2	2	no	—
17	#14579097	—	4	1	no	—
18	C₇₉⋊C₃₂	—	2	2	no	—
19	C₂×Q₁₆×C₇₉	—	2	1	no	—
20	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
21	C₂²⋊Q₈×C₇₉	—	4	1	no	—
22	Q₃₂×C₇₉	—	2	2	no	—
23	M₃₂×C₇₉	—	2	2	no	—
24	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
26	#107×C₇₉	—	2	1	no	—
27	#103×C₇₉	—	4	1	no	—
28	#11929608	—	4	2	no	—
29	#11929608	—	8	1	no	—
30	C₂×C₇₉⋊C₈	—	8	1	no	—
31	C₂×Q₈×C₇₉	—	4	1	no	—
32	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
33	Q₁₆×C₇₉	—	4	1	no	—
34	Q₁₆×C₇₉	—	4	1	no	—
35	C₂×C₈×C₇₉	—	2	1	no	—
36	C₂×C₈×C₇₉	—	2	1	no	—
37	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
38	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
39	C₄⋊C₄×C₇₉	—	4	1	no	—
40	C₄⋊C₄×C₇₉	—	8	1	no	—
41	C₂²⋊C₄×C₇₉	—	8	1	no	—
42	C₁₆×C₇₉	—	2	1	no	—
43	C₁₆×C₇₉	#41	1	2	yes	—
44	C₂³×C₇₉	SD₃₂	1	1	yes	characteristic
45	Q₈×C₇₉	—	8	1	no	—
46	Q₈×C₇₉	—	8	1	no	—
47	C₂×C₄×C₇₉	—	2	1	no	—
48	C₂×C₄×C₇₉	—	2	1	no	—
49	C₂×C₄×C₇₉	—	8	1	no	—
50	C₂×C₄×C₇₉	—	8	1	no	—
51	C₂×C₄×C₇₉	SD₃₂	1	1	yes	characteristic
52	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
53	C₈×C₇₉	—	2	1	no	—
54	C₈×C₇₉	#123	1	1	yes	characteristic
55	C₈×C₇₉	#137	1	1	yes	characteristic
56	C₇₉⋊C₈	—	8	2	no	—
57	C₂²×C₇₉	—	2	1	no	—
58	C₂²×C₇₉	—	4	1	no	—
59	C₂²×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
60	C₄×C₇₉	—	2	1	no	—
61	C₄×C₇₉	—	16	1	no	—
62	C₄×C₇₉	—	16	1	no	—
63	C₄×C₇₉	#480	1	1	yes	characteristic
64	C₄×C₇₉	#509	1	1	yes	characteristic
65	#13718	—	79	1	no	maximal, Sylow
66	C₂×C₇₉	—	2	1	no	—
67	C₂×C₇₉	—	4	1	no	—
68	C₂×C₇₉	#2233	1	1	yes	characteristic
69	#2219	D₇₉	1	1	yes	characteristic
70	#2161	—	79	1	no	—
71	#1426	—	79	2	no	—
72	#1420	—	79	2	no	—
73	#1400	—	79	1	no	—
74	C₇₉	#13718	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₃₂	C₂×D₇₉	1	1	yes	characteristic
76	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
77	#467	—	2	1	no	—
78	C₂×#108	—	158	1	no	—
79	M₆₄	—	79	2	no	—
80	#368	—	79	2	no	—
81	#368	—	158	1	no	—
82	#367	C₂×D₇₉	1	1	yes	characteristic
83	#358	—	2	1	no	—
84	C₂×Q₁₆	—	2	1	no	—
85	C₂×M₁₆	—	316	1	no	—
86	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
87	C₂²⋊Q₈	—	4	1	no	—
88	Q₃₂	—	2	2	no	—
89	M₃₂	—	2	2	no	—
90	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
91	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
92	#108	—	316	1	no	—
93	#108	—	158	2	no	—
94	#107	—	2	1	no	—
95	#103	—	4	1	no	—
96	C₃₂	—	158	2	no	—
97	C₂×Q₈	—	4	1	no	—
98	C₂²×C₄	#11929609	1	1	yes	characteristic
99	Q₁₆	—	4	1	no	—
100	Q₁₆	—	4	1	no	—
101	M₁₆	—	316	2	no	—
102	M₁₆	—	632	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	632	1	no	—
106	C₂×C₈	#11929609	1	1	yes	characteristic
107	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
108	C₄⋊C₄	—	4	1	no	—
109	C₄⋊C₄	—	8	1	no	—
110	C₂²⋊C₄	—	8	1	no	—
111	C₁₆	—	2	1	no	—
112	C₁₆	#11929623	1	2	yes	—
113	C₂³	#14579105	1	1	yes	characteristic
114	Q₈	—	8	1	no	—
115	Q₈	—	8	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	8	1	no	—
119	C₂×C₄	—	8	1	no	—
120	C₂×C₄	#14579105	1	1	yes	characteristic
121	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
122	C₈	—	2	1	no	—
123	C₈	—	632	2	no	—
124	C₈	#14579142	1	1	yes	characteristic
125	C₈	#14579191	1	1	yes	characteristic
126	C₂²	—	2	1	no	—
127	C₂²	—	4	1	no	—
128	C₂²	C₂×#14579105	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	16	1	no	—
131	C₄	—	16	1	no	—
132	C₄	#19098874	1	1	yes	characteristic
133	C₄	#19098998	1	1	yes	characteristic
134	C₂	—	2	1	no	—
135	C₂	—	4	1	no	—
136	C₂	#23765583	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765195

C₂

yes

maximal, characteristic

#23761605

C₂

yes

maximal, complemented

#23761593

C₂

yes

maximal, complemented

#23761540

C₂

yes

maximal, characteristic

C₇₉×#2219

C₂

yes

maximal, characteristic

C₂×#14579097

—

#19098397

C₂²

yes

—

#19098393

—

#19098393

C₂²

yes

—

C₂×Q₃₂×C₇₉

C₂²

yes

characteristic

C₂×M₃₂×C₇₉

C₂²

yes

characteristic

#467×C₇₉

—

#367×C₇₉

C₂²

yes

characteristic

#358×C₇₉

—

C₂×#11929608

—

#14579097

—

#14579097

—

C₇₉⋊C₃₂

—

C₂×Q₁₆×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²⋊Q₈×C₇₉

—

Q₃₂×C₇₉

—

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂³

yes

characteristic

#107×C₇₉

—

#103×C₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

D₈

yes

characteristic

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₁₆×C₇₉

—

C₁₆×C₇₉

#41

yes

—

C₂³×C₇₉

SD₃₂

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

SD₃₂

yes

characteristic

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

yes

C₈×C₇₉

yes

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×SD₃₂

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#2233

yes

characteristic

#2219

D₇₉

yes

—

—

—

—

C₇₉

#13718

yes

characteristic, complemented, Sylow

C₂×Q₃₂

C₂×D₇₉

yes

characteristic

C₂×M₃₂

C₂×D₇₉

yes

characteristic

#467

—

C₂×#108

—

158

—

M₆₄

—

#368

—

#368

—

158

—

#367

C₂×D₇₉

yes

characteristic

#358

—

C₂×Q₁₆

—

C₂×M₁₆

—

316

—

C₂²×C₈

C₂²⋊D₇₉

yes

characteristic

C₂²⋊Q₈

—

Q₃₂

—

M₃₂

—

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

C₂×C₁₆

C₂²×D₇₉

yes

characteristic

#108

—

316

—

#108

—

158

—

#107

—

#103

—

C₃₂

—

158

—

C₂×Q₈

—

C₂²×C₄

#11929609

yes

characteristic

Q₁₆

—

100

Q₁₆

—

101

M₁₆

—

316

—

102

M₁₆

—

632

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

632

—

106

C₂×C₈

#11929609

yes

characteristic

107

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₂²⋊C₄

—

111

C₁₆

—

112

C₁₆

#11929623

yes

—

113

C₂³

#14579105

yes

characteristic

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

#14579105

yes

characteristic

121

C₂×C₄

C₂×#11929609

yes

characteristic

122

C₈

—

123

C₈

—

632

—

124

C₈

#14579142

yes

characteristic

125

C₈

#14579191

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

C₂×#14579105

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#19098874

yes

characteristic

133

C₄

#19098998

yes

characteristic

134

C₂

—

135

C₂

—

136

C₂

#23765583

yes

characteristic

	subgroup	G/H
center	C₂	#23765583
commutator, cosocle	C₂×C₁₆×C₇₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting	C₇₉×#2219	C₂
socle	C₂×C₇₉	#2233
prime radical	C₂³×C₇₉	SD₃₂
hypercenter	#2219	D₇₉
hypocenter	C₇₉	#13718
LCS	G ▷ C₂×C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂ ◁ #2219

subgroup

G/H

center

C₂

#23765583

commutator, cosocle

C₂×C₁₆×C₇₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₉

Fitting

C₇₉×#2219

C₂

socle

C₂×C₇₉

#2233

prime radical

C₂³×C₇₉

SD₃₂

D₇₉

C₇₉

G ▷ C₂×C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂ ◁ #2219

N	H ≅ G/N	type
2. #23761605	C₂	semidirect
3. #23761593	C₂	semidirect
74. C₇₉	#13718	semidirect

H ≅ G/N

type

2. #23761605

C₂

semidirect

3. #23761593

C₂

semidirect

74. C₇₉

#13718

semidirect

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

subgroup	index
#23765195	2
#23761605	2
#23761593	2
#23761540	2
C₇₉×#2219	2
#13718	79

subgroup

index

C₇₉×#2219

Subgroups — #34209231

Conjugacy classes of subgroups

Complements

Maximal subgroups