Group #34200030 — subgroups

Subgroups — #34200030

Order 20224.

212 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 91 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	#23765237	C₂	1	2	yes	maximal, complemented
2	#23761476	C₂	1	1	yes	maximal, characteristic, complemented
3	#23761373	C₂	1	2	yes	maximal, complemented
4	#23761316	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#2178	C₂	1	1	yes	maximal, characteristic
6	#19098980	—	2	2	no	—
7	#19098384	C₂²	1	2	yes	—
8	#19098347	—	2	2	no	—
9	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
11	#508×C₇₉	C₄	1	1	yes	characteristic
12	#443×C₇₉	C₂²	1	2	yes	—
13	#368×C₇₉	C₄	1	1	yes	characteristic
14	#359×C₇₉	C₄	1	2	yes	—
15	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
16	#14579101	—	2	2	no	—
17	#14579101	—	4	2	no	—
18	C₂×C₇₉⋊C₁₆	—	4	2	no	—
19	C₂×C₇₉⋊C₁₆	D₄	1	1	yes	characteristic
20	C₂×C₇₉⋊C₁₆	D₄	1	1	yes	characteristic
21	C₄×C₇₉⋊C₈	—	2	1	no	—
22	#135×C₇₉	C₂×C₄	1	2	yes	—
23	#131×C₇₉	—	2	2	no	—
24	Q₃₂×C₇₉	C₈	1	1	yes	characteristic, complemented
25	SD₃₂×C₇₉	C₈	1	2	yes	complemented
26	D₁₆×C₇₉	C₈	1	1	yes	characteristic, complemented
27	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
29	#108×C₇₉	C₂×C₄	1	2	yes	—
30	#104×C₇₉	—	2	2	no	—
31	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
32	C₂×C₇₉⋊C₈	—	2	2	no	—
33	C₇₉⋊C₁₆	—	2	1	no	—
34	C₇₉⋊C₁₆	—	4	2	no	—
35	C₇₉⋊C₁₆	D₈	1	2	yes	—
36	#41×C₇₉	—	2	2	no	—
37	Q₁₆×C₇₉	C₂×C₈	1	2	yes	—
38	SD₁₆×C₇₉	—	2	2	no	—
39	D₈×C₇₉	C₂×C₈	1	2	yes	—
40	M₁₆×C₇₉	—	2	2	no	—
41	M₁₆×C₇₉	—	4	2	no	—
42	C₂×C₈×C₇₉	—	4	2	no	—
43	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₄²×C₇₉	#41	1	1	yes	characteristic
46	C₁₆×C₇₉	—	2	1	no	—
47	C₁₆×C₇₉	C₂×C₈	1	1	yes	characteristic
48	C₁₆×C₇₉	C₂×C₈	1	1	yes	characteristic
49	Q₈×C₇₉	—	2	2	no	—
50	D₄×C₇₉	—	2	2	no	—
51	D₄×C₇₉	—	4	2	no	—
52	C₂×C₄×C₇₉	—	2	1	no	—
53	C₂×C₄×C₇₉	—	4	2	no	—
54	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
55	C₈×C₇₉	—	4	2	no	—
56	C₈×C₇₉	C₂²×C₈	1	1	yes	characteristic
57	C₈×C₇₉	#131	1	1	yes	characteristic
58	C₈×C₇₉	C₂×D₈	1	1	yes	characteristic
59	C₈×C₇₉	#135	1	1	yes	characteristic
60	C₇₉⋊C₈	—	4	2	no	—
61	C₂²×C₇₉	—	4	2	no	—
62	C₂²×C₇₉	#443	1	1	yes	characteristic
63	C₄×C₇₉	—	2	2	no	—
64	C₄×C₇₉	—	4	2	no	—
65	C₄×C₇₉	C₈×D₄	1	1	yes	characteristic
66	C₄×C₇₉	C₄×D₈	1	1	yes	characteristic
67	#12047	—	79	1	no	maximal, Sylow
68	C₂×C₇₉	—	2	1	no	—
69	C₂×C₇₉	—	8	2	no	—
70	C₂×C₇₉	C₈×D₈	1	1	yes	characteristic
71	#2178	D₇₉	1	1	yes	characteristic
72	#2170	—	79	2	no	—
73	#1373	—	79	1	no	—
74	#1335	—	79	2	no	—
75	#1311	—	79	1	no	—
76	C₇₉	#12047	1	1	yes	characteristic, complemented, Sylow
77	#508	Dic₇₉	1	1	yes	characteristic
78	#504	—	158	2	no	—
79	#443	C₂×D₇₉	1	2	yes	—
80	#368	Dic₇₉	1	1	yes	characteristic
81	#364	—	79	2	no	—
82	#359	Dic₇₉	1	2	yes	—
83	#350	—	158	2	no	—
84	C₄×C₁₆	—	79	1	no	—
85	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
86	C₈²	—	79	1	no	—
87	#135	C₂×Dic₇₉	1	2	yes	—
88	#131	—	2	2	no	—
89	Q₃₂	C₇₉⋊C₈	1	1	yes	characteristic, complemented
90	SD₃₂	C₇₉⋊C₈	1	2	yes	complemented
91	D₁₆	C₇₉⋊C₈	1	1	yes	characteristic, complemented
92	M₃₂	—	316	2	no	—
93	M₃₂	—	158	2	no	—
94	C₂×C₁₆	—	316	2	no	—
95	C₂×C₁₆	—	79	1	no	—
96	C₂×C₁₆	—	79	1	no	—
97	C₂×C₁₆	C₂×Dic₇₉	1	1	yes	characteristic
98	C₂×C₁₆	C₂×Dic₇₉	1	1	yes	characteristic
99	#108	C₂×Dic₇₉	1	2	yes	—
100	#104	—	2	2	no	—
101	C₄×C₈	—	158	1	no	—
102	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
103	#41	—	2	2	no	—
104	Q₁₆	C₂×C₇₉⋊C₈	1	2	yes	—
105	SD₁₆	—	2	2	no	—
106	D₈	C₂×C₇₉⋊C₈	1	2	yes	—
107	M₁₆	—	2	2	no	—
108	M₁₆	—	4	2	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	—	158	2	no	—
111	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
112	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
113	C₄²	#11929623	1	1	yes	characteristic
114	C₁₆	—	2	1	no	—
115	C₁₆	—	316	2	no	—
116	C₁₆	—	79	2	no	—
117	C₁₆	—	158	1	no	—
118	C₁₆	C₂×C₇₉⋊C₈	1	1	yes	characteristic
119	C₁₆	C₂×C₇₉⋊C₈	1	1	yes	characteristic
120	Q₈	—	2	2	no	—
121	D₄	—	2	2	no	—
122	D₄	—	4	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
126	C₈	—	4	2	no	—
127	C₈	—	316	2	no	—
128	C₈	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
129	C₈	#14579168	1	1	yes	characteristic
130	C₈	D₈×D₇₉	1	1	yes	characteristic
131	C₈	#14579180	1	1	yes	characteristic
132	C₂²	—	4	2	no	—
133	C₂²	#19098680	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	4	2	no	—
136	C₄	D₄×C₇₉⋊C₈	1	1	yes	characteristic
137	C₄	D₈×Dic₇₉	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	8	2	no	—
140	C₂	D₈×C₇₉⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765237

C₂

yes

maximal, complemented

#23761476

C₂

yes

maximal, characteristic, complemented

#23761373

C₂

yes

maximal, complemented

#23761316

C₂

yes

maximal, characteristic, complemented

C₇₉×#2178

C₂

yes

maximal, characteristic

#19098980

—

#19098384

C₂²

yes

—

#19098347

—

C₄×C₇₉⋊C₁₆

C₂²

yes

characteristic

C₈×C₇₉⋊C₈

C₂²

yes

characteristic

#508×C₇₉

C₄

yes

characteristic

#443×C₇₉

C₂²

yes

—

#368×C₇₉

C₄

yes

characteristic

#359×C₇₉

C₄

yes

—

C₄×C₁₆×C₇₉

C₂²

yes

characteristic

#14579101

—

#14579101

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

D₄

yes

characteristic

C₂×C₇₉⋊C₁₆

D₄

yes

characteristic

C₄×C₇₉⋊C₈

—

#135×C₇₉

C₂×C₄

yes

—

#131×C₇₉

—

Q₃₂×C₇₉

C₈

yes

characteristic, complemented

SD₃₂×C₇₉

C₈

yes

complemented

D₁₆×C₇₉

C₈

yes

characteristic, complemented

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₉

C₂×C₄

yes

characteristic

#108×C₇₉

C₂×C₄

yes

—

#104×C₇₉

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₇₉⋊C₈

—

C₇₉⋊C₁₆

—

C₇₉⋊C₁₆

—

C₇₉⋊C₁₆

D₈

yes

—

#41×C₇₉

—

Q₁₆×C₇₉

C₂×C₈

yes

—

SD₁₆×C₇₉

—

D₈×C₇₉

C₂×C₈

yes

—

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

#41

yes

characteristic

C₁₆×C₇₉

—

C₁₆×C₇₉

C₂×C₈

yes

characteristic

C₁₆×C₇₉

C₂×C₈

yes

characteristic

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄×D₄

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

C₂²×C₈

yes

characteristic

C₈×C₇₉

#131

yes

characteristic

C₈×C₇₉

C₂×D₈

yes

characteristic

C₈×C₇₉

#135

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#443

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₈×D₄

yes

characteristic

C₄×C₇₉

C₄×D₈

yes

characteristic

#12047

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

C₈×D₈

yes

characteristic

#2178

D₇₉

yes

—

—

—

—

C₇₉

#12047

yes

characteristic, complemented, Sylow

#508

Dic₇₉

yes

characteristic

#504

—

158

—

#443

C₂×D₇₉

yes

—

#368

Dic₇₉

yes

characteristic

#364

—

#359

Dic₇₉

yes

—

#350

—

158

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₉

yes

characteristic

C₈²

—

#135

C₂×Dic₇₉

yes

—

#131

—

Q₃₂

C₇₉⋊C₈

yes

characteristic, complemented

SD₃₂

C₇₉⋊C₈

yes

complemented

D₁₆

C₇₉⋊C₈

yes

characteristic, complemented

M₃₂

—

316

—

M₃₂

—

158

—

C₂×C₁₆

—

316

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₇₉

yes

characteristic

C₂×C₁₆

C₂×Dic₇₉

yes

characteristic

#108

C₂×Dic₇₉

yes

—

100

#104

—

101

C₄×C₈

—

158

—

102

C₄×C₈

C₂²×D₇₉

yes

characteristic

103

#41

—

104

Q₁₆

C₂×C₇₉⋊C₈

yes

—

105

SD₁₆

—

106

D₈

C₂×C₇₉⋊C₈

yes

—

107

M₁₆

—

108

M₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

158

—

111

C₂×C₈

C₂²×Dic₇₉

yes

characteristic

112

C₂×C₈

D₄×D₇₉

yes

characteristic

113

C₄²

#11929623

yes

characteristic

114

C₁₆

—

115

C₁₆

—

316

—

116

C₁₆

—

117

C₁₆

—

158

—

118

C₁₆

C₂×C₇₉⋊C₈

yes

characteristic

119

C₁₆

C₂×C₇₉⋊C₈

yes

characteristic

120

Q₈

—

121

D₄

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

D₄×Dic₇₉

yes

characteristic

126

C₈

—

127

C₈

—

316

—

128

C₈

C₂²×C₇₉⋊C₈

yes

characteristic

129

C₈

#14579168

yes

characteristic

130

C₈

D₈×D₇₉

yes

characteristic

131

C₈

#14579180

yes

characteristic

132

C₂²

—

133

C₂²

#19098680

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

D₄×C₇₉⋊C₈

yes

characteristic

137

C₄

D₈×Dic₇₉

yes

characteristic

138

C₂

—

139

C₂

—

140

C₂

D₈×C₇₉⋊C₈

yes

characteristic

	subgroup	G/H
center	C₈	D₈×D₇₉
commutator	C₈×C₇₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#2178	C₂
socle	C₂×C₇₉	C₈×D₈
prime radical	#508×C₇₉	C₄
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	#2178	D₇₉
hypocenter	C₇₉	#12047
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2178

subgroup

G/H

center

C₈

D₈×D₇₉

commutator

C₈×C₇₉

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#2178

C₂

socle

C₂×C₇₉

C₈×D₈

prime radical

#508×C₇₉

C₄

cosocle

C₄×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2178

N	H ≅ G/N	type
1. #23765237	C₂	semidirect
2. #23761476	C₂	semidirect
3. #23761373	C₂	semidirect
4. #23761316	C₂	semidirect
24. Q₃₂×C₇₉	C₈	semidirect
25. SD₃₂×C₇₉	C₈	semidirect
26. D₁₆×C₇₉	C₈	semidirect
76. C₇₉	#12047	semidirect
89. Q₃₂	C₇₉⋊C₈	semidirect
90. SD₃₂	C₇₉⋊C₈	semidirect
91. D₁₆	C₇₉⋊C₈	semidirect

H ≅ G/N

type

1. #23765237

C₂

semidirect

2. #23761476

C₂

semidirect

3. #23761373

C₂

semidirect

4. #23761316

C₂

semidirect

24. Q₃₂×C₇₉

C₈

semidirect

25. SD₃₂×C₇₉

C₈

semidirect

26. D₁₆×C₇₉

C₈

semidirect

76. C₇₉

#12047

semidirect

89. Q₃₂

C₇₉⋊C₈

semidirect

90. SD₃₂

C₇₉⋊C₈

semidirect

91. D₁₆

C₇₉⋊C₈

semidirect

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

subgroup	index
#23765237	2
#23761476	2
#23761373	2
#23761316	2
C₇₉×#2178	2
#12047	79

subgroup

index

C₇₉×#2178

Subgroups — #34200030

Conjugacy classes of subgroups

Complements

Maximal subgroups