Group #34208770 — subgroups

Subgroups — #34208770

Order 20224.

190 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 107 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	C₂²×C₇₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	#23765184	C₂	1	1	yes	maximal, characteristic
3	#23761601	C₂	1	2	yes	maximal, complemented
4	#23761599	C₂	1	2	yes	maximal, complemented
5	C₇₉×#2158	C₂	1	1	yes	maximal, characteristic
6	#19098601	—	2	1	no	—
7	C₂×C₇₉⋊C₃₂	C₄	1	4	yes	complemented
8	C₂×C₇₉⋊C₃₂	C₂²	1	1	yes	characteristic
9	C₂×C₇₉⋊C₃₂	C₂²	1	1	yes	characteristic
10	#19098391	—	2	1	no	—
11	#19098389	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
13	#427×C₇₉	—	2	1	no	—
14	#367×C₇₉	—	2	1	no	—
15	#366×C₇₉	C₂²	1	2	yes	—
16	#14579120	—	4	1	no	—
17	#14579095	—	2	2	no	—
18	#14579093	—	4	1	no	—
19	C₇₉⋊C₃₂	C₂×C₄	1	2	yes	—
20	C₇₉⋊C₃₂	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
22	#117×C₇₉	—	4	1	no	—
23	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₉	Q₈	1	2	yes	—
26	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
27	#107×C₇₉	—	2	2	no	—
28	#106×C₇₉	—	4	1	no	—
29	C₄⋊Dic₇₉	—	4	2	no	—
30	C₂²⋊Dic₇₉	—	8	1	no	—
31	C₄×Dic₇₉	—	8	1	no	—
32	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
33	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
34	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
35	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
36	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
37	C₄⋊C₄×C₇₉	—	4	2	no	—
38	C₂²⋊C₄×C₇₉	—	8	1	no	—
39	C₄²×C₇₉	—	8	1	no	—
40	C₁₆×C₇₉	C₄⋊C₄	1	2	yes	—
41	C₁₆×C₇₉	C₂²×C₄	1	1	yes	characteristic
42	C₁₆×C₇₉	C₂×Q₈	1	1	yes	characteristic
43	C₂³×C₇₉	D₁₆	1	1	yes	characteristic
44	C₂×C₄×C₇₉	—	8	2	no	—
45	C₂×C₄×C₇₉	#107	1	2	yes	—
46	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
47	C₂×C₄×C₇₉	Q₃₂	1	2	yes	—
48	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
49	C₂×Dic₇₉	—	8	2	no	—
50	C₈×C₇₉	#107	1	2	yes	—
51	C₈×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₈×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
53	C₂²×C₇₉	—	2	2	no	—
54	C₂²×C₇₉	#366	1	2	yes	—
55	C₂²×C₇₉	C₂×D₁₆	1	1	yes	characteristic
56	C₄×C₇₉	—	16	2	no	—
57	C₄×C₇₉	#366	1	2	yes	—
58	C₄×C₇₉	C₂×#107	1	1	yes	characteristic
59	C₄×C₇₉	C₂×Q₃₂	1	1	yes	characteristic
60	Dic₇₉	—	16	2	no	—
61	#13633	—	79	1	no	maximal, Sylow
62	C₂×C₇₉	—	2	2	no	—
63	C₂×C₇₉	C₂×#366	1	1	yes	characteristic
64	C₂×C₇₉	#2262	1	2	yes	—
65	C₂²×C₃₂	—	79	1	no	—
66	#2158	—	79	1	no	—
67	#2158	D₇₉	1	1	yes	characteristic
68	#1424	—	79	2	no	—
69	#1423	—	79	2	no	—
70	C₇₉	#13633	1	1	yes	characteristic, complemented, Sylow
71	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
72	#427	—	2	1	no	—
73	#427	—	158	1	no	—
74	C₂×C₃₂	—	79	1	no	—
75	C₂×C₃₂	—	79	1	no	—
76	C₂×C₃₂	—	79	4	no	—
77	#367	—	2	1	no	—
78	#367	—	158	1	no	—
79	#366	—	79	2	no	—
80	#366	C₂×D₇₉	1	2	yes	—
81	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
82	#117	—	4	1	no	—
83	#117	—	316	1	no	—
84	C₂×C₁₆	Dic₁₅₈	1	2	yes	—
85	C₂×C₁₆	C₄×D₇₉	1	2	yes	—
86	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
87	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
88	#107	—	2	2	no	—
89	#107	—	158	2	no	—
90	#106	—	4	1	no	—
91	#106	—	316	1	no	—
92	C₃₂	—	79	2	no	—
93	C₃₂	—	79	2	no	—
94	C₂²×C₄	#11929609	1	1	yes	characteristic
95	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
96	C₂×C₈	#11929609	1	1	yes	characteristic
97	C₂×C₈	#11929614	1	2	yes	—
98	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	316	2	no	—
101	C₂²⋊C₄	—	8	1	no	—
102	C₂²⋊C₄	—	632	1	no	—
103	C₄²	—	8	1	no	—
104	C₄²	—	632	1	no	—
105	C₁₆	Dic₇₉⋊C₄	1	2	yes	—
106	C₁₆	C₂×C₄×D₇₉	1	1	yes	characteristic
107	C₁₆	C₂×Dic₁₅₈	1	1	yes	characteristic
108	C₂³	#14579102	1	1	yes	characteristic
109	C₂×C₄	—	8	2	no	—
110	C₂×C₄	—	632	2	no	—
111	C₂×C₄	#14579094	1	2	yes	—
112	C₂×C₄	#14579102	1	1	yes	characteristic
113	C₂×C₄	#14579107	1	2	yes	—
114	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
115	C₈	#14579094	1	2	yes	—
116	C₈	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
117	C₈	C₂×#11929614	1	1	yes	characteristic
118	C₂²	—	2	2	no	—
119	C₂²	#19098388	1	2	yes	—
120	C₂²	C₂×#14579102	1	1	yes	characteristic
121	C₄	—	16	2	no	—
122	C₄	—	1264	2	no	—
123	C₄	#19098388	1	2	yes	—
124	C₄	C₂×#14579094	1	1	yes	characteristic
125	C₄	C₂×#14579107	1	1	yes	characteristic
126	C₂	—	2	2	no	—
127	C₂	C₂×#19098388	1	1	yes	characteristic
128	C₂	#23765707	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₇₉⋊C₃₂

C₂

yes

maximal, characteristic

#23765184

C₂

yes

maximal, characteristic

#23761601

C₂

yes

maximal, complemented

#23761599

C₂

yes

maximal, complemented

C₇₉×#2158

C₂

yes

maximal, characteristic

#19098601

—

C₂×C₇₉⋊C₃₂

C₄

yes

complemented

C₂×C₇₉⋊C₃₂

C₂²

yes

characteristic

C₂×C₇₉⋊C₃₂

C₂²

yes

characteristic

#19098391

—

#19098389

C₂²

yes

—

C₂²×C₁₆×C₇₉

C₂²

yes

characteristic

#427×C₇₉

—

#367×C₇₉

—

#366×C₇₉

C₂²

yes

—

#14579120

—

#14579095

—

#14579093

—

C₇₉⋊C₃₂

C₂×C₄

yes

—

C₇₉⋊C₃₂

C₂×C₄

yes

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

#117×C₇₉

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

—

C₂×C₁₆×C₇₉

D₄

yes

characteristic

C₂×C₁₆×C₇₉

Q₈

yes

—

C₂×C₁₆×C₇₉

C₂³

yes

characteristic

#107×C₇₉

—

#106×C₇₉

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

Q₁₆

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₁₆×C₇₉

C₄⋊C₄

yes

—

C₁₆×C₇₉

C₂²×C₄

yes

characteristic

C₁₆×C₇₉

C₂×Q₈

yes

characteristic

C₂³×C₇₉

D₁₆

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#107

yes

—

C₂×C₄×C₇₉

D₁₆

yes

characteristic

C₂×C₄×C₇₉

Q₃₂

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×Dic₇₉

—

C₈×C₇₉

#107

yes

—

C₈×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₈×C₇₉

C₂×Q₁₆

yes

characteristic

C₂²×C₇₉

—

C₂²×C₇₉

#366

yes

—

C₂²×C₇₉

C₂×D₁₆

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

#366

yes

—

C₄×C₇₉

C₂×#107

yes

characteristic

C₄×C₇₉

C₂×Q₃₂

yes

characteristic

Dic₇₉

—

#13633

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

C₂×#366

yes

characteristic

C₂×C₇₉

#2262

yes

—

C₂²×C₃₂

—

#2158

—

#2158

D₇₉

yes

characteristic

#1424

—

#1423

—

C₇₉

#13633

yes

characteristic, complemented, Sylow

C₂²×C₁₆

C₂×D₇₉

yes

characteristic

#427

—

#427

—

158

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

#367

—

#367

—

158

—

#366

—

#366

C₂×D₇₉

yes

—

C₂²×C₈

C₂²⋊D₇₉

yes

characteristic

#117

—

#117

—

316

—

C₂×C₁₆

Dic₁₅₈

yes

—

C₂×C₁₆

C₄×D₇₉

yes

—

C₂×C₁₆

C₂²⋊D₇₉

yes

characteristic

C₂×C₁₆

C₂²×D₇₉

yes

characteristic

#107

—

#107

—

158

—

#106

—

#106

—

316

—

C₃₂

—

C₃₂

—

C₂²×C₄

#11929609

yes

characteristic

C₂×C₈

Dic₇₉⋊C₄

yes

—

C₂×C₈

#11929609

yes

characteristic

C₂×C₈

#11929614

yes

—

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

316

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

632

—

103

C₄²

—

104

C₄²

—

632

—

105

C₁₆

Dic₇₉⋊C₄

yes

—

106

C₁₆

C₂×C₄×D₇₉

yes

characteristic

107

C₁₆

C₂×Dic₁₅₈

yes

characteristic

108

C₂³

#14579102

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

632

—

111

C₂×C₄

#14579094

yes

—

112

C₂×C₄

#14579102

yes

characteristic

113

C₂×C₄

#14579107

yes

—

114

C₂×C₄

C₂×#11929609

yes

characteristic

115

C₈

#14579094

yes

—

116

C₈

C₂×Dic₇₉⋊C₄

yes

characteristic

117

C₈

C₂×#11929614

yes

characteristic

118

C₂²

—

119

C₂²

#19098388

yes

—

120

C₂²

C₂×#14579102

yes

characteristic

121

C₄

—

122

C₄

—

1264

—

123

C₄

#19098388

yes

—

124

C₄

C₂×#14579094

yes

characteristic

125

C₄

C₂×#14579107

yes

characteristic

126

C₂

—

127

C₂

C₂×#19098388

yes

characteristic

128

C₂

#23765707

yes

—

	subgroup	G/H
center	C₂×C₄	#14579102
commutator	C₁₆×C₇₉	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting	C₇₉×#2158	C₂
socle	C₂²×C₇₉	C₂×D₁₆
prime radical	C₂³×C₇₉	D₁₆
cosocle	C₂×C₁₆×C₇₉	C₂³
hypercenter	#2158	D₇₉
hypocenter	C₇₉	#13633
LCS	G ▷ C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ #2158

subgroup

G/H

center

C₂×C₄

#14579102

commutator

C₁₆×C₇₉

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₇₉

Fitting

C₇₉×#2158

C₂

socle

C₂²×C₇₉

C₂×D₁₆

prime radical

C₂³×C₇₉

D₁₆

cosocle

C₂×C₁₆×C₇₉

C₂³

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ #2158

N	H ≅ G/N	type
3. #23761601	C₂	semidirect
4. #23761599	C₂	semidirect
7. C₂×C₇₉⋊C₃₂	C₄	semidirect
70. C₇₉	#13633	semidirect

H ≅ G/N

type

3. #23761601

C₂

semidirect

4. #23761599

C₂

semidirect

7. C₂×C₇₉⋊C₃₂

C₄

semidirect

70. C₇₉

#13633

semidirect

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

subgroup	index
C₂²×C₇₉⋊C₃₂	2
#23765184	2
#23761601	2
#23761599	2
C₇₉×#2158	2
#13633	79

subgroup

index

C₂²×C₇₉⋊C₃₂

C₇₉×#2158

Subgroups — #34208770

Conjugacy classes of subgroups

Complements

Maximal subgroups