Group #34174922 — subgroups

Subgroups — #34174922

Order 20224.

146 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 61 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	C₈×Dic₃₁₆	C₂	1	1	yes	maximal, characteristic
2	#23761470	C₂	1	1	yes	maximal, characteristic
3	C₈×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₁₅₈	—	2	1	no	—
5	C₄×Dic₃₁₆	C₄	1	1	yes	characteristic
6	C₄⋊(C₇₉⋊C₁₆)	—	2	1	no	—
7	#19098316	C₄	1	1	yes	characteristic
8	#19098295	—	2	1	no	—
9	C₄×C₁₆×C₇₉	—	2	1	no	—
10	C₈²×C₇₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₁₆	C₈	1	1	yes	characteristic
12	C₄×Dic₁₅₈	—	2	1	no	—
13	C₂×C₇₉⋊C₁₆	—	4	1	no	—
14	#14579095	C₈	1	1	yes	characteristic
15	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
16	Dic₇₉⋊C₈	—	4	1	no	—
17	#14579086	—	2	1	no	—
18	C₈×Dic₇₉	—	4	1	no	—
19	C₂×C₁₆×C₇₉	—	2	2	no	—
20	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
22	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
23	C₂×Dic₁₅₈	—	2	1	no	—
24	Dic₃₁₆	—	2	1	no	—
25	Dic₃₁₆	C₁₆	1	2	yes	complemented
26	C₂×C₇₉⋊C₈	—	4	1	no	—
27	C₄⋊Dic₇₉	—	2	1	no	—
28	Dic₇₉⋊C₄	—	4	1	no	—
29	C₄×Dic₇₉	—	4	1	no	—
30	C₇₉⋊C₁₆	—	8	1	no	—
31	C₂×C₈×C₇₉	—	2	1	no	—
32	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₇₉	M₁₆	1	1	yes	characteristic
34	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
35	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
36	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₁₆×C₇₉	—	2	4	no	—
38	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
39	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
40	C₂×C₄×C₇₉	#104	1	1	yes	characteristic
41	C₂×Dic₇₉	—	4	1	no	—
42	C₂×Dic₇₉	—	4	1	no	—
43	Dic₁₅₈	—	2	2	no	—
44	Dic₁₅₈	—	4	1	no	—
45	C₈×C₇₉	—	2	2	no	—
46	C₈×C₇₉	—	2	1	no	—
47	C₈×C₇₉	—	2	1	no	—
48	C₈×C₇₉	C₂×C₁₆	1	1	yes	characteristic
49	C₈×C₇₉	M₃₂	1	1	yes	characteristic
50	C₈×C₇₉	SD₃₂	1	1	yes	characteristic
51	C₈×C₇₉	Q₃₂	1	1	yes	characteristic
52	C₇₉⋊C₈	—	8	1	no	—
53	C₂²×C₇₉	D₄⋊C₈	1	1	yes	characteristic
54	C₄×C₇₉	—	2	1	no	—
55	C₄×C₇₉	C₂²⋊C₁₆	1	1	yes	characteristic
56	C₄×C₇₉	#350	1	1	yes	characteristic
57	C₄×C₇₉	#358	1	1	yes	characteristic
58	C₄×C₇₉	#359	1	1	yes	characteristic
59	Dic₇₉	—	4	2	no	—
60	Dic₇₉	—	8	1	no	—
61	#7069	—	79	1	no	maximal, Sylow
62	C₂×C₇₉	D₄⋊C₁₆	1	1	yes	characteristic
63	C₂×C₇₉	#1332	1	1	yes	characteristic
64	C₂×C₇₉	#1335	1	1	yes	characteristic
65	C₈×Q₁₆	—	79	1	no	—
66	#1370	—	79	1	no	—
67	C₈×C₁₆	D₇₉	1	1	yes	characteristic
68	C₇₉	#7069	1	1	yes	characteristic, complemented, Sylow
69	C₈×Q₈	—	158	1	no	—
70	C₄×Q₁₆	—	79	1	no	—
71	C₄⋊C₁₆	—	158	1	no	—
72	C₄×C₁₆	—	2	1	no	—
73	#335	—	79	1	no	—
74	Q₈⋊C₈	—	158	1	no	—
75	C₈²	C₂×D₇₉	1	1	yes	characteristic
76	C₂×Q₁₆	—	79	1	no	—
77	C₄×Q₈	—	158	1	no	—
78	C₂×C₁₆	—	2	2	no	—
79	C₂×C₁₆	—	316	1	no	—
80	#107	—	79	1	no	—
81	C₄⋊C₈	—	316	1	no	—
82	C₄⋊C₈	—	158	1	no	—
83	#103	—	158	1	no	—
84	C₄×C₈	—	316	1	no	—
85	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
86	C₄×C₈	D₃₁₆	1	1	yes	characteristic
87	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
88	C₂×Q₈	—	158	1	no	—
89	Q₁₆	—	79	2	no	—
90	Q₁₆	—	158	1	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	316	1	no	—
93	C₂×C₈	C₈×D₇₉	1	1	yes	characteristic
94	C₂×C₈	#11929607	1	1	yes	characteristic
95	C₂×C₈	D₆₃₂	1	1	yes	characteristic
96	C₂×C₈	#11929613	1	1	yes	characteristic
97	C₄⋊C₄	—	316	1	no	—
98	C₄⋊C₄	—	158	1	no	—
99	C₄²	—	316	1	no	—
100	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
101	C₁₆	—	2	4	no	—
102	C₁₆	—	632	1	no	—
103	Q₈	—	316	1	no	—
104	Q₈	—	158	2	no	—
105	C₂×C₄	—	316	1	no	—
106	C₂×C₄	—	316	1	no	—
107	C₂×C₄	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
108	C₂×C₄	#14579083	1	1	yes	characteristic
109	C₂×C₄	#14579089	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	—	2	1	no	—
112	C₈	—	2	1	no	—
113	C₈	—	632	1	no	—
114	C₈	C₁₆×D₇₉	1	1	yes	characteristic
115	C₈	#14579100	1	1	yes	characteristic
116	C₈	#14579106	1	1	yes	characteristic
117	C₈	Dic₆₃₂	1	1	yes	characteristic
118	C₂²	#19098292	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	316	2	no	—
121	C₄	—	632	1	no	—
122	C₄	(C₂×D₇₉)⋊C₁₆	1	1	yes	characteristic
123	C₄	#19098346	1	1	yes	characteristic
124	C₄	#19098368	1	1	yes	characteristic
125	C₄	#19098371	1	1	yes	characteristic
126	C₂	#23761356	1	1	yes	characteristic
127	C₂	#23761365	1	1	yes	characteristic
128	C₂	#23761374	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Dic₃₁₆

C₂

yes

maximal, characteristic

#23761470

C₂

yes

maximal, characteristic

C₈×C₁₆×C₇₉

C₂

yes

maximal, characteristic

C₈×Dic₁₅₈

—

C₄×Dic₃₁₆

C₄

yes

characteristic

C₄⋊(C₇₉⋊C₁₆)

—

#19098316

C₄

yes

characteristic

#19098295

—

C₄×C₁₆×C₇₉

—

C₈²×C₇₉

C₂²

yes

characteristic

C₂×Dic₃₁₆

C₈

yes

characteristic

C₄×Dic₁₅₈

—

C₂×C₇₉⋊C₁₆

—

#14579095

C₈

yes

characteristic

C₄⋊(C₇₉⋊C₈)

—

Dic₇₉⋊C₈

—

#14579086

—

C₈×Dic₇₉

—

C₂×C₁₆×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₂×Dic₁₅₈

—

Dic₃₁₆

—

Dic₃₁₆

C₁₆

yes

complemented

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×C₈

yes

characteristic

C₂×C₈×C₇₉

M₁₆

yes

characteristic

C₂×C₈×C₇₉

D₈

yes

characteristic

C₂×C₈×C₇₉

SD₁₆

yes

characteristic

C₄²×C₇₉

C₂²⋊C₄

yes

characteristic

C₁₆×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₈

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₂×C₁₆

yes

characteristic

C₈×C₇₉

M₃₂

yes

characteristic

C₈×C₇₉

SD₃₂

yes

characteristic

C₈×C₇₉

Q₃₂

yes

characteristic

C₇₉⋊C₈

—

C₂²×C₇₉

D₄⋊C₈

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

C₂²⋊C₁₆

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

#7069

—

maximal, Sylow

C₂×C₇₉

D₄⋊C₁₆

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₈×Q₁₆

—

#1370

—

C₈×C₁₆

D₇₉

yes

characteristic

C₇₉

#7069

yes

characteristic, complemented, Sylow

C₈×Q₈

—

158

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

158

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

158

—

C₈²

C₂×D₇₉

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

158

—

C₂×C₁₆

—

C₂×C₁₆

—

316

—

#107

—

C₄⋊C₈

—

316

—

C₄⋊C₈

—

158

—

#103

—

158

—

C₄×C₈

—

316

—

C₄×C₈

C₄×D₇₉

yes

characteristic

C₄×C₈

D₃₁₆

yes

characteristic

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

C₂×Q₈

—

158

—

Q₁₆

—

Q₁₆

—

158

—

C₂×C₈

—

C₂×C₈

—

316

—

C₂×C₈

C₈×D₇₉

yes

characteristic

C₂×C₈

#11929607

yes

characteristic

C₂×C₈

D₆₃₂

yes

characteristic

C₂×C₈

#11929613

yes

characteristic

C₄⋊C₄

—

316

—

C₄⋊C₄

—

158

—

C₄²

—

316

—

100

C₄²

(C₂×D₇₉)⋊C₄

yes

characteristic

101

C₁₆

—

102

C₁₆

—

632

—

103

Q₈

—

316

—

104

Q₈

—

158

—

105

C₂×C₄

—

316

—

106

C₂×C₄

—

316

—

107

C₂×C₄

(C₂×D₇₉)⋊C₈

yes

characteristic

108

C₂×C₄

#14579083

yes

characteristic

109

C₂×C₄

#14579089

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

632

—

114

C₈

C₁₆×D₇₉

yes

characteristic

115

C₈

#14579100

yes

characteristic

116

C₈

#14579106

yes

characteristic

117

C₈

Dic₆₃₂

yes

characteristic

118

C₂²

#19098292

yes

characteristic

119

C₄

—

120

C₄

—

316

—

121

C₄

—

632

—

122

C₄

(C₂×D₇₉)⋊C₁₆

yes

characteristic

123

C₄

#19098346

yes

characteristic

124

C₄

#19098368

yes

characteristic

125

C₄

#19098371

yes

characteristic

126

C₂

#23761356

yes

characteristic

127

C₂

#23761365

yes

characteristic

128

C₂

#23761374

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₆₃₂
commutator	C₈×C₇₉	C₂×C₁₆
Frattini	C₈²	C₂×D₇₉
Fitting	C₈×C₁₆×C₇₉	C₂
socle, prime radical	C₂²×C₇₉	D₄⋊C₈
cosocle	C₈²×C₇₉	C₂²
hypercenter	C₈×C₁₆	D₇₉
hypocenter	C₇₉	#7069
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂×C₈

D₆₃₂

commutator

C₈×C₇₉

C₂×C₁₆

Frattini

C₈²

C₂×D₇₉

Fitting

C₈×C₁₆×C₇₉

C₂

socle, prime radical

C₂²×C₇₉

D₄⋊C₈

cosocle

C₈²×C₇₉

C₂²

hypercenter

C₈×C₁₆

D₇₉

hypocenter

C₇₉

#7069

LCS

G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
25. Dic₃₁₆	C₁₆	semidirect
68. C₇₉	#7069	semidirect

H ≅ G/N

type

25. Dic₃₁₆

C₁₆

semidirect

68. C₇₉

#7069

semidirect

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

subgroup	index
C₈×Dic₃₁₆	2
#23761470	2
C₈×C₁₆×C₇₉	2
#7069	79

subgroup

index

C₈×Dic₃₁₆

#23761470

C₈×C₁₆×C₇₉

#7069

Subgroups — #34174922

Conjugacy classes of subgroups

Complements

Maximal subgroups