Group #34182073 — subgroups

Subgroups — #34182073

Order 20224.

354 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 203 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	#23763572	C₂	1	2	yes	maximal
2	#23762159	C₂	1	4	yes	maximal
3	C₂×C₈²×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579097	—	2	4	no	—
5	#19098587	C₂²	1	4	yes	—
6	#19098316	—	2	8	no	—
7	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
9	C₈²×C₇₉	C₄	1	1	yes	characteristic
10	C₈²×C₇₉	C₄	1	2	yes	—
11	C₈²×C₇₉	C₄	1	1	yes	characteristic
12	C₂×#11929608	—	2	4	no	—
13	#14579097	—	4	8	no	—
14	C₄⋊(C₇₉⋊C₈)	—	2	8	no	—
15	C₂²×C₈×C₇₉	D₄	1	2	yes	—
16	C₂²×C₈×C₇₉	Q₈	1	4	yes	—
17	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₇₉	D₄	1	4	yes	—
23	C₄×C₈×C₇₉	Q₈	1	2	yes	—
24	#11929608	—	4	8	no	—
25	C₂×C₇₉⋊C₈	—	4	4	no	—
26	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
28	C₂×C₈×C₇₉	—	2	8	no	—
29	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
30	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
32	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
33	C₂×C₈×C₇₉	D₈	1	4	yes	—
34	C₂×C₈×C₇₉	Q₁₆	1	4	yes	—
35	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₇₉	C₂×D₄	1	2	yes	—
37	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
38	C₂³×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₄×C₇₉	—	2	4	no	—
40	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
43	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
44	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
45	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
46	C₈×C₇₉	—	2	4	no	—
47	C₈×C₇₉	#107	1	4	yes	—
48	C₈×C₇₉	#107	1	4	yes	—
49	C₈×C₇₉	#108	1	8	yes	complemented
50	C₇₉⋊C₈	—	8	4	no	—
51	C₂²×C₇₉	—	2	2	no	—
52	C₂²×C₇₉	#384	1	1	yes	characteristic
53	C₂²×C₇₉	#500	1	2	yes	—
54	C₄×C₇₉	—	2	2	no	—
55	C₄×C₇₉	C₂×#107	1	2	yes	—
56	C₄×C₇₉	C₂×#108	1	4	yes	—
57	C₄×C₇₉	#493	1	1	yes	characteristic
58	C₄×C₇₉	#494	1	1	yes	characteristic
59	#8823	—	79	1	no	maximal, Sylow
60	C₂×C₇₉	—	2	2	no	—
61	C₂×C₇₉	#1845	1	1	yes	characteristic
62	C₂×C₇₉	#1848	1	2	yes	—
63	#1848	—	79	2	no	—
64	#1567	—	79	4	no	—
65	C₂×C₈²	D₇₉	1	1	yes	characteristic
66	C₇₉	#8823	1	1	yes	characteristic, complemented, Sylow
67	C₂×#108	—	158	4	no	—
68	#423	—	79	4	no	—
69	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
70	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
71	#335	—	158	8	no	—
72	C₈²	Dic₇₉	1	1	yes	characteristic
73	C₈²	Dic₇₉	1	2	yes	—
74	C₈²	Dic₇₉	1	1	yes	characteristic
75	C₂×M₁₆	—	158	4	no	—
76	C₂²×C₈	Dic₁₅₈	1	4	yes	—
77	C₂²×C₈	D₃₁₆	1	2	yes	—
78	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
79	#108	—	316	8	no	—
80	C₄⋊C₈	—	158	8	no	—
81	C₄×C₈	Dic₁₅₈	1	2	yes	—
82	C₄×C₈	D₃₁₆	1	4	yes	—
83	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
84	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
85	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
86	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
87	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
88	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
89	M₁₆	—	316	8	no	—
90	C₂×C₈	—	2	8	no	—
91	C₂×C₈	—	316	4	no	—
92	C₂×C₈	C₄⋊Dic₇₉	1	4	yes	—
93	C₂×C₈	C₄⋊Dic₇₉	1	2	yes	—
94	C₂×C₈	C₄⋊Dic₇₉	1	2	yes	—
95	C₂×C₈	C₄⋊Dic₇₉	1	4	yes	—
96	C₂×C₈	D₆₃₂	1	4	yes	—
97	C₂×C₈	Dic₃₁₆	1	4	yes	—
98	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
99	C₄²	C₂×D₃₁₆	1	2	yes	—
100	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
101	C₂³	#14579161	1	1	yes	characteristic
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
104	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
105	C₂×C₄	#14579158	1	1	yes	characteristic
106	C₂×C₄	#14579161	1	2	yes	—
107	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
108	C₂×C₄	C₂×Dic₃₁₆	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₈	—	632	4	no	—
111	C₈	#14579095	1	4	yes	—
112	C₈	#14579095	1	4	yes	—
113	C₈	#14579097	1	8	yes	complemented
114	C₂²	—	2	2	no	—
115	C₂²	#19098433	1	1	yes	characteristic
116	C₂²	#19098968	1	2	yes	—
117	C₄	—	2	2	no	—
118	C₄	C₂×#14579095	1	2	yes	—
119	C₄	C₂×#14579097	1	4	yes	—
120	C₄	#19098940	1	1	yes	characteristic
121	C₄	#19098943	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#23763560	1	1	yes	characteristic
124	C₂	#23763572	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23763572

C₂

yes

maximal

#23762159

C₂

yes

maximal

C₂×C₈²×C₇₉

C₂

yes

maximal, characteristic

C₂×#14579097

—

#19098587

C₂²

yes

—

#19098316

—

C₂×C₄×C₈×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₈²×C₇₉

C₄

yes

characteristic

C₈²×C₇₉

C₄

yes

—

C₈²×C₇₉

C₄

yes

characteristic

C₂×#11929608

—

#14579097

—

C₄⋊(C₇₉⋊C₈)

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂²×C₈×C₇₉

Q₈

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

Q₈

yes

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

Q₁₆

yes

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

—

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₂³×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

—

C₂×C₄×C₇₉

C₂×Q₁₆

yes

—

C₈×C₇₉

—

C₈×C₇₉

#107

yes

—

C₈×C₇₉

#107

yes

—

C₈×C₇₉

#108

yes

complemented

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#384

yes

characteristic

C₂²×C₇₉

#500

yes

—

C₄×C₇₉

—

C₄×C₇₉

C₂×#107

yes

—

C₄×C₇₉

C₂×#108

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

#1845

yes

characteristic

C₂×C₇₉

#1848

yes

—

#1848

—

#1567

—

C₂×C₈²

D₇₉

yes

characteristic

C₇₉

#8823

yes

characteristic, complemented, Sylow

C₂×#108

—

158

—

#423

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

C₂×C₄×C₈

C₂×D₇₉

yes

—

#335

—

158

—

C₈²

Dic₇₉

yes

characteristic

C₈²

Dic₇₉

yes

—

C₈²

Dic₇₉

yes

characteristic

C₂×M₁₆

—

158

—

C₂²×C₈

Dic₁₅₈

yes

—

C₂²×C₈

D₃₁₆

yes

—

C₂×C₄²

C₂²×D₇₉

yes

characteristic

#108

—

316

—

C₄⋊C₈

—

158

—

C₄×C₈

Dic₁₅₈

yes

—

C₄×C₈

D₃₁₆

yes

—

C₄×C₈

C₂×Dic₇₉

yes

—

C₄×C₈

C₂×Dic₇₉

yes

characteristic

C₄×C₈

C₂×Dic₇₉

yes

characteristic

C₄×C₈

C₂×Dic₇₉

yes

—

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

C₂²×C₄

C₂×Dic₁₅₈

yes

—

M₁₆

—

316

—

C₂×C₈

—

C₂×C₈

—

316

—

C₂×C₈

C₄⋊Dic₇₉

yes

—

C₂×C₈

C₄⋊Dic₇₉

yes

—

C₂×C₈

C₄⋊Dic₇₉

yes

—

C₂×C₈

C₄⋊Dic₇₉

yes

—

C₂×C₈

D₆₃₂

yes

—

C₂×C₈

Dic₃₁₆

yes

—

C₄²

C₂²×Dic₇₉

yes

characteristic

C₄²

C₂×D₃₁₆

yes

—

100

C₄²

C₂×Dic₁₅₈

yes

characteristic

101

C₂³

#14579161

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

characteristic

104

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

—

105

C₂×C₄

#14579158

yes

characteristic

106

C₂×C₄

#14579161

yes

—

107

C₂×C₄

C₂×D₆₃₂

yes

—

108

C₂×C₄

C₂×Dic₃₁₆

yes

—

109

C₈

—

110

C₈

—

632

—

111

C₈

#14579095

yes

—

112

C₈

#14579095

yes

—

113

C₈

#14579097

yes

complemented

114

C₂²

—

115

C₂²

#19098433

yes

characteristic

116

C₂²

#19098968

yes

—

117

C₄

—

118

C₄

C₂×#14579095

yes

—

119

C₄

C₂×#14579097

yes

—

120

C₄

#19098940

yes

characteristic

121

C₄

#19098943

yes

characteristic

122

C₂

—

123

C₂

#23763560

yes

characteristic

124

C₂

#23763572

yes

—

	subgroup	G/H
center	C₂×C₄	#14579158
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×C₈²×C₇₉	C₂
socle	C₂²×C₇₉	#384
prime radical	C₂³×C₇₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×C₈²	D₇₉
hypocenter	C₇₉	#8823
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂×C₄

#14579158

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₂×C₈²×C₇₉

C₂

socle

C₂²×C₇₉

#384

prime radical

C₂³×C₇₉

C₄⋊Q₈

cosocle

C₂×C₄²×C₇₉

C₂³

hypercenter

C₂×C₈²

D₇₉

hypocenter

C₇₉

#8823

LCS

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
49. C₈×C₇₉	#108	semidirect
66. C₇₉	#8823	semidirect
113. C₈	#14579097	semidirect

H ≅ G/N

type

49. C₈×C₇₉

#108

semidirect

66. C₇₉

#8823

semidirect

113. C₈

#14579097

semidirect

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

subgroup	index
#23763572	2
#23762159	2
C₂×C₈²×C₇₉	2
#8823	79

subgroup

index

#23763572

#23762159

C₂×C₈²×C₇₉

#8823

Subgroups — #34182073

Conjugacy classes of subgroups

Complements

Maximal subgroups