Group #34253892 — subgroups

Subgroups — C₂×#23761699 #34253892

Order 20224.

802 conjugacy classes of subgroups; only the 379 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 379 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	C₂×#19098614	C₂	1	3	yes	maximal
2	#23761699	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19098285	C₂	1	3	yes	maximal, complemented
4	C₂×C₈²×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄²×D₇₉	C₂²	1	1	yes	characteristic
6	#19098614	C₄	1	12	yes	—
7	#19098614	C₂²	1	12	yes	—
8	C₂×Dic₇₉⋊C₈	C₄	1	6	yes	—
9	C₂×(C₂×D₇₉)⋊C₈	C₄	1	6	yes	—
10	C₂×C₄×C₇₉⋊C₈	C₂²	1	3	yes	—
11	#19098285	C₂²	1	12	yes	complemented
12	C₂×C₄×C₈×C₇₉	C₂²	1	3	yes	—
13	C₈²×C₇₉	C₂²	1	4	yes	complemented
14	C₂²×C₄×D₇₉	C₂×C₄	1	3	yes	—
15	C₂²×C₇₉⋊C₈	C₂×C₄	1	6	yes	—
16	C₂×C₄×Dic₇₉	C₂×C₄	1	3	yes	—
17	C₄²×D₇₉	C₂×C₄	1	6	yes	—
18	C₄²×D₇₉	C₂³	1	2	yes	—
19	Dic₇₉⋊C₈	C₂×C₄	1	24	yes	—
20	(C₂×D₇₉)⋊C₈	C₂×C₄	1	24	yes	—
21	C₄×C₇₉⋊C₈	C₂×C₄	1	6	yes	—
22	C₄×C₇₉⋊C₈	C₂³	1	6	yes	—
23	C₂²×C₈×C₇₉	C₂×C₄	1	6	yes	—
24	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
25	C₄×C₈×C₇₉	C₂×C₄	1	6	yes	—
26	C₄×C₈×C₇₉	C₂³	1	6	yes	—
27	C₂³×D₇₉	C₄²	1	1	yes	characteristic
28	C₂²×Dic₇₉	C₄²	1	3	yes	—
29	C₂×C₄×D₇₉	—	2	12	no	—
30	C₂×C₄×D₇₉	C₄²	1	6	yes	—
31	C₂×C₄×D₇₉	C₂²×C₄	1	6	yes	—
32	C₂×C₇₉⋊C₈	—	2	12	no	—
33	C₂×C₇₉⋊C₈	C₂²×C₄	1	12	yes	—
34	C₄×Dic₇₉	C₄²	1	6	yes	—
35	C₄×Dic₇₉	C₂²×C₄	1	6	yes	—
36	C₂²×C₄×C₇₉	C₂²×C₄	1	3	yes	—
37	C₂×C₈×C₇₉	—	2	12	no	—
38	C₂×C₈×C₇₉	C₂²×C₄	1	12	yes	—
39	C₄²×C₇₉	C₂²×C₄	1	3	yes	—
40	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
41	C₂³×C₇₉	C₂×C₄²	1	1	yes	characteristic
42	C₂²×D₇₉	—	2	6	no	—
43	C₂²×D₇₉	C₂×C₄²	1	2	yes	—
44	C₂×C₄×C₇₉	C₂×C₄²	1	3	yes	—
45	C₂×C₄×C₇₉	#131	1	12	yes	—
46	C₂×C₄×C₇₉	C₂³×C₄	1	3	yes	—
47	C₂×Dic₇₉	—	2	6	no	—
48	C₂×Dic₇₉	C₂×C₄²	1	6	yes	—
49	C₄×D₇₉	—	2	12	no	—
50	C₄×D₇₉	—	2	12	no	—
51	C₈×C₇₉	—	2	12	no	—
52	C₇₉⋊C₈	—	2	12	no	—
53	C₂²×C₇₉	#405	1	6	yes	—
54	C₂²×C₇₉	C₂²×C₄²	1	1	yes	characteristic
55	C₂×D₇₉	—	2	6	no	—
56	C₂×D₇₉	—	4	4	no	—
57	C₄×C₇₉	#405	1	6	yes	—
58	C₄×C₇₉	C₂×#131	1	6	yes	—
59	Dic₇₉	—	2	6	no	—
60	C₂×#1454	—	79	1	no	maximal, Sylow
61	C₂×C₇₉	#1454	1	4	yes	complemented
62	C₂×C₇₉	C₂×#405	1	3	yes	—
63	D₇₉	—	4	2	no	—
64	C₂×#431	—	79	3	no	—
65	#1454	—	79	8	no	—
66	C₂×#322	—	79	3	no	—
67	C₂×C₈²	D₇₉	1	1	yes	characteristic, complemented
68	C₇₉	C₂×#1454	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₄²	—	79	1	no	—
70	#431	—	79	12	no	—
71	#431	—	79	12	no	—
72	C₂×C₄⋊C₈	—	79	6	no	—
73	C₂×C₂²⋊C₈	—	79	6	no	—
74	C₂×C₄×C₈	—	79	3	no	—
75	C₂×C₄×C₈	C₂×D₇₉	1	3	yes	—
76	#322	—	79	12	no	—
77	C₈²	C₂×D₇₉	1	4	yes	complemented
78	C₂³×C₄	—	79	3	no	—
79	C₂²×C₈	—	79	6	no	—
80	C₂²×C₈	C₄×D₇₉	1	6	yes	—
81	C₂×C₄²	—	79	3	no	—
82	C₂×C₄²	—	79	6	no	—
83	C₂×C₄²	—	79	2	no	—
84	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
85	C₄⋊C₈	—	79	24	no	—
86	C₂²⋊C₈	—	79	24	no	—
87	C₄×C₈	—	79	6	no	—
88	C₄×C₈	—	79	6	no	—
89	C₄×C₈	C₄×D₇₉	1	6	yes	—
90	C₄×C₈	C₂²×D₇₉	1	6	yes	—
91	C₂⁴	—	79	1	no	—
92	C₂²×C₄	—	79	3	no	—
93	C₂²×C₄	—	79	6	no	—
94	C₂²×C₄	—	79	6	no	—
95	C₂²×C₄	—	158	12	no	—
96	C₂²×C₄	C₂×C₄×D₇₉	1	3	yes	—
97	C₂×C₈	—	2	12	no	—
98	C₂×C₈	—	79	12	no	—
99	C₂×C₈	—	158	12	no	—
100	C₂×C₈	C₂×C₄×D₇₉	1	12	yes	—
101	C₄²	—	79	6	no	—
102	C₄²	—	79	6	no	—
103	C₄²	C₂×C₄×D₇₉	1	3	yes	—
104	C₄²	C₂³×D₇₉	1	1	yes	characteristic
105	C₂³	—	79	2	no	—
106	C₂³	—	158	6	no	—
107	C₂³	C₄²×D₇₉	1	1	yes	characteristic
108	C₂×C₄	—	79	6	no	—
109	C₂×C₄	—	158	6	no	—
110	C₂×C₄	—	158	12	no	—
111	C₂×C₄	—	158	12	no	—
112	C₂×C₄	C₄²×D₇₉	1	3	yes	—
113	C₂×C₄	#14579167	1	12	yes	—
114	C₂×C₄	C₂²×C₄×D₇₉	1	3	yes	—
115	C₈	—	2	12	no	—
116	C₈	—	158	12	no	—
117	C₂²	—	316	4	no	—
118	C₂²	—	158	6	no	—
119	C₂²	#19098517	1	6	yes	—
120	C₂²	C₂×C₄²×D₇₉	1	1	yes	characteristic
121	C₄	—	158	6	no	—
122	C₄	#19098517	1	6	yes	—
123	C₄	C₂×#14579167	1	6	yes	—
124	C₂	—	316	2	no	—
125	C₂	#23761699	1	4	yes	direct product factor
126	C₂	C₂×#19098517	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098614

C₂

yes

maximal

#23761699

C₂

yes

maximal, direct product factor

C₂×#19098285

C₂

yes

maximal, complemented

C₂×C₈²×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×C₄²×D₇₉

C₂²

yes

characteristic

#19098614

C₄

yes

—

#19098614

C₂²

yes

—

C₂×Dic₇₉⋊C₈

C₄

yes

—

C₂×(C₂×D₇₉)⋊C₈

C₄

yes

—

C₂×C₄×C₇₉⋊C₈

C₂²

yes

—

#19098285

C₂²

yes

complemented

C₂×C₄×C₈×C₇₉

C₂²

yes

—

C₈²×C₇₉

C₂²

yes

complemented

C₂²×C₄×D₇₉

C₂×C₄

yes

—

C₂²×C₇₉⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₇₉

C₂×C₄

yes

—

C₄²×D₇₉

C₂×C₄

yes

—

C₄²×D₇₉

C₂³

yes

—

Dic₇₉⋊C₈

C₂×C₄

yes

—

(C₂×D₇₉)⋊C₈

C₂×C₄

yes

—

C₄×C₇₉⋊C₈

C₂×C₄

yes

—

C₄×C₇₉⋊C₈

C₂³

yes

—

C₂²×C₈×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

C₂³

yes

—

C₂³×D₇₉

C₄²

yes

characteristic

C₂²×Dic₇₉

C₄²

yes

—

C₂×C₄×D₇₉

—

C₂×C₄×D₇₉

C₄²

yes

—

C₂×C₄×D₇₉

C₂²×C₄

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂²×C₄

yes

—

C₄×Dic₇₉

C₄²

yes

—

C₄×Dic₇₉

C₂²×C₄

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²×C₄

yes

—

C₄²×C₇₉

C₂²×C₄

yes

—

C₄²×C₇₉

C₂⁴

yes

characteristic

C₂³×C₇₉

C₂×C₄²

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

C₂×C₄²

yes

—

C₂×C₄×C₇₉

C₂×C₄²

yes

—

C₂×C₄×C₇₉

#131

yes

—

C₂×C₄×C₇₉

C₂³×C₄

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×C₄²

yes

—

C₄×D₇₉

—

C₄×D₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

#405

yes

—

C₂²×C₇₉

C₂²×C₄²

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

#405

yes

—

C₄×C₇₉

C₂×#131

yes

—

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

—

C₂×#431

—

#1454

—

C₂×#322

—

C₂×C₈²

D₇₉

yes

characteristic, complemented

C₇₉

C₂×#1454

yes

characteristic, complemented, Sylow

C₂²×C₄²

—

#431

—

#431

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₉

yes

—

#322

—

C₈²

C₂×D₇₉

yes

complemented

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

C₄×D₇₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₇₉

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₇₉

yes

—

C₄×C₈

C₂²×D₇₉

yes

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

158

—

C₂²×C₄

C₂×C₄×D₇₉

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

158

—

100

C₂×C₈

C₂×C₄×D₇₉

yes

—

101

C₄²

—

102

C₄²

—

103

C₄²

C₂×C₄×D₇₉

yes

—

104

C₄²

C₂³×D₇₉

yes

characteristic

105

C₂³

—

106

C₂³

—

158

—

107

C₂³

C₄²×D₇₉

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

158

—

110

C₂×C₄

—

158

—

111

C₂×C₄

—

158

—

112

C₂×C₄

C₄²×D₇₉

yes

—

113

C₂×C₄

#14579167

yes

—

114

C₂×C₄

C₂²×C₄×D₇₉

yes

—

115

C₈

—

116

C₈

—

158

—

117

C₂²

—

316

—

118

C₂²

—

158

—

119

C₂²

#19098517

yes

—

120

C₂²

C₂×C₄²×D₇₉

yes

characteristic

121

C₄

—

158

—

122

C₄

#19098517

yes

—

123

C₄

C₂×#14579167

yes

—

124

C₂

—

316

—

125

C₂

#23761699

yes

direct product factor

126

C₂

C₂×#19098517

yes

—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₇₉
commutator	C₂²×C₇₉	C₂²×C₄²
Frattini	C₄²	C₂³×D₇₉
Fitting	C₂×C₈²×C₇₉	C₂
socle	C₂³×C₇₉	C₂×C₄²
prime radical	C₂³×D₇₉	C₄²
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	C₂×C₈²	D₇₉
hypocenter	C₇₉	C₂×#1454
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂×C₄²

C₂²×D₇₉

commutator

C₂²×C₇₉

C₂²×C₄²

Frattini

C₄²

C₂³×D₇₉

Fitting

C₂×C₈²×C₇₉

C₂

socle

C₂³×C₇₉

C₂×C₄²

prime radical

C₂³×D₇₉

C₄²

cosocle

C₄²×C₇₉

C₂⁴

hypercenter

C₂×C₈²

D₇₉

C₇₉

G ▷ C₂²×C₇₉ ▷ C₇₉

C₁ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
2. #23761699	C₂	direct (N × H)
3. C₂×#19098285	C₂	semidirect
4. C₂×C₈²×C₇₉	C₂	semidirect
11. #19098285	C₂²	semidirect
13. C₈²×C₇₉	C₂²	semidirect
61. C₂×C₇₉	#1454	semidirect
67. C₂×C₈²	D₇₉	semidirect
68. C₇₉	C₂×#1454	semidirect
77. C₈²	C₂×D₇₉	semidirect
125. C₂	#23761699	direct (N × H)

H ≅ G/N

type

2. #23761699

C₂

direct (N × H)

3. C₂×#19098285

C₂

semidirect

4. C₂×C₈²×C₇₉

C₂

semidirect

11. #19098285

C₂²

semidirect

13. C₈²×C₇₉

C₂²

semidirect

61. C₂×C₇₉

#1454

semidirect

67. C₂×C₈²

D₇₉

semidirect

68. C₇₉

C₂×#1454

semidirect

77. C₈²

C₂×D₇₉

semidirect

125. C₂

#23761699

direct (N × H)

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

subgroup	index
C₂×#19098614	2
#23761699	2
C₂×#19098285	2
C₂×C₈²×C₇₉	2
C₂×#1454	79

subgroup

index

C₂×C₈²×C₇₉

Subgroups — C₂×#23761699 #34253892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#23761699 #34253892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23761699 #34253892