Group #28371658 — subgroups

Subgroups — C₂×#20295087 #28371658

Order 13952.

412 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 229 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	#20295087	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊#11724784	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#11724784	C₂	1	2	yes	maximal
4	C₂×#16812610	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×#11724784	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#117×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11724784	C₂²	1	2	yes	—
8	C₂²×#11724784	C₂²	1	2	yes	—
9	C₄⋊#11724784	C₂²	1	8	yes	complemented
10	C₂²⋊#11724784	C₂²	1	8	yes	—
11	#16812610	C₂²	1	4	yes	complemented
12	C₄×#11724784	C₂²	1	4	yes	complemented
13	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
14	#117×C₁₀₉	C₄	1	4	yes	—
15	#117×C₁₀₉	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₁₀₉	C₄	1	2	yes	—
17	C₂×C₂²⋊C₄×C₁₀₉	C₄	1	2	yes	—
18	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×#11724784	—	2	4	no	—
21	C₂×#11724784	—	2	4	no	—
22	C₂×#11724784	C₂³	1	4	yes	—
23	C₂×#11724784	C₂³	1	4	yes	—
24	C₂⁴×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₀₉	—	2	4	no	—
26	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
31	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₁₀₉	C₂×C₄	1	8	yes	—
33	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	8	yes	—
34	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
35	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
36	C₄²×C₁₀₉	C₂³	1	2	yes	—
37	C₄²×C₁₀₉	C₂³	1	2	yes	—
38	C₂³×C₁₀₉	—	2	2	no	—
39	C₂³×C₁₀₉	—	2	4	no	—
40	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
41	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	—	2	2	no	—
43	C₂×C₄×C₁₀₉	—	2	4	no	—
44	C₂×C₄×C₁₀₉	—	2	4	no	—
45	C₂×C₄×C₁₀₉	—	2	4	no	—
46	C₂×C₄×C₁₀₉	—	2	4	no	—
47	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₀₉	#41	1	4	yes	—
52	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
53	#11724784	—	2	4	no	—
54	#11724784	—	2	4	no	—
55	C₂²×C₁₀₉	—	2	2	no	—
56	C₂²×C₁₀₉	—	2	4	no	—
57	C₂²×C₁₀₉	—	4	4	no	—
58	C₂²×C₁₀₉	#117	1	2	yes	—
59	C₂²×C₁₀₉	#131	1	4	yes	—
60	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
61	C₄×C₁₀₉	—	2	2	no	—
62	C₄×C₁₀₉	—	2	4	no	—
63	C₄×C₁₀₉	—	2	4	no	—
64	C₄×C₁₀₉	#117	1	2	yes	—
65	C₄×C₁₀₉	C₂×#41	1	2	yes	—
66	C₂×C₁₀₉	—	4	2	no	—
67	C₂×C₁₀₉	#433	1	4	yes	complemented
68	C₂×C₁₀₉	C₂×#117	1	1	yes	characteristic
69	C₂×C₁₀₉	C₂×#131	1	2	yes	—
70	C₂×#433	—	109	1	no	maximal, Sylow
71	C₁₀₉	C₂×#433	1	1	yes	characteristic, complemented, Sylow
72	C₂×#117	D₁₀₉	1	1	yes	characteristic
73	#433	—	109	8	no	—
74	C₂×C₄⋊C₈	—	109	2	no	—
75	C₂×C₂²⋊C₈	—	109	2	no	—
76	C₂×#97	—	109	1	no	—
77	C₂×C₄×C₈	—	109	1	no	—
78	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
79	C₂²×C₈	—	109	2	no	—
80	C₂²×C₈	—	109	2	no	—
81	#117	Dic₁₀₉	1	4	yes	—
82	#117	C₂×D₁₀₉	1	4	yes	—
83	C₂×C₄⋊C₄	Dic₁₀₉	1	2	yes	—
84	C₂×C₂²⋊C₄	Dic₁₀₉	1	2	yes	—
85	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
86	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
87	C₄⋊C₈	—	109	8	no	—
88	C₂²⋊C₈	—	109	8	no	—
89	#97	—	109	4	no	—
90	C₄×C₈	—	109	4	no	—
91	C₂⁴	C₂×Dic₁₀₉	1	1	yes	characteristic
92	C₂²×C₄	—	2	4	no	—
93	C₂²×C₄	C₂×Dic₁₀₉	1	2	yes	—
94	C₂²×C₄	C₂×Dic₁₀₉	1	2	yes	—
95	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
96	C₂²×C₄	C₂×Dic₁₀₉	1	2	yes	—
97	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
98	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
99	C₂×C₈	—	109	4	no	—
100	C₂×C₈	—	109	4	no	—
101	C₂×C₈	—	218	4	no	—
102	C₂×C₈	—	218	4	no	—
103	C₄⋊C₄	C₂×Dic₁₀₉	1	8	yes	—
104	C₂²⋊C₄	C₂×Dic₁₀₉	1	8	yes	—
105	C₄²	C₂×Dic₁₀₉	1	2	yes	—
106	C₄²	C₂×Dic₁₀₉	1	2	yes	—
107	C₄²	C₂²×D₁₀₉	1	2	yes	—
108	C₄²	C₂²×D₁₀₉	1	2	yes	—
109	C₂³	—	2	2	no	—
110	C₂³	—	2	4	no	—
111	C₂³	C₂²×Dic₁₀₉	1	2	yes	—
112	C₂³	C₂²×Dic₁₀₉	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	C₂²×Dic₁₀₉	1	4	yes	—
119	C₂×C₄	C₂²×Dic₁₀₉	1	4	yes	—
120	C₂×C₄	C₂²×Dic₁₀₉	1	2	yes	—
121	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
122	C₂×C₄	#13182029	1	4	yes	—
123	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
124	C₈	—	218	4	no	—
125	C₈	—	218	4	no	—
126	C₂²	—	2	2	no	—
127	C₂²	—	2	4	no	—
128	C₂²	—	4	4	no	—
129	C₂²	#16812658	1	2	yes	—
130	C₂²	#16812706	1	4	yes	—
131	C₂²	C₂³×Dic₁₀₉	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	2	4	no	—
134	C₄	—	2	4	no	—
135	C₄	#16812658	1	2	yes	—
136	C₄	C₂×#13182029	1	2	yes	—
137	C₂	—	4	2	no	—
138	C₂	#20295087	1	4	yes	direct product factor
139	C₂	C₂×#16812658	1	1	yes	characteristic
140	C₂	C₂×#16812706	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295087

C₂

yes

maximal, direct product factor

C₂×C₄⋊#11724784

C₂

yes

maximal, complemented

C₂×C₂²⋊#11724784

C₂

yes

maximal

C₂×#16812610

C₂

yes

maximal, characteristic, complemented

C₂×C₄×#11724784

C₂

yes

maximal, characteristic, complemented

C₂×#117×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

—

C₂²×#11724784

C₂²

yes

—

C₄⋊#11724784

C₂²

yes

complemented

C₂²⋊#11724784

C₂²

yes

—

#16812610

C₂²

yes

complemented

C₄×#11724784

C₂²

yes

complemented

C₂³×C₄×C₁₀₉

C₂²

yes

characteristic

#117×C₁₀₉

C₄

yes

—

#117×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

C₄

yes

—

C₂×C₂²⋊C₄×C₁₀₉

C₄

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂×C₄²×C₁₀₉

C₂²

yes

—

—

C₂³

yes

—

C₂×#11724784

C₂³

yes

—

C₂⁴×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₂²⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂³

yes

—

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂²×C₄

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#117

yes

—

C₂²×C₁₀₉

#131

yes

—

C₂²×C₁₀₉

C₂³×C₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#117

yes

—

C₄×C₁₀₉

C₂×#41

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

C₂×#433

—

109

maximal, Sylow

C₁₀₉

C₂×#433

yes

characteristic, complemented, Sylow

C₂×#117

D₁₀₉

yes

characteristic

#433

—

109

—

C₂×C₄⋊C₈

—

109

—

C₂×C₂²⋊C₈

—

109

—

C₂×#97

—

109

—

C₂×C₄×C₈

—

109

—

C₂³×C₄

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

#117

Dic₁₀₉

yes

—

#117

C₂×D₁₀₉

yes

—

C₂×C₄⋊C₄

Dic₁₀₉

yes

—

C₂×C₂²⋊C₄

Dic₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

109

—

C₂²⋊C₈

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

C₂⁴

C₂×Dic₁₀₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₀₉

yes

—

C₂²×C₄

C₂×Dic₁₀₉

yes

—

C₂²×C₄

C₂×Dic₁₀₉

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

109

—

100

C₂×C₈

—

109

—

101

C₂×C₈

—

218

—

102

C₂×C₈

—

218

—

103

C₄⋊C₄

C₂×Dic₁₀₉

yes

—

104

C₂²⋊C₄

C₂×Dic₁₀₉

yes

—

105

C₄²

C₂×Dic₁₀₉

yes

—

106

C₄²

C₂×Dic₁₀₉

yes

—

107

C₄²

C₂²×D₁₀₉

yes

—

108

C₄²

C₂²×D₁₀₉

yes

—

109

C₂³

—

110

C₂³

—

111

C₂³

C₂²×Dic₁₀₉

yes

—

112

C₂³

C₂²×Dic₁₀₉

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₂²×Dic₁₀₉

yes

—

119

C₂×C₄

C₂²×Dic₁₀₉

yes

—

120

C₂×C₄

C₂²×Dic₁₀₉

yes

—

121

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

122

C₂×C₄

#13182029

yes

—

123

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

124

C₈

—

218

—

125

C₈

—

218

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#16812658

yes

—

130

C₂²

#16812706

yes

—

131

C₂²

C₂³×Dic₁₀₉

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#16812658

yes

—

136

C₄

C₂×#13182029

yes

—

137

C₂

—

138

C₂

#20295087

yes

direct product factor

139

C₂

C₂×#16812658

yes

characteristic

140

C₂

C₂×#16812706

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₉
commutator	C₂²×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×#117×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂²×C₄
prime radical	C₂⁴×C₁₀₉	C₂×C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₂×#117	D₁₀₉
hypocenter	C₁₀₉	C₂×#433
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

subgroup

G/H

center

C₂²×C₄

C₂²×D₁₀₉

commutator

C₂²×C₁₀₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

C₂×#117×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂²×C₄

prime radical

C₂⁴×C₁₀₉

C₂×C₄

cosocle

C₂×C₄×C₁₀₉

C₂⁴

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂²×C₄ ◁ C₂×#117

N	H ≅ G/N	type
1. #20295087	C₂	direct (N × H)
2. C₂×C₄⋊#11724784	C₂	semidirect
4. C₂×#16812610	C₂	semidirect
5. C₂×C₄×#11724784	C₂	semidirect
9. C₄⋊#11724784	C₂²	semidirect
11. #16812610	C₂²	semidirect
12. C₄×#11724784	C₂²	semidirect
67. C₂×C₁₀₉	#433	semidirect
71. C₁₀₉	C₂×#433	semidirect
138. C₂	#20295087	direct (N × H)

H ≅ G/N

type

1. #20295087

C₂

direct (N × H)

2. C₂×C₄⋊#11724784

C₂

semidirect

4. C₂×#16812610

C₂

semidirect

5. C₂×C₄×#11724784

C₂

semidirect

9. C₄⋊#11724784

C₂²

semidirect

11. #16812610

C₂²

semidirect

12. C₄×#11724784

C₂²

semidirect

67. C₂×C₁₀₉

#433

semidirect

71. C₁₀₉

C₂×#433

semidirect

138. C₂

#20295087

direct (N × H)

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

subgroup	index
#20295087	2
C₂×C₄⋊#11724784	2
C₂×C₂²⋊#11724784	2
C₂×#16812610	2
C₂×C₄×#11724784	2
C₂×#117×C₁₀₉	2
C₂×#433	109

subgroup

index

C₂×C₂²⋊#11724784

C₂×#117×C₁₀₉

109

Subgroups — C₂×#20295087 #28371658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20295087 #28371658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295087 #28371658