Group #28371592 — subgroups

Subgroups — C₂×#20295054 #28371592

Order 13952.

512 conjugacy classes of subgroups; only the 269 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 269 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	C₂³×#11724784	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812658	C₂	1	1	yes	maximal, characteristic
3	#20295054	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16812628	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11724784	—	2	4	no	—
7	C₂²×#11724784	C₂²	1	2	yes	—
8	C₂²×#11724784	C₂²	1	4	yes	—
9	#16812658	C₄	1	4	yes	—
10	#16812658	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₁₀₉	C₄	1	2	yes	—
12	C₂×C₂²⋊Dic₁₀₉	C₄	1	2	yes	—
13	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
14	#16812628	C₂²	1	16	yes	complemented
15	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₁₀₉	—	2	4	no	—
17	C₂×M₁₆×C₁₀₉	C₂²	1	4	yes	—
18	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
19	C₂²×Dic₁₀₉	C₂×C₄	1	4	yes	—
20	C₂×#11724784	—	2	4	no	—
21	C₂×#11724784	—	2	8	no	—
22	C₂×#11724784	C₂³	1	4	yes	—
23	C₄⋊Dic₁₀₉	C₂×C₄	1	8	yes	—
24	C₂²⋊Dic₁₀₉	C₂×C₄	1	8	yes	—
25	C₄×Dic₁₀₉	C₂×C₄	1	4	yes	—
26	C₄×Dic₁₀₉	C₂³	1	4	yes	—
27	C₂⁴×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₀₉	D₄	1	4	yes	—
31	C₂²×C₄×C₁₀₉	Q₈	1	4	yes	—
32	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
33	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
34	M₁₆×C₁₀₉	—	2	8	no	—
35	C₂×C₈×C₁₀₉	—	2	4	no	—
36	C₂×C₈×C₁₀₉	C₂³	1	4	yes	—
37	C₂³×C₁₀₉	—	2	4	no	—
38	C₂³×C₁₀₉	C₄⋊C₄	1	4	yes	—
39	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
40	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	C₄⋊C₄	1	8	yes	—
42	C₂×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
43	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
47	C₂×C₄×C₁₀₉	C₂×Q₈	1	4	yes	—
48	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₁₀₉	—	2	8	no	—
51	C₂×Dic₁₀₉	C₂²×C₄	1	8	yes	—
52	C₈×C₁₀₉	—	2	4	no	—
53	#11724784	—	2	4	no	—
54	C₂²×C₁₀₉	—	2	4	no	—
55	C₂²×C₁₀₉	—	2	8	no	—
56	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
57	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
58	C₂²×C₁₀₉	#131	1	4	yes	—
59	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
60	C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
61	C₄×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
62	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
64	Dic₁₀₉	—	2	8	no	—
65	C₂×C₁₀₉	—	2	4	no	—
66	C₂×C₁₀₉	#424	1	4	yes	complemented
67	C₂×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₁₀₉	C₂×#131	1	2	yes	—
69	C₂×#424	—	109	1	no	maximal, Sylow
70	C₁₀₉	C₂×#424	1	1	yes	characteristic, complemented, Sylow
71	C₂²×M₁₆	D₁₀₉	1	1	yes	characteristic
72	C₂³×C₈	—	109	1	no	—
73	C₂×#117	—	109	1	no	—
74	#424	—	109	8	no	—
75	C₂×C₄⋊C₈	—	109	4	no	—
76	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
77	C₂×M₁₆	—	2	4	no	—
78	C₂×M₁₆	C₂×D₁₀₉	1	4	yes	—
79	C₂²×C₈	—	109	4	no	—
80	C₂²×C₈	—	109	2	no	—
81	C₂²×C₈	—	218	4	no	—
82	C₂²×C₈	C₂×D₁₀₉	1	2	yes	—
83	#117	—	109	4	no	—
84	#117	—	109	4	no	—
85	C₂×C₄⋊C₄	—	109	2	no	—
86	C₂×C₂²⋊C₄	—	109	2	no	—
87	C₂×C₄²	—	109	2	no	—
88	C₄⋊C₈	—	109	16	no	—
89	C₂⁴	C₄×D₁₀₉	1	1	yes	characteristic
90	C₂²×C₄	—	109	4	no	—
91	C₂²×C₄	Dic₂₁₈	1	4	yes	—
92	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
93	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
94	C₂²×C₄	C₂²⋊D₁₀₉	1	4	yes	—
95	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
96	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
97	M₁₆	—	2	8	no	—
98	C₂×C₈	—	2	4	no	—
99	C₂×C₈	—	109	4	no	—
100	C₂×C₈	—	218	4	no	—
101	C₂×C₈	—	218	8	no	—
102	C₂×C₈	C₂²×D₁₀₉	1	4	yes	—
103	C₄⋊C₄	—	109	8	no	—
104	C₂²⋊C₄	—	109	8	no	—
105	C₄²	—	109	4	no	—
106	C₄²	—	109	4	no	—
107	C₂³	—	2	4	no	—
108	C₂³	#13182008	1	4	yes	—
109	C₂³	C₂×C₄×D₁₀₉	1	2	yes	—
110	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
111	C₂×C₄	—	109	8	no	—
112	C₂×C₄	—	218	8	no	—
113	C₂×C₄	#13182008	1	8	yes	—
114	C₂×C₄	#13182008	1	4	yes	—
115	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
116	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
117	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	4	yes	—
119	C₂×C₄	C₂×Dic₂₁₈	1	4	yes	—
120	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
121	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
122	C₈	—	2	4	no	—
123	C₈	—	218	4	no	—
124	C₂²	—	2	4	no	—
125	C₂²	—	2	8	no	—
126	C₂²	C₂×#13182008	1	4	yes	—
127	C₂²	C₂×#13182008	1	2	yes	—
128	C₂²	#16812707	1	4	yes	—
129	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
130	C₄	—	218	8	no	—
131	C₄	C₂×#13182008	1	2	yes	—
132	C₄	C₂×#13182008	1	4	yes	—
133	C₄	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
134	C₄	C₂²×Dic₂₁₈	1	1	yes	characteristic
135	C₂	—	2	4	no	—
136	C₂	#20295054	1	4	yes	direct product factor
137	C₂	C₂²×#13182008	1	1	yes	characteristic
138	C₂	C₂×#16812707	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×#11724784

C₂

yes

maximal, characteristic

C₂×#16812658

C₂

yes

maximal, characteristic

#20295054

C₂

yes

maximal, direct product factor

C₂×#16812628

C₂

yes

maximal, complemented

C₂²×M₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

—

C₂²×#11724784

C₂²

yes

—

C₂²×#11724784

C₂²

yes

—

#16812658

C₄

yes

—

#16812658

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₄

yes

—

C₂×C₂²⋊Dic₁₀₉

C₄

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

#16812628

C₂²

yes

complemented

C₂³×C₄×C₁₀₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₉

—

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

—

C₂²×Dic₁₀₉

C₂×C₄

yes

—

C₂×#11724784

—

C₂×#11724784

—

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

C₂²⋊Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂³

yes

—

C₂⁴×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

Q₈

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₄⋊C₄

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂²×C₄

yes

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

—

C₂²×C₁₀₉

#131

yes

—

C₂²×C₁₀₉

C₂³×C₄

yes

characteristic

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

—

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

—

C₄×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

C₂²×Q₈

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#424

yes

complemented

C₂×C₁₀₉

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₁₀₉

C₂×#131

yes

—

C₂×#424

—

109

maximal, Sylow

C₁₀₉

C₂×#424

yes

characteristic, complemented, Sylow

C₂²×M₁₆

D₁₀₉

yes

characteristic

C₂³×C₈

—

109

—

C₂×#117

—

109

—

#424

—

109

—

C₂×C₄⋊C₈

—

109

—

C₂³×C₄

C₂×D₁₀₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₀₉

yes

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

218

—

C₂²×C₈

C₂×D₁₀₉

yes

—

#117

—

109

—

#117

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₂⁴

C₄×D₁₀₉

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

Dic₂₁₈

yes

—

C₂²×C₄

C₄×D₁₀₉

yes

characteristic

C₂²×C₄

C₄×D₁₀₉

yes

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

109

—

100

C₂×C₈

—

218

—

101

C₂×C₈

—

218

—

102

C₂×C₈

C₂²×D₁₀₉

yes

—

103

C₄⋊C₄

—

109

—

104

C₂²⋊C₄

—

109

—

105

C₄²

—

109

—

106

C₄²

—

109

—

107

C₂³

—

108

C₂³

#13182008

yes

—

109

C₂³

C₂×C₄×D₁₀₉

yes

—

110

C₂³

C₂×C₄×D₁₀₉

yes

characteristic

111

C₂×C₄

—

109

—

112

C₂×C₄

—

218

—

113

C₂×C₄

#13182008

yes

—

114

C₂×C₄

#13182008

yes

—

115

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

116

C₂×C₄

C₂×C₄×D₁₀₉

yes

—

117

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

118

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

119

C₂×C₄

C₂×Dic₂₁₈

yes

—

120

C₂×C₄

C₂×Dic₂₁₈

yes

—

121

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

122

C₈

—

123

C₈

—

218

—

124

C₂²

—

125

C₂²

—

126

C₂²

C₂×#13182008

yes

—

127

C₂²

C₂×#13182008

yes

—

128

C₂²

#16812707

yes

—

129

C₂²

C₂²×C₄×D₁₀₉

yes

characteristic

130

C₄

—

218

—

131

C₄

C₂×#13182008

yes

—

132

C₄

C₂×#13182008

yes

—

133

C₄

C₂²×C₂²⋊D₁₀₉

yes

characteristic

134

C₄

C₂²×Dic₂₁₈

yes

characteristic

135

C₂

—

136

C₂

#20295054

yes

direct product factor

137

C₂

C₂²×#13182008

yes

characteristic

138

C₂

C₂×#16812707

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₉
commutator	C₂²×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂²×M₁₆×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂²×C₄
prime radical	C₂⁴×C₁₀₉	C₂×C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₂²×M₁₆	D₁₀₉
hypocenter	C₁₀₉	C₂×#424
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

subgroup

G/H

center

C₂²×C₄

C₂²×D₁₀₉

commutator

C₂²×C₁₀₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

C₂²×M₁₆×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂²×C₄

prime radical

C₂⁴×C₁₀₉

C₂×C₄

cosocle

C₂×C₄×C₁₀₉

C₂⁴

hypercenter

C₂²×M₁₆

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

N	H ≅ G/N	type
3. #20295054	C₂	direct (N × H)
4. C₂×#16812628	C₂	semidirect
14. #16812628	C₂²	semidirect
66. C₂×C₁₀₉	#424	semidirect
70. C₁₀₉	C₂×#424	semidirect
136. C₂	#20295054	direct (N × H)

H ≅ G/N

type

3. #20295054

C₂

direct (N × H)

4. C₂×#16812628

C₂

semidirect

14. #16812628

C₂²

semidirect

66. C₂×C₁₀₉

#424

semidirect

70. C₁₀₉

C₂×#424

semidirect

136. C₂

#20295054

direct (N × H)

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

subgroup	index
C₂³×#11724784	2
C₂×#16812658	2
#20295054	2
C₂×#16812628	2
C₂²×M₁₆×C₁₀₉	2
C₂×#424	109

subgroup

index

C₂²×M₁₆×C₁₀₉

109

Subgroups — C₂×#20295054 #28371592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20295054 #28371592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295054 #28371592