Group #28376433 — subgroups

Subgroups — C₂×#20295657 #28376433

Order 13952.

1080 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1080 conjugacy classes of subgroups — too many to draw. Only its 565 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	#20295657	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16812698	C₂	1	4	yes	maximal, complemented
3	C₂×#16812690	C₂	1	4	yes	maximal, complemented
4	C₂×#16812678	C₂	1	4	yes	maximal
5	C₂×#16812658	C₂	1	2	yes	maximal
6	C₂²×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₁₈	C₂²	1	4	yes	—
8	#16812698	C₂²	1	32	yes	complemented
9	#16812690	C₂²	1	32	yes	complemented
10	#16812678	C₂²	1	32	yes	—
11	#16812658	C₂²	1	16	yes	—
12	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₁₀₉	C₂²	1	4	yes	—
15	C₂×#13182008	C₂²	1	8	yes	—
16	C₂×C₂²⋊Dic₁₀₉	C₂²	1	4	yes	—
17	C₂×C₄×Dic₁₀₉	C₂²	1	4	yes	—
18	C₂³×C₄×C₁₀₉	C₂²	1	2	yes	—
19	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
21	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	8	yes	—
22	C₂×Dic₂₁₈	—	4	8	no	—
23	C₂×Dic₂₁₈	C₂³	1	16	yes	—
24	C₂²×Dic₁₀₉	C₂³	1	4	yes	—
25	C₂²×Dic₁₀₉	C₂³	1	4	yes	—
26	C₄⋊Dic₁₀₉	C₂³	1	8	yes	—
27	C₄⋊Dic₁₀₉	C₂³	1	8	yes	—
28	C₄⋊Dic₁₀₉	C₂³	1	16	yes	—
29	#13182008	C₂³	1	32	yes	—
30	C₂²⋊Dic₁₀₉	C₂³	1	16	yes	—
31	C₄×Dic₁₀₉	C₂³	1	16	yes	—
32	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₀₉	—	4	4	no	—
34	C₂²×C₄×C₁₀₉	Q₈	1	8	yes	complemented
35	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
36	C₂²×C₄×C₁₀₉	C₂³	1	8	yes	—
37	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
39	C₄⋊C₄×C₁₀₉	C₂³	1	16	yes	—
40	C₂³×C₁₀₉	—	2	2	no	—
41	C₂³×C₁₀₉	—	2	2	no	—
42	C₂³×C₁₀₉	C₂×Q₈	1	4	yes	—
43	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
44	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	—	2	8	no	—
46	C₂×C₄×C₁₀₉	—	4	8	no	—
47	C₂×C₄×C₁₀₉	C₂×Q₈	1	8	yes	—
48	C₂×C₄×C₁₀₉	C₂×Q₈	1	16	yes	complemented
49	C₂×C₄×C₁₀₉	C₂⁴	1	8	yes	—
50	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
51	C₂×Dic₁₀₉	—	2	8	no	—
52	C₂×Dic₁₀₉	—	2	8	no	—
53	C₂×Dic₁₀₉	C₂⁴	1	8	yes	—
54	C₂×Dic₁₀₉	C₂⁴	1	8	yes	—
55	Dic₂₁₈	—	4	16	no	—
56	C₂²×C₁₀₉	—	2	8	no	—
57	C₂²×C₁₀₉	—	2	2	no	—
58	C₂²×C₁₀₉	—	2	2	no	—
59	C₂²×C₁₀₉	C₂²×Q₈	1	4	yes	—
60	C₂²×C₁₀₉	C₂²×Q₈	1	2	yes	—
61	C₂²×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
62	C₂²×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
63	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
64	C₄×C₁₀₉	—	2	8	no	—
65	C₄×C₁₀₉	C₂²×Q₈	1	8	yes	—
66	Dic₁₀₉	—	2	8	no	—
67	Dic₁₀₉	—	2	8	no	—
68	C₂×C₁₀₉	—	2	4	no	—
69	C₂×C₁₀₉	#544	1	4	yes	complemented
70	C₂×C₁₀₉	C₂³×Q₈	1	1	yes	characteristic
71	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
72	C₂×C₁₀₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
73	C₂×#544	—	109	1	no	maximal, Sylow
74	C₁₀₉	C₂×#544	1	1	yes	characteristic, complemented, Sylow
75	#544	—	109	16	no	—
76	C₂×C₄⋊Q₈	—	109	4	no	—
77	C₂×#125	—	109	4	no	—
78	C₂×C₂²⋊Q₈	—	109	4	no	—
79	C₂×#117	—	109	2	no	—
80	C₂²×C₄⋊C₄	D₁₀₉	1	1	yes	characteristic
81	C₂²×Q₈	—	109	4	no	—
82	C₂³×C₄	C₂×D₁₀₉	1	2	yes	—
83	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
84	C₄⋊Q₈	—	109	32	no	—
85	#125	—	109	32	no	—
86	C₂²⋊Q₈	—	109	32	no	—
87	#117	—	109	16	no	—
88	C₂×C₄⋊C₄	—	109	2	no	—
89	C₂×C₄⋊C₄	—	109	2	no	—
90	C₂×C₄⋊C₄	—	109	4	no	—
91	C₂×C₄⋊C₄	—	109	8	no	—
92	C₂×C₄⋊C₄	C₂×D₁₀₉	1	4	yes	—
93	C₂×C₄⋊C₄	C₂×D₁₀₉	1	8	yes	—
94	C₂×C₂²⋊C₄	—	109	4	no	—
95	C₂×C₄²	—	109	4	no	—
96	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
97	C₂×Q₈	—	109	16	no	—
98	C₂×Q₈	—	436	8	no	—
99	C₂²×C₄	—	4	4	no	—
100	C₂²×C₄	—	109	4	no	—
101	C₂²×C₄	—	109	4	no	—
102	C₂²×C₄	Dic₂₁₈	1	8	yes	complemented
103	C₂²×C₄	C₂²×D₁₀₉	1	4	yes	—
104	C₂²×C₄	C₂²×D₁₀₉	1	8	yes	—
105	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
106	C₂²×C₄	C₂²×D₁₀₉	1	4	yes	—
107	C₄⋊C₄	—	109	32	no	—
108	C₄⋊C₄	—	109	8	no	—
109	C₄⋊C₄	—	109	8	no	—
110	C₄⋊C₄	—	109	16	no	—
111	C₄⋊C₄	C₂²×D₁₀₉	1	16	yes	—
112	C₂²⋊C₄	—	109	16	no	—
113	C₄²	—	109	16	no	—
114	C₂³	—	2	2	no	—
115	C₂³	—	2	2	no	—
116	C₂³	C₂×Dic₂₁₈	1	4	yes	—
117	C₂³	C₂³×D₁₀₉	1	2	yes	—
118	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
119	Q₈	—	436	16	no	—
120	C₂×C₄	—	2	8	no	—
121	C₂×C₄	—	4	8	no	—
122	C₂×C₄	—	109	8	no	—
123	C₂×C₄	—	109	8	no	—
124	C₂×C₄	—	218	8	no	—
125	C₂×C₄	—	218	8	no	—
126	C₂×C₄	C₂×Dic₂₁₈	1	8	yes	—
127	C₂×C₄	C₂×Dic₂₁₈	1	16	yes	complemented
128	C₂×C₄	C₂³×D₁₀₉	1	8	yes	—
129	C₂×C₄	C₂³×D₁₀₉	1	4	yes	—
130	C₂²	—	2	2	no	—
131	C₂²	—	2	2	no	—
132	C₂²	—	2	8	no	—
133	C₂²	C₂²×Dic₂₁₈	1	4	yes	—
134	C₂²	C₂²×Dic₂₁₈	1	2	yes	—
135	C₂²	#16812744	1	2	yes	—
136	C₂²	#16812747	1	2	yes	—
137	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
138	C₄	—	2	8	no	—
139	C₄	—	218	8	no	—
140	C₄	—	218	8	no	—
141	C₄	C₂²×Dic₂₁₈	1	8	yes	—
142	C₂	—	2	4	no	—
143	C₂	#20295657	1	4	yes	direct product factor
144	C₂	C₂³×Dic₂₁₈	1	1	yes	characteristic
145	C₂	C₂×#16812744	1	1	yes	characteristic
146	C₂	C₂×#16812747	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295657

C₂

yes

maximal, direct product factor

C₂×#16812698

C₂

yes

maximal, complemented

C₂×#16812690

C₂

yes

maximal, complemented

C₂×#16812678

C₂

yes

maximal

C₂×#16812658

C₂

yes

maximal

C₂²×C₄⋊C₄×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

C₂²

yes

—

#16812698

C₂²

yes

complemented

#16812690

C₂²

yes

complemented

#16812678

C₂²

yes

—

#16812658

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×#13182008

C₂²

yes

—

C₂×C₂²⋊Dic₁₀₉

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

C₂³×C₄×C₁₀₉

C₂²

yes

—

C₂³×C₄×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂³

yes

—

C₂²×Dic₁₀₉

C₂³

yes

—

C₂²×Dic₁₀₉

C₂³

yes

—

C₄⋊Dic₁₀₉

C₂³

yes

—

C₄⋊Dic₁₀₉

C₂³

yes

—

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182008

C₂³

yes

—

C₂²⋊Dic₁₀₉

C₂³

yes

—

C₄×Dic₁₀₉

C₂³

yes

—

C₂⁴×C₁₀₉

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

Q₈

yes

complemented

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂×Q₈

yes

—

C₂³×C₁₀₉

C₂⁴

yes

—

C₂³×C₁₀₉

C₂⁴

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

complemented

C₂×C₄×C₁₀₉

C₂⁴

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂⁴

yes

—

C₂×Dic₁₀₉

C₂⁴

yes

—

Dic₂₁₈

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²×Q₈

yes

—

C₂²×C₁₀₉

C₂²×Q₈

yes

—

C₂²×C₁₀₉

H(5,𝔽₂)

yes

—

C₂²×C₁₀₉

2₋¹⁺⁴

yes

—

C₂²×C₁₀₉

C₂⁵

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×Q₈

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#544

yes

complemented

C₂×C₁₀₉

C₂³×Q₈

yes

characteristic

C₂×C₁₀₉

C₂×H(5,𝔽₂)

yes

characteristic

C₂×C₁₀₉

C₂×2₋¹⁺⁴

yes

characteristic

C₂×#544

—

109

maximal, Sylow

C₁₀₉

C₂×#544

yes

characteristic, complemented, Sylow

#544

—

109

—

C₂×C₄⋊Q₈

—

109

—

C₂×#125

—

109

—

C₂×C₂²⋊Q₈

—

109

—

C₂×#117

—

109

—

C₂²×C₄⋊C₄

D₁₀₉

yes

characteristic

C₂²×Q₈

—

109

—

C₂³×C₄

C₂×D₁₀₉

yes

—

C₂³×C₄

C₂×D₁₀₉

yes

characteristic

C₄⋊Q₈

—

109

—

#125

—

109

—

C₂²⋊Q₈

—

109

—

#117

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₄²

—

109

—

C₂⁴

C₂²×D₁₀₉

yes

characteristic

C₂×Q₈

—

109

—

C₂×Q₈

—

436

—

C₂²×C₄

—

100

C₂²×C₄

—

109

—

101

C₂²×C₄

—

109

—

102

C₂²×C₄

Dic₂₁₈

yes

complemented

103

C₂²×C₄

C₂²×D₁₀₉

yes

—

104

C₂²×C₄

C₂²×D₁₀₉

yes

—

105

C₂²×C₄

C₂²×D₁₀₉

yes

—

106

C₂²×C₄

C₂²×D₁₀₉

yes

—

107

C₄⋊C₄

—

109

—

108

C₄⋊C₄

—

109

—

109

C₄⋊C₄

—

109

—

110

C₄⋊C₄

—

109

—

111

C₄⋊C₄

C₂²×D₁₀₉

yes

—

112

C₂²⋊C₄

—

109

—

113

C₄²

—

109

—

114

C₂³

—

115

C₂³

—

116

C₂³

C₂×Dic₂₁₈

yes

—

117

C₂³

C₂³×D₁₀₉

yes

—

118

C₂³

C₂³×D₁₀₉

yes

characteristic

119

Q₈

—

436

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

109

—

123

C₂×C₄

—

109

—

124

C₂×C₄

—

218

—

125

C₂×C₄

—

218

—

126

C₂×C₄

C₂×Dic₂₁₈

yes

—

127

C₂×C₄

C₂×Dic₂₁₈

yes

complemented

128

C₂×C₄

C₂³×D₁₀₉

yes

—

129

C₂×C₄

C₂³×D₁₀₉

yes

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

C₂²×Dic₂₁₈

yes

—

134

C₂²

C₂²×Dic₂₁₈

yes

—

135

C₂²

#16812744

yes

—

136

C₂²

#16812747

yes

—

137

C₂²

C₂⁴×D₁₀₉

yes

characteristic

138

C₄

—

139

C₄

—

218

—

140

C₄

—

218

—

141

C₄

C₂²×Dic₂₁₈

yes

—

142

C₂

—

143

C₂

#20295657

yes

direct product factor

144

C₂

C₂³×Dic₂₁₈

yes

characteristic

145

C₂

C₂×#16812744

yes

characteristic

146

C₂

C₂×#16812747

yes

characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₀₉
commutator, cosocle	C₂²×C₁₀₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	C₂²×C₄⋊C₄×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂⁴
prime radical	C₂⁴×C₁₀₉	C₂³
hypercenter	C₂²×C₄⋊C₄	D₁₀₉
hypocenter	C₁₀₉	C₂×#544
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂³×D₁₀₉

commutator, cosocle

C₂²×C₁₀₉

C₂⁵

Frattini

C₂²

C₂⁴×D₁₀₉

Fitting

C₂²×C₄⋊C₄×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂⁴

prime radical

C₂⁴×C₁₀₉

C₂³

hypercenter

C₂²×C₄⋊C₄

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
1. #20295657	C₂	direct (N × H)
2. C₂×#16812698	C₂	semidirect
3. C₂×#16812690	C₂	semidirect
8. #16812698	C₂²	semidirect
9. #16812690	C₂²	semidirect
34. C₂²×C₄×C₁₀₉	Q₈	semidirect
48. C₂×C₄×C₁₀₉	C₂×Q₈	semidirect
69. C₂×C₁₀₉	#544	semidirect
74. C₁₀₉	C₂×#544	semidirect
102. C₂²×C₄	Dic₂₁₈	semidirect
127. C₂×C₄	C₂×Dic₂₁₈	semidirect
143. C₂	#20295657	direct (N × H)

H ≅ G/N

type

1. #20295657

C₂

direct (N × H)

2. C₂×#16812698

C₂

semidirect

3. C₂×#16812690

C₂

semidirect

8. #16812698

C₂²

semidirect

9. #16812690

C₂²

semidirect

34. C₂²×C₄×C₁₀₉

Q₈

semidirect

48. C₂×C₄×C₁₀₉

C₂×Q₈

semidirect

69. C₂×C₁₀₉

#544

semidirect

74. C₁₀₉

C₂×#544

semidirect

102. C₂²×C₄

Dic₂₁₈

semidirect

127. C₂×C₄

C₂×Dic₂₁₈

semidirect

143. C₂

#20295657

direct (N × H)

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

subgroup	index
#20295657	2
C₂×#16812698	2
C₂×#16812690	2
C₂×#16812678	2
C₂×#16812658	2
C₂²×C₄⋊C₄×C₁₀₉	2
C₂×#544	109

subgroup

index

C₂²×C₄⋊C₄×C₁₀₉

109

Subgroups — C₂×#20295657 #28376433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20295657 #28376433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295657 #28376433