Group #28371319 — subgroups

Subgroups — #28371319

Order 13952.

1068 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 237 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×#16812678	C₂	1	3	yes	maximal, complemented
2	#20294899	C₂	1	6	yes	maximal, complemented
3	#20294886	C₂	1	3	yes	maximal, complemented
4	#20294882	C₂	1	2	yes	maximal
5	C₂⁴×C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₈	C₂²	1	1	yes	characteristic, complemented
7	#16812678	—	2	12	no	—
8	C₂×C₄⋊Dic₁₀₉	C₂²	1	3	yes	complemented
9	C₂×#13182008	C₂²	1	6	yes	complemented
10	C₂×C₂²⋊Dic₁₀₉	C₂²	1	6	yes	—
11	C₂×C₂²⋊Dic₁₀₉	C₂²	1	6	yes	—
12	#16812606	C₂²	1	6	yes	complemented
13	C₂⁵×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₀₉	—	2	12	no	—
15	C₂³×C₄×C₁₀₉	C₂²	1	3	yes	—
16	C₂³×C₄×C₁₀₉	C₂²	1	3	yes	—
17	C₂×Dic₂₁₈	—	2	6	no	—
18	C₂²×Dic₁₀₉	C₂³	1	6	yes	—
19	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
20	C₄⋊Dic₁₀₉	—	2	6	no	—
21	#13182008	—	2	12	no	—
22	C₂²⋊Dic₁₀₉	—	2	12	no	—
23	C₂²⋊Dic₁₀₉	—	2	12	no	—
24	C₂⁴×C₁₀₉	—	2	6	no	—
25	C₂⁴×C₁₀₉	—	2	6	no	—
26	C₂⁴×C₁₀₉	Q₈	1	4	yes	complemented
27	C₂⁴×C₁₀₉	C₂³	1	3	yes	—
28	C₂²×C₄×C₁₀₉	—	2	32	no	—
29	C₂²×C₄×C₁₀₉	—	2	6	no	—
30	C₂²×C₄×C₁₀₉	—	2	12	no	—
31	C₂²×C₄×C₁₀₉	—	2	12	no	—
32	C₂²×C₄×C₁₀₉	D₄	1	12	yes	—
33	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
34	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
35	C₂³×C₁₀₉	—	2	8	no	—
36	C₂³×C₁₀₉	—	2	24	no	—
37	C₂³×C₁₀₉	—	2	24	no	—
38	C₂³×C₁₀₉	—	2	6	no	—
39	C₂³×C₁₀₉	—	2	6	no	—
40	C₂³×C₁₀₉	C₂×D₄	1	6	yes	—
41	C₂³×C₁₀₉	C₂×Q₈	1	6	yes	—
42	C₂³×C₁₀₉	#41	1	6	yes	—
43	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	—	2	24	no	—
45	C₂×C₄×C₁₀₉	—	2	6	no	—
46	C₂×C₄×C₁₀₉	—	2	24	no	—
47	C₂×C₄×C₁₀₉	C₂×D₄	1	6	yes	—
48	C₂×C₄×C₁₀₉	C₂×D₄	1	6	yes	—
49	C₂×Dic₁₀₉	—	2	6	no	—
50	C₂×Dic₁₀₉	—	2	6	no	—
51	C₂×Dic₁₀₉	—	2	12	no	—
52	Dic₂₁₈	—	4	4	no	—
53	C₂²×C₁₀₉	—	2	32	no	—
54	C₂²×C₁₀₉	—	2	6	no	—
55	C₂²×C₁₀₉	—	2	12	no	—
56	C₂²×C₁₀₉	—	2	6	no	—
57	C₂²×C₁₀₉	—	2	12	no	—
58	C₂²×C₁₀₉	C₂²⋊Q₈	1	12	yes	complemented
59	C₂²×C₁₀₉	C₂²×D₄	1	3	yes	—
60	C₂²×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
61	C₂²×C₁₀₉	C₂×#41	1	3	yes	—
62	C₄×C₁₀₉	—	2	6	no	—
63	C₄×C₁₀₉	#120	1	4	yes	—
64	Dic₁₀₉	—	4	2	no	—
65	Dic₁₀₉	—	4	6	no	—
66	C₂×C₁₀₉	—	2	12	no	—
67	C₂×C₁₀₉	C₂×#120	1	1	yes	characteristic
68	C₂×C₁₀₉	C₂×C₂²⋊Q₈	1	3	yes	—
69	C₂×C₁₀₉	#525	1	3	yes	—
70	#2848	—	109	1	no	maximal, Sylow
71	C₁₀₉	#2848	1	1	yes	characteristic, complemented, Sylow
72	C₂⁴×C₄	D₁₀₉	1	1	yes	characteristic
73	C₂×C₂²⋊Q₈	—	109	3	no	—
74	#385	—	109	6	no	—
75	#380	—	109	3	no	—
76	#379	—	109	2	no	—
77	C₂⁵	C₂×D₁₀₉	1	1	yes	characteristic
78	C₂²×Q₈	—	109	1	no	—
79	C₂³×C₄	—	2	12	no	—
80	C₂³×C₄	C₂×D₁₀₉	1	3	yes	—
81	C₂³×C₄	C₂×D₁₀₉	1	3	yes	—
82	C₂²⋊Q₈	—	218	12	no	—
83	C₂×C₄⋊C₄	—	109	6	no	—
84	C₂×C₄⋊C₄	—	109	3	no	—
85	C₂×C₂²⋊C₄	—	109	6	no	—
86	C₂×C₂²⋊C₄	—	109	6	no	—
87	#95	—	109	6	no	—
88	C₂⁴	—	2	6	no	—
89	C₂⁴	—	2	6	no	—
90	C₂⁴	Dic₂₁₈	1	4	yes	complemented
91	C₂⁴	C₂²×D₁₀₉	1	3	yes	—
92	C₂×Q₈	—	218	6	no	—
93	C₂²×C₄	—	2	32	no	—
94	C₂²×C₄	—	2	6	no	—
95	C₂²×C₄	—	2	12	no	—
96	C₂²×C₄	—	2	12	no	—
97	C₂²×C₄	—	109	6	no	—
98	C₂²×C₄	—	109	2	no	—
99	C₂²×C₄	C₂²⋊D₁₀₉	1	12	yes	—
100	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
101	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
102	C₄⋊C₄	—	218	12	no	—
103	C₄⋊C₄	—	218	6	no	—
104	C₂²⋊C₄	—	218	12	no	—
105	C₂²⋊C₄	—	218	12	no	—
106	C₂³	—	2	8	no	—
107	C₂³	—	2	24	no	—
108	C₂³	—	2	24	no	—
109	C₂³	—	2	6	no	—
110	C₂³	—	2	6	no	—
111	C₂³	C₂×C₂²⋊D₁₀₉	1	6	yes	—
112	C₂³	C₂×Dic₂₁₈	1	6	yes	—
113	C₂³	#13182029	1	6	yes	—
114	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
115	Q₈	—	436	4	no	—
116	C₂×C₄	—	2	24	no	—
117	C₂×C₄	—	2	6	no	—
118	C₂×C₄	—	2	24	no	—
119	C₂×C₄	—	218	6	no	—
120	C₂×C₄	—	218	6	no	—
121	C₂×C₄	—	218	12	no	—
122	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	6	yes	—
123	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	6	yes	—
124	C₂²	—	2	32	no	—
125	C₂²	—	2	6	no	—
126	C₂²	—	2	12	no	—
127	C₂²	—	2	6	no	—
128	C₂²	—	2	12	no	—
129	C₂²	#16812678	1	12	yes	complemented
130	C₂²	C₂²×C₂²⋊D₁₀₉	1	3	yes	—
131	C₂²	C₂²×Dic₂₁₈	1	1	yes	characteristic
132	C₂²	C₂×#13182029	1	3	yes	—
133	C₄	—	2	6	no	—
134	C₄	—	436	2	no	—
135	C₄	—	436	6	no	—
136	C₄	#16812668	1	4	yes	—
137	C₂	—	2	12	no	—
138	C₂	C₂×#16812668	1	1	yes	characteristic
139	C₂	C₂×#16812678	1	3	yes	—
140	C₂	#20295537	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812678

C₂

yes

maximal, complemented

#20294899

C₂

yes

maximal, complemented

#20294886

C₂

yes

maximal, complemented

#20294882

C₂

yes

maximal

C₂⁴×C₄×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

C₂²

yes

characteristic, complemented

#16812678

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

complemented

C₂×#13182008

C₂²

yes

complemented

C₂×C₂²⋊Dic₁₀₉

C₂²

yes

—

C₂×C₂²⋊Dic₁₀₉

C₂²

yes

—

#16812606

C₂²

yes

complemented

C₂⁵×C₁₀₉

C₂²

yes

characteristic

C₂³×C₄×C₁₀₉

—

C₂³×C₄×C₁₀₉

C₂²

yes

—

C₂³×C₄×C₁₀₉

C₂²

yes

—

C₂×Dic₂₁₈

—

C₂²×Dic₁₀₉

C₂³

yes

—

C₂²×Dic₁₀₉

C₂³

yes

—

C₄⋊Dic₁₀₉

—

#13182008

—

C₂²⋊Dic₁₀₉

—

C₂²⋊Dic₁₀₉

—

C₂⁴×C₁₀₉

—

C₂⁴×C₁₀₉

—

C₂⁴×C₁₀₉

Q₈

yes

complemented

C₂⁴×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

—

C₂³×C₁₀₉

C₂×Q₈

yes

—

C₂³×C₁₀₉

#41

yes

—

C₂³×C₁₀₉

C₂⁴

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²⋊Q₈

yes

complemented

C₂²×C₁₀₉

C₂²×D₄

yes

—

C₂²×C₁₀₉

C₂²×Q₈

yes

characteristic

C₂²×C₁₀₉

C₂×#41

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#120

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×#120

yes

characteristic

C₂×C₁₀₉

C₂×C₂²⋊Q₈

yes

—

C₂×C₁₀₉

#525

yes

—

#2848

—

109

maximal, Sylow

C₁₀₉

#2848

yes

characteristic, complemented, Sylow

C₂⁴×C₄

D₁₀₉

yes

characteristic

C₂×C₂²⋊Q₈

—

109

—

#385

—

109

—

#380

—

109

—

#379

—

109

—

C₂⁵

C₂×D₁₀₉

yes

characteristic

C₂²×Q₈

—

109

—

C₂³×C₄

—

C₂³×C₄

C₂×D₁₀₉

yes

—

C₂³×C₄

C₂×D₁₀₉

yes

—

C₂²⋊Q₈

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

#95

—

109

—

C₂⁴

—

C₂⁴

—

C₂⁴

Dic₂₁₈

yes

complemented

C₂⁴

C₂²×D₁₀₉

yes

—

C₂×Q₈

—

218

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

100

C₂²×C₄

C₂²×D₁₀₉

yes

—

101

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

102

C₄⋊C₄

—

218

—

103

C₄⋊C₄

—

218

—

104

C₂²⋊C₄

—

218

—

105

C₂²⋊C₄

—

218

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

C₂×C₂²⋊D₁₀₉

yes

—

112

C₂³

C₂×Dic₂₁₈

yes

—

113

C₂³

#13182029

yes

—

114

C₂³

C₂³×D₁₀₉

yes

characteristic

115

Q₈

—

436

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

218

—

120

C₂×C₄

—

218

—

121

C₂×C₄

—

218

—

122

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

123

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#16812678

yes

complemented

130

C₂²

C₂²×C₂²⋊D₁₀₉

yes

—

131

C₂²

C₂²×Dic₂₁₈

yes

characteristic

132

C₂²

C₂×#13182029

yes

—

133

C₄

—

134

C₄

—

436

—

135

C₄

—

436

—

136

C₄

#16812668

yes

—

137

C₂

—

138

C₂

C₂×#16812668

yes

characteristic

139

C₂

C₂×#16812678

yes

—

140

C₂

#20295537

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₉
commutator, socle, cosocle	C₂³×C₁₀₉	C₂⁴
Fitting	C₂⁴×C₄×C₁₀₉	C₂
prime radical	C₂⁵×C₁₀₉	C₂²
hypercenter	C₂⁴×C₄	D₁₀₉
hypocenter	C₁₀₉	#2848
LCS	G ▷ C₂³×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₀₉

commutator, socle, cosocle

C₂³×C₁₀₉

C₂⁴

Fitting

C₂⁴×C₄×C₁₀₉

C₂

prime radical

C₂⁵×C₁₀₉

C₂²

hypercenter

C₂⁴×C₄

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ C₂⁴×C₄

N	H ≅ G/N	type
1. C₂×#16812678	C₂	semidirect
2. #20294899	C₂	semidirect
3. #20294886	C₂	semidirect
6. C₂²×Dic₂₁₈	C₂²	semidirect
8. C₂×C₄⋊Dic₁₀₉	C₂²	semidirect
9. C₂×#13182008	C₂²	semidirect
12. #16812606	C₂²	semidirect
26. C₂⁴×C₁₀₉	Q₈	semidirect
58. C₂²×C₁₀₉	C₂²⋊Q₈	semidirect
71. C₁₀₉	#2848	semidirect
90. C₂⁴	Dic₂₁₈	semidirect
129. C₂²	#16812678	semidirect

H ≅ G/N

type

1. C₂×#16812678

C₂

semidirect

2. #20294899

C₂

semidirect

3. #20294886

C₂

semidirect

6. C₂²×Dic₂₁₈

C₂²

semidirect

8. C₂×C₄⋊Dic₁₀₉

C₂²

semidirect

9. C₂×#13182008

C₂²

semidirect

12. #16812606

C₂²

semidirect

26. C₂⁴×C₁₀₉

Q₈

semidirect

58. C₂²×C₁₀₉

C₂²⋊Q₈

semidirect

71. C₁₀₉

#2848

semidirect

90. C₂⁴

Dic₂₁₈

semidirect

129. C₂²

#16812678

semidirect

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

subgroup	index
C₂×#16812678	2
#20294899	2
#20294886	2
#20294882	2
C₂⁴×C₄×C₁₀₉	2
#2848	109

subgroup

index

C₂⁴×C₄×C₁₀₉

109

Subgroups — #28371319

Conjugacy classes of subgroups

Complements

Maximal subgroups