Group #28371378 — subgroups

Subgroups — #28371378

Order 13952.

924 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 209 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×#16812696	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16812674	C₂	1	6	yes	maximal, complemented
3	C₂×C₄²×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294927	C₂	1	6	yes	maximal, complemented
5	#384×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄₃₆	C₂²	1	6	yes	—
7	C₂²×D₄₃₆	C₂²	1	4	yes	—
8	C₂²×C₄×D₁₀₉	C₂²	1	3	yes	—
9	#16812696	—	2	4	no	—
10	#16812674	—	2	24	no	—
11	C₂×#13182006	C₂²	1	12	yes	—
12	C₂×C₄×Dic₁₀₉	C₂²	1	3	yes	—
13	C₄²×D₁₀₉	—	2	4	no	—
14	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	6	yes	—
15	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂³×D₁₀₉	C₂³	1	4	yes	—
17	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
18	C₂×D₄₃₆	—	2	24	no	—
19	C₂×D₄₃₆	—	2	12	no	—
20	C₂×D₄₃₆	—	2	12	no	—
21	C₂×D₄₃₆	—	2	12	no	—
22	C₂²×Dic₁₀₉	C₂³	1	3	yes	—
23	C₂×C₄×D₁₀₉	—	2	6	no	—
24	C₂×C₄×D₁₀₉	—	2	6	no	—
25	C₂×C₄×D₁₀₉	D₄	1	12	yes	complemented
26	#13182006	—	2	24	no	—
27	C₄×Dic₁₀₉	—	2	6	no	—
28	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
29	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
30	C₄⋊C₄×C₁₀₉	—	2	12	no	—
31	C₄²×C₁₀₉	D₄	1	4	yes	complemented
32	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
33	C₂²×D₁₀₉	—	2	1	no	—
34	C₂²×D₁₀₉	—	2	3	no	—
35	C₂²×D₁₀₉	—	2	12	no	—
36	C₂²×D₁₀₉	—	2	4	no	—
37	C₂²×D₁₀₉	—	2	12	no	—
38	C₂²×D₁₀₉	C₂×D₄	1	6	yes	—
39	C₂×C₄×C₁₀₉	—	2	12	no	—
40	C₂×C₄×C₁₀₉	C₂×D₄	1	6	yes	—
41	C₂×C₄×C₁₀₉	C₂×D₄	1	6	yes	—
42	C₂×C₄×C₁₀₉	#41	1	6	yes	—
43	C₂×Dic₁₀₉	—	2	3	no	—
44	C₂×Dic₁₀₉	—	2	3	no	—
45	C₂×Dic₁₀₉	C₂×D₄	1	6	yes	—
46	D₄₃₆	—	4	16	no	—
47	D₄₃₆	—	4	24	no	—
48	C₄×D₁₀₉	—	2	24	no	—
49	C₂²×C₁₀₉	C₂²×D₄	1	3	yes	—
50	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
51	C₂²×C₁₀₉	C₂×#41	1	3	yes	—
52	C₂×D₁₀₉	—	2	6	no	—
53	C₂×D₁₀₉	—	2	6	no	—
54	C₂×D₁₀₉	—	4	12	no	—
55	C₂×D₁₀₉	—	4	4	no	—
56	C₂×D₁₀₉	—	4	12	no	—
57	C₂×D₁₀₉	#127	1	4	yes	complemented
58	C₄×C₁₀₉	—	4	4	no	—
59	C₄×C₁₀₉	#121	1	12	yes	complemented
60	Dic₁₀₉	—	2	6	no	—
61	C₂×C₁₀₉	C₂×#121	1	3	yes	—
62	C₂×C₁₀₉	C₂×#127	1	1	yes	characteristic
63	C₂×C₁₀₉	#532	1	3	yes	—
64	D₁₀₉	—	2	4	no	—
65	D₁₀₉	—	8	4	no	—
66	#2856	—	109	1	no	maximal, Sylow
67	C₁₀₉	#2856	1	1	yes	characteristic, complemented, Sylow
68	C₂×#127	—	109	1	no	—
69	C₂×#121	—	109	6	no	—
70	C₂²×C₄²	—	109	1	no	—
71	#390	—	109	6	no	—
72	#384	D₁₀₉	1	1	yes	characteristic, complemented
73	C₂²×D₄	—	109	4	no	—
74	C₂²×D₄	—	109	6	no	—
75	C₂³×C₄	—	109	3	no	—
76	#127	—	218	4	no	—
77	#121	—	218	24	no	—
78	C₂×C₄⋊C₄	C₂×D₁₀₉	1	6	yes	—
79	C₂×C₂²⋊C₄	—	109	12	no	—
80	C₂×C₄²	—	109	3	no	—
81	C₂×C₄²	—	218	4	no	—
82	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
83	C₂⁴	—	109	1	no	—
84	C₂⁴	—	109	4	no	—
85	C₂×D₄	—	218	24	no	—
86	C₂×D₄	—	218	12	no	—
87	C₂×D₄	—	218	12	no	—
88	C₂×D₄	—	218	12	no	—
89	C₂²×C₄	—	109	3	no	—
90	C₂²×C₄	—	109	12	no	—
91	C₂²×C₄	—	218	6	no	—
92	C₂²×C₄	—	218	6	no	—
93	C₂²×C₄	C₂²×D₁₀₉	1	4	yes	—
94	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
95	C₄⋊C₄	—	2	12	no	—
96	C₂²⋊C₄	—	218	24	no	—
97	C₄²	—	218	6	no	—
98	C₄²	D₄₃₆	1	4	yes	complemented
99	C₂³	—	109	6	no	—
100	C₂³	—	218	12	no	—
101	C₂³	—	218	4	no	—
102	C₂³	—	218	12	no	—
103	C₂³	—	218	1	no	—
104	C₂³	—	218	3	no	—
105	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
106	D₄	—	436	16	no	—
107	D₄	—	436	24	no	—
108	C₂×C₄	—	2	12	no	—
109	C₂×C₄	—	109	6	no	—
110	C₂×C₄	—	218	3	no	—
111	C₂×C₄	—	218	3	no	—
112	C₂×C₄	—	218	24	no	—
113	C₂×C₄	D₄×D₁₀₉	1	6	yes	—
114	C₂×C₄	C₂×D₄₃₆	1	6	yes	—
115	C₂×C₄	#13182030	1	6	yes	—
116	C₂²	—	109	4	no	—
117	C₂²	—	436	12	no	—
118	C₂²	—	436	4	no	—
119	C₂²	—	436	12	no	—
120	C₂²	—	218	6	no	—
121	C₂²	—	218	6	no	—
122	C₂²	C₂×D₄×D₁₀₉	1	3	yes	—
123	C₂²	C₂²×D₄₃₆	1	1	yes	characteristic
124	C₂²	C₂×#13182030	1	3	yes	—
125	C₄	—	4	4	no	—
126	C₄	—	218	6	no	—
127	C₄	#16812674	1	12	yes	complemented
128	C₂	—	872	4	no	—
129	C₂	—	218	4	no	—
130	C₂	C₂×#16812674	1	3	yes	—
131	C₂	#127×D₁₀₉	1	1	yes	characteristic
132	C₂	#20295577	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812696

C₂

yes

maximal, characteristic, complemented

C₂×#16812674

C₂

yes

maximal, complemented

C₂×C₄²×D₁₀₉

C₂

yes

maximal, characteristic, complemented

#20294927

C₂

yes

maximal, complemented

#384×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₃₆

C₂²

yes

—

C₂²×D₄₃₆

C₂²

yes

—

C₂²×C₄×D₁₀₉

C₂²

yes

—

#16812696

—

#16812674

—

C₂×#13182006

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

C₄²×D₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂³×D₁₀₉

C₂³

yes

—

C₂³×D₁₀₉

C₂³

yes

characteristic

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂²×Dic₁₀₉

C₂³

yes

—

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

D₄

yes

complemented

#13182006

—

C₄×Dic₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

complemented

C₂³×C₁₀₉

C₂⁴

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×D₄

yes

—

D₄₃₆

—

D₄₃₆

—

C₄×D₁₀₉

—

C₂²×C₁₀₉

C₂²×D₄

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₂²×C₁₀₉

C₂×#41

yes

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

#127

yes

complemented

C₄×C₁₀₉

—

C₄×C₁₀₉

#121

yes

complemented

Dic₁₀₉

—

C₂×C₁₀₉

C₂×#121

yes

—

C₂×C₁₀₉

C₂×#127

yes

characteristic

C₂×C₁₀₉

#532

yes

—

D₁₀₉

—

D₁₀₉

—

#2856

—

109

maximal, Sylow

C₁₀₉

#2856

yes

characteristic, complemented, Sylow

C₂×#127

—

109

—

C₂×#121

—

109

—

C₂²×C₄²

—

109

—

#390

—

109

—

#384

D₁₀₉

yes

characteristic, complemented

C₂²×D₄

—

109

—

C₂²×D₄

—

109

—

C₂³×C₄

—

109

—

#127

—

218

—

#121

—

218

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

218

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₂⁴

—

109

—

C₂⁴

—

109

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

218

—

C₄²

—

218

—

C₄²

D₄₃₆

yes

complemented

C₂³

—

109

—

100

C₂³

—

218

—

101

C₂³

—

218

—

102

C₂³

—

218

—

103

C₂³

—

218

—

104

C₂³

—

218

—

105

C₂³

C₂³×D₁₀₉

yes

characteristic

106

D₄

—

436

—

107

D₄

—

436

—

108

C₂×C₄

—

109

C₂×C₄

—

109

—

110

C₂×C₄

—

218

—

111

C₂×C₄

—

218

—

112

C₂×C₄

—

218

—

113

C₂×C₄

D₄×D₁₀₉

yes

—

114

C₂×C₄

C₂×D₄₃₆

yes

—

115

C₂×C₄

#13182030

yes

—

116

C₂²

—

109

—

117

C₂²

—

436

—

118

C₂²

—

436

—

119

C₂²

—

436

—

120

C₂²

—

218

—

121

C₂²

—

218

—

122

C₂²

C₂×D₄×D₁₀₉

yes

—

123

C₂²

C₂²×D₄₃₆

yes

characteristic

124

C₂²

C₂×#13182030

yes

—

125

C₄

—

126

C₄

—

218

—

127

C₄

#16812674

yes

complemented

128

C₂

—

872

—

129

C₂

—

218

—

130

C₂

C₂×#16812674

yes

—

131

C₂

#127×D₁₀₉

yes

characteristic

132

C₂

#20295577

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₉
commutator, socle, cosocle	C₂³×C₁₀₉	C₂⁴
Fitting	#384×C₁₀₉	C₂
hypercenter	#384	D₁₀₉
hypocenter	C₁₀₉	#2856
LCS	G ▷ C₂³×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₀₉

commutator, socle, cosocle

C₂³×C₁₀₉

C₂⁴

Fitting

#384×C₁₀₉

C₂

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
1. C₂×#16812696	C₂	semidirect
2. C₂×#16812674	C₂	semidirect
3. C₂×C₄²×D₁₀₉	C₂	semidirect
4. #20294927	C₂	semidirect
5. #384×C₁₀₉	C₂	semidirect
25. C₂×C₄×D₁₀₉	D₄	semidirect
31. C₄²×C₁₀₉	D₄	semidirect
57. C₂×D₁₀₉	#127	semidirect
59. C₄×C₁₀₉	#121	semidirect
67. C₁₀₉	#2856	semidirect
72. #384	D₁₀₉	semidirect
98. C₄²	D₄₃₆	semidirect
127. C₄	#16812674	semidirect

H ≅ G/N

type

1. C₂×#16812696

C₂

semidirect

2. C₂×#16812674

C₂

semidirect

3. C₂×C₄²×D₁₀₉

C₂

semidirect

4. #20294927

C₂

semidirect

5. #384×C₁₀₉

C₂

semidirect

25. C₂×C₄×D₁₀₉

D₄

semidirect

31. C₄²×C₁₀₉

D₄

semidirect

57. C₂×D₁₀₉

#127

semidirect

59. C₄×C₁₀₉

#121

semidirect

67. C₁₀₉

#2856

semidirect

72. #384

D₁₀₉

semidirect

98. C₄²

D₄₃₆

semidirect

127. C₄

#16812674

semidirect

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

subgroup	index
C₂×#16812696	2
C₂×#16812674	2
C₂×C₄²×D₁₀₉	2
#20294927	2
#384×C₁₀₉	2
#2856	109

subgroup

index

C₂×C₄²×D₁₀₉

#384×C₁₀₉

109

Subgroups — #28371378

Conjugacy classes of subgroups

Complements

Maximal subgroups