Group #28362882 — subgroups

Subgroups — C₄×#16812628 #28362882

Order 13952.

316 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 181 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×#16812628	C₂	1	2	yes	maximal
2	C₂×C₄×#11724784	C₂	1	1	yes	maximal, characteristic
3	C₄²×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#20294780	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11724784	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₀₉	C₄	1	2	yes	—
8	C₂×C₄×Dic₁₀₉	C₄	1	2	yes	—
9	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
11	#16812628	C₄	1	8	yes	direct product factor
12	C₄×#11724784	—	2	2	no	—
13	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₀₉	—	2	2	no	—
16	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	—
18	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂×#11724784	—	2	2	no	—
20	C₂×#11724784	—	2	2	no	—
21	C₂×#11724784	C₂×C₄	1	4	yes	—
22	C₄×Dic₁₀₉	—	2	4	no	—
23	C₄×Dic₁₀₉	—	2	2	no	—
24	C₄×Dic₁₀₉	C₈	1	8	yes	complemented
25	C₄×Dic₁₀₉	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₀₉	—	2	2	no	—
29	C₂×C₈×C₁₀₉	—	2	2	no	—
30	C₂×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
31	C₄²×C₁₀₉	D₄	1	2	yes	—
32	C₄²×C₁₀₉	Q₈	1	2	yes	—
33	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
35	C₂×C₄×C₁₀₉	C₂×C₈	1	4	yes	—
36	C₂×C₄×C₁₀₉	M₁₆	1	4	yes	—
37	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	#41	1	2	yes	—
41	C₂×Dic₁₀₉	—	2	2	no	—
42	C₂×Dic₁₀₉	—	2	2	no	—
43	C₂×Dic₁₀₉	—	2	2	no	—
44	C₂×Dic₁₀₉	C₄²	1	2	yes	—
45	C₂×Dic₁₀₉	C₄²	1	2	yes	—
46	C₂×Dic₁₀₉	C₂×C₈	1	4	yes	—
47	C₂×Dic₁₀₉	C₂×C₈	1	4	yes	—
48	C₈×C₁₀₉	—	2	4	no	—
49	#11724784	—	2	4	no	—
50	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
51	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₁₀₉	#117	1	1	yes	characteristic
53	C₂²×C₁₀₉	C₂²×C₈	1	2	yes	—
54	C₂²×C₁₀₉	C₂×M₁₆	1	2	yes	—
55	C₄×C₁₀₉	C₄⋊C₈	1	8	yes	complemented
56	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
57	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
58	Dic₁₀₉	—	2	2	no	—
59	Dic₁₀₉	—	2	4	no	—
60	Dic₁₀₉	C₄×C₈	1	4	yes	complemented
61	C₂×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
63	C₂×C₁₀₉	C₄×M₁₆	1	1	yes	characteristic
64	C₂×C₁₀₉	C₂×C₄⋊C₈	1	2	yes	—
65	C₂×C₁₀₉	#431	1	2	yes	—
66	C₄×C₄⋊C₈	—	109	1	no	maximal, Sylow
67	C₁₀₉	C₄×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₄⋊C₈	—	109	2	no	—
69	C₂×C₄×C₈	—	109	1	no	—
70	C₂×C₄×C₈	D₁₀₉	1	1	yes	characteristic
71	C₄³	—	109	1	no	—
72	#336	—	109	2	no	—
73	C₂²×C₈	—	109	2	no	—
74	C₂²×C₈	C₂×D₁₀₉	1	2	yes	—
75	C₂×C₄²	—	109	2	no	—
76	C₂×C₄²	—	109	1	no	—
77	C₂×C₄²	—	109	2	no	—
78	C₂×C₄²	—	109	1	no	—
79	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
80	C₄⋊C₈	—	109	8	no	—
81	C₄×C₈	—	2	2	no	—
82	C₄×C₈	—	218	2	no	—
83	C₂²×C₄	—	109	1	no	—
84	C₂²×C₄	—	109	1	no	—
85	C₂²×C₄	—	109	2	no	—
86	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
87	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	109	4	no	—
91	C₂×C₈	—	218	2	no	—
92	C₂×C₈	—	218	2	no	—
93	C₂×C₈	C₄×D₁₀₉	1	4	yes	—
94	C₄²	—	109	8	no	—
95	C₄²	—	109	4	no	—
96	C₄²	—	218	4	no	—
97	C₄²	—	218	2	no	—
98	C₄²	Dic₂₁₈	1	2	yes	—
99	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
100	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
101	C₂×C₄	—	109	2	no	—
102	C₂×C₄	—	109	4	no	—
103	C₂×C₄	—	109	2	no	—
104	C₂×C₄	—	109	4	no	—
105	C₂×C₄	—	218	2	no	—
106	C₂×C₄	—	218	2	no	—
107	C₂×C₄	—	218	2	no	—
108	C₂×C₄	#13182008	1	4	yes	—
109	C₂×C₄	C₈×D₁₀₉	1	4	yes	—
110	C₂×C₄	#13182012	1	4	yes	—
111	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
113	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
114	C₂×C₄	#13182029	1	2	yes	—
115	C₈	—	2	4	no	—
116	C₈	—	218	4	no	—
117	C₂²	C₄²×D₁₀₉	1	1	yes	characteristic
118	C₂²	C₂×#13182008	1	1	yes	characteristic
119	C₂²	#16812656	1	1	yes	characteristic
120	C₂²	C₂×C₈×D₁₀₉	1	2	yes	—
121	C₂²	C₂×#13182012	1	2	yes	—
122	C₄	—	109	4	no	—
123	C₄	—	218	2	no	—
124	C₄	—	218	4	no	—
125	C₄	#16812628	1	8	yes	direct product factor
126	C₄	C₄×C₂²⋊D₁₀₉	1	2	yes	—
127	C₄	C₄×Dic₂₁₈	1	2	yes	—
128	C₂	C₄×#13182008	1	1	yes	characteristic
129	C₂	C₄×C₈×D₁₀₉	1	1	yes	characteristic
130	C₂	C₄×#13182012	1	1	yes	characteristic
131	C₂	C₂×#16812628	1	2	yes	—
132	C₂	#20295077	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812628

C₂

yes

maximal

C₂×C₄×#11724784

C₂

yes

maximal, characteristic

C₄²×Dic₁₀₉

C₂

yes

maximal, characteristic

#20294780

C₂

yes

maximal

C₂×C₄×C₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₄

yes

—

C₂×C₄×Dic₁₀₉

C₄

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812628

C₄

yes

direct product factor

C₄×#11724784

—

C₂²×C₈×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

—

C₂²×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₉

C₂×C₄

yes

—

C₂²×Dic₁₀₉

C₂×C₄

yes

—

—

C₂×C₄

yes

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

C₈

yes

complemented

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

Q₈

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

M₁₆

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄²

yes

—

C₂×Dic₁₀₉

C₄²

yes

—

C₂×Dic₁₀₉

C₂×C₈

yes

—

C₂×Dic₁₀₉

C₂×C₈

yes

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×C₄²

yes

characteristic

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

#117

yes

characteristic

C₂²×C₁₀₉

C₂²×C₈

yes

—

C₂²×C₁₀₉

C₂×M₁₆

yes

—

C₄×C₁₀₉

C₄⋊C₈

yes

complemented

C₄×C₁₀₉

C₄×D₄

yes

—

C₄×C₁₀₉

C₄×Q₈

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₄×C₈

yes

complemented

C₂×C₁₀₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₀₉

C₂×C₄×C₈

yes

characteristic

C₂×C₁₀₉

C₄×M₁₆

yes

characteristic

C₂×C₁₀₉

C₂×C₄⋊C₈

yes

—

C₂×C₁₀₉

#431

yes

—

C₄×C₄⋊C₈

—

109

maximal, Sylow

C₁₀₉

C₄×C₄⋊C₈

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

109

—

C₂×C₄×C₈

—

109

—

C₂×C₄×C₈

D₁₀₉

yes

characteristic

C₄³

—

109

—

#336

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

109

—

C₄×C₈

—

C₄×C₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₄×D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₄×D₁₀₉

yes

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

218

—

C₄²

—

218

—

C₄²

Dic₂₁₈

yes

—

C₄²

C₂²⋊D₁₀₉

yes

—

100

C₂³

C₂×C₄×D₁₀₉

yes

characteristic

101

C₂×C₄

—

109

—

102

C₂×C₄

—

109

—

103

C₂×C₄

—

109

—

104

C₂×C₄

—

109

—

105

C₂×C₄

—

218

—

106

C₂×C₄

—

218

—

107

C₂×C₄

—

218

—

108

C₂×C₄

#13182008

yes

—

109

C₂×C₄

C₈×D₁₀₉

yes

—

110

C₂×C₄

#13182012

yes

—

111

C₂×C₄

C₂×C₄×D₁₀₉

yes

—

112

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

113

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

114

C₂×C₄

#13182029

yes

—

115

C₈

—

116

C₈

—

218

—

117

C₂²

C₄²×D₁₀₉

yes

characteristic

118

C₂²

C₂×#13182008

yes

characteristic

119

C₂²

#16812656

yes

characteristic

120

C₂²

C₂×C₈×D₁₀₉

yes

—

121

C₂²

C₂×#13182012

yes

—

122

C₄

—

109

—

123

C₄

—

218

—

124

C₄

—

218

—

125

C₄

#16812628

yes

direct product factor

126

C₄

C₄×C₂²⋊D₁₀₉

yes

—

127

C₄

C₄×Dic₂₁₈

yes

—

128

C₂

C₄×#13182008

yes

characteristic

129

C₂

C₄×C₈×D₁₀₉

yes

characteristic

130

C₂

C₄×#13182012

yes

characteristic

131

C₂

C₂×#16812628

yes

—

132

C₂

#20295077

yes

—

	subgroup	G/H
center	C₂×C₄²	C₂×D₁₀₉
commutator	C₂×C₁₀₉	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×C₄×C₈×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×C₄×C₈	D₁₀₉
hypocenter	C₁₀₉	C₄×C₄⋊C₈
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄²

C₂×D₁₀₉

commutator

C₂×C₁₀₉

C₂×C₄×C₈

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂×C₄×C₈×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂²×C₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

hypercenter

C₂×C₄×C₈

D₁₀₉

hypocenter

C₁₀₉

C₄×C₄⋊C₈

LCS

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
11. #16812628	C₄	direct (N × H)
24. C₄×Dic₁₀₉	C₈	semidirect
55. C₄×C₁₀₉	C₄⋊C₈	semidirect
60. Dic₁₀₉	C₄×C₈	semidirect
67. C₁₀₉	C₄×C₄⋊C₈	semidirect
125. C₄	#16812628	direct (N × H)

H ≅ G/N

type

11. #16812628

C₄

direct (N × H)

24. C₄×Dic₁₀₉

C₈

semidirect

55. C₄×C₁₀₉

C₄⋊C₈

semidirect

60. Dic₁₀₉

C₄×C₈

semidirect

67. C₁₀₉

C₄×C₄⋊C₈

semidirect

125. C₄

#16812628

direct (N × H)

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

subgroup	index
C₂×#16812628	2
C₂×C₄×#11724784	2
C₄²×Dic₁₀₉	2
#20294780	2
C₂×C₄×C₈×C₁₀₉	2
C₄×C₄⋊C₈	109

subgroup

index

C₂×#16812628

C₂×C₄×#11724784

C₄²×Dic₁₀₉

#20294780

C₂×C₄×C₈×C₁₀₉

C₄×C₄⋊C₈

109

Subgroups — C₄×#16812628 #28362882

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#16812628 #28362882

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16812628 #28362882