Group #28371966 — subgroups

Subgroups — #28371966

Order 13952.

364 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 197 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	#20295583	C₂	1	1	yes	maximal, characteristic
2	#20295156	C₂	1	4	yes	maximal, complemented
3	C₈×Dic₂₁₈	C₂	1	4	yes	maximal, complemented
4	#20295052	C₂	1	2	yes	maximal
5	#20295050	C₂	1	1	yes	maximal, characteristic
6	#20294981	C₂	1	2	yes	maximal
7	C₂×C₄×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13182013	C₂²	1	2	yes	—
9	#16812699	C₄	1	2	yes	—
10	#16812691	C₄	1	2	yes	—
11	#16812678	C₄	1	4	yes	—
12	C₄×Dic₂₁₈	C₂²	1	4	yes	—
13	#16812658	C₂²	1	2	yes	—
14	C₄⋊#11724784	C₂²	1	4	yes	—
15	#16812628	C₂²	1	8	yes	—
16	#16812609	C₂²	1	4	yes	—
17	C₈×Dic₁₀₉	C₂²	1	4	yes	—
18	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
22	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
23	C₂×Dic₂₁₈	C₂×C₄	1	4	yes	—
24	#13182013	—	2	4	no	—
25	C₂×#11724784	C₂³	1	4	yes	—
26	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
27	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
28	#13182008	C₂×C₄	1	8	yes	—
29	C₂²⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
30	C₄×Dic₁₀₉	C₂³	1	4	yes	—
31	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₁₀₉	—	2	2	no	—
34	C₂×C₈×C₁₀₉	Q₈	1	4	yes	complemented
35	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
36	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
38	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₁₀₉	C₂³	1	2	yes	—
41	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	—	2	4	no	—
43	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₁₀₉	#41	1	2	yes	—
48	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
49	C₂×Dic₁₀₉	C₂²×C₄	1	8	yes	—
50	Dic₂₁₈	—	2	8	no	—
51	C₈×C₁₀₉	—	2	2	no	—
52	C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
53	C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
54	#11724784	—	2	4	no	—
55	C₂²×C₁₀₉	—	2	2	no	—
56	C₂²×C₁₀₉	C₄×Q₈	1	2	yes	—
57	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
58	C₄×C₁₀₉	—	2	2	no	—
59	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
60	C₄×C₁₀₉	#131	1	4	yes	—
61	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
62	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
63	Dic₁₀₉	—	2	8	no	—
64	C₂×C₁₀₉	—	2	2	no	—
65	C₂×C₁₀₉	C₂×C₄×Q₈	1	1	yes	characteristic
66	C₂×C₁₀₉	C₂×#131	1	2	yes	—
67	#2960	—	109	1	no	maximal, Sylow
68	C₁₀₉	#2960	1	1	yes	characteristic, complemented, Sylow
69	#533	—	109	1	no	—
70	#446	—	109	4	no	—
71	C₈×Q₈	—	109	4	no	—
72	#424	—	109	2	no	—
73	#423	—	109	1	no	—
74	#405	—	109	2	no	—
75	C₂×C₄×C₈	D₁₀₉	1	1	yes	characteristic
76	C₂×M₁₆	—	109	2	no	—
77	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
78	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
79	C₄⋊Q₈	—	109	2	no	—
80	#125	—	109	2	no	—
81	C₂²⋊Q₈	—	109	4	no	—
82	C₄×Q₈	—	109	4	no	—
83	#117	—	109	2	no	—
84	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
85	C₄⋊C₈	—	109	8	no	—
86	C₄⋊C₈	—	109	4	no	—
87	#97	—	109	4	no	—
88	C₄×C₈	—	109	4	no	—
89	C₄×C₈	C₂×D₁₀₉	1	2	yes	—
90	C₄×C₈	C₂×D₁₀₉	1	2	yes	—
91	C₂×Q₈	—	109	4	no	—
92	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
93	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
94	M₁₆	—	218	4	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	109	4	no	—
97	C₂×C₈	Dic₂₁₈	1	4	yes	complemented
98	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
99	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
100	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
101	C₄⋊C₄	—	109	8	no	—
102	C₄⋊C₄	—	109	4	no	—
103	C₄⋊C₄	—	109	4	no	—
104	C₂²⋊C₄	—	109	4	no	—
105	C₄²	—	109	4	no	—
106	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
107	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
108	C₄²	C₂²×D₁₀₉	1	2	yes	—
109	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
110	Q₈	—	218	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	109	8	no	—
113	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
114	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
116	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
117	C₂×C₄	#13182029	1	2	yes	—
118	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	—	218	4	no	—
121	C₈	C₂×Dic₂₁₈	1	2	yes	—
122	C₈	C₂×Dic₂₁₈	1	2	yes	—
123	C₂²	—	2	2	no	—
124	C₂²	C₄×Dic₂₁₈	1	2	yes	—
125	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	218	8	no	—
128	C₄	C₄×Dic₂₁₈	1	2	yes	—
129	C₄	#16812705	1	4	yes	—
130	C₄	C₂²×Dic₂₁₈	1	1	yes	characteristic
131	C₄	C₂×#13182029	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×C₄×Dic₂₁₈	1	1	yes	characteristic
134	C₂	C₂×#16812705	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295583

C₂

yes

maximal, characteristic

#20295156

C₂

yes

maximal, complemented

C₈×Dic₂₁₈

C₂

yes

maximal, complemented

#20295052

C₂

yes

maximal

#20295050

C₂

yes

maximal, characteristic

#20294981

C₂

yes

maximal

C₂×C₄×C₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

#16812699

C₄

yes

—

#16812691

C₄

yes

—

#16812678

C₄

yes

—

C₄×Dic₂₁₈

C₂²

yes

—

#16812658

C₂²

yes

—

C₄⋊#11724784

C₂²

yes

—

#16812628

C₂²

yes

—

#16812609

C₂²

yes

—

C₈×Dic₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₂²

yes

—

C₄×C₈×C₁₀₉

C₂²

yes

—

C₂×Dic₂₁₈

C₂×C₄

yes

—

#13182013

—

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182008

C₂×C₄

yes

—

C₂²⋊Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

Q₈

yes

complemented

C₂×C₈×C₁₀₉

C₂³

yes

—

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

C₂²×C₄

yes

—

Dic₂₁₈

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂×Q₈

yes

—

C₈×C₁₀₉

C₂×Q₈

yes

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄×Q₈

yes

—

C₂²×C₁₀₉

C₂³×C₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×Q₈

yes

—

C₄×C₁₀₉

#131

yes

—

C₄×C₁₀₉

C₂²×Q₈

yes

characteristic

C₄×C₁₀₉

C₂×#41

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₀₉

C₂×#131

yes

—

#2960

—

109

maximal, Sylow

C₁₀₉

#2960

yes

characteristic, complemented, Sylow

#533

—

109

—

#446

—

109

—

C₈×Q₈

—

109

—

#424

—

109

—

#423

—

109

—

#405

—

109

—

C₂×C₄×C₈

D₁₀₉

yes

characteristic

C₂×M₁₆

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₄⋊Q₈

—

109

—

#125

—

109

—

C₂²⋊Q₈

—

109

—

C₄×Q₈

—

109

—

#117

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

C₄×C₈

C₂×D₁₀₉

yes

—

C₄×C₈

C₂×D₁₀₉

yes

—

C₂×Q₈

—

109

—

C₂²×C₄

C₄×D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

218

—

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

Dic₂₁₈

yes

complemented

C₂×C₈

C₂²×D₁₀₉

yes

—

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

100

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

101

C₄⋊C₄

—

109

—

102

C₄⋊C₄

—

109

—

103

C₄⋊C₄

—

109

—

104

C₂²⋊C₄

—

109

—

105

C₄²

—

109

—

106

C₄²

C₄×D₁₀₉

yes

characteristic

107

C₄²

C₄×D₁₀₉

yes

characteristic

108

C₄²

C₂²×D₁₀₉

yes

—

109

C₂³

C₂×C₄×D₁₀₉

yes

characteristic

110

Q₈

—

218

—

111

C₂×C₄

—

112

C₂×C₄

—

109

—

113

C₂×C₄

C₂×C₄×D₁₀₉

yes

—

114

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

115

C₂×C₄

C₂×C₄×D₁₀₉

yes

—

116

C₂×C₄

C₂×Dic₂₁₈

yes

—

117

C₂×C₄

#13182029

yes

—

118

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

119

C₈

—

120

C₈

—

218

—

121

C₈

C₂×Dic₂₁₈

yes

—

122

C₈

C₂×Dic₂₁₈

yes

—

123

C₂²

—

124

C₂²

C₄×Dic₂₁₈

yes

—

125

C₂²

C₂²×C₄×D₁₀₉

yes

characteristic

126

C₄

—

127

C₄

—

218

—

128

C₄

C₄×Dic₂₁₈

yes

—

129

C₄

#16812705

yes

—

130

C₄

C₂²×Dic₂₁₈

yes

characteristic

131

C₄

C₂×#13182029

yes

characteristic

132

C₂

—

133

C₂

C₂×C₄×Dic₂₁₈

yes

characteristic

134

C₂

C₂×#16812705

yes

—

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×C₄×C₈×C₁₀₉	C₂
prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₂×C₄×C₈	D₁₀₉
hypocenter	C₁₀₉	#2960
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₈

C₂²×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

C₂×C₄×C₈×C₁₀₉

C₂

prime radical

C₂³×C₁₀₉

C₂²×C₄

cosocle

C₂×C₄×C₁₀₉

C₂⁴

hypercenter

C₂×C₄×C₈

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #20295156	C₂	semidirect
3. C₈×Dic₂₁₈	C₂	semidirect
34. C₂×C₈×C₁₀₉	Q₈	semidirect
68. C₁₀₉	#2960	semidirect
97. C₂×C₈	Dic₂₁₈	semidirect

H ≅ G/N

type

2. #20295156

C₂

semidirect

3. C₈×Dic₂₁₈

C₂

semidirect

34. C₂×C₈×C₁₀₉

Q₈

semidirect

68. C₁₀₉

#2960

semidirect

97. C₂×C₈

Dic₂₁₈

semidirect

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

subgroup	index
#20295583	2
#20295156	2
C₈×Dic₂₁₈	2
#20295052	2
#20295050	2
#20294981	2
C₂×C₄×C₈×C₁₀₉	2
#2960	109

subgroup

index

C₈×Dic₂₁₈

C₂×C₄×C₈×C₁₀₉

109

Subgroups — #28371966

Conjugacy classes of subgroups

Complements

Maximal subgroups