Group #28373555 — subgroups

Subgroups — #28373555

Order 13952.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#20295579	C₂	1	2	yes	maximal
2	#20295434	C₂	1	4	yes	maximal, complemented
3	#20295433	C₂	1	4	yes	maximal, complemented
4	#20295064	C₂	1	1	yes	maximal, characteristic
5	#20295062	C₂	1	2	yes	maximal
6	C₂×#16812609	C₂	1	1	yes	maximal, characteristic
7	#427×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11724784	C₂²	1	1	yes	characteristic
9	#16812698	C₂²	1	4	yes	—
10	#16812690	C₂²	1	4	yes	—
11	#16812676	—	2	4	no	—
12	C₄×Dic₂₁₈	—	2	4	no	—
13	#16812658	C₂²	1	2	yes	—
14	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
15	#16812633	C₂²	1	2	yes	—
16	#16812632	C₂²	1	4	yes	—
17	#16812631	C₂²	1	2	yes	—
18	#16812630	C₂²	1	4	yes	—
19	#16812609	C₂²	1	2	yes	—
20	#16812609	C₂²	1	2	yes	—
21	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
22	#117×C₁₀₉	C₂²	1	2	yes	—
23	#107×C₁₀₉	C₂²	1	2	yes	—
24	#106×C₁₀₉	C₂²	1	2	yes	—
25	C₂×Dic₂₁₈	—	2	4	no	—
26	C₂²×Dic₁₀₉	D₄	1	2	yes	—
27	C₂×#11724784	Q₈	1	4	yes	complemented
28	C₂×#11724784	C₂³	1	2	yes	—
29	C₄⋊Dic₁₀₉	—	2	4	no	—
30	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
31	#13182008	—	2	4	no	—
32	#13182008	—	2	4	no	—
33	C₂²⋊Dic₁₀₉	—	2	2	no	—
34	C₄×Dic₁₀₉	—	2	2	no	—
35	C₄×Dic₁₀₉	D₄	1	2	yes	—
36	C₄×Dic₁₀₉	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₁₀₉	Q₈	1	4	yes	complemented
39	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
42	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
43	C₄²×C₁₀₉	—	2	2	no	—
44	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	—	2	4	no	—
46	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₁₀₉	—	2	4	no	—
51	C₂×Dic₁₀₉	—	4	2	no	—
52	C₂×Dic₁₀₉	C₂×D₄	1	4	yes	—
53	Dic₂₁₈	—	4	8	no	—
54	C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
55	C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
56	#11724784	C₂×Q₈	1	4	yes	—
57	C₂²×C₁₀₉	—	2	2	no	—
58	C₂²×C₁₀₉	C₄⋊Q₈	1	2	yes	—
59	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
60	C₄×C₁₀₉	—	4	4	no	—
61	C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
62	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
63	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
64	Dic₁₀₉	—	2	4	no	—
65	Dic₁₀₉	—	4	4	no	—
66	C₂×C₁₀₉	—	2	2	no	—
67	C₂×C₁₀₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₁₀₉	#575	1	2	yes	—
69	#3162	—	109	1	no	maximal, Sylow
70	C₁₀₉	#3162	1	1	yes	characteristic, complemented, Sylow
71	#533	—	109	2	no	—
72	#501	—	109	4	no	—
73	#501	—	109	4	no	—
74	#427	—	109	1	no	—
75	#427	—	109	2	no	—
76	#427	D₁₀₉	1	1	yes	characteristic
77	C₂×#97	—	109	1	no	—
78	C₂²×C₈	—	109	1	no	—
79	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
80	C₄⋊Q₈	—	109	4	no	—
81	#125	—	109	4	no	—
82	C₂²⋊Q₈	—	218	4	no	—
83	C₄×Q₈	—	218	4	no	—
84	#117	—	109	2	no	—
85	#117	C₂×D₁₀₉	1	2	yes	—
86	C₂×C₄²	—	109	1	no	—
87	#107	—	109	2	no	—
88	#107	—	109	4	no	—
89	#107	C₂×D₁₀₉	1	2	yes	—
90	#106	—	109	2	no	—
91	#106	—	109	4	no	—
92	#106	C₂×D₁₀₉	1	2	yes	—
93	#97	—	109	2	no	—
94	#97	—	109	2	no	—
95	C₂×Q₈	—	218	4	no	—
96	C₂²×C₄	—	109	2	no	—
97	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
98	C₂×C₈	—	109	4	no	—
99	C₂×C₈	—	109	2	no	—
100	C₂×C₈	Dic₂₁₈	1	4	yes	complemented
101	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
103	C₄⋊C₄	—	109	4	no	—
104	C₄⋊C₄	—	218	4	no	—
105	C₄⋊C₄	—	218	4	no	—
106	C₄⋊C₄	—	218	4	no	—
107	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₂²⋊C₄	—	218	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	109	2	no	—
112	C₄²	—	109	2	no	—
113	C₄²	—	218	2	no	—
114	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
115	Q₈	—	436	8	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	109	4	no	—
118	C₂×C₄	—	436	2	no	—
119	C₂×C₄	—	218	4	no	—
120	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
121	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
122	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
123	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
124	C₈	—	109	4	no	—
125	C₈	C₂×Dic₂₁₈	1	2	yes	—
126	C₈	C₂×Dic₂₁₈	1	2	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	#16812698	1	2	yes	—
129	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
130	C₄	—	4	4	no	—
131	C₄	—	436	4	no	—
132	C₄	—	218	4	no	—
133	C₄	#16812698	1	2	yes	—
134	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
135	C₄	C₂²×Dic₂₁₈	1	1	yes	characteristic
136	C₂	—	2	2	no	—
137	C₂	C₂×#16812698	1	1	yes	characteristic
138	C₂	#20295823	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295579

C₂

yes

maximal

#20295434

C₂

yes

maximal, complemented

#20295433

C₂

yes

maximal, complemented

#20295064

C₂

yes

maximal, characteristic

#20295062

C₂

yes

maximal

C₂×#16812609

C₂

yes

maximal, characteristic

#427×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

characteristic

#16812698

C₂²

yes

—

#16812690

C₂²

yes

—

#16812676

—

C₄×Dic₂₁₈

—

#16812658

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812633

C₂²

yes

—

#16812632

C₂²

yes

—

#16812631

C₂²

yes

—

#16812630

C₂²

yes

—

#16812609

C₂²

yes

—

#16812609

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

#117×C₁₀₉

C₂²

yes

—

#107×C₁₀₉

C₂²

yes

—

#106×C₁₀₉

C₂²

yes

—

C₂×Dic₂₁₈

—

C₂²×Dic₁₀₉

D₄

yes

—

C₂×#11724784

Q₈

yes

complemented

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182008

—

#13182008

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

—

C₄×Dic₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

Q₈

yes

complemented

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×D₄

yes

—

Dic₂₁₈

—

C₈×C₁₀₉

C₂×Q₈

yes

—

C₈×C₁₀₉

C₂×Q₈

yes

—

#11724784

C₂×Q₈

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄⋊Q₈

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄⋊Q₈

yes

—

C₄×C₁₀₉

C₂²×Q₈

yes

characteristic

C₄×C₁₀₉

C₂²×Q₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₀₉

#575

yes

—

#3162

—

109

maximal, Sylow

C₁₀₉

#3162

yes

characteristic, complemented, Sylow

#533

—

109

—

#501

—

109

—

#501

—

109

—

#427

—

109

—

#427

—

109

—

#427

D₁₀₉

yes

characteristic

C₂×#97

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₄⋊Q₈

—

109

—

#125

—

109

—

C₂²⋊Q₈

—

218

—

C₄×Q₈

—

218

—

#117

—

109

—

#117

C₂×D₁₀₉

yes

—

C₂×C₄²

—

109

—

#107

—

109

—

#107

—

109

—

#107

C₂×D₁₀₉

yes

—

#106

—

109

—

#106

—

109

—

#106

C₂×D₁₀₉

yes

—

#97

—

109

—

#97

—

109

—

C₂×Q₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

109

—

100

C₂×C₈

Dic₂₁₈

yes

complemented

101

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

102

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

103

C₄⋊C₄

—

109

—

104

C₄⋊C₄

—

218

—

105

C₄⋊C₄

—

218

—

106

C₄⋊C₄

—

218

—

107

C₄⋊C₄

C₂²×D₁₀₉

yes

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

218

—

110

C₄²

—

111

C₄²

—

109

—

112

C₄²

—

109

—

113

C₄²

—

218

—

114

C₂³

D₄×D₁₀₉

yes

characteristic

115

Q₈

—

436

—

116

C₂×C₄

—

117

C₂×C₄

—

109

—

118

C₂×C₄

—

436

—

119

C₂×C₄

—

218

—

120

C₂×C₄

D₄×D₁₀₉

yes

characteristic

121

C₂×C₄

Q₈×D₁₀₉

yes

—

122

C₂×C₄

C₂×Dic₂₁₈

yes

—

123

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

124

C₈

—

109

—

125

C₈

C₂×Dic₂₁₈

yes

—

126

C₈

C₂×Dic₂₁₈

yes

—

127

C₂²

—

128

C₂²

#16812698

yes

—

129

C₂²

C₂×D₄×D₁₀₉

yes

characteristic

130

C₄

—

131

C₄

—

436

—

132

C₄

—

218

—

133

C₄

#16812698

yes

—

134

C₄

C₂×Q₈×D₁₀₉

yes

characteristic

135

C₄

C₂²×Dic₂₁₈

yes

characteristic

136

C₂

—

137

C₂

C₂×#16812698

yes

characteristic

138

C₂

#20295823

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#427×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂³×C₁₀₉	C₂×D₄
hypercenter	#427	D₁₀₉
hypocenter	C₁₀₉	#3162
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

#427×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

prime radical

C₂³×C₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #20295434	C₂	semidirect
3. #20295433	C₂	semidirect
27. C₂×#11724784	Q₈	semidirect
38. C₂×C₈×C₁₀₉	Q₈	semidirect
70. C₁₀₉	#3162	semidirect
100. C₂×C₈	Dic₂₁₈	semidirect

H ≅ G/N

type

2. #20295434

C₂

semidirect

3. #20295433

C₂

semidirect

27. C₂×#11724784

Q₈

semidirect

38. C₂×C₈×C₁₀₉

Q₈

semidirect

70. C₁₀₉

#3162

semidirect

100. C₂×C₈

Dic₂₁₈

semidirect

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

subgroup	index
#20295579	2
#20295434	2
#20295433	2
#20295064	2
#20295062	2
C₂×#16812609	2
#427×C₁₀₉	2
#3162	109

subgroup

index

#427×C₁₀₉

109

Subgroups — #28373555

Conjugacy classes of subgroups

Complements

Maximal subgroups