Group #28361713 — subgroups

Subgroups — #28361713

Order 13952.

240 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 101 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	#20295079	C₂	1	1	yes	maximal, characteristic
2	#20295050	C₂	1	1	yes	maximal, characteristic
3	#20294775	C₂	1	4	yes	maximal, complemented
4	C₂×C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182013	—	2	1	no	—
6	C₄⋊#11724784	—	2	1	no	—
7	C₄⋊#11724784	C₂²	1	2	yes	—
8	C₄⋊#11724784	C₂²	1	2	yes	—
9	C₂²⋊#11724784	—	2	1	no	—
10	C₄×#11724784	—	2	1	no	—
11	C₂²×Q₈×C₁₀₉	C₄	1	2	yes	—
12	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
13	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
14	C₄⋊Q₈×C₁₀₉	C₄	1	2	yes	—
15	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
17	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
18	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
19	#13182013	—	4	2	no	—
20	C₂×#11724784	—	2	2	no	—
21	C₂×#11724784	—	2	2	no	—
22	C₂×Q₈×C₁₀₉	—	2	2	no	—
23	C₂×Q₈×C₁₀₉	—	4	4	no	—
24	C₂×Q₈×C₁₀₉	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₁₀₉	—	2	2	no	—
26	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
29	C₄⋊C₄×C₁₀₉	—	2	2	no	—
30	C₄⋊C₄×C₁₀₉	—	2	2	no	—
31	C₄⋊C₄×C₁₀₉	—	2	2	no	—
32	C₄⋊C₄×C₁₀₉	—	2	2	no	—
33	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
35	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
36	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₂³×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
38	Q₈×C₁₀₉	—	4	4	no	—
39	Q₈×C₁₀₉	—	4	4	no	—
40	C₂×C₄×C₁₀₉	—	2	2	no	—
41	C₂×C₄×C₁₀₉	—	2	2	no	—
42	C₂×C₄×C₁₀₉	—	2	4	no	—
43	C₂×C₄×C₁₀₉	—	4	4	no	—
44	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	SD₁₆	1	2	yes	—
48	C₂×C₄×C₁₀₉	Q₁₆	1	2	yes	—
49	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
52	#11724784	—	4	2	no	—
53	#11724784	—	4	2	no	—
54	C₂²×C₁₀₉	—	2	1	no	—
55	C₂²×C₁₀₉	—	2	1	no	—
56	C₂²×C₁₀₉	#103	1	2	yes	—
57	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₁₀₉	—	2	1	no	—
59	C₄×C₁₀₉	—	2	1	no	—
60	C₄×C₁₀₉	—	4	4	no	—
61	C₄×C₁₀₉	#100	1	2	yes	—
62	C₄×C₁₀₉	#103	1	2	yes	—
63	C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
64	C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
65	C₄×C₁₀₉	#137	1	2	yes	—
66	C₂×C₁₀₉	—	2	2	no	—
67	C₂×C₁₀₉	C₂×#100	1	1	yes	characteristic
68	C₂×C₁₀₉	C₂×#103	1	1	yes	characteristic
69	C₂×C₁₀₉	#419	1	1	yes	characteristic
70	#1548	—	109	1	no	maximal, Sylow
71	C₁₀₉	#1548	1	1	yes	characteristic, complemented, Sylow
72	C₂×C₄⋊Q₈	D₁₀₉	1	1	yes	characteristic
73	#431	—	109	1	no	—
74	#423	—	109	1	no	—
75	#333	—	109	4	no	—
76	C₂²×Q₈	Dic₁₀₉	1	2	yes	—
77	C₂×M₁₆	—	218	1	no	—
78	C₄⋊Q₈	—	2	1	no	—
79	C₄⋊Q₈	—	2	1	no	—
80	C₄⋊Q₈	Dic₁₀₉	1	2	yes	—
81	C₄⋊Q₈	C₂×D₁₀₉	1	2	yes	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
85	C₄⋊C₈	—	109	2	no	—
86	C₄⋊C₈	—	109	2	no	—
87	C₄⋊C₈	—	218	1	no	—
88	C₂²⋊C₈	—	218	1	no	—
89	C₄×C₈	—	218	1	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	4	4	no	—
92	C₂×Q₈	C₂×Dic₁₀₉	1	4	yes	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
95	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
97	M₁₆	—	436	2	no	—
98	C₂×C₈	—	218	2	no	—
99	C₂×C₈	—	218	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
105	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
106	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
107	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
108	C₂³	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
109	Q₈	—	4	4	no	—
110	Q₈	—	4	4	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	4	no	—
115	C₂×C₄	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
116	C₂×C₄	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
117	C₂×C₄	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
118	C₂×C₄	#13182016	1	2	yes	—
119	C₂×C₄	#13182019	1	2	yes	—
120	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
122	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
123	C₈	—	436	2	no	—
124	C₈	—	436	2	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	#16812623	1	2	yes	—
128	C₂²	C₂×C₂²⋊Dic₁₀₉	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	4	4	no	—
132	C₄	#16812616	1	2	yes	—
133	C₄	#16812623	1	2	yes	—
134	C₄	C₂×#13182016	1	1	yes	characteristic
135	C₄	C₂×#13182019	1	1	yes	characteristic
136	C₄	#16812730	1	2	yes	—
137	C₂	—	2	2	no	—
138	C₂	C₂×#16812616	1	1	yes	characteristic
139	C₂	C₂×#16812623	1	1	yes	characteristic
140	C₂	#20295035	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295079

C₂

yes

maximal, characteristic

#20295050

C₂

yes

maximal, characteristic

#20294775

C₂

yes

maximal, complemented

C₂×C₄⋊Q₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

—

C₄⋊#11724784

—

C₄⋊#11724784

C₂²

yes

—

C₄⋊#11724784

C₂²

yes

—

C₂²⋊#11724784

—

C₄×#11724784

—

C₂²×Q₈×C₁₀₉

C₄

yes

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

C₄

yes

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂²

yes

—

—

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

C₂²⋊C₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

SD₁₆

yes

—

C₂×C₄×C₁₀₉

Q₁₆

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#103

yes

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#100

yes

—

C₄×C₁₀₉

#103

yes

—

C₄×C₁₀₉

C₂×SD₁₆

yes

characteristic

C₄×C₁₀₉

C₂×Q₁₆

yes

characteristic

C₄×C₁₀₉

#137

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1548

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₁₀₉

yes

characteristic

#431

—

109

—

#423

—

109

—

#333

—

109

—

C₂²×Q₈

Dic₁₀₉

yes

—

C₂×M₁₆

—

218

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₁₀₉

yes

—

C₄⋊Q₈

C₂×D₁₀₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₂²⋊C₈

—

218

—

C₄×C₈

—

218

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₀₉

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₀₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

M₁₆

—

436

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄²

C₂×Dic₁₀₉

yes

characteristic

105

C₄²

C₂²⋊D₁₀₉

yes

characteristic

106

C₄²

C₂²⋊D₁₀₉

yes

characteristic

107

C₄²

C₂²×D₁₀₉

yes

characteristic

108

C₂³

C₂²⋊Dic₁₀₉

yes

characteristic

109

Q₈

—

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

C₂²⋊Dic₁₀₉

yes

characteristic

116

C₂×C₄

C₂²⋊Dic₁₀₉

yes

characteristic

117

C₂×C₄

C₂²⋊Dic₁₀₉

yes

characteristic

118

C₂×C₄

#13182016

yes

—

119

C₂×C₄

#13182019

yes

—

120

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

122

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

123

C₈

—

436

—

124

C₈

—

436

—

125

C₂²

—

126

C₂²

—

127

C₂²

#16812623

yes

—

128

C₂²

C₂×C₂²⋊Dic₁₀₉

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#16812616

yes

—

133

C₄

#16812623

yes

—

134

C₄

C₂×#13182016

yes

characteristic

135

C₄

C₂×#13182019

yes

characteristic

136

C₄

#16812730

yes

—

137

C₂

—

138

C₂

C₂×#16812616

yes

characteristic

139

C₂

C₂×#16812623

yes

characteristic

140

C₂

#20295035

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₀₉
Fitting	C₂×C₄⋊Q₈×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₀₉	C₂²⋊C₄
cosocle	C₄²×C₁₀₉	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₀₉
hypocenter	C₁₀₉	#1548
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₄²

C₂²×D₁₀₉

Fitting

C₂×C₄⋊Q₈×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₀₉

C₂²⋊C₄

cosocle

C₄²×C₁₀₉

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₀₉

hypocenter

C₁₀₉

#1548

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #20294775	C₂	semidirect
71. C₁₀₉	#1548	semidirect

H ≅ G/N

type

3. #20294775

C₂

semidirect

71. C₁₀₉

#1548

semidirect

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

subgroup	index
#20295079	2
#20295050	2
#20294775	2
C₂×C₄⋊Q₈×C₁₀₉	2
#1548	109

subgroup

index

C₂×C₄⋊Q₈×C₁₀₉

109

Subgroups — #28361713

Conjugacy classes of subgroups

Complements

Maximal subgroups