Group #28378619 — subgroups

Subgroups — #28378619

Order 13952.

164 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 66 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	#20296075	C₂	1	1	yes	maximal, characteristic
2	Q₈×#11724786	C₂	1	1	yes	maximal, characteristic
3	#20295890	C₂	1	1	yes	maximal, characteristic
4	#20295876	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊#11724786	C₂	1	1	yes	maximal, characteristic
6	C₈×#11724786	C₂	1	1	yes	maximal, characteristic
7	#20295118	C₂	1	1	yes	maximal, characteristic
8	#16812766	—	2	1	no	—
9	C₄⋊#11724786	—	2	1	no	—
10	C₄⋊#11724786	C₂²	1	1	yes	characteristic
11	C₄⋊#11724786	C₂²	1	1	yes	characteristic
12	C₄×#11724786	—	2	1	no	—
13	C₄×#11724786	—	2	1	no	—
14	C₄×#11724786	C₂²	1	1	yes	characteristic
15	C₄×#11724786	C₂²	1	1	yes	characteristic
16	C₂×#13182019	C₄	1	1	yes	characteristic
17	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
18	#16812664	C₂²	1	1	yes	characteristic
19	#16812632	C₄	1	1	yes	characteristic
20	#16812624	C₄	1	1	yes	characteristic
21	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
22	#103×C₁₀₉	C₄	1	1	yes	characteristic
23	C₂×#11724786	—	2	1	no	—
24	C₂×#11724786	—	2	1	no	—
25	C₂×#11724786	—	2	1	no	—
26	C₂×#11724786	D₄	1	1	yes	characteristic
27	C₂×#11724786	D₄	1	1	yes	characteristic
28	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
29	#13182019	C₈	1	4	yes	complemented
30	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
31	C₄⋊Dic₁₀₉	—	2	1	no	—
32	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
33	#13182008	—	2	1	no	—
34	C₄×Dic₁₀₉	—	2	1	no	—
35	C₄×Dic₁₀₉	—	2	1	no	—
36	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
37	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
39	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
40	Q₈×C₁₀₉	—	2	1	no	—
41	Q₈×C₁₀₉	C₂×C₈	1	2	yes	—
42	C₂×C₄×C₁₀₉	—	2	1	no	—
43	C₂×C₄×C₁₀₉	—	2	1	no	—
44	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₁₀₉	—	2	1	no	—
46	C₂×Dic₁₀₉	—	2	1	no	—
47	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₁₀₉	#41	1	1	yes	characteristic
49	Dic₂₁₈	—	2	1	no	—
50	Dic₂₁₈	C₂×C₈	1	2	yes	—
51	#11724786	—	2	2	no	—
52	#11724786	—	2	1	no	—
53	#11724786	—	4	1	no	—
54	#11724786	—	4	1	no	—
55	#11724786	Q₁₆	1	2	yes	complemented
56	C₈×C₁₀₉	—	2	1	no	—
57	#11724784	C₂×C₈	1	2	yes	—
58	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
59	C₄×C₁₀₉	—	2	2	no	—
60	C₄×C₁₀₉	—	4	1	no	—
61	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
62	C₄×C₁₀₉	#131	1	1	yes	characteristic
63	Dic₁₀₉	—	2	2	no	—
64	Dic₁₀₉	—	2	1	no	—
65	Dic₁₀₉	—	4	1	no	—
66	Dic₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
67	Dic₁₀₉	#135	1	1	yes	characteristic
68	C₂×C₁₀₉	C₈×D₄	1	1	yes	characteristic
69	C₂×C₁₀₉	C₄×Q₁₆	1	1	yes	characteristic
70	C₂×C₁₀₉	#443	1	1	yes	characteristic
71	C₈×Q₁₆	—	109	1	no	maximal, Sylow
72	C₁₀₉	C₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
73	C₈×Q₈	—	109	1	no	—
74	C₈×Q₈	—	109	1	no	—
75	C₄×Q₁₆	—	109	1	no	—
76	#335	—	109	1	no	—
77	Q₈⋊C₈	—	109	1	no	—
78	Q₈⋊C₈	—	109	1	no	—
79	C₈²	—	109	1	no	—
80	C₂×Q₁₆	—	109	1	no	—
81	C₄×Q₈	—	109	1	no	—
82	C₄×Q₈	—	109	1	no	—
83	#107	—	109	1	no	—
84	C₄⋊C₈	—	109	1	no	—
85	C₄⋊C₈	—	109	1	no	—
86	C₄⋊C₈	—	218	1	no	—
87	C₄⋊C₈	—	218	1	no	—
88	#103	—	109	1	no	—
89	#103	#89137	1	1	yes	characteristic
90	C₄×C₈	—	109	1	no	—
91	C₄×C₈	—	109	1	no	—
92	C₄×C₈	—	109	1	no	—
93	C₄×C₈	—	218	1	no	—
94	C₄×C₈	—	218	1	no	—
95	C₂×Q₈	—	109	1	no	—
96	C₂×Q₈	C₂×#89137	1	1	yes	characteristic
97	Q₁₆	—	109	4	no	—
98	C₂×C₈	—	109	1	no	—
99	C₂×C₈	—	109	1	no	—
100	C₂×C₈	—	109	1	no	—
101	C₂×C₈	—	218	1	no	—
102	C₂×C₈	—	218	1	no	—
103	C₂×C₈	—	218	1	no	—
104	C₂×C₈	C₂×#89137	1	1	yes	characteristic
105	C₄⋊C₄	—	109	1	no	—
106	C₄⋊C₄	—	218	1	no	—
107	C₄⋊C₄	—	218	1	no	—
108	C₄⋊C₄	C₂×#89137	1	1	yes	characteristic
109	C₄²	—	109	1	no	—
110	C₄²	—	218	1	no	—
111	C₄²	—	218	1	no	—
112	Q₈	—	2	1	no	—
113	Q₈	—	109	2	no	—
114	Q₈	—	218	1	no	—
115	Q₈	D₁₀₉⋊C₈	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	109	1	no	—
119	C₂×C₄	—	109	1	no	—
120	C₂×C₄	—	218	1	no	—
121	C₂×C₄	—	218	1	no	—
122	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	109	2	no	—
125	C₈	—	109	2	no	—
126	C₈	—	436	1	no	—
127	C₈	—	436	1	no	—
128	C₈	—	218	1	no	—
129	C₈	—	218	2	no	—
130	C₂²	D₄×#89137	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	4	1	no	—
133	C₄	—	109	1	no	—
134	C₄	—	109	1	no	—
135	C₄	—	436	1	no	—
136	C₄	—	218	2	no	—
137	C₄	—	218	1	no	—
138	C₄	C₂×D₁₀₉⋊C₈	1	1	yes	characteristic
139	C₄	#16812786	1	1	yes	characteristic
140	C₂	#20296053	1	1	yes	characteristic
141	C₂	Q₁₆×#89137	1	1	yes	characteristic
142	C₂	#20296069	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296075

C₂

yes

maximal, characteristic

Q₈×#11724786

C₂

yes

maximal, characteristic

#20295890

C₂

yes

maximal, characteristic

#20295876

C₂

yes

maximal, characteristic

Q₈⋊#11724786

C₂

yes

maximal, characteristic

C₈×#11724786

C₂

yes

maximal, characteristic

#20295118

C₂

yes

maximal, characteristic

#16812766

—

C₄⋊#11724786

—

C₄⋊#11724786

C₂²

yes

characteristic

C₄⋊#11724786

C₂²

yes

—

—

C₂²

yes

characteristic

C₄×#11724786

C₂²

yes

characteristic

C₂×#13182019

C₄

yes

characteristic

Q₈×Dic₁₀₉

C₂²

yes

characteristic

#16812664

C₂²

yes

characteristic

#16812632

C₄

yes

characteristic

#16812624

C₄

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

#103×C₁₀₉

C₄

yes

—

—

—

D₄

yes

characteristic

C₂×#11724786

D₄

yes

characteristic

C₂×Dic₂₁₈

C₂×C₄

yes

characteristic

#13182019

C₈

yes

complemented

C₂×#11724784

C₂×C₄

yes

characteristic

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

characteristic

#13182008

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

#41

yes

characteristic

Dic₂₁₈

—

Dic₂₁₈

C₂×C₈

yes

—

—

—

—

—

Q₁₆

yes

complemented

C₈×C₁₀₉

—

#11724784

C₂×C₈

yes

—

C₂²×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×C₈

yes

characteristic

C₄×C₁₀₉

#131

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂×Q₁₆

yes

characteristic

Dic₁₀₉

#135

yes

characteristic

C₂×C₁₀₉

C₈×D₄

yes

characteristic

C₂×C₁₀₉

C₄×Q₁₆

yes

characteristic

C₂×C₁₀₉

#443

yes

characteristic

C₈×Q₁₆

—

109

maximal, Sylow

C₁₀₉

C₈×Q₁₆

yes

characteristic, complemented, Sylow

C₈×Q₈

—

109

—

C₈×Q₈

—

109

—

C₄×Q₁₆

—

109

—

#335

—

109

—

Q₈⋊C₈

—

109

—

Q₈⋊C₈

—

109

—

C₈²

—

109

—

C₂×Q₁₆

—

109

—

C₄×Q₈

—

109

—

C₄×Q₈

—

109

—

#107

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

—

218

—

#103

—

109

—

#103

#89137

yes

characteristic

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

218

—

C₄×C₈

—

218

—

C₂×Q₈

—

109

—

C₂×Q₈

C₂×#89137

yes

characteristic

Q₁₆

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

100

C₂×C₈

—

109

—

101

C₂×C₈

—

218

—

102

C₂×C₈

—

218

—

103

C₂×C₈

—

218

—

104

C₂×C₈

C₂×#89137

yes

characteristic

105

C₄⋊C₄

—

109

—

106

C₄⋊C₄

—

218

—

107

C₄⋊C₄

—

218

—

108

C₄⋊C₄

C₂×#89137

yes

characteristic

109

C₄²

—

109

—

110

C₄²

—

218

—

111

C₄²

—

218

—

112

Q₈

—

113

Q₈

—

109

—

114

Q₈

—

218

—

115

Q₈

D₁₀₉⋊C₈

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

109

—

119

C₂×C₄

—

109

—

120

C₂×C₄

—

218

—

121

C₂×C₄

—

218

—

122

C₂×C₄

C₂²×#89137

yes

characteristic

123

C₈

—

124

C₈

—

109

—

125

C₈

—

109

—

126

C₈

—

436

—

127

C₈

—

436

—

128

C₈

—

218

—

129

C₈

—

218

—

130

C₂²

D₄×#89137

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

109

—

134

C₄

—

109

—

135

C₄

—

436

—

136

C₄

—

218

—

137

C₄

—

218

—

138

C₄

C₂×D₁₀₉⋊C₈

yes

characteristic

139

C₄

#16812786

yes

characteristic

140

C₂

#20296053

yes

characteristic

141

C₂

Q₁₆×#89137

yes

characteristic

142

C₂

#20296069

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#89137
commutator	C₄×C₁₀₉	C₂²×C₈
Frattini	C₂×C₄	C₂²×#89137
Fitting	#103×C₁₀₉	C₄
socle, prime radical	C₂²×C₁₀₉	C₄×D₄
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	#103	#89137
hypocenter	C₁₀₉	C₈×Q₁₆
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

subgroup

G/H

center

C₂²

D₄×#89137

commutator

C₄×C₁₀₉

C₂²×C₈

Frattini

C₂×C₄

C₂²×#89137

Fitting

#103×C₁₀₉

C₄

socle, prime radical

C₂²×C₁₀₉

C₄×D₄

cosocle

C₄×Dic₁₀₉

C₂³

C₁₀₉

C₈×Q₁₆

LCS

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

N	H ≅ G/N	type
29. #13182019	C₈	semidirect
55. #11724786	Q₁₆	semidirect
72. C₁₀₉	C₈×Q₁₆	semidirect

H ≅ G/N

type

29. #13182019

C₈

semidirect

55. #11724786

Q₁₆

semidirect

72. C₁₀₉

C₈×Q₁₆

semidirect

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

subgroup	index
#20296075	2
Q₈×#11724786	2
#20295890	2
#20295876	2
Q₈⋊#11724786	2
C₈×#11724786	2
#20295118	2
C₈×Q₁₆	109

subgroup

index

C₈×Q₁₆

109

Subgroups — #28378619

Conjugacy classes of subgroups

Complements

Maximal subgroups