Group #28377861 — subgroups

Subgroups — #28377861

Order 13952.

180 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 74 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	D₁₀₉⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×D₁₀₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×D₁₀₉⋊C₈	—	2	1	no	—
5	C₂×D₁₀₉⋊C₈	C₄	1	2	yes	—
6	#16812766	—	2	1	no	—
7	C₄⋊#11724786	C₄	1	2	yes	—
8	C₄×#11724786	—	2	1	no	—
9	C₄×#11724786	—	2	1	no	—
10	C₂×C₈×D₁₀₉	—	2	1	no	—
11	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
12	C₄⋊#11724784	C₄	1	1	yes	characteristic
13	#16812628	—	2	1	no	—
14	C₄⋊C₈×C₁₀₉	C₄	1	1	yes	characteristic
15	D₁₀₉⋊C₈	—	2	2	no	—
16	D₁₀₉⋊C₈	—	4	1	no	—
17	D₁₀₉⋊C₈	—	4	1	no	—
18	D₁₀₉⋊C₈	C₈	1	4	yes	complemented
19	C₂×#11724786	—	2	2	no	—
20	C₂×#11724786	—	2	2	no	—
21	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
23	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
24	C₂×#11724784	—	2	1	no	—
25	C₈×D₁₀₉	—	4	1	no	—
26	C₈×D₁₀₉	—	4	1	no	—
27	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
29	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
30	C₂×C₈×C₁₀₉	—	2	1	no	—
31	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
39	C₄×D₁₀₉	—	2	2	no	—
40	C₄×D₁₀₉	C₂×C₈	1	2	yes	—
41	C₄×D₁₀₉	M₁₆	1	2	yes	—
42	C₄×D₁₀₉	D₈	1	1	yes	characteristic
43	C₄×D₁₀₉	SD₁₆	1	1	yes	characteristic
44	C₄×D₁₀₉	SD₁₆	1	1	yes	characteristic
45	C₄×D₁₀₉	Q₁₆	1	1	yes	characteristic
46	#11724786	—	2	4	no	—
47	#11724786	—	4	2	no	—
48	C₈×C₁₀₉	—	4	1	no	—
49	#11724784	—	4	1	no	—
50	C₂²×C₁₀₉	#95	1	1	yes	characteristic
51	C₂×D₁₀₉	C₄⋊C₈	1	2	yes	—
52	C₂×D₁₀₉	#106	1	1	yes	characteristic
53	C₂×D₁₀₉	#107	1	1	yes	characteristic
54	C₂×D₁₀₉	#108	1	2	yes	—
55	C₄×C₁₀₉	—	2	1	no	—
56	C₄×C₁₀₉	C₄×C₈	1	1	yes	characteristic
57	C₄×C₁₀₉	#97	1	1	yes	characteristic
58	C₄×C₁₀₉	#102	1	1	yes	characteristic
59	C₄×C₁₀₉	#103	1	1	yes	characteristic
60	Dic₁₀₉	—	2	1	no	—
61	Dic₁₀₉	C₂²⋊C₈	1	2	yes	—
62	Dic₁₀₉	#102	1	1	yes	characteristic
63	Dic₁₀₉	#103	1	1	yes	characteristic
64	C₂×C₁₀₉	#336	1	1	yes	characteristic
65	C₂×C₁₀₉	#340	1	1	yes	characteristic
66	C₂×C₁₀₉	#341	1	1	yes	characteristic
67	D₁₀₉	#334	1	2	yes	—
68	D₁₀₉	#335	1	2	yes	—
69	#1277	—	109	1	no	maximal, Sylow
70	C₁₀₉	#1277	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄⋊C₈	—	109	1	no	—
72	C₂×C₄⋊C₈	—	109	1	no	—
73	C₂×C₄×C₈	—	109	1	no	—
74	C₂²×C₈	—	109	2	no	—
75	C₂²×C₈	—	218	1	no	—
76	C₂²×C₈	—	218	1	no	—
77	C₂×C₄²	—	109	1	no	—
78	C₄⋊C₈	—	109	2	no	—
79	C₄⋊C₈	—	109	1	no	—
80	C₄⋊C₈	—	218	1	no	—
81	C₄⋊C₈	—	218	1	no	—
82	C₄⋊C₈	#89137	1	1	yes	characteristic
83	C₄×C₈	—	218	1	no	—
84	C₄×C₈	—	218	1	no	—
85	C₂²×C₄	—	109	1	no	—
86	C₂²×C₄	—	109	1	no	—
87	C₂²×C₄	—	109	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	109	4	no	—
90	C₂×C₈	—	436	1	no	—
91	C₂×C₈	—	436	1	no	—
92	C₂×C₈	—	436	1	no	—
93	C₂×C₈	—	436	1	no	—
94	C₂×C₈	—	218	2	no	—
95	C₂×C₈	—	218	1	no	—
96	C₂×C₈	—	218	2	no	—
97	C₂×C₈	—	218	2	no	—
98	C₄²	—	109	1	no	—
99	C₄²	—	109	1	no	—
100	C₄²	—	109	1	no	—
101	C₄²	C₂×#89137	1	1	yes	characteristic
102	C₂³	—	109	1	no	—
103	C₂×C₄	—	109	1	no	—
104	C₂×C₄	—	109	1	no	—
105	C₂×C₄	—	109	2	no	—
106	C₂×C₄	—	109	2	no	—
107	C₂×C₄	—	109	1	no	—
108	C₂×C₄	—	109	1	no	—
109	C₂×C₄	—	109	1	no	—
110	C₂×C₄	—	109	1	no	—
111	C₂×C₄	—	109	1	no	—
112	C₂×C₄	—	218	2	no	—
113	C₂×C₄	C₄×#89137	1	1	yes	characteristic
114	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
115	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
116	C₈	—	4	1	no	—
117	C₈	—	436	2	no	—
118	C₈	—	436	1	no	—
119	C₈	—	218	4	no	—
120	C₂²	—	109	2	no	—
121	C₂²	—	109	1	no	—
122	C₂²	—	109	1	no	—
123	C₂²	—	109	2	no	—
124	C₂²	(C₂×#89137)⋊C₄	1	1	yes	characteristic
125	C₄	—	2	1	no	—
126	C₄	—	109	2	no	—
127	C₄	—	109	1	no	—
128	C₄	—	109	1	no	—
129	C₄	—	218	1	no	—
130	C₄	C₈×#89137	1	1	yes	characteristic
131	C₄	#16812753	1	1	yes	characteristic
132	C₄	#16812761	1	1	yes	characteristic
133	C₄	#16812762	1	1	yes	characteristic
134	C₂	—	109	2	no	—
135	C₂	—	109	2	no	—
136	C₂	(C₂×#89137)⋊C₈	1	1	yes	characteristic
137	C₂	#20295899	1	1	yes	characteristic
138	C₂	#20295901	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₁₀₉⋊(C₄⋊C₈)

C₂

yes

maximal, characteristic

C₄×D₁₀₉⋊C₈

C₂

yes

maximal, characteristic

C₄⋊C₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×D₁₀₉⋊C₈

—

C₂×D₁₀₉⋊C₈

C₄

yes

—

#16812766

—

C₄⋊#11724786

C₄

yes

—

C₄×#11724786

—

C₄×#11724786

—

C₂×C₈×D₁₀₉

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄⋊#11724784

C₄

yes

characteristic

#16812628

—

C₄⋊C₈×C₁₀₉

C₄

yes

characteristic

D₁₀₉⋊C₈

—

D₁₀₉⋊C₈

—

D₁₀₉⋊C₈

—

D₁₀₉⋊C₈

C₈

yes

complemented

C₂×#11724786

—

C₂×#11724786

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

Q₈

yes

characteristic

C₂×#11724784

—

C₈×D₁₀₉

—

C₈×D₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×C₈

yes

—

C₄×D₁₀₉

M₁₆

yes

—

C₄×D₁₀₉

D₈

yes

characteristic

C₄×D₁₀₉

SD₁₆

yes

characteristic

C₄×D₁₀₉

SD₁₆

yes

characteristic

C₄×D₁₀₉

Q₁₆

yes

characteristic

#11724786

—

#11724786

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

#95

yes

characteristic

C₂×D₁₀₉

C₄⋊C₈

yes

—

C₂×D₁₀₉

yes

C₂×D₁₀₉

yes

C₂×D₁₀₉

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×C₈

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

Dic₁₀₉

C₂²⋊C₈

yes

—

Dic₁₀₉

yes

Dic₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

yes

—

D₁₀₉

#335

yes

—

#1277

—

109

maximal, Sylow

C₁₀₉

#1277

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

109

—

C₂×C₄⋊C₈

—

109

—

C₂×C₄×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

218

—

C₂²×C₈

—

218

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

—

218

—

C₄⋊C₈

#89137

yes

characteristic

C₄×C₈

—

218

—

C₄×C₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

—

436

—

C₂×C₈

—

436

—

C₂×C₈

—

436

—

C₂×C₈

—

436

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄²

—

109

—

C₄²

—

109

—

100

C₄²

—

109

—

101

C₄²

C₂×#89137

yes

characteristic

102

C₂³

—

109

—

103

C₂×C₄

—

109

—

104

C₂×C₄

—

109

—

105

C₂×C₄

—

109

—

106

C₂×C₄

—

109

—

107

C₂×C₄

—

109

—

108

C₂×C₄

—

109

—

109

C₂×C₄

—

109

—

110

C₂×C₄

—

109

—

111

C₂×C₄

—

109

—

112

C₂×C₄

—

218

—

113

C₂×C₄

C₄×#89137

yes

characteristic

114

C₂×C₄

C₂²⋊#89137

yes

characteristic

115

C₂×C₄

C₄⋊#89137

yes

characteristic

116

C₈

—

117

C₈

—

436

—

118

C₈

—

436

—

119

C₈

—

218

—

120

C₂²

—

109

—

121

C₂²

—

109

—

122

C₂²

—

109

—

123

C₂²

—

109

—

124

C₂²

(C₂×#89137)⋊C₄

yes

characteristic

125

C₄

—

126

C₄

—

109

—

127

C₄

—

109

—

128

C₄

—

109

—

129

C₄

—

218

—

130

C₄

C₈×#89137

yes

characteristic

131

C₄

#16812753

yes

characteristic

132

C₄

#16812761

yes

characteristic

133

C₄

#16812762

yes

characteristic

134

C₂

—

109

—

135

C₂

—

109

—

136

C₂

(C₂×#89137)⋊C₈

yes

characteristic

137

C₂

#20295899

yes

characteristic

138

C₂

#20295901

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#89137
commutator	C₄×C₁₀₉	C₄×C₈
Frattini	C₄²	C₂×#89137
Fitting	C₄⋊C₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	#95
prime radical	C₂²×D₁₀₉	C₄⋊C₄
cosocle	C₄²×D₁₀₉	C₂²
hypercenter	C₄⋊C₈	#89137
hypocenter	C₁₀₉	#1277
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂²⋊#89137

commutator

C₄×C₁₀₉

C₄×C₈

Frattini

C₄²

C₂×#89137

Fitting

C₄⋊C₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

#95

prime radical

C₂²×D₁₀₉

C₄⋊C₄

cosocle

C₄²×D₁₀₉

C₂²

C₄⋊C₈

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

N	H ≅ G/N	type
18. D₁₀₉⋊C₈	C₈	semidirect
70. C₁₀₉	#1277	semidirect

H ≅ G/N

type

18. D₁₀₉⋊C₈

C₈

semidirect

70. C₁₀₉

#1277

semidirect

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

subgroup	index
D₁₀₉⋊(C₄⋊C₈)	2
C₄×D₁₀₉⋊C₈	2
C₄⋊C₈×D₁₀₉	2
#1277	109

subgroup

index

D₁₀₉⋊(C₄⋊C₈)

C₄×D₁₀₉⋊C₈

C₄⋊C₈×D₁₀₉

#1277

109

Subgroups — #28377861

Conjugacy classes of subgroups

Complements

Maximal subgroups