Group #28363984 — subgroups

Subgroups — #28363984

Order 13952.

292 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 125 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	C₂×#16812699	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11724784	C₂	1	1	yes	maximal, characteristic
3	#20294788	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₁₈	—	2	1	no	—
6	C₂²×#11724784	C₂²	1	2	yes	—
7	#16812699	—	2	2	no	—
8	#16812699	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₀₉	—	2	1	no	—
10	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
11	C₄⋊#11724784	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₀₉	—	2	1	no	—
14	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
15	#95×C₁₀₉	—	2	1	no	—
16	C₂×Dic₂₁₈	—	2	4	no	—
17	C₂×Dic₂₁₈	—	4	2	no	—
18	C₂×Dic₂₁₈	—	4	2	no	—
19	C₂²×Dic₁₀₉	—	2	2	no	—
20	C₂×#11724784	—	2	2	no	—
21	C₂×#11724784	—	2	4	no	—
22	C₄⋊Dic₁₀₉	—	2	4	no	—
23	C₄⋊Dic₁₀₉	—	2	2	no	—
24	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₁₀₉	—	2	2	no	—
26	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₀₉	—	2	2	no	—
30	C₄⋊C₄×C₁₀₉	Q₈	1	4	yes	complemented
31	C₄²×C₁₀₉	—	4	2	no	—
32	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
33	C₄²×C₁₀₉	D₄	1	2	yes	—
34	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	—	2	2	no	—
36	C₂×C₄×C₁₀₉	—	2	4	no	—
37	C₂×C₄×C₁₀₉	—	4	2	no	—
38	C₂×C₄×C₁₀₉	—	4	2	no	—
39	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₀₉	D₈	1	2	yes	—
42	C₂×C₄×C₁₀₉	SD₁₆	1	2	yes	—
43	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₁₀₉	#41	1	2	yes	—
47	C₂×Dic₁₀₉	—	2	4	no	—
48	C₂×Dic₁₀₉	—	4	2	no	—
49	C₂×Dic₁₀₉	—	4	2	no	—
50	Dic₂₁₈	—	4	8	no	—
51	#11724784	—	4	2	no	—
52	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
54	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
55	C₂²×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
56	C₂²×C₁₀₉	#137	1	2	yes	—
57	C₄×C₁₀₉	—	2	2	no	—
58	C₄×C₁₀₉	—	4	4	no	—
59	C₄×C₁₀₉	#102	1	4	yes	—
60	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
61	C₄×C₁₀₉	#124	1	1	yes	characteristic
62	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
63	Dic₁₀₉	—	4	4	no	—
64	C₂×C₁₀₉	#391	1	1	yes	characteristic
65	C₂×C₁₀₉	C₂×#102	1	1	yes	characteristic
66	C₂×C₁₀₉	#419	1	1	yes	characteristic
67	C₂×C₁₀₉	#474	1	2	yes	—
68	C₂×C₁₀₉	#475	1	2	yes	—
69	#1988	—	109	1	no	maximal, Sylow
70	C₁₀₉	#1988	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄⋊Q₈	—	109	1	no	—
72	C₂×C₄⋊C₈	—	109	1	no	—
73	C₄×C₄⋊C₄	D₁₀₉	1	1	yes	characteristic
74	#340	—	109	4	no	—
75	C₂²×Q₈	—	218	1	no	—
76	C₂²×C₈	—	109	2	no	—
77	C₄⋊Q₈	—	109	4	no	—
78	C₄⋊Q₈	—	218	2	no	—
79	C₂×C₄⋊C₄	—	109	2	no	—
80	C₂×C₄⋊C₄	—	218	1	no	—
81	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
82	C₂×C₄²	—	2	1	no	—
83	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
84	C₄⋊C₈	—	218	2	no	—
85	#95	—	2	1	no	—
86	C₂×Q₈	—	436	2	no	—
87	C₂×Q₈	—	436	2	no	—
88	C₂×Q₈	—	218	4	no	—
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	—	218	2	no	—
91	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
92	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
94	C₂×C₈	—	218	2	no	—
95	C₂×C₈	—	218	4	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	109	4	no	—
98	C₄⋊C₄	—	218	4	no	—
99	C₄⋊C₄	—	218	2	no	—
100	C₄⋊C₄	Dic₂₁₈	1	4	yes	complemented
101	C₄²	—	4	2	no	—
102	C₄²	C₄×D₁₀₉	1	2	yes	—
103	C₄²	D₄₃₆	1	2	yes	—
104	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
105	Q₈	—	436	8	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	436	2	no	—
111	C₂×C₄	—	436	2	no	—
112	C₂×C₄	—	218	4	no	—
113	C₂×C₄	#13182006	1	2	yes	—
114	C₂×C₄	#13182006	1	2	yes	—
115	C₂×C₄	#13182014	1	2	yes	—
116	C₂×C₄	#13182016	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
118	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
119	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
120	C₂×C₄	#13182029	1	2	yes	—
121	C₈	—	436	2	no	—
122	C₂²	C₂×#13182006	1	1	yes	characteristic
123	C₂²	#16812678	1	2	yes	—
124	C₂²	C₂×#13182014	1	1	yes	characteristic
125	C₂²	C₂×#13182016	1	1	yes	characteristic
126	C₂²	#16812729	1	2	yes	—
127	C₄	—	2	2	no	—
128	C₄	—	4	4	no	—
129	C₄	—	436	4	no	—
130	C₄	#16812619	1	4	yes	—
131	C₄	C₄×Dic₂₁₈	1	2	yes	—
132	C₄	#16812687	1	1	yes	characteristic
133	C₄	#16812699	1	1	yes	characteristic
134	C₂	#20294932	1	1	yes	characteristic
135	C₂	C₂×#16812619	1	1	yes	characteristic
136	C₂	#20295034	1	1	yes	characteristic
137	C₂	#20295306	1	2	yes	—
138	C₂	#20295311	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812699

C₂

yes

maximal, characteristic

C₂×C₄⋊#11724784

C₂

yes

maximal, characteristic

#20294788

C₂

yes

maximal

C₄×C₄⋊C₄×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

—

C₂²×#11724784

C₂²

yes

—

#16812699

—

#16812699

C₄

yes

complemented

C₂×C₄⋊Dic₁₀₉

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₄⋊#11724784

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

#95×C₁₀₉

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂²×Dic₁₀₉

—

C₂×#11724784

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

Q₈

yes

complemented

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

D₈

yes

—

C₂×C₄×C₁₀₉

SD₁₆

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

#11724784

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₂²⋊Q₈

yes

—

C₂²×C₁₀₉

C₂×D₈

yes

characteristic

C₂²×C₁₀₉

C₂×SD₁₆

yes

characteristic

C₂²×C₁₀₉

#137

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#102

yes

—

C₄×C₁₀₉

C₄×Q₈

yes

—

C₄×C₁₀₉

#124

yes

characteristic

C₄×C₁₀₉

C₄⋊Q₈

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

C₂×C₁₀₉

#475

yes

—

#1988

—

109

maximal, Sylow

C₁₀₉

#1988

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

109

—

C₂×C₄⋊C₈

—

109

—

C₄×C₄⋊C₄

D₁₀₉

yes

characteristic

#340

—

109

—

C₂²×Q₈

—

218

—

C₂²×C₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

218

—

#95

—

C₂×Q₈

—

436

—

C₂×Q₈

—

436

—

C₂×Q₈

—

218

—

C₂²×C₄

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

100

C₄⋊C₄

Dic₂₁₈

yes

complemented

101

C₄²

—

102

C₄²

C₄×D₁₀₉

yes

—

103

C₄²

D₄₃₆

yes

—

104

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

105

Q₈

—

436

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

436

—

111

C₂×C₄

—

436

—

112

C₂×C₄

—

218

—

113

C₂×C₄

#13182006

yes

—

114

C₂×C₄

#13182006

yes

—

115

C₂×C₄

#13182014

yes

—

116

C₂×C₄

#13182016

yes

—

117

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

118

C₂×C₄

C₂×D₄₃₆

yes

characteristic

119

C₂×C₄

C₂×Dic₂₁₈

yes

—

120

C₂×C₄

#13182029

yes

—

121

C₈

—

436

—

122

C₂²

C₂×#13182006

yes

characteristic

123

C₂²

#16812678

yes

—

124

C₂²

C₂×#13182014

yes

characteristic

125

C₂²

C₂×#13182016

yes

characteristic

126

C₂²

#16812729

yes

—

127

C₄

—

128

C₄

—

129

C₄

—

436

—

130

C₄

#16812619

yes

—

131

C₄

C₄×Dic₂₁₈

yes

—

132

C₄

#16812687

yes

characteristic

133

C₄

#16812699

yes

characteristic

134

C₂

#20294932

yes

characteristic

135

C₂

C₂×#16812619

yes

characteristic

136

C₂

#20295034

yes

characteristic

137

C₂

#20295306

yes

—

138

C₂

#20295311

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₄×C₄⋊C₄×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₀₉
hypocenter	C₁₀₉	#1988
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₄×C₄⋊C₄×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

hypercenter

C₄×C₄⋊C₄

D₁₀₉

hypocenter

C₁₀₉

#1988

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
8. #16812699	C₄	semidirect
30. C₄⋊C₄×C₁₀₉	Q₈	semidirect
70. C₁₀₉	#1988	semidirect
100. C₄⋊C₄	Dic₂₁₈	semidirect

H ≅ G/N

type

8. #16812699

C₄

semidirect

30. C₄⋊C₄×C₁₀₉

Q₈

semidirect

70. C₁₀₉

#1988

semidirect

100. C₄⋊C₄

Dic₂₁₈

semidirect

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

subgroup	index
C₂×#16812699	2
C₂×C₄⋊#11724784	2
#20294788	2
C₄×C₄⋊C₄×C₁₀₉	2
#1988	109

subgroup

index

C₄×C₄⋊C₄×C₁₀₉

109

Subgroups — #28363984

Conjugacy classes of subgroups

Complements

Maximal subgroups