Group #28373408 — subgroups

Subgroups — #28373408

Order 13952.

484 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16812696	C₂	1	1	yes	maximal, characteristic
2	#20295392	C₂	1	4	yes	maximal, complemented
3	#20295374	C₂	1	4	yes	maximal, complemented
4	#20295030	C₂	1	4	yes	maximal, complemented
5	C₄×#13182013	C₂	1	1	yes	maximal, characteristic, complemented
6	#533×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₄₃₆	—	2	2	no	—
8	C₂²×D₄₃₆	C₂²	1	2	yes	—
9	C₂×#13182013	C₂²	1	2	yes	—
10	#16812696	—	2	2	no	—
11	#16812696	C₂²	1	2	yes	—
12	#16812696	C₂²	1	2	yes	—
13	#16812619	C₂²	1	16	yes	complemented
14	#16812610	C₂²	1	2	yes	complemented
15	C₄×#11724784	C₂²	1	2	yes	complemented
16	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	complemented
17	#125×C₁₀₉	C₂²	1	2	yes	complemented
18	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
19	C₄×Q₈×C₁₀₉	—	2	2	no	—
20	#117×C₁₀₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₂³×D₁₀₉	—	2	2	no	—
23	C₂×D₄₃₆	—	2	8	no	—
24	C₂×D₄₃₆	—	2	2	no	—
25	C₂×D₄₃₆	—	2	4	no	—
26	C₂×D₄₃₆	—	2	4	no	—
27	C₂×D₄₃₆	—	4	8	no	—
28	C₂×D₄₃₆	C₂³	1	4	yes	—
29	#13182013	—	2	4	no	—
30	C₂×#11724784	C₂³	1	4	yes	—
31	C₂×Q₈×C₁₀₉	—	2	2	no	—
32	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
33	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₀₉	—	2	4	no	—
35	C₄⋊C₄×C₁₀₉	—	2	2	no	—
36	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
37	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
38	C₄²×C₁₀₉	—	2	2	no	—
39	C₄²×C₁₀₉	D₄	1	2	yes	—
40	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
41	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
42	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×D₁₀₉	—	2	4	no	—
44	C₂²×D₁₀₉	—	4	8	no	—
45	C₂²×D₁₀₉	—	4	4	no	—
46	Q₈×C₁₀₉	—	4	4	no	—
47	C₂×C₄×C₁₀₉	—	2	4	no	—
48	C₂×C₄×C₁₀₉	—	2	4	no	—
49	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₀₉	#41	1	4	yes	—
53	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
54	D₄₃₆	—	2	8	no	—
55	D₄₃₆	—	4	8	no	—
56	D₄₃₆	—	4	4	no	—
57	D₄₃₆	—	4	8	no	—
58	#11724784	—	2	4	no	—
59	C₂²×C₁₀₉	—	2	2	no	—
60	C₂²×C₁₀₉	#124	1	2	yes	—
61	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
62	C₂×D₁₀₉	—	4	4	no	—
63	C₂×D₁₀₉	—	4	8	no	—
64	C₂×D₁₀₉	—	8	8	no	—
65	C₄×C₁₀₉	—	2	2	no	—
66	C₄×C₁₀₉	—	4	4	no	—
67	C₄×C₁₀₉	#124	1	2	yes	—
68	C₄×C₁₀₉	AGL(1,ℤ/8)	1	4	yes	—
69	C₄×C₁₀₉	C₂×#41	1	2	yes	—
70	C₂×C₁₀₉	—	2	2	no	—
71	C₂×C₁₀₉	C₂×#124	1	1	yes	characteristic
72	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
73	D₁₀₉	—	8	4	no	—
74	#3138	—	109	1	no	maximal, Sylow
75	C₁₀₉	#3138	1	1	yes	characteristic, complemented, Sylow
76	#533	D₁₀₉	1	1	yes	characteristic, complemented
77	C₂×#127	—	109	1	no	—
78	#490	—	109	4	no	—
79	#486	—	109	4	no	—
80	#418	—	109	4	no	—
81	C₄×M₁₆	—	109	1	no	—
82	C₂²×D₄	—	109	2	no	—
83	C₂²×D₄	—	218	2	no	—
84	C₂×M₁₆	—	109	2	no	—
85	C₄⋊Q₈	C₂×D₁₀₉	1	2	yes	complemented
86	#127	—	109	2	no	—
87	#127	—	109	2	no	—
88	#127	—	218	2	no	—
89	#125	C₂×D₁₀₉	1	2	yes	complemented
90	C₂²⋊Q₈	—	2	2	no	—
91	C₄×Q₈	—	2	2	no	—
92	#117	C₂×D₁₀₉	1	2	yes	—
93	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
94	#102	—	109	16	no	—
95	#97	—	109	2	no	—
96	C₄×C₈	—	109	2	no	—
97	C₂⁴	—	218	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×D₄	—	109	4	no	—
100	C₂×D₄	—	436	8	no	—
101	C₂×D₄	—	218	4	no	—
102	C₂×D₄	—	218	4	no	—
103	C₂×D₄	—	218	2	no	—
104	C₂×D₄	—	218	8	no	—
105	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
106	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
107	M₁₆	—	218	4	no	—
108	C₂×C₈	—	109	4	no	—
109	C₄⋊C₄	—	2	4	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
115	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
116	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
117	C₂³	—	436	8	no	—
118	C₂³	—	436	4	no	—
119	C₂³	—	218	4	no	—
120	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
121	Q₈	—	4	4	no	—
122	D₄	—	436	8	no	—
123	D₄	—	436	4	no	—
124	D₄	—	436	8	no	—
125	D₄	—	218	8	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
129	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
131	C₂×C₄	#13182030	1	4	yes	—
132	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
133	C₈	—	218	4	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	872	8	no	—
136	C₂²	—	436	4	no	—
137	C₂²	—	436	8	no	—
138	C₂²	#16812688	1	2	yes	—
139	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	4	4	no	—
142	C₄	#16812688	1	2	yes	—
143	C₄	#16812724	1	4	yes	—
144	C₄	C₂×#13182030	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	872	4	no	—
147	C₂	C₂×#16812688	1	1	yes	characteristic
148	C₂	C₂×#16812724	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812696

C₂

yes

maximal, characteristic

#20295392

C₂

yes

maximal, complemented

#20295374

C₂

yes

maximal, complemented

#20295030

C₂

yes

maximal, complemented

C₄×#13182013

C₂

yes

maximal, characteristic, complemented

#533×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₃₆

—

C₂²×D₄₃₆

C₂²

yes

—

C₂×#13182013

C₂²

yes

—

#16812696

—

#16812696

C₂²

yes

—

#16812696

C₂²

yes

—

#16812619

C₂²

yes

complemented

#16812610

C₂²

yes

complemented

C₄×#11724784

C₂²

yes

complemented

C₄⋊Q₈×C₁₀₉

C₂²

yes

complemented

#125×C₁₀₉

C₂²

yes

complemented

C₂²⋊Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

—

#117×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂³×D₁₀₉

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

C₂³

yes

—

#13182013

—

C₂×#11724784

C₂³

yes

—

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#124

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#124

yes

—

C₄×C₁₀₉

AGL(1,ℤ/8)

yes

—

C₄×C₁₀₉

C₂×#41

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×#124

yes

characteristic

C₂×C₁₀₉

C₂×AGL(1,ℤ/8)

yes

—

D₁₀₉

—

#3138

—

109

maximal, Sylow

C₁₀₉

#3138

yes

characteristic, complemented, Sylow

#533

D₁₀₉

yes

characteristic, complemented

C₂×#127

—

109

—

#490

—

109

—

#486

—

109

—

#418

—

109

—

C₄×M₁₆

—

109

—

C₂²×D₄

—

109

—

C₂²×D₄

—

218

—

C₂×M₁₆

—

109

—

C₄⋊Q₈

C₂×D₁₀₉

yes

complemented

#127

—

109

—

#127

—

109

—

#127

—

218

—

#125

C₂×D₁₀₉

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#102

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

C₂⁴

—

218

—

C₂×Q₈

—

C₂×D₄

—

109

—

100

C₂×D₄

—

436

—

101

C₂×D₄

—

218

—

102

C₂×D₄

—

218

—

103

C₂×D₄

—

218

—

104

C₂×D₄

—

218

—

105

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

106

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

107

M₁₆

—

218

—

108

C₂×C₈

—

109

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂²×D₁₀₉

yes

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

C₂²⋊D₁₀₉

yes

—

115

C₄²

C₂²×D₁₀₉

yes

characteristic

116

C₄²

C₂²×D₁₀₉

yes

characteristic

117

C₂³

—

436

—

118

C₂³

—

436

—

119

C₂³

—

218

—

120

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

121

Q₈

—

122

D₄

—

436

—

123

D₄

—

436

—

124

D₄

—

436

—

125

D₄

—

218

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

129

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

131

C₂×C₄

#13182030

yes

—

132

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

133

C₈

—

218

—

134

C₂²

—

135

C₂²

—

872

—

136

C₂²

—

436

—

137

C₂²

—

436

—

138

C₂²

#16812688

yes

—

139

C₂²

C₂²×C₂²⋊D₁₀₉

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

#16812688

yes

—

143

C₄

#16812724

yes

—

144

C₄

C₂×#13182030

yes

—

145

C₂

—

146

C₂

—

872

—

147

C₂

C₂×#16812688

yes

characteristic

148

C₂

C₂×#16812724

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#533×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#16812696	C₂
hypercenter	#533	D₁₀₉
hypocenter	C₁₀₉	#3138
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

#533×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

C₂

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #20295392	C₂	semidirect
3. #20295374	C₂	semidirect
4. #20295030	C₂	semidirect
5. C₄×#13182013	C₂	semidirect
6. #533×C₁₀₉	C₂	semidirect
13. #16812619	C₂²	semidirect
14. #16812610	C₂²	semidirect
15. C₄×#11724784	C₂²	semidirect
16. C₄⋊Q₈×C₁₀₉	C₂²	semidirect
17. #125×C₁₀₉	C₂²	semidirect
75. C₁₀₉	#3138	semidirect
76. #533	D₁₀₉	semidirect
85. C₄⋊Q₈	C₂×D₁₀₉	semidirect
89. #125	C₂×D₁₀₉	semidirect

H ≅ G/N

type

2. #20295392

C₂

semidirect

3. #20295374

C₂

semidirect

4. #20295030

C₂

semidirect

5. C₄×#13182013

C₂

semidirect

6. #533×C₁₀₉

C₂

semidirect

13. #16812619

C₂²

semidirect

14. #16812610

C₂²

semidirect

15. C₄×#11724784

C₂²

semidirect

16. C₄⋊Q₈×C₁₀₉

C₂²

semidirect

17. #125×C₁₀₉

C₂²

semidirect

75. C₁₀₉

#3138

semidirect

76. #533

D₁₀₉

semidirect

85. C₄⋊Q₈

C₂×D₁₀₉

semidirect

89. #125

C₂×D₁₀₉

semidirect

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

subgroup	index
C₂×#16812696	2
#20295392	2
#20295374	2
#20295030	2
C₄×#13182013	2
#533×C₁₀₉	2
#3138	109

subgroup

index

#533×C₁₀₉

109

Subgroups — #28373408

Conjugacy classes of subgroups

Complements

Maximal subgroups