Group #28375706 — subgroups

Subgroups — #28375706

Order 13952.

336 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 139 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	Q₈×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic
2	#20295376	C₂	1	3	yes	maximal
3	#20295321	C₂	1	3	yes	maximal
4	#20295233	C₂	1	3	yes	maximal
5	Q₈×#11724784	C₂	1	1	yes	maximal, characteristic
6	C₄×#13182019	C₂	1	3	yes	maximal
7	#557×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13182019	C₂²	1	3	yes	—
9	#16812699	C₂²	1	3	yes	—
10	#16812698	—	2	3	no	—
11	Q₈×Dic₁₀₉	—	2	1	no	—
12	C₄×Dic₂₁₈	C₂²	1	3	yes	—
13	#16812632	C₂²	1	3	yes	—
14	C₄⋊#11724784	C₂²	1	3	yes	—
15	#16812623	C₂²	1	6	yes	—
16	#16812623	C₂²	1	3	yes	—
17	#16812622	C₂²	1	1	yes	characteristic
18	C₄×#11724784	C₂²	1	3	yes	—
19	C₄⋊Q₈×C₁₀₉	C₂²	1	3	yes	—
20	#125×C₁₀₉	—	2	3	no	—
21	C₄×Q₈×C₁₀₉	—	2	1	no	—
22	C₄×Q₈×C₁₀₉	C₂²	1	3	yes	—
23	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₂×Dic₂₁₈	—	2	3	no	—
25	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
26	#13182019	—	2	6	no	—
27	C₂×#11724784	C₂³	1	1	yes	characteristic
28	C₂×#11724784	C₂³	1	3	yes	—
29	C₄⋊Dic₁₀₉	—	2	3	no	—
30	C₄⋊Dic₁₀₉	C₂³	1	3	yes	—
31	#13182008	—	2	3	no	—
32	C₄×Dic₁₀₉	—	2	3	no	—
33	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
34	C₂×Q₈×C₁₀₉	C₂³	1	3	yes	—
35	C₄⋊C₄×C₁₀₉	—	2	3	no	—
36	C₄⋊C₄×C₁₀₉	—	2	3	no	—
37	C₄⋊C₄×C₁₀₉	—	2	3	no	—
38	C₄⋊C₄×C₁₀₉	D₄	1	3	yes	—
39	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
40	C₄²×C₁₀₉	—	2	3	no	—
41	C₄²×C₁₀₉	C₂³	1	3	yes	—
42	Q₈×C₁₀₉	—	2	3	no	—
43	Q₈×C₁₀₉	—	2	3	no	—
44	Q₈×C₁₀₉	Q₁₆	1	4	yes	complemented
45	C₂×C₄×C₁₀₉	—	2	1	no	—
46	C₂×C₄×C₁₀₉	—	2	3	no	—
47	C₂×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
48	C₂×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
49	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₁₀₉	—	2	1	no	—
51	C₂×Dic₁₀₉	—	2	3	no	—
52	Dic₂₁₈	—	2	1	no	—
53	Dic₂₁₈	—	8	3	no	—
54	Dic₂₁₈	#41	1	2	yes	—
55	#11724784	—	2	3	no	—
56	#11724784	—	2	1	no	—
57	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₁₀₉	—	2	3	no	—
59	C₄×C₁₀₉	—	4	3	no	—
60	C₄×C₁₀₉	—	4	1	no	—
61	C₄×C₁₀₉	C₂×Q₁₆	1	6	yes	—
62	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
63	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
64	Dic₁₀₉	—	2	2	no	—
65	Dic₁₀₉	—	4	3	no	—
66	C₂×C₁₀₉	#548	1	1	yes	characteristic
67	C₂×C₁₀₉	C₂²×Q₁₆	1	1	yes	characteristic
68	C₂×C₁₀₉	#577	1	1	yes	characteristic
69	#3363	—	109	1	no	maximal, Sylow
70	C₁₀₉	#3363	1	1	yes	characteristic, complemented, Sylow
71	Q₈²	—	109	1	no	—
72	#557	D₁₀₉	1	1	yes	characteristic
73	#487	—	109	3	no	—
74	#477	—	109	3	no	—
75	#462	—	109	3	no	—
76	C₈×Q₈	—	109	1	no	—
77	C₄×Q₁₆	—	109	3	no	—
78	C₂×Q₁₆	—	109	3	no	—
79	C₄⋊Q₈	—	109	3	no	—
80	C₄⋊Q₈	—	218	3	no	—
81	C₄⋊Q₈	C₂×D₁₀₉	1	3	yes	—
82	#125	—	2	3	no	—
83	C₄×Q₈	—	2	1	no	—
84	C₄×Q₈	—	109	3	no	—
85	C₄×Q₈	—	218	1	no	—
86	C₄×Q₈	C₂×D₁₀₉	1	3	yes	—
87	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
88	#107	—	109	3	no	—
89	C₄⋊C₈	—	109	3	no	—
90	#103	—	109	3	no	—
91	#103	—	109	6	no	—
92	#103	—	109	1	no	—
93	C₄×C₈	—	109	3	no	—
94	C₂×Q₈	—	109	1	no	—
95	C₂×Q₈	—	218	3	no	—
96	C₂×Q₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
97	C₂×Q₈	C₂²×D₁₀₉	1	3	yes	—
98	Q₁₆	—	218	6	no	—
99	C₂×C₈	—	109	1	no	—
100	C₂×C₈	—	109	3	no	—
101	C₄⋊C₄	—	2	3	no	—
102	C₄⋊C₄	—	2	3	no	—
103	C₄⋊C₄	—	2	3	no	—
104	C₄⋊C₄	—	109	3	no	—
105	C₄⋊C₄	—	218	3	no	—
106	C₄⋊C₄	—	218	3	no	—
107	C₄⋊C₄	C₂²⋊D₁₀₉	1	3	yes	—
108	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
109	C₄²	—	2	3	no	—
110	C₄²	—	218	3	no	—
111	C₄²	C₂²×D₁₀₉	1	3	yes	—
112	Q₈	—	2	3	no	—
113	Q₈	—	2	3	no	—
114	Q₈	—	872	3	no	—
115	Q₈	—	109	2	no	—
116	Q₈	—	218	1	no	—
117	Q₈	#13182019	1	4	yes	complemented
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	218	3	no	—
121	C₂×C₄	—	218	1	no	—
122	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	3	yes	—
123	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	3	yes	—
124	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
125	C₈	—	218	3	no	—
126	C₈	—	218	1	no	—
127	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
128	C₄	—	2	3	no	—
129	C₄	—	4	3	no	—
130	C₄	—	4	1	no	—
131	C₄	—	436	3	no	—
132	C₄	—	218	2	no	—
133	C₄	C₂×#13182019	1	6	yes	—
134	C₄	#41×D₁₀₉	1	1	yes	characteristic
135	C₄	#16812746	1	1	yes	characteristic
136	C₂	#20295683	1	1	yes	characteristic
137	C₂	C₂²×#13182019	1	1	yes	characteristic
138	C₂	#20295834	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₂₁₈

C₂

yes

maximal, characteristic

#20295376

C₂

yes

maximal

#20295321

C₂

yes

maximal

#20295233

C₂

yes

maximal

Q₈×#11724784

C₂

yes

maximal, characteristic

C₄×#13182019

C₂

yes

maximal

#557×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182019

C₂²

yes

—

#16812699

C₂²

yes

—

#16812698

—

Q₈×Dic₁₀₉

—

C₄×Dic₂₁₈

C₂²

yes

—

#16812632

C₂²

yes

—

C₄⋊#11724784

C₂²

yes

—

#16812623

C₂²

yes

—

#16812623

C₂²

yes

—

#16812622

C₂²

yes

characteristic

C₄×#11724784

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

—

#125×C₁₀₉

—

C₄×Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

C₂²

yes

—

C₄×Q₈×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂³

yes

characteristic

#13182019

—

C₂×#11724784

C₂³

yes

characteristic

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182008

—

C₄×Dic₁₀₉

—

C₂×Q₈×C₁₀₉

D₄

yes

characteristic

C₂×Q₈×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

D₄

yes

—

C₄⋊C₄×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂³

yes

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

Q₁₆

yes

complemented

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

#41

yes

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×Q₁₆

yes

—

C₄×C₁₀₉

C₂×#41

yes

characteristic

C₄×C₁₀₉

2₋¹⁺⁴

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#548

yes

characteristic

C₂×C₁₀₉

C₂²×Q₁₆

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#3363

yes

characteristic, complemented, Sylow

Q₈²

—

109

—

#557

D₁₀₉

yes

characteristic

#487

—

109

—

#477

—

109

—

#462

—

109

—

C₈×Q₈

—

109

—

C₄×Q₁₆

—

109

—

C₂×Q₁₆

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

218

—

C₄⋊Q₈

C₂×D₁₀₉

yes

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

109

—

C₄×Q₈

—

218

—

C₄×Q₈

C₂×D₁₀₉

yes

—

C₄×Q₈

C₂×D₁₀₉

yes

characteristic

#107

—

109

—

C₄⋊C₈

—

109

—

#103

—

109

—

#103

—

109

—

#103

—

109

—

C₄×C₈

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂×Q₈

C₂²⋊D₁₀₉

yes

characteristic

C₂×Q₈

C₂²×D₁₀₉

yes

—

Q₁₆

—

218

—

C₂×C₈

—

109

—

100

C₂×C₈

—

109

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

109

—

105

C₄⋊C₄

—

218

—

106

C₄⋊C₄

—

218

—

107

C₄⋊C₄

C₂²⋊D₁₀₉

yes

—

108

C₄⋊C₄

C₂²×D₁₀₉

yes

characteristic

109

C₄²

—

110

C₄²

—

218

—

111

C₄²

C₂²×D₁₀₉

yes

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

872

—

115

Q₈

—

109

—

116

Q₈

—

218

—

117

Q₈

#13182019

yes

complemented

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

218

—

121

C₂×C₄

—

218

—

122

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

123

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

124

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

125

C₈

—

218

—

126

C₈

—

218

—

127

C₂²

C₂²×C₂²⋊D₁₀₉

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

436

—

132

C₄

—

218

—

133

C₄

C₂×#13182019

yes

—

134

C₄

#41×D₁₀₉

yes

characteristic

135

C₄

#16812746

yes

characteristic

136

C₂

#20295683

yes

characteristic

137

C₂

C₂²×#13182019

yes

characteristic

138

C₂

#20295834

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#557×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	C₂²×D₄
hypercenter	#557	D₁₀₉
hypocenter	C₁₀₉	#3363
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

#557×C₁₀₉

C₂

socle, prime radical

C₂²×C₁₀₉

C₂²×D₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

N	H ≅ G/N	type
44. Q₈×C₁₀₉	Q₁₆	semidirect
70. C₁₀₉	#3363	semidirect
117. Q₈	#13182019	semidirect

H ≅ G/N

type

44. Q₈×C₁₀₉

Q₁₆

semidirect

70. C₁₀₉

#3363

semidirect

117. Q₈

#13182019

semidirect

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

subgroup	index
Q₈×Dic₂₁₈	2
#20295376	2
#20295321	2
#20295233	2
Q₈×#11724784	2
C₄×#13182019	2
#557×C₁₀₉	2
#3363	109

subgroup

index

Q₈×Dic₂₁₈

#557×C₁₀₉

109

Subgroups — #28375706

Conjugacy classes of subgroups

Complements

Maximal subgroups