Group #28371409 — subgroups

Subgroups — #28371409

Order 13952.

504 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 197 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	#20294933	C₂	1	4	yes	maximal
3	#20294931	C₂	1	4	yes	maximal
4	#20294891	C₂	1	4	yes	maximal
5	C₄²×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₁₈	C₂²	1	2	yes	—
8	#16812699	—	2	4	no	—
9	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
11	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
12	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
13	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
14	#16812606	C₂²	1	16	yes	—
15	C₂²×Q₈×C₁₀₉	C₂²	1	2	yes	—
16	C₄⋊Q₈×C₁₀₉	—	2	4	no	—
17	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
19	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×Dic₂₁₈	—	2	4	no	—
21	C₂×Dic₂₁₈	—	2	8	no	—
22	C₂²×Dic₁₀₉	C₂³	1	4	yes	—
23	C₂²×Dic₁₀₉	C₂³	1	4	yes	—
24	C₄⋊Dic₁₀₉	—	2	4	no	—
25	C₄⋊Dic₁₀₉	—	2	4	no	—
26	C₄×Dic₁₀₉	—	2	4	no	—
27	C₄×Dic₁₀₉	—	2	4	no	—
28	C₄×Dic₁₀₉	Q₈	1	8	yes	complemented
29	C₂×Q₈×C₁₀₉	—	2	4	no	—
30	C₂×Q₈×C₁₀₉	—	2	8	no	—
31	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
32	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₀₉	—	2	4	no	—
36	C₄⋊C₄×C₁₀₉	—	2	4	no	—
37	C₄²×C₁₀₉	D₄	1	4	yes	—
38	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
39	Q₈×C₁₀₉	—	4	8	no	—
40	C₂×C₄×C₁₀₉	—	2	4	no	—
41	C₂×C₄×C₁₀₉	—	2	8	no	—
42	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₀₉	C₂×Q₈	1	4	yes	—
46	C₂×C₄×C₁₀₉	#41	1	4	yes	—
47	C₂×C₄×C₁₀₉	#41	1	4	yes	—
48	C₂×Dic₁₀₉	—	2	4	no	—
49	C₂×Dic₁₀₉	—	2	4	no	—
50	C₂×Dic₁₀₉	—	2	4	no	—
51	C₂×Dic₁₀₉	—	2	8	no	—
52	C₂×Dic₁₀₉	C₂×Q₈	1	8	yes	—
53	Dic₂₁₈	—	4	8	no	—
54	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
55	C₂²×C₁₀₉	C₂²×Q₈	1	2	yes	—
56	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
57	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
58	C₄×C₁₀₉	—	4	4	no	—
59	C₄×C₁₀₉	#124	1	4	yes	—
60	C₄×C₁₀₉	#124	1	4	yes	—
61	C₄×C₁₀₉	C₄⋊Q₈	1	4	yes	—
62	Dic₁₀₉	—	2	8	no	—
63	Dic₁₀₉	—	4	4	no	—
64	C₂×C₁₀₉	C₂×#124	1	1	yes	characteristic
65	C₂×C₁₀₉	C₂×#124	1	1	yes	characteristic
66	C₂×C₁₀₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₁₀₉	#533	1	4	yes	—
68	#2862	—	109	1	no	maximal, Sylow
69	C₁₀₉	#2862	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₄⋊Q₈	—	109	1	no	—
71	C₂×C₄⋊Q₈	D₁₀₉	1	1	yes	characteristic
72	#391	—	109	4	no	—
73	#391	—	109	4	no	—
74	#382	—	109	4	no	—
75	C₄³	—	109	1	no	—
76	C₂²×Q₈	—	109	2	no	—
77	C₂²×Q₈	C₂×D₁₀₉	1	2	yes	—
78	C₄⋊Q₈	—	2	4	no	—
79	C₄⋊Q₈	—	218	4	no	—
80	C₂×C₄⋊C₄	—	109	2	no	—
81	C₂×C₄⋊C₄	—	109	2	no	—
82	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
83	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
84	C₂×C₄²	—	109	2	no	—
85	C₂×C₄²	—	109	2	no	—
86	C₂×C₄²	—	109	2	no	—
87	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
88	#95	—	109	16	no	—
89	C₂×Q₈	—	2	4	no	—
90	C₂×Q₈	—	2	8	no	—
91	C₂×Q₈	—	218	4	no	—
92	C₂×Q₈	—	218	8	no	—
93	C₂²×C₄	—	109	4	no	—
94	C₂²×C₄	—	109	4	no	—
95	C₂²×C₄	C₂²×D₁₀₉	1	4	yes	—
96	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	218	4	no	—
102	C₄⋊C₄	—	218	4	no	—
103	C₄²	—	109	8	no	—
104	C₄²	—	218	4	no	—
105	C₄²	—	218	4	no	—
106	C₄²	C₂²⋊D₁₀₉	1	4	yes	—
107	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
108	Q₈	—	4	8	no	—
109	Q₈	—	436	8	no	—
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	2	8	no	—
112	C₂×C₄	—	109	8	no	—
113	C₂×C₄	—	218	4	no	—
114	C₂×C₄	—	218	8	no	—
115	C₂×C₄	—	218	4	no	—
116	C₂×C₄	—	218	4	no	—
117	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
120	C₂×C₄	Q₈×D₁₀₉	1	4	yes	—
121	C₂×C₄	#13182028	1	4	yes	—
122	C₂×C₄	#13182030	1	4	yes	—
123	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
124	C₂²	C₂×Q₈×D₁₀₉	1	2	yes	—
125	C₂²	C₂×#13182028	1	2	yes	—
126	C₂²	C₂×#13182030	1	2	yes	—
127	C₄	—	4	4	no	—
128	C₄	—	436	4	no	—
129	C₄	—	218	8	no	—
130	C₄	#16812686	1	4	yes	—
131	C₄	#16812688	1	4	yes	—
132	C₄	#16812697	1	4	yes	—
133	C₂	C₂×#16812686	1	1	yes	characteristic
134	C₂	C₂×#16812688	1	1	yes	characteristic
135	C₂	C₂×#16812697	1	1	yes	characteristic
136	C₂	#20295581	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812699

C₂

yes

maximal, characteristic

#20294933

C₂

yes

maximal

#20294931

C₂

yes

maximal

#20294891

C₂

yes

maximal

C₄²×Dic₁₀₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

C₂²

yes

—

#16812699

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

#16812606

C₂²

yes

—

C₂²×Q₈×C₁₀₉

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂²×Dic₁₀₉

C₂³

yes

—

C₂²×Dic₁₀₉

C₂³

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

Q₈

yes

complemented

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

C₂⁴

yes

characteristic

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×Q₈

yes

—

Dic₂₁₈

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₂²×C₁₀₉

C₂²×Q₈

yes

—

C₂²×C₁₀₉

C₂×#41

yes

—

C₂²×C₁₀₉

C₂×#41

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#124

yes

—

C₄×C₁₀₉

#124

yes

—

C₄×C₁₀₉

C₄⋊Q₈

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₀₉

#533

yes

—

#2862

—

109

maximal, Sylow

C₁₀₉

#2862

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

109

—

C₂×C₄⋊Q₈

D₁₀₉

yes

characteristic

#391

—

109

—

#391

—

109

—

#382

—

109

—

C₄³

—

109

—

C₂²×Q₈

—

109

—

C₂²×Q₈

C₂×D₁₀₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#95

—

109

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

218

—

102

C₄⋊C₄

—

218

—

103

C₄²

—

109

—

104

C₄²

—

218

—

105

C₄²

—

218

—

106

C₄²

C₂²⋊D₁₀₉

yes

—

107

C₂³

C₂³×D₁₀₉

yes

characteristic

108

Q₈

—

109

Q₈

—

436

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

109

—

113

C₂×C₄

—

218

—

114

C₂×C₄

—

218

—

115

C₂×C₄

—

218

—

116

C₂×C₄

—

218

—

117

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

118

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

119

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

120

C₂×C₄

Q₈×D₁₀₉

yes

—

121

C₂×C₄

#13182028

yes

—

122

C₂×C₄

#13182030

yes

—

123

C₂²

C₂²×C₂²⋊D₁₀₉

yes

characteristic

124

C₂²

C₂×Q₈×D₁₀₉

yes

—

125

C₂²

C₂×#13182028

yes

—

126

C₂²

C₂×#13182030

yes

—

127

C₄

—

128

C₄

—

436

—

129

C₄

—

218

—

130

C₄

#16812686

yes

—

131

C₄

#16812688

yes

—

132

C₄

#16812697

yes

—

133

C₂

C₂×#16812686

yes

characteristic

134

C₂

C₂×#16812688

yes

characteristic

135

C₂

C₂×#16812697

yes

characteristic

136

C₂

#20295581

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₉
commutator, socle, prime radical, cosocle	C₂³×C₁₀₉	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₁₀₉	C₂
hypercenter	C₂×C₄⋊Q₈	D₁₀₉
hypocenter	C₁₀₉	#2862
LCS	G ▷ C₂³×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₀₉

commutator, socle, prime radical, cosocle

C₂³×C₁₀₉

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₁₀₉

C₂

hypercenter

C₂×C₄⋊Q₈

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
28. C₄×Dic₁₀₉	Q₈	semidirect
69. C₁₀₉	#2862	semidirect

H ≅ G/N

type

28. C₄×Dic₁₀₉

Q₈

semidirect

69. C₁₀₉

#2862

semidirect

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

subgroup	index
C₂×#16812699	2
#20294933	2
#20294931	2
#20294891	2
C₄²×Dic₁₀₉	2
C₂×C₄⋊Q₈×C₁₀₉	2
#2862	109

subgroup

index

C₄²×Dic₁₀₉

C₂×C₄⋊Q₈×C₁₀₉

#2862

109

Subgroups — #28371409

Conjugacy classes of subgroups

Complements

Maximal subgroups