Group #28364180 — subgroups

Subgroups — #28364180

Order 13952.

258 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 87 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×#16812622	C₂	1	2	yes	maximal
2	#20294932	C₂	1	1	yes	maximal, characteristic
3	#20294788	C₂	1	2	yes	maximal
4	#20294781	C₂	1	1	yes	maximal, characteristic
5	C₂×#125×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₈	C₂²	1	1	yes	characteristic
7	C₂²×#11724784	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
9	#16812622	—	2	4	no	—
10	#16812606	—	2	2	no	—
11	#125×C₁₀₉	—	2	4	no	—
12	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₁₈	—	2	2	no	—
16	C₂×Dic₂₁₈	—	2	2	no	—
17	C₂×Dic₂₁₈	—	2	1	no	—
18	C₂×Dic₂₁₈	Q₈	1	2	yes	—
19	C₂²×Dic₁₀₉	—	2	1	no	—
20	C₂²×Dic₁₀₉	—	2	1	no	—
21	C₂×#11724784	—	2	2	no	—
22	C₂×#11724784	—	2	2	no	—
23	C₂×#11724784	—	2	2	no	—
24	C₄⋊Dic₁₀₉	—	2	2	no	—
25	C₂²×C₄×C₁₀₉	—	2	2	no	—
26	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
27	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₀₉	—	2	8	no	—
29	C₄⋊C₄×C₁₀₉	—	2	2	no	—
30	C₄⋊C₄×C₁₀₉	D₄	1	4	yes	—
31	C₄²×C₁₀₉	—	2	2	no	—
32	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	—	2	2	no	—
34	C₂×C₄×C₁₀₉	—	2	4	no	—
35	C₂×C₄×C₁₀₉	—	2	2	no	—
36	C₂×C₄×C₁₀₉	—	2	2	no	—
37	C₂×C₄×C₁₀₉	—	4	2	no	—
38	C₂×C₄×C₁₀₉	—	4	2	no	—
39	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	#41	1	2	yes	—
42	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
43	C₂×Dic₁₀₉	—	2	2	no	—
44	C₂×Dic₁₀₉	—	4	1	no	—
45	C₂×Dic₁₀₉	—	4	1	no	—
46	C₂×Dic₁₀₉	—	4	1	no	—
47	C₂×Dic₁₀₉	—	4	2	no	—
48	Dic₂₁₈	—	2	4	no	—
49	Dic₂₁₈	—	4	2	no	—
50	#11724784	—	4	2	no	—
51	C₂²×C₁₀₉	#120	1	1	yes	characteristic
52	C₂²×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
53	C₂²×C₁₀₉	#124	1	1	yes	characteristic
54	C₂²×C₁₀₉	#135	1	2	yes	—
55	C₂²×C₁₀₉	#137	1	2	yes	—
56	C₄×C₁₀₉	—	4	2	no	—
57	C₄×C₁₀₉	—	4	4	no	—
58	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₁₀₉	#124	1	2	yes	—
60	Dic₁₀₉	—	4	2	no	—
61	Dic₁₀₉	—	8	1	no	—
62	C₂×C₁₀₉	#393	1	1	yes	characteristic
63	C₂×C₁₀₉	#452	1	2	yes	—
64	C₂×C₁₀₉	#479	1	2	yes	—
65	C₂×C₁₀₉	#488	1	1	yes	characteristic
66	C₂×C₁₀₉	#491	1	1	yes	characteristic
67	#2026	—	109	1	no	maximal, Sylow
68	C₁₀₉	#2026	1	1	yes	characteristic, complemented, Sylow
69	C₂×#125	D₁₀₉	1	1	yes	characteristic
70	C₂×#103	—	109	2	no	—
71	#391	—	109	1	no	—
72	#340	—	109	2	no	—
73	#336	—	109	1	no	—
74	C₂²×Q₈	—	109	1	no	—
75	C₂²×C₈	—	109	2	no	—
76	#125	—	2	4	no	—
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	—	109	1	no	—
79	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
80	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
81	#103	—	218	4	no	—
82	#95	—	218	2	no	—
83	C₂×Q₈	—	109	2	no	—
84	C₂×Q₈	—	218	2	no	—
85	C₂×Q₈	—	218	2	no	—
86	C₂×Q₈	—	218	1	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	—	218	1	no	—
89	C₂²×C₄	—	218	1	no	—
90	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
91	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
92	C₂×C₈	—	218	2	no	—
93	C₂×C₈	—	218	2	no	—
94	C₂×C₈	—	218	2	no	—
95	C₄⋊C₄	—	2	8	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	218	2	no	—
98	C₄⋊C₄	C₂²⋊D₁₀₉	1	4	yes	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
101	Q₈	—	436	2	no	—
102	Q₈	—	218	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	436	1	no	—
110	C₂×C₄	—	436	2	no	—
111	C₂×C₄	—	436	1	no	—
112	C₂×C₄	—	436	1	no	—
113	C₂×C₄	—	218	2	no	—
114	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
115	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
116	C₂×C₄	#13182030	1	2	yes	—
117	C₂×C₄	#13182030	1	1	yes	characteristic
118	C₈	—	436	2	no	—
119	C₂²	#16812668	1	1	yes	characteristic
120	C₂²	#16812675	1	1	yes	characteristic
121	C₂²	#16812688	1	1	yes	characteristic
122	C₂²	#16812719	1	2	yes	—
123	C₂²	#16812729	1	2	yes	—
124	C₄	—	4	2	no	—
125	C₄	—	4	4	no	—
126	C₄	—	872	1	no	—
127	C₄	—	436	2	no	—
128	C₄	#16812675	1	2	yes	—
129	C₄	#16812688	1	2	yes	—
130	C₂	#20294937	1	1	yes	characteristic
131	C₂	#20295188	1	2	yes	—
132	C₂	#20295330	1	2	yes	—
133	C₂	#20295383	1	1	yes	characteristic
134	C₂	#20295396	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812622

C₂

yes

maximal

#20294932

C₂

yes

maximal, characteristic

#20294788

C₂

yes

maximal

#20294781

C₂

yes

maximal, characteristic

C₂×#125×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

C₂²

yes

characteristic

C₂²×#11724784

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

characteristic

#16812622

—

#16812606

—

#125×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

Q₈

yes

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

—

C₂×#11724784

—

C₂×#11724784

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

#41

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

#11724784

—

C₂²×C₁₀₉

#120

yes

characteristic

C₂²×C₁₀₉

C₂²⋊Q₈

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

—

C₂²×C₁₀₉

#137

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

C₄×C₁₀₉

#124

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#393

yes

characteristic

C₂×C₁₀₉

#452

yes

—

C₂×C₁₀₉

#479

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#2026

yes

characteristic, complemented, Sylow

C₂×#125

D₁₀₉

yes

characteristic

C₂×#103

—

109

—

#391

—

109

—

#340

—

109

—

#336

—

109

—

C₂²×Q₈

—

109

—

C₂²×C₈

—

109

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#103

—

218

—

#95

—

218

—

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂²×C₄

—

C₂²×C₄

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

218

—

C₄⋊C₄

C₂²⋊D₁₀₉

yes

—

C₄²

—

100

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

101

Q₈

—

436

—

102

Q₈

—

218

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

436

—

110

C₂×C₄

—

436

—

111

C₂×C₄

—

436

—

112

C₂×C₄

—

436

—

113

C₂×C₄

—

218

—

114

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

115

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

116

C₂×C₄

#13182030

yes

—

117

C₂×C₄

#13182030

yes

characteristic

118

C₈

—

436

—

119

C₂²

#16812668

yes

characteristic

120

C₂²

#16812675

yes

characteristic

121

C₂²

#16812688

yes

characteristic

122

C₂²

#16812719

yes

—

123

C₂²

#16812729

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

872

—

127

C₄

—

436

—

128

C₄

#16812675

yes

—

129

C₄

#16812688

yes

—

130

C₂

#20294937

yes

characteristic

131

C₂

#20295188

yes

—

132

C₂

#20295330

yes

—

133

C₂

#20295383

yes

characteristic

134

C₂

#20295396

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₉
commutator, cosocle	C₂²×C₄×C₁₀₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×#125×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
hypercenter	C₂×#125	D₁₀₉
hypocenter	C₁₀₉	#2026
LCS	G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₉

commutator, cosocle

C₂²×C₄×C₁₀₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂×#125×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
68. C₁₀₉	#2026	semidirect

H ≅ G/N

type

68. C₁₀₉

#2026

semidirect

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

subgroup	index
C₂×#16812622	2
#20294932	2
#20294788	2
#20294781	2
C₂×#125×C₁₀₉	2
#2026	109

subgroup

index

C₂×#125×C₁₀₉

109

Subgroups — #28364180

Conjugacy classes of subgroups

Complements

Maximal subgroups