Group #28361327 — subgroups

Subgroups — #28361327

Order 13952.

190 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 105 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	#20295079	C₂	1	1	yes	maximal, characteristic
2	C₄×#13182013	C₂	1	1	yes	maximal, characteristic
3	#20294748	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294748	C₂	1	1	yes	maximal, characteristic, complemented
5	#20294747	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#11724784	C₂	1	1	yes	maximal, characteristic, complemented
7	#431×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13182013	C₄	1	2	yes	—
9	C₄⋊#11724784	C₄	1	2	yes	—
10	C₂²⋊#11724784	C₄	1	2	yes	—
11	#16812610	C₄	1	2	yes	—
12	C₄×#11724784	C₂²	1	1	yes	characteristic
13	C₄×#11724784	C₂²	1	1	yes	characteristic
14	C₄×#11724784	C₂²	1	1	yes	characteristic
15	C₄×#11724784	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₁₀₉	C₄	1	2	yes	—
18	C₂²⋊C₈×C₁₀₉	C₄	1	2	yes	—
19	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	#13182013	—	2	4	no	—
22	C₂×#11724784	C₂×C₄	1	2	yes	—
23	C₂×#11724784	C₂×C₄	1	2	yes	—
24	C₂×#11724784	C₂×C₄	1	2	yes	—
25	C₂×#11724784	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
31	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
35	C₂³×C₁₀₉	C₄²	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	—	2	2	no	—
37	C₂×C₄×C₁₀₉	—	2	2	no	—
38	C₂×C₄×C₁₀₉	—	2	2	no	—
39	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
45	C₈×C₁₀₉	—	2	2	no	—
46	C₈×C₁₀₉	—	2	2	no	—
47	#11724784	—	2	2	no	—
48	#11724784	—	2	2	no	—
49	#11724784	—	2	2	no	—
50	#11724784	M₁₆	1	4	yes	—
51	C₂²×C₁₀₉	—	2	1	no	—
52	C₂²×C₁₀₉	—	2	1	no	—
53	C₂²×C₁₀₉	—	2	1	no	—
54	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
55	C₄×C₁₀₉	—	2	1	no	—
56	C₄×C₁₀₉	—	2	1	no	—
57	C₄×C₁₀₉	—	2	1	no	—
58	C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
59	C₄×C₁₀₉	#131	1	2	yes	—
60	C₄×C₁₀₉	#131	1	2	yes	—
61	C₂×C₁₀₉	—	4	1	no	—
62	C₂×C₁₀₉	C₄×M₁₆	1	1	yes	characteristic
63	C₂×C₁₀₉	#405	1	1	yes	characteristic
64	C₂×C₁₀₉	#405	1	1	yes	characteristic
65	#1453	—	109	1	no	maximal, Sylow
66	C₁₀₉	#1453	1	1	yes	characteristic, complemented, Sylow
67	#431	—	109	1	no	—
68	#431	D₁₀₉	1	1	yes	characteristic
69	C₄×M₁₆	—	109	1	no	—
70	#322	—	109	1	no	—
71	#322	—	109	1	no	—
72	#322	—	109	1	no	—
73	C₈²	—	109	1	no	—
74	C₂×M₁₆	—	109	2	no	—
75	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
76	C₄⋊C₈	—	109	2	no	—
77	C₄⋊C₈	Dic₁₀₉	1	2	yes	—
78	C₂²⋊C₈	—	109	2	no	—
79	C₂²⋊C₈	Dic₁₀₉	1	2	yes	—
80	#97	—	109	2	no	—
81	C₄×C₈	—	109	1	no	—
82	C₄×C₈	—	109	1	no	—
83	C₄×C₈	—	109	1	no	—
84	C₄×C₈	—	109	1	no	—
85	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
86	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
87	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
88	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
89	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
90	M₁₆	—	218	4	no	—
91	C₂×C₈	—	109	2	no	—
92	C₂×C₈	—	109	2	no	—
93	C₂×C₈	—	109	2	no	—
94	C₂×C₈	—	109	2	no	—
95	C₂×C₈	C₂×Dic₁₀₉	1	2	yes	—
96	C₂×C₈	C₂×Dic₁₀₉	1	2	yes	—
97	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
98	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
99	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
100	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
101	C₂³	C₄×Dic₁₀₉	1	1	yes	characteristic
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₄×Dic₁₀₉	1	1	yes	characteristic
106	C₂×C₄	C₄×Dic₁₀₉	1	1	yes	characteristic
107	C₂×C₄	C₄×Dic₁₀₉	1	1	yes	characteristic
108	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
109	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
110	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	2	2	no	—
113	C₈	—	109	4	no	—
114	C₈	—	218	2	no	—
115	C₈	—	218	2	no	—
116	C₈	—	218	2	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂×C₄×Dic₁₀₉	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	M₁₆×D₁₀₉	1	2	yes	—
125	C₄	#16812705	1	2	yes	—
126	C₄	#16812706	1	2	yes	—
127	C₂	—	4	1	no	—
128	C₂	M₁₆×Dic₁₀₉	1	1	yes	characteristic
129	C₂	#20294981	1	1	yes	characteristic
130	C₂	#20294982	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295079

C₂

yes

maximal, characteristic

C₄×#13182013

C₂

yes

maximal, characteristic

#20294748

C₂

yes

maximal, characteristic, complemented

#20294748

C₂

yes

maximal, characteristic, complemented

#20294747

C₂

yes

maximal, characteristic, complemented

C₈×#11724784

C₂

yes

maximal, characteristic, complemented

#431×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₄

yes

—

C₄⋊#11724784

C₄

yes

—

C₂²⋊#11724784

C₄

yes

—

#16812610

C₄

yes

—

C₄×#11724784

C₂²

yes

characteristic

C₄×#11724784

C₂²

yes

characteristic

C₄×#11724784

C₂²

yes

characteristic

C₄×#11724784

C₂²

yes

characteristic

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

C₄

yes

—

C₂²⋊C₈×C₁₀₉

C₄

yes

—

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

#13182013

—

C₂×#11724784

C₂×C₄

yes

—

C₂×#11724784

C₂×C₄

yes

—

C₂×#11724784

C₂×C₄

yes

—

C₂×#11724784

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₈×C₁₀₉

—

C₈×C₁₀₉

—

—

—

—

M₁₆

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₄²

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×M₁₆

yes

—

C₄×C₁₀₉

#131

yes

—

C₄×C₁₀₉

#131

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₄×M₁₆

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1453

yes

characteristic, complemented, Sylow

#431

—

109

—

#431

D₁₀₉

yes

characteristic

C₄×M₁₆

—

109

—

#322

—

109

—

#322

—

109

—

#322

—

109

—

C₈²

—

109

—

C₂×M₁₆

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

109

—

C₄⋊C₈

Dic₁₀₉

yes

—

C₂²⋊C₈

—

109

—

C₂²⋊C₈

Dic₁₀₉

yes

—

#97

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

C₄×D₁₀₉

yes

characteristic

C₂²×C₄

C₄×D₁₀₉

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₉

yes

characteristic

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

C₂×Dic₁₀₉

yes

—

C₂×C₈

C₂×Dic₁₀₉

yes

—

C₄²

C₄×D₁₀₉

yes

characteristic

C₄²

C₄×D₁₀₉

yes

characteristic

C₄²

C₂×Dic₁₀₉

yes

characteristic

100

C₄²

C₂²×D₁₀₉

yes

characteristic

101

C₂³

C₄×Dic₁₀₉

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₄×Dic₁₀₉

yes

characteristic

106

C₂×C₄

C₄×Dic₁₀₉

yes

characteristic

107

C₂×C₄

C₄×Dic₁₀₉

yes

characteristic

108

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

109

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

110

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

109

—

114

C₈

—

218

—

115

C₈

—

218

—

116

C₈

—

218

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×C₄×Dic₁₀₉

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

M₁₆×D₁₀₉

yes

—

125

C₄

#16812705

yes

—

126

C₄

#16812706

yes

—

127

C₂

—

128

C₂

M₁₆×Dic₁₀₉

yes

characteristic

129

C₂

#20294981

yes

characteristic

130

C₂

#20294982

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂×C₄²
Fitting	#431×C₁₀₉	C₂
prime radical	C₂³×C₁₀₉	C₄²
cosocle	C₄²×C₁₀₉	C₂³
hypercenter	#431	D₁₀₉
hypocenter	C₁₀₉	#1453
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄² ◁ #431

subgroup

G/H

center, Frattini

C₄²

C₂²×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂×C₄²

Fitting

#431×C₁₀₉

C₂

prime radical

C₂³×C₁₀₉

C₄²

cosocle

C₄²×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₄² ◁ #431

N	H ≅ G/N	type
3. #20294748	C₂	semidirect
4. #20294748	C₂	semidirect
5. #20294747	C₂	semidirect
6. C₈×#11724784	C₂	semidirect
66. C₁₀₉	#1453	semidirect

H ≅ G/N

type

3. #20294748

C₂

semidirect

4. #20294748

C₂

semidirect

5. #20294747

C₂

semidirect

6. C₈×#11724784

C₂

semidirect

66. C₁₀₉

#1453

semidirect

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

subgroup	index
#20295079	2
C₄×#13182013	2
#20294748	2
#20294747	2
C₈×#11724784	2
#431×C₁₀₉	2
#1453	109

subgroup

index

#431×C₁₀₉

109

Subgroups — #28361327

Conjugacy classes of subgroups

Complements

Maximal subgroups