Group #28378242 — subgroups

Subgroups — #28378242

Order 13952.

208 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 94 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	#20296045	C₂	1	1	yes	maximal, characteristic
2	C₄×#13182039	C₂	1	1	yes	maximal, characteristic
3	#11724786⋊C₈	C₂	1	2	yes	maximal, complemented
4	#20295867	C₂	1	2	yes	maximal, complemented
5	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182039	C₄	1	2	yes	—
7	C₄⋊#11724786	C₄	1	2	yes	—
8	#16812756	C₄	1	2	yes	—
9	#16812754	C₄	1	2	yes	—
10	C₄×#11724786	C₂²	1	2	yes	—
11	C₄×#11724786	C₂²	1	2	yes	—
12	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
13	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
14	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
15	C₄×#11724784	C₄	1	1	yes	characteristic
16	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
17	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
18	C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
19	#13182039	—	2	4	no	—
20	C₂×#11724786	C₂×C₄	1	4	yes	—
21	C₂×#11724786	C₂×C₄	1	4	yes	—
22	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
26	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
27	C₈×D₁₀₉	—	2	2	no	—
28	C₈×D₁₀₉	—	2	2	no	—
29	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×D₁₀₉	C₄²	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×Dic₁₀₉	C₄²	1	1	yes	characteristic
40	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
42	C₄×D₁₀₉	—	2	1	no	—
43	C₄×D₁₀₉	—	2	1	no	—
44	C₄×D₁₀₉	—	2	2	no	—
45	C₄×D₁₀₉	M₁₆	1	4	yes	—
46	#11724786	—	2	4	no	—
47	#11724786	—	2	4	no	—
48	C₈×C₁₀₉	—	2	1	no	—
49	C₈×C₁₀₉	M₁₆	1	2	yes	complemented
50	#11724784	—	2	1	no	—
51	#11724784	M₁₆	1	2	yes	complemented
52	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
53	C₂×D₁₀₉	—	2	1	no	—
54	C₂×D₁₀₉	—	2	1	no	—
55	C₂×D₁₀₉	#97	1	2	yes	—
56	C₄×C₁₀₉	—	2	1	no	—
57	C₄×C₁₀₉	#97	1	2	yes	—
58	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
59	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
60	Dic₁₀₉	—	2	1	no	—
61	Dic₁₀₉	C₂×M₁₆	1	2	yes	—
62	Dic₁₀₉	#131	1	2	yes	—
63	C₂×C₁₀₉	C₂×#97	1	1	yes	characteristic
64	C₂×C₁₀₉	C₄×M₁₆	1	1	yes	characteristic
65	C₂×C₁₀₉	#405	1	1	yes	characteristic
66	D₁₀₉	—	2	2	no	—
67	#1451	—	109	1	no	maximal, Sylow
68	C₁₀₉	#1451	1	1	yes	characteristic, complemented, Sylow
69	#431	—	109	1	no	—
70	C₄×M₁₆	—	109	1	no	—
71	C₂×C₄×C₈	—	109	1	no	—
72	#322	—	109	2	no	—
73	#322	—	109	2	no	—
74	C₂×M₁₆	—	109	2	no	—
75	C₂²×C₈	—	109	1	no	—
76	C₂²×C₈	—	109	1	no	—
77	C₂×C₄²	—	109	1	no	—
78	C₄⋊C₈	—	109	2	no	—
79	C₂²⋊C₈	—	109	2	no	—
80	#97	—	109	2	no	—
81	C₄×C₈	—	109	2	no	—
82	C₄×C₈	—	109	2	no	—
83	C₄×C₈	—	109	1	no	—
84	C₄×C₈	—	109	1	no	—
85	C₄×C₈	—	109	1	no	—
86	C₄×C₈	#89137	1	1	yes	characteristic
87	C₂²×C₄	—	109	1	no	—
88	C₂²×C₄	—	109	1	no	—
89	C₂²×C₄	—	109	1	no	—
90	M₁₆	—	218	4	no	—
91	C₂×C₈	—	109	1	no	—
92	C₂×C₈	—	109	4	no	—
93	C₂×C₈	—	109	4	no	—
94	C₂×C₈	—	109	1	no	—
95	C₂×C₈	—	218	2	no	—
96	C₂×C₈	—	218	2	no	—
97	C₂×C₈	C₂×#89137	1	1	yes	characteristic
98	C₂×C₈	C₂×#89137	1	1	yes	characteristic
99	C₄²	—	109	1	no	—
100	C₄²	—	109	1	no	—
101	C₄²	—	109	1	no	—
102	C₄²	C₂×#89137	1	1	yes	characteristic
103	C₂³	—	109	1	no	—
104	C₂×C₄	—	109	1	no	—
105	C₂×C₄	—	109	1	no	—
106	C₂×C₄	—	109	4	no	—
107	C₂×C₄	—	109	1	no	—
108	C₂×C₄	—	218	2	no	—
109	C₂×C₄	—	218	1	no	—
110	C₂×C₄	—	218	1	no	—
111	C₂×C₄	C₄×#89137	1	1	yes	characteristic
112	C₂×C₄	C₄×#89137	1	1	yes	characteristic
113	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
114	C₈	—	2	1	no	—
115	C₈	—	109	2	no	—
116	C₈	—	218	4	no	—
117	C₈	—	218	4	no	—
118	C₈	—	218	1	no	—
119	C₈	#13182039	1	2	yes	complemented
120	C₂²	—	109	2	no	—
121	C₂²	—	218	1	no	—
122	C₂²	—	218	1	no	—
123	C₂²	C₂×C₄×#89137	1	1	yes	characteristic
124	C₄	—	2	1	no	—
125	C₄	—	109	2	no	—
126	C₄	—	109	2	no	—
127	C₄	—	218	1	no	—
128	C₄	#16812753	1	2	yes	—
129	C₄	D₁₀₉⋊M₁₆	1	1	yes	characteristic
130	C₄	C₂×#13182039	1	1	yes	characteristic
131	C₂	—	218	2	no	—
132	C₂	C₂×#16812753	1	1	yes	characteristic
133	C₂	C₄×#13182039	1	1	yes	characteristic
134	C₂	#20295989	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296045

C₂

yes

maximal, characteristic

C₄×#13182039

C₂

yes

maximal, characteristic

#11724786⋊C₈

C₂

yes

maximal, complemented

#20295867

C₂

yes

maximal, complemented

C₄×C₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182039

C₄

yes

—

C₄⋊#11724786

C₄

yes

—

#16812756

C₄

yes

—

#16812754

C₄

yes

—

C₄×#11724786

C₂²

yes

—

C₄×#11724786

C₂²

yes

—

C₂×C₈×D₁₀₉

C₄

yes

characteristic

C₂×C₈×D₁₀₉

C₄

yes

characteristic

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄×#11724784

C₄

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₄

yes

characteristic

#13182039

—

C₂×#11724786

C₂×C₄

yes

—

C₂×#11724786

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₈×D₁₀₉

—

C₈×D₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₄²

yes

characteristic

C₂×Dic₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²×C₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

M₁₆

yes

—

#11724786

—

#11724786

—

C₈×C₁₀₉

—

C₈×C₁₀₉

M₁₆

yes

complemented

#11724784

—

#11724784

M₁₆

yes

complemented

C₂²×C₁₀₉

C₂×C₄²

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

#97

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#97

yes

—

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

C₂×M₁₆

yes

—

Dic₁₀₉

#131

yes

—

C₂×C₁₀₉

C₂×#97

yes

characteristic

C₂×C₁₀₉

C₄×M₁₆

yes

characteristic

C₂×C₁₀₉

#405

yes

characteristic

D₁₀₉

—

#1451

—

109

maximal, Sylow

C₁₀₉

#1451

yes

characteristic, complemented, Sylow

#431

—

109

—

C₄×M₁₆

—

109

—

C₂×C₄×C₈

—

109

—

#322

—

109

—

#322

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₂²⋊C₈

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

#89137

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

109

—

100

C₄²

—

109

—

101

C₄²

—

109

—

102

C₄²

C₂×#89137

yes

characteristic

103

C₂³

—

109

—

104

C₂×C₄

—

109

—

105

C₂×C₄

—

109

—

106

C₂×C₄

—

109

—

107

C₂×C₄

—

109

—

108

C₂×C₄

—

218

—

109

C₂×C₄

—

218

—

110

C₂×C₄

—

218

—

111

C₂×C₄

C₄×#89137

yes

characteristic

112

C₂×C₄

C₄×#89137

yes

characteristic

113

C₂×C₄

C₂²×#89137

yes

characteristic

114

C₈

—

115

C₈

—

109

—

116

C₈

—

218

—

117

C₈

—

218

—

118

C₈

—

218

—

119

C₈

#13182039

yes

complemented

120

C₂²

—

109

—

121

C₂²

—

218

—

122

C₂²

—

218

—

123

C₂²

C₂×C₄×#89137

yes

characteristic

124

C₄

—

125

C₄

—

109

—

126

C₄

—

109

—

127

C₄

—

218

—

128

C₄

#16812753

yes

—

129

C₄

D₁₀₉⋊M₁₆

yes

characteristic

130

C₄

C₂×#13182039

yes

characteristic

131

C₂

—

218

—

132

C₂

C₂×#16812753

yes

characteristic

133

C₂

C₄×#13182039

yes

characteristic

134

C₂

#20295989

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#89137
commutator, socle	C₂²×C₁₀₉	C₂×C₄²
Fitting	C₄×C₈×C₁₀₉	C₄
prime radical	C₂²×D₁₀₉	C₄²
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	C₄×C₈	#89137
hypocenter	C₁₀₉	#1451
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#89137

commutator, socle

C₂²×C₁₀₉

C₂×C₄²

Fitting

C₄×C₈×C₁₀₉

C₄

prime radical

C₂²×D₁₀₉

C₄²

cosocle

C₄×Dic₁₀₉

C₂³

C₄×C₈

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₄ ◁ C₄×C₈

N	H ≅ G/N	type
3. #11724786⋊C₈	C₂	semidirect
4. #20295867	C₂	semidirect
49. C₈×C₁₀₉	M₁₆	semidirect
51. #11724784	M₁₆	semidirect
68. C₁₀₉	#1451	semidirect
119. C₈	#13182039	semidirect

H ≅ G/N

type

3. #11724786⋊C₈

C₂

semidirect

4. #20295867

C₂

semidirect

49. C₈×C₁₀₉

M₁₆

semidirect

51. #11724784

M₁₆

semidirect

68. C₁₀₉

#1451

semidirect

119. C₈

#13182039

semidirect

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

subgroup	index
#20296045	2
C₄×#13182039	2
#11724786⋊C₈	2
#20295867	2
C₄×C₈×D₁₀₉	2
#1451	109

subgroup

index

C₄×C₈×D₁₀₉

109

Subgroups — #28378242

Conjugacy classes of subgroups

Complements

Maximal subgroups