Group #28364786 — subgroups

Subgroups — #28364786

Order 13952.

172 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 85 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×#16812638	C₂	1	2	yes	maximal
2	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#13182004	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294844	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
7	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
8	C₄²×D₁₀₉	C₄	1	1	yes	characteristic
9	#16812638	—	2	4	no	—
10	C₂×#13182004	C₂²	1	2	yes	—
11	C₄×#11724784	C₄	1	1	yes	characteristic
12	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic
13	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₁₀₉	C₂²	1	2	yes	—
15	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₁₀₉	C₈	1	1	yes	characteristic
17	C₂×C₄×D₁₀₉	C₈	1	1	yes	characteristic
18	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
20	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
21	C₈×D₁₀₉	—	2	2	no	—
22	C₈×D₁₀₉	D₄	1	2	yes	—
23	C₈×D₁₀₉	Q₈	1	2	yes	—
24	C₄×Dic₁₀₉	C₈	1	1	yes	characteristic
25	C₄×Dic₁₀₉	C₈	1	1	yes	characteristic
26	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
27	#13182004	—	2	2	no	—
28	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
30	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₁₆×C₁₀₉	—	2	2	no	—
32	C₂²×D₁₀₉	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
37	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
38	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
39	C₄×D₁₀₉	—	2	2	no	—
40	C₄×D₁₀₉	—	2	2	no	—
41	C₄×D₁₀₉	C₄⋊C₄	1	2	yes	—
42	C₄×D₁₀₉	M₁₆	1	2	yes	—
43	C₈×C₁₀₉	—	2	1	no	—
44	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
45	C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
46	#11724784	—	2	1	no	—
47	#11724784	C₄⋊C₄	1	2	yes	—
48	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
49	C₂×D₁₀₉	—	2	1	no	—
50	C₂×D₁₀₉	—	2	1	no	—
51	C₂×D₁₀₉	C₄⋊C₈	1	2	yes	—
52	C₄×C₁₀₉	—	2	1	no	—
53	C₄×C₁₀₉	M₃₂	1	2	yes	complemented
54	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
56	Dic₁₀₉	—	2	1	no	—
57	Dic₁₀₉	C₄⋊C₈	1	2	yes	—
58	Dic₁₀₉	M₃₂	1	2	yes	complemented
59	C₂×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₁₀₉	C₂×M₃₂	1	1	yes	characteristic
61	C₂×C₁₀₉	C₂×M₃₂	1	1	yes	characteristic
62	D₁₀₉	—	2	2	no	—
63	#2150	—	109	1	no	maximal, Sylow
64	C₁₀₉	#2150	1	1	yes	characteristic, complemented, Sylow
65	C₂×M₃₂	—	109	2	no	—
66	C₂×C₄×C₈	—	109	1	no	—
67	C₄⋊C₁₆	—	109	2	no	—
68	C₄⋊C₁₆	—	109	1	no	—
69	C₄⋊C₁₆	D₁₀₉	1	1	yes	characteristic, complemented
70	C₂²×C₈	—	109	1	no	—
71	C₂²×C₈	—	109	1	no	—
72	C₂×C₄²	—	109	1	no	—
73	M₃₂	—	218	4	no	—
74	C₂×C₁₆	—	109	2	no	—
75	C₂×C₁₆	C₂×D₁₀₉	1	2	yes	—
76	C₄×C₈	—	109	1	no	—
77	C₄×C₈	—	109	1	no	—
78	C₄×C₈	—	109	1	no	—
79	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
80	C₂²×C₄	—	109	1	no	—
81	C₂²×C₄	—	109	1	no	—
82	C₂²×C₄	—	109	1	no	—
83	C₂×C₈	—	109	1	no	—
84	C₂×C₈	—	109	2	no	—
85	C₂×C₈	—	109	2	no	—
86	C₂×C₈	—	109	1	no	—
87	C₂×C₈	—	218	2	no	—
88	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
89	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
90	C₄²	—	109	1	no	—
91	C₄²	—	109	1	no	—
92	C₄²	—	109	1	no	—
93	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
94	C₁₆	—	2	2	no	—
95	C₁₆	—	218	2	no	—
96	C₂³	—	109	1	no	—
97	C₂×C₄	—	109	1	no	—
98	C₂×C₄	—	109	1	no	—
99	C₂×C₄	—	109	2	no	—
100	C₂×C₄	—	109	2	no	—
101	C₂×C₄	—	109	1	no	—
102	C₂×C₄	—	218	2	no	—
103	C₂×C₄	—	218	2	no	—
104	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
105	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
106	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	109	2	no	—
109	C₈	—	218	1	no	—
110	C₈	D₄×D₁₀₉	1	1	yes	characteristic
111	C₈	Q₈×D₁₀₉	1	1	yes	characteristic
112	C₂²	—	109	2	no	—
113	C₂²	—	218	1	no	—
114	C₂²	—	218	1	no	—
115	C₂²	C₂×C₈×D₁₀₉	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	109	2	no	—
118	C₄	—	109	2	no	—
119	C₄	—	218	1	no	—
120	C₄	#16812638	1	2	yes	complemented
121	C₄	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
122	C₄	M₁₆×D₁₀₉	1	1	yes	characteristic
123	C₂	—	218	2	no	—
124	C₂	C₄⋊C₈×D₁₀₉	1	1	yes	characteristic
125	C₂	M₃₂×D₁₀₉	1	1	yes	characteristic
126	C₂	C₂×#16812638	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812638

C₂

yes

maximal

C₄×C₈×D₁₀₉

C₂

yes

maximal, characteristic

C₄⋊#13182004

C₂

yes

maximal, characteristic, complemented

#20294844

C₂

yes

maximal, complemented

C₄⋊C₁₆×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₀₉

C₄

yes

characteristic

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

C₄²×D₁₀₉

C₄

yes

characteristic

#16812638

—

C₂×#13182004

C₂²

yes

—

C₄×#11724784

C₄

yes

characteristic

C₈×Dic₁₀₉

C₄

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₀₉

C₂²

yes

—

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄×D₁₀₉

C₈

yes

characteristic

C₂×C₄×D₁₀₉

C₈

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

D₄

yes

—

C₈×D₁₀₉

Q₈

yes

—

C₄×Dic₁₀₉

C₈

yes

characteristic

C₄×Dic₁₀₉

C₈

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

#13182004

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₁₆×C₁₀₉

—

C₂²×D₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₄⋊C₄

yes

—

C₄×D₁₀₉

M₁₆

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂×D₄

yes

characteristic

C₈×C₁₀₉

C₂×Q₈

yes

characteristic

#11724784

—

#11724784

C₄⋊C₄

yes

—

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₄⋊C₈

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

M₃₂

yes

complemented

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

C₄⋊C₈

yes

—

Dic₁₀₉

M₃₂

yes

complemented

C₂×C₁₀₉

C₂×C₄⋊C₈

yes

characteristic

C₂×C₁₀₉

C₂×M₃₂

yes

characteristic

C₂×C₁₀₉

C₂×M₃₂

yes

characteristic

D₁₀₉

—

#2150

—

109

maximal, Sylow

C₁₀₉

#2150

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

109

—

C₂×C₄×C₈

—

109

—

C₄⋊C₁₆

—

109

—

C₄⋊C₁₆

—

109

—

C₄⋊C₁₆

D₁₀₉

yes

characteristic, complemented

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

M₃₂

—

218

—

C₂×C₁₆

—

109

—

C₂×C₁₆

C₂×D₁₀₉

yes

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

C₄×D₁₀₉

yes

characteristic

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₄×D₁₀₉

yes

characteristic

C₁₆

—

C₁₆

—

218

—

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

100

C₂×C₄

—

109

—

101

C₂×C₄

—

109

—

102

C₂×C₄

—

218

—

103

C₂×C₄

—

218

—

104

C₂×C₄

C₈×D₁₀₉

yes

characteristic

105

C₂×C₄

C₈×D₁₀₉

yes

characteristic

106

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

107

C₈

—

108

C₈

—

109

—

109

C₈

—

218

—

110

C₈

D₄×D₁₀₉

yes

characteristic

111

C₈

Q₈×D₁₀₉

yes

characteristic

112

C₂²

—

109

—

113

C₂²

—

218

—

114

C₂²

—

218

—

115

C₂²

C₂×C₈×D₁₀₉

yes

characteristic

116

C₄

—

117

C₄

—

109

—

118

C₄

—

109

—

119

C₄

—

218

—

120

C₄

#16812638

yes

complemented

121

C₄

C₄⋊C₄×D₁₀₉

yes

characteristic

122

C₄

M₁₆×D₁₀₉

yes

characteristic

123

C₂

—

218

—

124

C₂

C₄⋊C₈×D₁₀₉

yes

characteristic

125

C₂

M₃₂×D₁₀₉

yes

characteristic

126

C₂

C₂×#16812638

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂²×C₈
Fitting	C₄⋊C₁₆×C₁₀₉	C₂
prime radical	C₂²×D₁₀₉	C₂×C₈
cosocle	C₂×C₈×C₁₀₉	C₂³
hypercenter	C₄⋊C₁₆	D₁₀₉
hypocenter	C₁₀₉	#2150
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂²×C₈

Fitting

C₄⋊C₁₆×C₁₀₉

C₂

prime radical

C₂²×D₁₀₉

C₂×C₈

cosocle

C₂×C₈×C₁₀₉

C₂³

hypercenter

C₄⋊C₁₆

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
3. C₄⋊#13182004	C₂	semidirect
4. #20294844	C₂	semidirect
5. C₄⋊C₁₆×C₁₀₉	C₂	semidirect
53. C₄×C₁₀₉	M₃₂	semidirect
58. Dic₁₀₉	M₃₂	semidirect
64. C₁₀₉	#2150	semidirect
69. C₄⋊C₁₆	D₁₀₉	semidirect
120. C₄	#16812638	semidirect

H ≅ G/N

type

3. C₄⋊#13182004

C₂

semidirect

4. #20294844

C₂

semidirect

5. C₄⋊C₁₆×C₁₀₉

C₂

semidirect

53. C₄×C₁₀₉

M₃₂

semidirect

58. Dic₁₀₉

M₃₂

semidirect

64. C₁₀₉

#2150

semidirect

69. C₄⋊C₁₆

D₁₀₉

semidirect

120. C₄

#16812638

semidirect

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

subgroup	index
C₂×#16812638	2
C₄×C₈×D₁₀₉	2
C₄⋊#13182004	2
#20294844	2
C₄⋊C₁₆×C₁₀₉	2
#2150	109

subgroup

index

C₄×C₈×D₁₀₉

C₄⋊C₁₆×C₁₀₉

109

Subgroups — #28364786

Conjugacy classes of subgroups

Complements

Maximal subgroups