Group #28361777 — subgroups

Subgroups — #28361777

Order 13952.

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

normal subgroups only

This group has 180 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	#20295078	C₂	1	2	yes	maximal
2	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#20294778	C₂	1	2	yes	maximal, complemented
4	#20294777	C₂	1	1	yes	maximal, characteristic, complemented
5	#334×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
7	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
8	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
9	C₄⋊#11724784	C₂²	1	2	yes	—
10	#16812611	—	2	2	no	—
11	C₄×#11724784	C₂²	1	1	yes	characteristic
12	C₈×Dic₁₀₉	—	2	2	no	—
13	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic
14	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic
15	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
16	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
19	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
20	C₂×#11724784	—	2	2	no	—
21	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
22	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
23	C₈×D₁₀₉	—	2	2	no	—
24	C₈×D₁₀₉	C₈	1	4	yes	complemented
25	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
27	C₄×Dic₁₀₉	Q₈	1	1	yes	characteristic
28	C₂×C₈×C₁₀₉	—	2	2	no	—
29	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
32	C₂²×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
36	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
39	C₄×D₁₀₉	—	2	2	no	—
40	C₄×D₁₀₉	—	2	2	no	—
41	C₄×D₁₀₉	C₂×C₈	1	2	yes	—
42	C₄×D₁₀₉	M₁₆	1	2	yes	—
43	C₈×C₁₀₉	—	2	1	no	—
44	C₈×C₁₀₉	—	4	2	no	—
45	C₈×C₁₀₉	C₂×C₈	1	2	yes	—
46	#11724784	—	2	1	no	—
47	#11724784	—	4	2	no	—
48	#11724784	C₂×C₈	1	2	yes	—
49	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×D₁₀₉	—	2	1	no	—
51	C₂×D₁₀₉	—	2	1	no	—
52	C₂×D₁₀₉	C₄⋊C₈	1	2	yes	—
53	C₄×C₁₀₉	—	2	1	no	—
54	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
55	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
56	C₄×C₁₀₉	#135	1	2	yes	—
57	Dic₁₀₉	—	2	1	no	—
58	Dic₁₀₉	C₄⋊C₈	1	2	yes	—
59	Dic₁₀₉	#108	1	2	yes	—
60	C₂×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₁₀₉	#427	1	1	yes	characteristic
62	C₂×C₁₀₉	C₂×#108	1	1	yes	characteristic
63	D₁₀₉	—	2	2	no	—
64	#1564	—	109	1	no	maximal, Sylow
65	C₁₀₉	#1564	1	1	yes	characteristic, complemented, Sylow
66	#431	—	109	2	no	—
67	C₂×C₄×C₈	—	109	1	no	—
68	#335	—	109	2	no	—
69	#334	—	109	1	no	—
70	#334	D₁₀₉	1	1	yes	characteristic, complemented
71	C₂²×C₈	—	109	1	no	—
72	C₂²×C₈	—	109	1	no	—
73	C₂×C₄²	—	109	1	no	—
74	C₄⋊C₈	—	109	2	no	—
75	C₄⋊C₈	C₂×D₁₀₉	1	2	yes	—
76	C₂²⋊C₈	—	218	2	no	—
77	C₄×C₈	—	109	1	no	—
78	C₄×C₈	—	109	1	no	—
79	C₄×C₈	—	109	1	no	—
80	C₄×C₈	—	218	2	no	—
81	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
82	C₂²×C₄	—	109	1	no	—
83	C₂²×C₄	—	109	1	no	—
84	C₂²×C₄	—	109	1	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	109	1	no	—
87	C₂×C₈	—	109	4	no	—
88	C₂×C₈	—	109	1	no	—
89	C₂×C₈	—	218	2	no	—
90	C₂×C₈	—	218	2	no	—
91	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
92	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
93	C₄²	—	109	1	no	—
94	C₄²	—	109	1	no	—
95	C₄²	—	109	1	no	—
96	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
97	C₂³	—	109	1	no	—
98	C₂×C₄	—	109	1	no	—
99	C₂×C₄	—	109	1	no	—
100	C₂×C₄	—	109	2	no	—
101	C₂×C₄	—	109	2	no	—
102	C₂×C₄	—	109	1	no	—
103	C₂×C₄	—	218	2	no	—
104	C₂×C₄	—	218	2	no	—
105	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
106	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
107	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
108	C₈	—	2	1	no	—
109	C₈	—	4	2	no	—
110	C₈	—	109	2	no	—
111	C₈	—	436	2	no	—
112	C₈	—	218	1	no	—
113	C₈	C₈×D₁₀₉	1	2	yes	—
114	C₂²	—	109	2	no	—
115	C₂²	—	218	1	no	—
116	C₂²	—	218	1	no	—
117	C₂²	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
118	C₄	—	2	1	no	—
119	C₄	—	109	2	no	—
120	C₄	—	109	2	no	—
121	C₄	—	218	1	no	—
122	C₄	C₂×C₈×D₁₀₉	1	1	yes	characteristic
123	C₄	M₁₆×D₁₀₉	1	1	yes	characteristic
124	C₄	#16812720	1	2	yes	—
125	C₂	—	218	2	no	—
126	C₂	C₄⋊C₈×D₁₀₉	1	1	yes	characteristic
127	C₂	#20295061	1	1	yes	characteristic
128	C₂	#108×D₁₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295078

C₂

yes

maximal

C₄×C₈×D₁₀₉

C₂

yes

maximal, characteristic

#20294778

C₂

yes

maximal, complemented

#20294777

C₂

yes

maximal, characteristic, complemented

#334×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₀₉

C₄

yes

characteristic

C₂×C₈×D₁₀₉

C₄

yes

characteristic

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄⋊#11724784

C₂²

yes

—

#16812611

—

C₄×#11724784

C₂²

yes

characteristic

C₈×Dic₁₀₉

—

C₈×Dic₁₀₉

C₄

yes

characteristic

C₈×Dic₁₀₉

C₄

yes

characteristic

C₄⋊C₈×C₁₀₉

C₂²

yes

—

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

Q₈

yes

characteristic

C₂×#11724784

—

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₈×D₁₀₉

—

C₈×D₁₀₉

C₈

yes

complemented

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

Q₈

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×C₈

yes

—

C₄×D₁₀₉

M₁₆

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂×C₈

yes

—

#11724784

—

#11724784

—

#11724784

C₂×C₈

yes

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₄⋊C₈

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×C₈

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

C₄×C₁₀₉

#135

yes

—

Dic₁₀₉

—

Dic₁₀₉

C₄⋊C₈

yes

—

Dic₁₀₉

#108

yes

—

C₂×C₁₀₉

C₂×C₄⋊C₈

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

#1564

—

109

maximal, Sylow

C₁₀₉

#1564

yes

characteristic, complemented, Sylow

#431

—

109

—

C₂×C₄×C₈

—

109

—

#335

—

109

—

#334

—

109

—

#334

D₁₀₉

yes

characteristic, complemented

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

C₂×D₁₀₉

yes

—

C₂²⋊C₈

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

218

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₄×D₁₀₉

yes

characteristic

C₂×C₈

C₄×D₁₀₉

yes

characteristic

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₂²×D₁₀₉

yes

characteristic

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

100

C₂×C₄

—

109

—

101

C₂×C₄

—

109

—

102

C₂×C₄

—

109

—

103

C₂×C₄

—

218

—

104

C₂×C₄

—

218

—

105

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

106

C₂×C₄

D₄×D₁₀₉

yes

characteristic

107

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

109

—

111

C₈

—

436

—

112

C₈

—

218

—

113

C₈

C₈×D₁₀₉

yes

—

114

C₂²

—

109

—

115

C₂²

—

218

—

116

C₂²

—

218

—

117

C₂²

C₄⋊C₄×D₁₀₉

yes

characteristic

118

C₄

—

119

C₄

—

109

—

120

C₄

—

109

—

121

C₄

—

218

—

122

C₄

C₂×C₈×D₁₀₉

yes

characteristic

123

C₄

M₁₆×D₁₀₉

yes

characteristic

124

C₄

#16812720

yes

—

125

C₂

—

218

—

126

C₂

C₄⋊C₈×D₁₀₉

yes

characteristic

127

C₂

#20295061

yes

characteristic

128

C₂

#108×D₁₀₉

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂²×C₈
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#334×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂²×D₁₀₉	C₄⋊C₄
cosocle	C₄²×C₁₀₉	C₂³
hypercenter	#334	D₁₀₉
hypocenter	C₁₀₉	#1564
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×D₁₀₉

commutator

C₄×C₁₀₉

C₂²×C₈

Frattini

C₄²

C₂²×D₁₀₉

Fitting

#334×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂²×D₁₀₉

C₄⋊C₄

cosocle

C₄²×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
3. #20294778	C₂	semidirect
4. #20294777	C₂	semidirect
5. #334×C₁₀₉	C₂	semidirect
24. C₈×D₁₀₉	C₈	semidirect
65. C₁₀₉	#1564	semidirect
70. #334	D₁₀₉	semidirect

H ≅ G/N

type

3. #20294778

C₂

semidirect

4. #20294777

C₂

semidirect

5. #334×C₁₀₉

C₂

semidirect

24. C₈×D₁₀₉

C₈

semidirect

65. C₁₀₉

#1564

semidirect

70. #334

D₁₀₉

semidirect

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

subgroup	index
#20295078	2
C₄×C₈×D₁₀₉	2
#20294778	2
#20294777	2
#334×C₁₀₉	2
#1564	109

subgroup

index

C₄×C₈×D₁₀₉

#334×C₁₀₉

109

Subgroups — #28361777

Conjugacy classes of subgroups

Complements

Maximal subgroups