Group #28377862 — subgroups

Subgroups — #28377862

Order 13952.

180 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 90 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×C₄⋊#11724786	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11724786	—	2	2	no	—
4	C₄⋊#11724786	—	2	2	no	—
5	C₄⋊#11724786	C₄	1	4	yes	—
6	C₂²×#11724784	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
8	C₈×Dic₁₀₉	—	2	2	no	—
9	C₂²×C₈×C₁₀₉	C₄	1	1	yes	characteristic
10	C₂×#11724786	—	2	4	no	—
11	C₂×#11724786	—	4	2	no	—
12	C₂×#11724786	—	4	2	no	—
13	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
14	C₂²×Dic₁₀₉	Q₈	1	1	yes	characteristic
15	C₂×#11724784	—	2	1	no	—
16	C₂×#11724784	—	2	1	no	—
17	C₂×#11724784	C₈	1	2	yes	complemented
18	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
19	C₄×Dic₁₀₉	D₄	1	2	yes	—
20	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₀₉	—	2	1	no	—
22	C₂×C₈×C₁₀₉	—	2	1	no	—
23	C₂×C₈×C₁₀₉	C₈	1	2	yes	complemented
24	C₂³×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
26	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
31	C₂×Dic₁₀₉	—	2	1	no	—
32	C₂×Dic₁₀₉	—	2	1	no	—
33	C₂×Dic₁₀₉	C₂²⋊C₄	1	2	yes	—
34	C₂×Dic₁₀₉	C₄⋊C₄	1	2	yes	—
35	C₂×Dic₁₀₉	D₈	1	1	yes	characteristic
36	C₂×Dic₁₀₉	SD₁₆	1	2	yes	—
37	C₂×Dic₁₀₉	Q₁₆	1	1	yes	characteristic
38	#11724786	—	4	4	no	—
39	C₈×C₁₀₉	—	2	2	no	—
40	#11724784	—	2	2	no	—
41	C₂²×C₁₀₉	#95	1	1	yes	characteristic
42	C₂²×C₁₀₉	C₄⋊C₈	1	1	yes	characteristic
43	C₂²×C₁₀₉	C₄⋊C₈	1	1	yes	characteristic
44	C₂²×C₁₀₉	#106	1	1	yes	characteristic
45	C₂²×C₁₀₉	#107	1	1	yes	characteristic
46	C₂²×C₁₀₉	#108	1	1	yes	characteristic
47	C₂²×C₁₀₉	#108	1	1	yes	characteristic
48	C₄×C₁₀₉	C₄×C₈	1	1	yes	characteristic
49	C₄×C₁₀₉	#97	1	1	yes	characteristic
50	C₄×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
51	C₄×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
52	Dic₁₀₉	—	2	2	no	—
53	Dic₁₀₉	#102	1	2	yes	—
54	Dic₁₀₉	#103	1	2	yes	—
55	C₂×C₁₀₉	#334	1	1	yes	characteristic
56	C₂×C₁₀₉	#334	1	1	yes	characteristic
57	C₂×C₁₀₉	#335	1	1	yes	characteristic
58	C₂×C₁₀₉	#335	1	1	yes	characteristic
59	C₂×C₁₀₉	#336	1	1	yes	characteristic
60	C₂×C₁₀₉	#340	1	1	yes	characteristic
61	C₂×C₁₀₉	#341	1	1	yes	characteristic
62	#1277	—	109	1	no	maximal, Sylow
63	C₁₀₉	#1277	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊C₈	—	109	2	no	—
65	C₂×C₄×C₈	—	109	1	no	—
66	C₂²×C₈	—	109	1	no	—
67	C₂²×C₈	—	218	2	no	—
68	C₂²×C₈	#89137	1	1	yes	characteristic
69	C₂×C₄²	—	109	1	no	—
70	C₄⋊C₈	—	109	4	no	—
71	C₄⋊C₈	—	218	2	no	—
72	C₄×C₈	—	218	2	no	—
73	C₂²×C₄	—	109	1	no	—
74	C₂²×C₄	—	109	1	no	—
75	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	109	2	no	—
79	C₂×C₈	—	436	2	no	—
80	C₂×C₈	—	436	2	no	—
81	C₂×C₈	—	218	4	no	—
82	C₂×C₈	—	218	1	no	—
83	C₂×C₈	—	218	1	no	—
84	C₂×C₈	#11724786	1	2	yes	complemented
85	C₄²	—	109	2	no	—
86	C₄²	—	109	2	no	—
87	C₂³	C₄⋊#89137	1	1	yes	characteristic
88	C₂×C₄	—	109	2	no	—
89	C₂×C₄	—	109	2	no	—
90	C₂×C₄	—	109	1	no	—
91	C₂×C₄	—	109	1	no	—
92	C₂×C₄	—	109	2	no	—
93	C₂×C₄	—	218	1	no	—
94	C₂×C₄	—	218	1	no	—
95	C₂×C₄	C₄×#89137	1	1	yes	characteristic
96	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
97	C₂×C₄	C₂×#11724786	1	1	yes	characteristic
98	C₂×C₄	D₁₀₉⋊C₈	1	1	yes	characteristic
99	C₂×C₄	#13182038	1	1	yes	characteristic
100	C₂×C₄	#13182039	1	1	yes	characteristic
101	C₈	—	2	2	no	—
102	C₈	—	436	4	no	—
103	C₈	—	218	2	no	—
104	C₂²	(C₂×#89137)⋊C₄	1	1	yes	characteristic
105	C₂²	C₄⋊#11724786	1	1	yes	characteristic
106	C₂²	#16812766	1	1	yes	characteristic
107	C₂²	#16812767	1	1	yes	characteristic
108	C₂²	#16812768	1	1	yes	characteristic
109	C₂²	#16812769	1	1	yes	characteristic
110	C₂²	#16812770	1	1	yes	characteristic
111	C₄	—	109	2	no	—
112	C₄	—	109	2	no	—
113	C₄	—	218	2	no	—
114	C₄	C₄×#11724786	1	1	yes	characteristic
115	C₄	#16812754	1	1	yes	characteristic
116	C₄	C₂²⋊#11724786	1	1	yes	characteristic
117	C₄	#16812756	1	1	yes	characteristic
118	C₂	#20295887	1	1	yes	characteristic
119	C₂	#20295888	1	1	yes	characteristic
120	C₂	#20295889	1	1	yes	characteristic
121	C₂	#20295890	1	1	yes	characteristic
122	C₂	#20295892	1	1	yes	characteristic
123	C₂	#20295900	1	1	yes	characteristic
124	C₂	#20295902	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11724786

C₂

yes

maximal

C₂×C₈×Dic₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724786

—

C₄⋊#11724786

—

C₄⋊#11724786

C₄

yes

—

C₂²×#11724784

C₄

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

C₈×Dic₁₀₉

—

C₂²×C₈×C₁₀₉

C₄

yes

—

—

—

C₂²×Dic₁₀₉

D₄

yes

characteristic

C₂²×Dic₁₀₉

Q₈

yes

—

—

C₈

yes

complemented

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₈

yes

complemented

C₂³×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

M₁₆

yes

characteristic

C₂×C₄×C₁₀₉

M₁₆

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂²⋊C₄

yes

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₂×Dic₁₀₉

D₈

yes

characteristic

C₂×Dic₁₀₉

SD₁₆

yes

—

C₂×Dic₁₀₉

Q₁₆

yes

characteristic

#11724786

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

#95

yes

characteristic

C₂²×C₁₀₉

C₄⋊C₈

yes

characteristic

C₂²×C₁₀₉

C₄⋊C₈

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

C₄×C₈

yes

characteristic

C₄×C₁₀₉

#97

yes

characteristic

C₄×C₁₀₉

C₂²⋊C₈

yes

characteristic

C₄×C₁₀₉

C₂²⋊C₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

#102

yes

—

Dic₁₀₉

#103

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1277

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

109

—

C₂×C₄×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

218

—

C₂²×C₈

#89137

yes

characteristic

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₄×C₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×#89137

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

—

436

—

C₂×C₈

—

436

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

#11724786

yes

complemented

C₄²

—

109

—

C₄²

—

109

—

C₂³

C₄⋊#89137

yes

characteristic

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

D₁₀₉⋊C₈

yes

characteristic

C₂×C₄

#13182038

yes

characteristic

100

C₂×C₄

#13182039

yes

characteristic

101

C₈

—

102

C₈

—

436

—

103

C₈

—

218

—

104

C₂²

(C₂×#89137)⋊C₄

yes

characteristic

105

C₂²

C₄⋊#11724786

yes

characteristic

106

C₂²

#16812766

yes

characteristic

107

C₂²

#16812767

yes

characteristic

108

C₂²

#16812768

yes

characteristic

109

C₂²

#16812769

yes

characteristic

110

C₂²

#16812770

yes

characteristic

111

C₄

—

109

—

112

C₄

—

109

—

113

C₄

—

218

—

114

C₄

C₄×#11724786

yes

characteristic

115

C₄

#16812754

yes

characteristic

116

C₄

C₂²⋊#11724786

yes

characteristic

117

C₄

#16812756

yes

characteristic

118

C₂

#20295887

yes

characteristic

119

C₂

#20295888

yes

characteristic

120

C₂

#20295889

yes

characteristic

121

C₂

#20295890

yes

characteristic

122

C₂

#20295892

yes

characteristic

123

C₂

#20295900

yes

characteristic

124

C₂

#20295902

yes

characteristic

	subgroup	G/H
center	C₂³	C₄⋊#89137
commutator	C₄×C₁₀₉	C₄×C₈
Frattini	C₂²×C₄	C₂×#89137
Fitting	C₂²×C₈×C₁₀₉	C₄
socle, prime radical	C₂³×C₁₀₉	C₄⋊C₄
cosocle	C₂×C₄×Dic₁₀₉	C₂²
hypercenter	C₂²×C₈	#89137
hypocenter	C₁₀₉	#1277
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

subgroup

G/H

center

C₂³

C₄⋊#89137

commutator

C₄×C₁₀₉

C₄×C₈

Frattini

C₂²×C₄

C₂×#89137

Fitting

C₂²×C₈×C₁₀₉

C₄

socle, prime radical

C₂³×C₁₀₉

C₄⋊C₄

cosocle

C₂×C₄×Dic₁₀₉

C₂²

C₂²×C₈

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

N	H ≅ G/N	type
17. C₂×#11724784	C₈	semidirect
23. C₂×C₈×C₁₀₉	C₈	semidirect
63. C₁₀₉	#1277	semidirect
84. C₂×C₈	#11724786	semidirect

H ≅ G/N

type

17. C₂×#11724784

C₈

semidirect

23. C₂×C₈×C₁₀₉

C₈

semidirect

63. C₁₀₉

#1277

semidirect

84. C₂×C₈

#11724786

semidirect

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

subgroup	index
C₂×C₄⋊#11724786	2
C₂×C₈×Dic₁₀₉	2
#1277	109

subgroup

index

C₂×C₄⋊#11724786

C₂×C₈×Dic₁₀₉

#1277

109

Subgroups — #28377862

Conjugacy classes of subgroups

Complements

Maximal subgroups