Group #28377864 — subgroups

Subgroups — #28377864

Order 13952.

172 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 62 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	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#13182039	C₄	1	2	yes	—
5	C₄⋊#11724786	C₄	1	2	yes	—
6	#16812756	—	2	1	no	—
7	#16812754	—	2	1	no	—
8	C₄×#11724786	—	2	1	no	—
9	C₄×#11724786	—	2	1	no	—
10	C₂×C₈×D₁₀₉	—	2	1	no	—
11	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
12	C₄×#11724784	C₄	1	1	yes	characteristic
13	C₈×Dic₁₀₉	—	2	1	no	—
14	C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
15	#13182039	—	2	2	no	—
16	#13182039	C₈	1	4	yes	complemented
17	C₂×#11724786	—	2	2	no	—
18	C₂×#11724786	—	2	2	no	—
19	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
21	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
22	C₂×#11724784	—	2	1	no	—
23	C₈×D₁₀₉	—	2	4	no	—
24	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
26	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
27	C₂×C₈×C₁₀₉	—	2	1	no	—
28	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
36	C₄×D₁₀₉	—	2	1	no	—
37	C₄×D₁₀₉	—	2	1	no	—
38	C₄×D₁₀₉	—	2	2	no	—
39	C₄×D₁₀₉	C₂×C₈	1	2	yes	—
40	C₄×D₁₀₉	M₁₆	1	2	yes	—
41	#11724786	—	2	4	no	—
42	#11724786	—	4	2	no	—
43	C₈×C₁₀₉	—	2	2	no	—
44	#11724784	—	2	2	no	—
45	C₂²×C₁₀₉	#95	1	1	yes	characteristic
46	C₂×D₁₀₉	—	2	1	no	—
47	C₂×D₁₀₉	—	2	1	no	—
48	C₂×D₁₀₉	C₄⋊C₈	1	2	yes	—
49	C₄×C₁₀₉	—	2	1	no	—
50	C₄×C₁₀₉	C₄×C₈	1	1	yes	characteristic
51	C₄×C₁₀₉	#97	1	1	yes	characteristic
52	C₄×C₁₀₉	#108	1	1	yes	characteristic
53	C₄×C₁₀₉	#108	1	1	yes	characteristic
54	Dic₁₀₉	—	2	1	no	—
55	Dic₁₀₉	C₂²⋊C₈	1	2	yes	—
56	Dic₁₀₉	#104	1	2	yes	—
57	C₂×C₁₀₉	#336	1	1	yes	characteristic
58	C₂×C₁₀₉	#339	1	1	yes	characteristic
59	C₂×C₁₀₉	#341	1	1	yes	characteristic
60	D₁₀₉	—	2	2	no	—
61	#1278	—	109	1	no	maximal, Sylow
62	C₁₀₉	#1278	1	1	yes	characteristic, complemented, Sylow
63	#431	—	109	1	no	—
64	C₄×M₁₆	—	109	1	no	—
65	C₂×C₄×C₈	—	109	1	no	—
66	C₂×M₁₆	—	109	2	no	—
67	C₂²×C₈	—	218	1	no	—
68	C₂×C₄²	—	109	1	no	—
69	C₄⋊C₈	—	109	2	no	—
70	C₂²⋊C₈	—	218	1	no	—
71	#97	—	218	1	no	—
72	C₄×C₈	—	109	1	no	—
73	C₄×C₈	—	218	1	no	—
74	C₄×C₈	—	218	1	no	—
75	C₄×C₈	—	218	1	no	—
76	C₄×C₈	#89137	1	1	yes	characteristic
77	C₂²×C₄	—	109	1	no	—
78	C₂²×C₄	—	109	1	no	—
79	C₂²×C₄	—	109	1	no	—
80	M₁₆	—	109	4	no	—
81	M₁₆	—	218	2	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	218	4	no	—
84	C₂×C₈	—	218	2	no	—
85	C₂×C₈	—	218	2	no	—
86	C₂×C₈	—	218	1	no	—
87	C₄²	—	109	1	no	—
88	C₄²	—	109	1	no	—
89	C₄²	—	109	1	no	—
90	C₄²	C₂×#89137	1	1	yes	characteristic
91	C₂³	—	109	1	no	—
92	C₂×C₄	—	109	1	no	—
93	C₂×C₄	—	109	1	no	—
94	C₂×C₄	—	109	2	no	—
95	C₂×C₄	—	109	2	no	—
96	C₂×C₄	—	109	1	no	—
97	C₂×C₄	—	218	2	no	—
98	C₂×C₄	—	218	1	no	—
99	C₂×C₄	—	218	1	no	—
100	C₂×C₄	C₄×#89137	1	1	yes	characteristic
101	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
102	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
103	C₈	—	2	2	no	—
104	C₈	—	436	2	no	—
105	C₈	—	218	2	no	—
106	C₈	—	218	4	no	—
107	C₂²	—	109	2	no	—
108	C₂²	—	218	1	no	—
109	C₂²	—	218	1	no	—
110	C₂²	(C₂×#89137)⋊C₄	1	1	yes	characteristic
111	C₄	—	2	1	no	—
112	C₄	—	109	2	no	—
113	C₄	—	109	2	no	—
114	C₄	—	218	1	no	—
115	C₄	C₈×#89137	1	1	yes	characteristic
116	C₄	#16812753	1	1	yes	characteristic
117	C₄	#16812769	1	1	yes	characteristic
118	C₄	#16812770	1	1	yes	characteristic
119	C₂	—	218	2	no	—
120	C₂	(C₂×#89137)⋊C₈	1	1	yes	characteristic
121	C₂	#20295897	1	1	yes	characteristic
122	C₂	#20295902	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296045

C₂

yes

maximal, characteristic

C₄×#13182039

C₂

yes

maximal, characteristic

C₄×C₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182039

C₄

yes

—

C₄⋊#11724786

C₄

yes

—

—

—

—

—

C₂×C₈×D₁₀₉

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄×#11724784

C₄

yes

characteristic

C₈×Dic₁₀₉

—

C₄×C₈×C₁₀₉

C₄

yes

characteristic

#13182039

—

#13182039

C₈

yes

complemented

C₂×#11724786

—

C₂×#11724786

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

Q₈

yes

characteristic

C₂×#11724784

—

C₈×D₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×C₈

yes

—

C₄×D₁₀₉

M₁₆

yes

—

#11724786

—

#11724786

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

#95

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₄⋊C₈

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×C₈

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

Dic₁₀₉

C₂²⋊C₈

yes

—

Dic₁₀₉

#104

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

#1278

—

109

maximal, Sylow

C₁₀₉

#1278

yes

characteristic, complemented, Sylow

#431

—

109

—

C₄×M₁₆

—

109

—

C₂×C₄×C₈

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

218

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₂²⋊C₈

—

218

—

#97

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

218

—

C₄×C₈

—

218

—

C₄×C₈

—

218

—

C₄×C₈

#89137

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

M₁₆

—

109

—

M₁₆

—

218

—

C₂×C₈

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₂×#89137

yes

characteristic

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

100

C₂×C₄

C₄×#89137

yes

characteristic

101

C₂×C₄

C₂²⋊#89137

yes

characteristic

102

C₂×C₄

C₄⋊#89137

yes

characteristic

103

C₈

—

104

C₈

—

436

—

105

C₈

—

218

—

106

C₈

—

218

—

107

C₂²

—

109

—

108

C₂²

—

218

—

109

C₂²

—

218

—

110

C₂²

(C₂×#89137)⋊C₄

yes

characteristic

111

C₄

—

112

C₄

—

109

—

113

C₄

—

109

—

114

C₄

—

218

—

115

C₄

C₈×#89137

yes

characteristic

116

C₄

#16812753

yes

characteristic

117

C₄

#16812769

yes

characteristic

118

C₄

#16812770

yes

characteristic

119

C₂

—

218

—

120

C₂

(C₂×#89137)⋊C₈

yes

characteristic

121

C₂

#20295897

yes

characteristic

122

C₂

#20295902

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#89137
commutator	C₄×C₁₀₉	C₄×C₈
Frattini	C₄²	C₂×#89137
Fitting	C₄×C₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	#95
prime radical	C₂²×D₁₀₉	C₄⋊C₄
cosocle	C₄²×D₁₀₉	C₂²
hypercenter	C₄×C₈	#89137
hypocenter	C₁₀₉	#1278
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂×C₄

C₄⋊#89137

commutator

C₄×C₁₀₉

C₄×C₈

Frattini

C₄²

C₂×#89137

Fitting

C₄×C₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

#95

prime radical

C₂²×D₁₀₉

C₄⋊C₄

cosocle

C₄²×D₁₀₉

C₂²

C₄×C₈

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
16. #13182039	C₈	semidirect
62. C₁₀₉	#1278	semidirect

H ≅ G/N

type

16. #13182039

C₈

semidirect

62. C₁₀₉

#1278

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
C₄×C₈×D₁₀₉	2
#1278	109

subgroup

index

#20296045

C₄×#13182039

C₄×C₈×D₁₀₉

#1278

109

Subgroups — #28377864

Conjugacy classes of subgroups

Complements

Maximal subgroups