Group #28377863 — subgroups

Subgroups — #28377863

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	C₄×#13182039	C₂	1	1	yes	maximal, characteristic
2	C₄×D₁₀₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#20295078	C₂	1	1	yes	maximal, characteristic
4	C₂×#13182039	C₄	1	2	yes	—
5	C₂×D₁₀₉⋊C₈	—	2	1	no	—
6	#16812754	—	2	1	no	—
7	C₄×#11724786	—	2	1	no	—
8	C₄×#11724786	—	2	1	no	—
9	C₄×#11724786	C₄	1	2	yes	—
10	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
11	C₄⋊#11724784	C₄	1	1	yes	characteristic
12	#16812611	—	2	1	no	—
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	D₁₀₉⋊C₈	—	2	2	no	—
18	D₁₀₉⋊C₈	—	2	2	no	—
19	C₂×#11724786	—	2	2	no	—
20	C₂×#11724786	—	2	2	no	—
21	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
23	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
24	C₂×#11724784	—	2	1	no	—
25	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
27	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
28	C₂×C₈×C₁₀₉	—	2	1	no	—
29	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
37	C₄×D₁₀₉	—	2	1	no	—
38	C₄×D₁₀₉	—	2	1	no	—
39	C₄×D₁₀₉	—	2	2	no	—
40	C₄×D₁₀₉	C₂×C₈	1	2	yes	—
41	C₄×D₁₀₉	M₁₆	1	2	yes	—
42	#11724786	—	2	4	no	—
43	#11724786	—	2	4	no	—
44	C₈×C₁₀₉	—	4	1	no	—
45	#11724784	—	4	1	no	—
46	C₂²×C₁₀₉	#95	1	1	yes	characteristic
47	C₂×D₁₀₉	—	2	1	no	—
48	C₂×D₁₀₉	—	2	1	no	—
49	C₂×D₁₀₉	C₄⋊C₈	1	2	yes	—
50	C₄×C₁₀₉	—	2	1	no	—
51	C₄×C₁₀₉	C₄×C₈	1	1	yes	characteristic
52	C₄×C₁₀₉	#97	1	1	yes	characteristic
53	C₄×C₁₀₉	#104	1	1	yes	characteristic
54	C₄×C₁₀₉	#104	1	1	yes	characteristic
55	Dic₁₀₉	—	2	1	no	—
56	Dic₁₀₉	C₂²⋊C₈	1	2	yes	—
57	Dic₁₀₉	#108	1	2	yes	—
58	C₂×C₁₀₉	#336	1	1	yes	characteristic
59	C₂×C₁₀₉	#339	1	1	yes	characteristic
60	C₂×C₁₀₉	#341	1	1	yes	characteristic
61	D₁₀₉	—	2	2	no	—
62	#1278	—	109	1	no	maximal, Sylow
63	C₁₀₉	#1278	1	1	yes	characteristic, complemented, Sylow
64	#431	—	109	1	no	—
65	C₄×M₁₆	—	109	1	no	—
66	C₂×C₄×C₈	—	109	1	no	—
67	C₂×M₁₆	—	109	2	no	—
68	C₂²×C₈	—	218	1	no	—
69	C₂×C₄²	—	109	1	no	—
70	C₄⋊C₈	—	109	1	no	—
71	C₄⋊C₈	#89137	1	1	yes	characteristic
72	C₂²⋊C₈	—	218	1	no	—
73	#97	—	218	1	no	—
74	C₄×C₈	—	109	2	no	—
75	C₄×C₈	—	218	1	no	—
76	C₄×C₈	—	218	1	no	—
77	C₄×C₈	—	218	1	no	—
78	C₂²×C₄	—	109	1	no	—
79	C₂²×C₄	—	109	1	no	—
80	C₂²×C₄	—	109	1	no	—
81	M₁₆	—	109	4	no	—
82	M₁₆	—	218	2	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	218	1	no	—
85	C₂×C₈	—	218	2	no	—
86	C₂×C₈	—	218	2	no	—
87	C₂×C₈	—	218	2	no	—
88	C₂×C₈	—	218	2	no	—
89	C₄²	—	109	1	no	—
90	C₄²	—	109	1	no	—
91	C₄²	—	109	1	no	—
92	C₄²	C₂×#89137	1	1	yes	characteristic
93	C₂³	—	109	1	no	—
94	C₂×C₄	—	109	1	no	—
95	C₂×C₄	—	109	1	no	—
96	C₂×C₄	—	109	2	no	—
97	C₂×C₄	—	109	2	no	—
98	C₂×C₄	—	109	1	no	—
99	C₂×C₄	—	218	2	no	—
100	C₂×C₄	—	218	1	no	—
101	C₂×C₄	—	218	1	no	—
102	C₂×C₄	C₄×#89137	1	1	yes	characteristic
103	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
104	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
105	C₈	—	4	1	no	—
106	C₈	—	436	1	no	—
107	C₈	—	218	4	no	—
108	C₈	—	218	4	no	—
109	C₂²	—	109	2	no	—
110	C₂²	—	218	1	no	—
111	C₂²	—	218	1	no	—
112	C₂²	(C₂×#89137)⋊C₄	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	109	2	no	—
115	C₄	—	109	2	no	—
116	C₄	—	218	1	no	—
117	C₄	C₈×#89137	1	1	yes	characteristic
118	C₄	#16812753	1	1	yes	characteristic
119	C₄	#16812763	1	1	yes	characteristic
120	C₄	#16812764	1	1	yes	characteristic
121	C₂	—	218	2	no	—
122	C₂	(C₂×#89137)⋊C₈	1	1	yes	characteristic
123	C₂	#20295896	1	1	yes	characteristic
124	C₂	#20295901	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13182039

C₂

yes

maximal, characteristic

C₄×D₁₀₉⋊C₈

C₂

yes

maximal, characteristic

#20295078

C₂

yes

maximal, characteristic

C₂×#13182039

C₄

yes

—

C₂×D₁₀₉⋊C₈

—

—

—

—

C₄

yes

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄⋊#11724784

C₄

yes

characteristic

#16812611

—

C₈×Dic₁₀₉

—

C₄⋊C₈×C₁₀₉

C₄

yes

characteristic

#13182039

—

#13182039

C₈

yes

complemented

D₁₀₉⋊C₈

—

D₁₀₉⋊C₈

—

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

—

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

#108

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

—

C₄⋊C₈

#89137

yes

characteristic

C₂²⋊C₈

—

218

—

#97

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

218

—

C₄×C₈

—

218

—

C₄×C₈

—

218

—

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

—

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

—

100

C₂×C₄

—

218

—

101

C₂×C₄

—

218

—

102

C₂×C₄

C₄×#89137

yes

characteristic

103

C₂×C₄

C₂²⋊#89137

yes

characteristic

104

C₂×C₄

C₄⋊#89137

yes

characteristic

105

C₈

—

106

C₈

—

436

—

107

C₈

—

218

—

108

C₈

—

218

—

109

C₂²

—

109

—

110

C₂²

—

218

—

111

C₂²

—

218

—

112

C₂²

(C₂×#89137)⋊C₄

yes

characteristic

113

C₄

—

114

C₄

—

109

—

115

C₄

—

109

—

116

C₄

—

218

—

117

C₄

C₈×#89137

yes

characteristic

118

C₄

#16812753

yes

characteristic

119

C₄

#16812763

yes

characteristic

120

C₄

#16812764

yes

characteristic

121

C₂

—

218

—

122

C₂

(C₂×#89137)⋊C₈

yes

characteristic

123

C₂

#20295896

yes

characteristic

124

C₂

#20295901

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#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

C₂²⋊#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
63. C₁₀₉	#1278	semidirect

H ≅ G/N

type

16. #13182039

C₈

semidirect

63. C₁₀₉

#1278

semidirect

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

subgroup	index
C₄×#13182039	2
C₄×D₁₀₉⋊C₈	2
#20295078	2
#1278	109

subgroup

index

C₄×#13182039

C₄×D₁₀₉⋊C₈

#20295078

#1278

109

Subgroups — #28377863

Conjugacy classes of subgroups

Complements

Maximal subgroups