Group #28378950 — subgroups

Subgroups — #28378950

Order 13952.

216 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 82 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×#16812766	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11724786	C₂	1	1	yes	maximal, characteristic
3	#20295892	C₂	1	4	yes	maximal
4	#20294871	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11724786	C₂²	1	2	yes	—
6	C₂²×#11724786	C₂²	1	2	yes	—
7	#16812766	—	2	2	no	—
8	C₄⋊#11724786	—	2	2	no	—
9	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
11	#16812606	C₄	1	2	yes	—
12	#16812606	C₄	1	1	yes	characteristic
13	#95×C₁₀₉	C₄	1	1	yes	characteristic
14	C₂×#11724786	—	2	2	no	—
15	C₂×#11724786	—	2	2	no	—
16	C₂×#11724786	—	2	4	no	—
17	C₂×#11724786	—	2	4	no	—
18	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	—
19	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
21	C₄×Dic₁₀₉	—	2	2	no	—
22	C₄×Dic₁₀₉	—	2	2	no	—
23	C₄×Dic₁₀₉	D₄	1	2	yes	—
24	C₄×Dic₁₀₉	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	—	2	2	no	—
29	C₂×C₄×C₁₀₉	—	2	2	no	—
30	C₂×C₄×C₁₀₉	—	2	2	no	—
31	C₂×C₄×C₁₀₉	—	2	2	no	—
32	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
33	C₂×Dic₁₀₉	—	2	1	no	—
34	C₂×Dic₁₀₉	—	2	1	no	—
35	C₂×Dic₁₀₉	—	2	2	no	—
36	C₂×Dic₁₀₉	—	2	2	no	—
37	C₂×Dic₁₀₉	—	2	2	no	—
38	C₂×Dic₁₀₉	C₄⋊C₄	1	2	yes	—
39	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₁₀₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×Dic₁₀₉	#41	1	2	yes	—
42	C₂×Dic₁₀₉	#41	1	2	yes	—
43	#11724786	—	4	2	no	—
44	#11724786	—	4	2	no	—
45	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₀₉	#117	1	2	yes	—
47	C₂²×C₁₀₉	#131	1	2	yes	—
48	C₂²×C₁₀₉	#131	1	1	yes	characteristic
49	C₂²×C₁₀₉	#131	1	1	yes	characteristic
50	C₄×C₁₀₉	—	2	2	no	—
51	C₄×C₁₀₉	—	4	2	no	—
52	Dic₁₀₉	—	2	2	no	—
53	Dic₁₀₉	—	4	2	no	—
54	Dic₁₀₉	#124	1	1	yes	characteristic
55	Dic₁₀₉	#124	1	1	yes	characteristic
56	Dic₁₀₉	#125	1	2	yes	—
57	C₂×C₁₀₉	#382	1	1	yes	characteristic
58	C₂×C₁₀₉	#424	1	1	yes	characteristic
59	C₂×C₁₀₉	#424	1	1	yes	characteristic
60	C₂×C₁₀₉	#433	1	2	yes	—
61	C₂×C₁₀₉	#433	1	2	yes	—
62	#1832	—	109	1	no	maximal, Sylow
63	C₁₀₉	#1832	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊C₈	—	109	1	no	—
65	C₂×C₄⋊C₈	—	109	1	no	—
66	#376	—	109	1	no	—
67	#336	—	109	4	no	—
68	C₂²×C₈	—	109	2	no	—
69	C₂²×C₈	—	109	2	no	—
70	C₂×C₄²	—	109	1	no	—
71	C₂×C₄²	—	109	2	no	—
72	C₄⋊C₈	—	218	2	no	—
73	C₄⋊C₈	—	218	2	no	—
74	#95	—	109	2	no	—
75	#95	—	109	1	no	—
76	#95	#89137	1	1	yes	characteristic
77	C₂²×C₄	—	109	1	no	—
78	C₂²×C₄	—	109	1	no	—
79	C₂²×C₄	—	109	2	no	—
80	C₂²×C₄	C₂×#89137	1	2	yes	—
81	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
82	C₂×C₈	—	218	2	no	—
83	C₂×C₈	—	218	2	no	—
84	C₂×C₈	—	218	4	no	—
85	C₂×C₈	—	218	4	no	—
86	C₄²	—	109	2	no	—
87	C₄²	—	109	2	no	—
88	C₄²	—	218	2	no	—
89	C₄²	—	218	2	no	—
90	C₂³	C₂²×#89137	1	1	yes	characteristic
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	109	1	no	—
96	C₂×C₄	—	109	1	no	—
97	C₂×C₄	—	109	2	no	—
98	C₂×C₄	—	109	2	no	—
99	C₂×C₄	—	109	2	no	—
100	C₂×C₄	—	218	2	no	—
101	C₂×C₄	—	218	2	no	—
102	C₂×C₄	—	218	2	no	—
103	C₂×C₄	—	218	1	no	—
104	C₂×C₄	—	218	1	no	—
105	C₂×C₄	C₄⋊#89137	1	2	yes	—
106	C₈	—	436	2	no	—
107	C₈	—	436	2	no	—
108	C₂²	C₂×C₄⋊#89137	1	1	yes	characteristic
109	C₂²	#16812778	1	2	yes	—
110	C₂²	#16812786	1	2	yes	—
111	C₂²	#16812786	1	1	yes	characteristic
112	C₂²	#16812787	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	4	2	no	—
115	C₄	—	109	1	no	—
116	C₄	—	109	1	no	—
117	C₄	—	109	2	no	—
118	C₄	—	436	2	no	—
119	C₄	—	218	2	no	—
120	C₂	#20295970	1	1	yes	characteristic
121	C₂	#20296032	1	1	yes	characteristic
122	C₂	#20296033	1	1	yes	characteristic
123	C₂	#20296050	1	2	yes	—
124	C₂	#20296051	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812766

C₂

yes

maximal, characteristic

C₂×C₄⋊#11724786

C₂

yes

maximal, characteristic

#20295892

C₂

yes

maximal

#20294871

C₂

yes

maximal, characteristic

C₂²×#11724786

C₂²

yes

—

C₂²×#11724786

C₂²

yes

—

#16812766

—

C₄⋊#11724786

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812606

C₄

yes

—

#16812606

C₄

yes

characteristic

#95×C₁₀₉

C₄

yes

—

—

—

—

C₂²×Dic₁₀₉

C₂×C₄

yes

—

C₂²×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₉

C₂³

yes

characteristic

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

—

C₄×Dic₁₀₉

Q₈

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

C₂×Q₈

yes

characteristic

C₂×Dic₁₀₉

#41

yes

—

C₂×Dic₁₀₉

#41

yes

—

#11724786

—

#11724786

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

#117

yes

—

C₂²×C₁₀₉

#131

yes

—

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

yes

Dic₁₀₉

yes

Dic₁₀₉

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

C₂×C₁₀₉

#433

yes

—

#1832

—

109

maximal, Sylow

C₁₀₉

#1832

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

109

—

C₂×C₄⋊C₈

—

109

—

#376

—

109

—

#336

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

—

218

—

#95

—

109

—

#95

—

109

—

#95

#89137

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×#89137

yes

—

C₂²×C₄

C₂×#89137

yes

characteristic

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

218

—

C₄²

—

218

—

C₂³

C₂²×#89137

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

100

C₂×C₄

—

218

—

101

C₂×C₄

—

218

—

102

C₂×C₄

—

218

—

103

C₂×C₄

—

218

—

104

C₂×C₄

—

218

—

105

C₂×C₄

C₄⋊#89137

yes

—

106

C₈

—

436

—

107

C₈

—

436

—

108

C₂²

C₂×C₄⋊#89137

yes

characteristic

109

C₂²

#16812778

yes

—

110

C₂²

#16812786

yes

—

111

C₂²

#16812786

yes

characteristic

112

C₂²

#16812787

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

109

—

116

C₄

—

109

—

117

C₄

—

109

—

118

C₄

—

436

—

119

C₄

—

218

—

120

C₂

#20295970

yes

characteristic

121

C₂

#20296032

yes

characteristic

122

C₂

#20296033

yes

characteristic

123

C₂

#20296050

yes

—

124

C₂

#20296051

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#89137
commutator, socle, prime radical	C₂³×C₁₀₉	C₂²×C₄
Fitting	#95×C₁₀₉	C₄
cosocle	C₂²×Dic₁₀₉	C₂³
hypercenter	#95	#89137
hypocenter	C₁₀₉	#1832
LCS	G ▷ C₂³×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ #95

subgroup

G/H

center, Frattini

C₂³

C₂²×#89137

commutator, socle, prime radical

C₂³×C₁₀₉

C₂²×C₄

Fitting

#95×C₁₀₉

C₄

cosocle

C₂²×Dic₁₀₉

C₂³

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
63. C₁₀₉	#1832	semidirect

H ≅ G/N

type

63. C₁₀₉

#1832

semidirect

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

subgroup	index
C₂×#16812766	2
C₂×C₄⋊#11724786	2
#20295892	2
#20294871	2
#1832	109

subgroup

index

109

Subgroups — #28378950

Conjugacy classes of subgroups

Complements

Maximal subgroups