Group #28373414 — subgroups

Subgroups — #28373414

Order 13952.

380 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16812699	C₂	1	1	yes	maximal, characteristic
2	#20295396	C₂	1	4	yes	maximal, complemented
3	#20295378	C₂	1	4	yes	maximal, complemented
4	#20295034	C₂	1	4	yes	maximal
5	C₄×#13182013	C₂	1	1	yes	maximal, characteristic
6	#533×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₁₈	C₂²	1	2	yes	—
8	C₂×#13182013	C₂²	1	2	yes	—
9	#16812699	—	2	2	no	—
10	#16812699	C₂²	1	2	yes	—
11	#16812699	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₁₀₉	—	2	2	no	—
13	#16812622	C₂²	1	16	yes	—
14	#16812610	C₂²	1	2	yes	—
15	C₄×#11724784	C₂²	1	2	yes	—
16	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
17	#125×C₁₀₉	C₂²	1	2	yes	—
18	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
19	C₄×Q₈×C₁₀₉	—	2	2	no	—
20	#117×C₁₀₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₂×Dic₂₁₈	—	2	8	no	—
23	C₂×Dic₂₁₈	—	2	2	no	—
24	C₂×Dic₂₁₈	C₂³	1	4	yes	—
25	C₂²×Dic₁₀₉	—	2	2	no	—
26	#13182013	—	2	4	no	—
27	C₂×#11724784	C₂³	1	4	yes	—
28	C₄⋊Dic₁₀₉	—	2	4	no	—
29	C₄⋊Dic₁₀₉	—	2	4	no	—
30	C₂×Q₈×C₁₀₉	—	2	2	no	—
31	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
32	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₀₉	—	2	4	no	—
34	C₄⋊C₄×C₁₀₉	—	2	2	no	—
35	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
36	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
37	C₄²×C₁₀₉	—	2	2	no	—
38	C₄²×C₁₀₉	D₄	1	2	yes	—
39	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
40	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
41	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₁₀₉	—	4	4	no	—
43	C₂×C₄×C₁₀₉	—	2	4	no	—
44	C₂×C₄×C₁₀₉	—	2	4	no	—
45	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₀₉	#41	1	4	yes	—
49	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₁₀₉	—	2	4	no	—
51	C₂×Dic₁₀₉	—	4	4	no	—
52	Dic₂₁₈	—	2	8	no	—
53	Dic₂₁₈	—	4	4	no	—
54	#11724784	—	2	4	no	—
55	C₂²×C₁₀₉	—	2	2	no	—
56	C₂²×C₁₀₉	#124	1	2	yes	—
57	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₁₀₉	—	2	2	no	—
59	C₄×C₁₀₉	—	4	4	no	—
60	C₄×C₁₀₉	#124	1	2	yes	—
61	C₄×C₁₀₉	#137	1	4	yes	—
62	C₄×C₁₀₉	C₂×#41	1	2	yes	—
63	Dic₁₀₉	—	4	4	no	—
64	C₂×C₁₀₉	—	2	2	no	—
65	C₂×C₁₀₉	C₂×#124	1	1	yes	characteristic
66	C₂×C₁₀₉	C₂×#137	1	2	yes	—
67	#3139	—	109	1	no	maximal, Sylow
68	C₁₀₉	#3139	1	1	yes	characteristic, complemented, Sylow
69	#533	D₁₀₉	1	1	yes	characteristic
70	C₂×C₄⋊Q₈	—	109	1	no	—
71	#491	—	109	4	no	—
72	#487	—	109	4	no	—
73	#419	—	109	4	no	—
74	C₄×M₁₆	—	109	1	no	—
75	C₂²×Q₈	—	109	2	no	—
76	C₂×M₁₆	—	109	2	no	—
77	C₄⋊Q₈	—	109	2	no	—
78	C₄⋊Q₈	—	109	2	no	—
79	C₄⋊Q₈	—	218	2	no	—
80	C₄⋊Q₈	C₂×D₁₀₉	1	2	yes	—
81	#125	C₂×D₁₀₉	1	2	yes	—
82	C₂²⋊Q₈	—	2	2	no	—
83	C₄×Q₈	—	2	2	no	—
84	#117	C₂×D₁₀₉	1	2	yes	—
85	C₂×C₄⋊C₄	—	218	2	no	—
86	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
87	#103	—	109	16	no	—
88	#97	—	109	2	no	—
89	C₄×C₈	—	109	2	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	109	4	no	—
92	C₂×Q₈	—	218	8	no	—
93	C₂×Q₈	—	218	2	no	—
94	C₂²×C₄	—	218	2	no	—
95	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
96	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
97	M₁₆	—	218	4	no	—
98	C₂×C₈	—	109	4	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	218	4	no	—
102	C₄⋊C₄	—	218	4	no	—
103	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	—
104	C₂²⋊C₄	—	2	2	no	—
105	C₄²	—	2	2	no	—
106	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
107	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
108	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
109	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
110	Q₈	—	4	4	no	—
111	Q₈	—	436	4	no	—
112	Q₈	—	218	8	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	436	4	no	—
116	C₂×C₄	—	218	4	no	—
117	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
120	C₂×C₄	#13182030	1	4	yes	—
121	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
122	C₈	—	218	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#16812688	1	2	yes	—
125	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	4	4	no	—
128	C₄	—	436	4	no	—
129	C₄	#16812688	1	2	yes	—
130	C₄	#16812729	1	4	yes	—
131	C₄	C₂×#13182030	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₂×#16812688	1	1	yes	characteristic
134	C₂	C₂×#16812729	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812699

C₂

yes

maximal, characteristic

#20295396

C₂

yes

maximal, complemented

#20295378

C₂

yes

maximal, complemented

#20295034

C₂

yes

maximal

C₄×#13182013

C₂

yes

maximal, characteristic

#533×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

C₂²

yes

—

C₂×#13182013

C₂²

yes

—

#16812699

—

#16812699

C₂²

yes

—

#16812699

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

—

#16812622

C₂²

yes

—

#16812610

C₂²

yes

—

C₄×#11724784

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

—

#125×C₁₀₉

C₂²

yes

—

C₂²⋊Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

—

#117×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂³

yes

—

C₂²×Dic₁₀₉

—

#13182013

—

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#124

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#124

yes

—

C₄×C₁₀₉

#137

yes

—

C₄×C₁₀₉

C₂×#41

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×#124

yes

characteristic

C₂×C₁₀₉

C₂×#137

yes

—

#3139

—

109

maximal, Sylow

C₁₀₉

#3139

yes

characteristic, complemented, Sylow

#533

D₁₀₉

yes

characteristic

C₂×C₄⋊Q₈

—

109

—

#491

—

109

—

#487

—

109

—

#419

—

109

—

C₄×M₁₆

—

109

—

C₂²×Q₈

—

109

—

C₂×M₁₆

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

218

—

C₄⋊Q₈

C₂×D₁₀₉

yes

—

#125

C₂×D₁₀₉

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₁₀₉

yes

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#103

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

C₂×Q₈

—

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

218

—

C₂×C₈

—

109

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

218

—

102

C₄⋊C₄

—

218

—

103

C₄⋊C₄

C₂²×D₁₀₉

yes

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

C₂²⋊D₁₀₉

yes

—

107

C₄²

C₂²×D₁₀₉

yes

characteristic

108

C₄²

C₂²×D₁₀₉

yes

characteristic

109

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

110

Q₈

—

111

Q₈

—

436

—

112

Q₈

—

218

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

436

—

116

C₂×C₄

—

218

—

117

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

118

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

120

C₂×C₄

#13182030

yes

—

121

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

122

C₈

—

218

—

123

C₂²

—

124

C₂²

#16812688

yes

—

125

C₂²

C₂²×C₂²⋊D₁₀₉

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

436

—

129

C₄

#16812688

yes

—

130

C₄

#16812729

yes

—

131

C₄

C₂×#13182030

yes

—

132

C₂

—

133

C₂

C₂×#16812688

yes

characteristic

134

C₂

C₂×#16812729

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#533×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂³×C₁₀₉	C₂×D₄
hypercenter	#533	D₁₀₉
hypocenter	C₁₀₉	#3139
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

#533×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

prime radical

C₂³×C₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #20295396	C₂	semidirect
3. #20295378	C₂	semidirect
68. C₁₀₉	#3139	semidirect

H ≅ G/N

type

2. #20295396

C₂

semidirect

3. #20295378

C₂

semidirect

68. C₁₀₉

#3139

semidirect

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

subgroup	index
C₂×#16812699	2
#20295396	2
#20295378	2
#20295034	2
C₄×#13182013	2
#533×C₁₀₉	2
#3139	109

subgroup

index

#533×C₁₀₉

109

Subgroups — #28373414

Conjugacy classes of subgroups

Complements

Maximal subgroups