Group #28364288 — subgroups

Subgroups — #28364288

Order 13952.

296 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂²×D₈₇₂	C₂	1	1	yes	maximal, characteristic
2	#20295030	C₂	1	2	yes	maximal, complemented
3	C₂×#16812615	C₂	1	2	yes	maximal, complemented
4	#20294791	C₂	1	1	yes	maximal, characteristic, complemented
5	#424×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄₃₆	C₂²	1	2	yes	—
7	C₂×D₈₇₂	—	2	4	no	—
8	C₂×D₈₇₂	—	2	2	no	—
9	C₂×#13182013	C₂²	1	2	yes	—
10	#16812619	—	2	4	no	—
11	#16812615	—	2	4	no	—
12	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	#117×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₁₀₉	—	2	2	no	—
16	C₂³×D₁₀₉	—	2	2	no	—
17	C₂×D₄₃₆	—	2	2	no	—
18	C₂×D₄₃₆	—	2	2	no	—
19	C₂×D₄₃₆	—	2	2	no	—
20	C₂×D₄₃₆	—	2	2	no	—
21	C₂×D₄₃₆	—	2	2	no	—
22	C₂×D₄₃₆	D₄	1	4	yes	—
23	D₈₇₂	—	2	8	no	—
24	#13182013	—	2	2	no	—
25	#13182013	—	2	2	no	—
26	C₂×#11724784	—	2	2	no	—
27	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
28	M₁₆×C₁₀₉	—	2	2	no	—
29	C₂×C₈×C₁₀₉	—	2	2	no	—
30	C₂×C₈×C₁₀₉	—	2	1	no	—
31	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
32	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
33	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
34	C₄²×C₁₀₉	—	2	1	no	—
35	C₂³×C₁₀₉	#41	1	1	yes	characteristic
36	C₂²×D₁₀₉	—	2	4	no	—
37	C₂²×D₁₀₉	—	4	2	no	—
38	C₂²×D₁₀₉	—	4	2	no	—
39	C₂²×D₁₀₉	—	8	4	no	—
40	C₂×C₄×C₁₀₉	—	2	2	no	—
41	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	#41	1	2	yes	—
45	D₄₃₆	—	2	4	no	—
46	D₄₃₆	—	2	4	no	—
47	D₄₃₆	—	4	2	no	—
48	D₄₃₆	—	4	2	no	—
49	C₈×C₁₀₉	—	2	2	no	—
50	C₈×C₁₀₉	—	4	1	no	—
51	#11724784	—	4	2	no	—
52	C₂²×C₁₀₉	—	2	2	no	—
53	C₂²×C₁₀₉	#121	1	1	yes	characteristic
54	C₂²×C₁₀₉	#123	1	2	yes	—
55	C₂×D₁₀₉	—	4	4	no	—
56	C₂×D₁₀₉	—	8	4	no	—
57	C₂×D₁₀₉	—	8	4	no	—
58	C₂×D₁₀₉	—	8	4	no	—
59	C₄×C₁₀₉	—	4	2	no	—
60	C₄×C₁₀₉	#120	1	1	yes	characteristic
61	C₄×C₁₀₉	#121	1	2	yes	—
62	C₄×C₁₀₉	#124	1	1	yes	characteristic
63	C₂×C₁₀₉	—	2	2	no	—
64	C₂×C₁₀₉	#394	1	1	yes	characteristic
65	C₂×C₁₀₉	#472	1	2	yes	—
66	D₁₀₉	—	8	4	no	—
67	#2045	—	109	1	no	maximal, Sylow
68	C₁₀₉	#2045	1	1	yes	characteristic, complemented, Sylow
69	C₂²×D₈	—	109	1	no	—
70	#424	D₁₀₉	1	1	yes	characteristic, complemented
71	#418	—	109	2	no	—
72	C₂×#100	—	109	2	no	—
73	#341	—	109	1	no	—
74	C₂²×D₄	—	109	2	no	—
75	C₂×D₈	—	218	4	no	—
76	C₂×D₈	—	218	2	no	—
77	C₂×M₁₆	—	109	2	no	—
78	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
79	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
80	#117	C₂×D₁₀₉	1	1	yes	characteristic
81	C₄⋊C₈	—	2	2	no	—
82	#102	—	218	4	no	—
83	#100	—	218	4	no	—
84	C₂⁴	—	218	2	no	—
85	C₂×D₄	—	109	4	no	—
86	C₂×D₄	—	218	2	no	—
87	C₂×D₄	—	218	2	no	—
88	C₂×D₄	—	218	2	no	—
89	C₂×D₄	—	218	2	no	—
90	C₂×D₄	—	218	2	no	—
91	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
92	D₈	—	218	8	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	218	2	no	—
95	M₁₆	—	218	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	218	2	no	—
99	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
100	C₄⋊C₄	D₄₃₆	1	2	yes	—
101	C₂²⋊C₄	—	2	1	no	—
102	C₄²	—	2	1	no	—
103	C₂³	—	872	4	no	—
104	C₂³	—	436	2	no	—
105	C₂³	—	436	2	no	—
106	C₂³	—	218	4	no	—
107	C₂³	#13182029	1	1	yes	characteristic
108	D₄	—	436	2	no	—
109	D₄	—	436	2	no	—
110	D₄	—	218	4	no	—
111	D₄	—	218	4	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
114	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
115	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
116	C₂×C₄	#13182030	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	1	no	—
119	C₈	—	436	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	872	4	no	—
122	C₂²	—	872	4	no	—
123	C₂²	—	872	4	no	—
124	C₂²	—	436	4	no	—
125	C₂²	#16812673	1	1	yes	characteristic
126	C₂²	#16812684	1	2	yes	—
127	C₄	—	4	2	no	—
128	C₄	#16812667	1	1	yes	characteristic
129	C₄	#16812674	1	2	yes	—
130	C₄	#16812688	1	1	yes	characteristic
131	C₂	—	2	2	no	—
132	C₂	—	872	4	no	—
133	C₂	#20294943	1	1	yes	characteristic
134	C₂	#20295296	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₈₇₂

C₂

yes

maximal, characteristic

#20295030

C₂

yes

maximal, complemented

C₂×#16812615

C₂

yes

maximal, complemented

#20294791

C₂

yes

maximal, characteristic, complemented

#424×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₃₆

C₂²

yes

—

C₂×D₈₇₂

—

C₂×D₈₇₂

—

C₂×#13182013

C₂²

yes

—

#16812619

—

#16812615

—

C₂×M₁₆×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

#117×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₂³×D₁₀₉

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

D₄

yes

—

D₈₇₂

—

#13182013

—

#13182013

—

C₂×#11724784

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₄⋊C₄×C₁₀₉

D₄

yes

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

#41

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#121

yes

characteristic

C₂²×C₁₀₉

#123

yes

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#120

yes

characteristic

C₄×C₁₀₉

#121

yes

—

C₄×C₁₀₉

#124

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

#394

yes

characteristic

C₂×C₁₀₉

#472

yes

—

D₁₀₉

—

#2045

—

109

maximal, Sylow

C₁₀₉

#2045

yes

characteristic, complemented, Sylow

C₂²×D₈

—

109

—

#424

D₁₀₉

yes

characteristic, complemented

#418

—

109

—

C₂×#100

—

109

—

#341

—

109

—

C₂²×D₄

—

109

—

C₂×D₈

—

218

—

C₂×D₈

—

218

—

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

#117

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

#102

—

218

—

#100

—

218

—

C₂⁴

—

218

—

C₂×D₄

—

109

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

D₈

—

218

—

M₁₆

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

218

—

C₂×C₈

C₂²⋊D₁₀₉

yes

—

100

C₄⋊C₄

D₄₃₆

yes

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₂³

—

872

—

104

C₂³

—

436

—

105

C₂³

—

436

—

106

C₂³

—

218

—

107

C₂³

#13182029

yes

characteristic

108

D₄

—

436

—

109

D₄

—

436

—

110

D₄

—

218

—

111

D₄

—

218

—

112

C₂×C₄

—

113

C₂×C₄

D₄×D₁₀₉

yes

—

114

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

115

C₂×C₄

C₂×D₄₃₆

yes

characteristic

116

C₂×C₄

#13182030

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

436

—

120

C₂²

—

121

C₂²

—

872

—

122

C₂²

—

872

—

123

C₂²

—

872

—

124

C₂²

—

436

—

125

C₂²

#16812673

yes

characteristic

126

C₂²

#16812684

yes

—

127

C₄

—

128

C₄

#16812667

yes

characteristic

129

C₄

#16812674

yes

—

130

C₄

#16812688

yes

characteristic

131

C₂

—

132

C₂

—

872

—

133

C₂

#20294943

yes

characteristic

134

C₂

#20295296

yes

—

	subgroup	G/H
center	C₂²	#16812673
commutator, cosocle	C₂²×C₄×C₁₀₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#424×C₁₀₉	C₂
socle	C₂²×C₁₀₉	#121
prime radical	C₂²×D₈₇₂	C₂
hypercenter	#424	D₁₀₉
hypocenter	C₁₀₉	#2045
LCS	G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#16812673

commutator, cosocle

C₂²×C₄×C₁₀₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#424×C₁₀₉

C₂

socle

C₂²×C₁₀₉

#121

prime radical

C₂²×D₈₇₂

C₂

D₁₀₉

C₁₀₉

G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
2. #20295030	C₂	semidirect
3. C₂×#16812615	C₂	semidirect
4. #20294791	C₂	semidirect
5. #424×C₁₀₉	C₂	semidirect
68. C₁₀₉	#2045	semidirect
70. #424	D₁₀₉	semidirect

H ≅ G/N

type

2. #20295030

C₂

semidirect

3. C₂×#16812615

C₂

semidirect

4. #20294791

C₂

semidirect

5. #424×C₁₀₉

C₂

semidirect

68. C₁₀₉

#2045

semidirect

70. #424

D₁₀₉

semidirect

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

subgroup	index
C₂²×D₈₇₂	2
#20295030	2
C₂×#16812615	2
#20294791	2
#424×C₁₀₉	2
#2045	109

subgroup

index

C₂²×D₈₇₂

#424×C₁₀₉

109

Subgroups — #28364288

Conjugacy classes of subgroups

Complements

Maximal subgroups