Group #28363855 — subgroups

Subgroups — #28363855

Order 13952.

412 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 105 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×#16812696	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812619	C₂	1	4	yes	maximal, complemented
3	C₂×#16812610	C₂	1	1	yes	maximal, characteristic, complemented
4	#382×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄₃₆	—	2	2	no	—
6	C₂²×D₄₃₆	C₂²	1	2	yes	—
7	C₂²×#11724784	C₂²	1	2	yes	—
8	#16812696	—	2	2	no	—
9	#16812696	C₄	1	4	yes	complemented
10	#16812619	—	2	8	no	—
11	#16812610	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
14	#95×C₁₀₉	—	2	2	no	—
15	C₂³×D₁₀₉	—	2	2	no	—
16	C₂×D₄₃₆	—	2	4	no	—
17	C₂×D₄₃₆	—	2	4	no	—
18	C₂×D₄₃₆	—	2	8	no	—
19	C₂×D₄₃₆	—	2	2	no	—
20	C₂×D₄₃₆	—	4	8	no	—
21	C₂×D₄₃₆	C₂×C₄	1	4	yes	—
22	C₂×#11724784	—	2	2	no	—
23	C₂×#11724784	—	2	2	no	—
24	C₂×#11724784	D₄	1	4	yes	complemented
25	C₂²×C₄×C₁₀₉	—	2	2	no	—
26	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
27	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₀₉	—	2	4	no	—
29	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₄²×C₁₀₉	D₄	1	2	yes	—
31	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
32	C₂²×D₁₀₉	—	2	4	no	—
33	C₂²×D₁₀₉	—	4	2	no	—
34	C₂²×D₁₀₉	—	4	2	no	—
35	C₂²×D₁₀₉	—	4	8	no	—
36	C₂×C₄×C₁₀₉	—	2	4	no	—
37	C₂×C₄×C₁₀₉	—	4	2	no	—
38	C₂×C₄×C₁₀₉	—	4	4	no	—
39	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	4	yes	—
40	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	#41	1	2	yes	—
44	D₄₃₆	—	2	8	no	—
45	D₄₃₆	—	4	16	no	—
46	D₄₃₆	—	4	4	no	—
47	#11724784	—	2	4	no	—
48	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₀₉	#124	1	1	yes	characteristic
50	C₂²×C₁₀₉	#127	1	1	yes	characteristic
51	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	4	yes	—
52	C₂×D₁₀₉	—	4	4	no	—
53	C₂×D₁₀₉	—	4	8	no	—
54	C₂×D₁₀₉	—	8	4	no	—
55	C₂×D₁₀₉	—	8	4	no	—
56	C₄×C₁₀₉	—	2	4	no	—
57	C₄×C₁₀₉	—	8	2	no	—
58	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
59	C₄×C₁₀₉	#121	1	2	yes	—
60	C₂×C₁₀₉	#390	1	1	yes	characteristic
61	C₂×C₁₀₉	#418	1	2	yes	—
62	C₂×C₁₀₉	#490	1	2	yes	—
63	C₂×C₁₀₉	#496	1	2	yes	—
64	D₁₀₉	—	8	4	no	—
65	#1961	—	109	1	no	maximal, Sylow
66	C₁₀₉	#1961	1	1	yes	characteristic, complemented, Sylow
67	C₂×#127	—	109	1	no	—
68	C₂×#102	—	109	4	no	—
69	C₂×#97	—	109	1	no	—
70	#382	D₁₀₉	1	1	yes	characteristic, complemented
71	C₂²×D₄	—	109	2	no	—
72	C₂²×D₄	—	218	2	no	—
73	C₂²×C₈	—	109	2	no	—
74	#127	—	109	4	no	—
75	#127	—	218	2	no	—
76	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
77	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
78	#102	—	218	8	no	—
79	#97	—	218	2	no	—
80	#95	—	2	2	no	—
81	C₂⁴	—	218	2	no	—
82	C₂×D₄	—	109	4	no	—
83	C₂×D₄	—	436	8	no	—
84	C₂×D₄	—	218	8	no	—
85	C₂×D₄	—	218	4	no	—
86	C₂×D₄	—	218	2	no	—
87	C₂×D₄	—	218	4	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
91	C₂×C₈	—	109	4	no	—
92	C₂×C₈	—	218	2	no	—
93	C₂×C₈	—	218	2	no	—
94	C₄⋊C₄	—	2	4	no	—
95	C₄²	C₄×D₁₀₉	1	2	yes	—
96	C₄²	D₄₃₆	1	2	yes	—
97	C₂³	—	436	2	no	—
98	C₂³	—	436	2	no	—
99	C₂³	—	436	8	no	—
100	C₂³	—	218	4	no	—
101	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
102	D₄	—	436	16	no	—
103	D₄	—	436	4	no	—
104	D₄	—	218	8	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	4	4	no	—
108	C₂×C₄	#13182006	1	4	yes	—
109	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
110	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
111	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
112	C₂×C₄	#13182030	1	2	yes	—
113	C₈	—	218	4	no	—
114	C₂²	—	872	4	no	—
115	C₂²	—	872	4	no	—
116	C₂²	—	436	4	no	—
117	C₂²	—	436	8	no	—
118	C₂²	C₂×#13182006	1	1	yes	characteristic
119	C₂²	#16812688	1	1	yes	characteristic
120	C₂²	#16812695	1	1	yes	characteristic
121	C₂²	#16812724	1	4	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	8	2	no	—
124	C₄	#16812661	1	2	yes	—
125	C₄	#16812674	1	2	yes	—
126	C₂	—	872	4	no	—
127	C₂	#20294927	1	1	yes	characteristic
128	C₂	#20295030	1	2	yes	—
129	C₂	#20295392	1	2	yes	—
130	C₂	#20295416	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812696

C₂

yes

maximal, characteristic

C₂×#16812619

C₂

yes

maximal, complemented

C₂×#16812610

C₂

yes

maximal, characteristic, complemented

#382×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₃₆

—

C₂²×D₄₃₆

C₂²

yes

—

C₂²×#11724784

C₂²

yes

—

#16812696

—

#16812696

C₄

yes

complemented

#16812619

—

#16812610

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

#95×C₁₀₉

—

C₂³×D₁₀₉

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

C₂×C₄

yes

—

C₂×#11724784

—

C₂×#11724784

—

C₂×#11724784

D₄

yes

complemented

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

#11724784

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×D₄

yes

—

C₄×C₁₀₉

#121

yes

—

C₂×C₁₀₉

#390

yes

characteristic

C₂×C₁₀₉

#418

yes

—

C₂×C₁₀₉

#490

yes

—

C₂×C₁₀₉

#496

yes

—

D₁₀₉

—

#1961

—

109

maximal, Sylow

C₁₀₉

#1961

yes

characteristic, complemented, Sylow

C₂×#127

—

109

—

C₂×#102

—

109

—

C₂×#97

—

109

—

#382

D₁₀₉

yes

characteristic, complemented

C₂²×D₄

—

109

—

C₂²×D₄

—

218

—

C₂²×C₈

—

109

—

#127

—

109

—

#127

—

218

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#102

—

218

—

#97

—

218

—

#95

—

C₂⁴

—

218

—

C₂×D₄

—

109

—

C₂×D₄

—

436

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₄²

C₄×D₁₀₉

yes

—

C₄²

D₄₃₆

yes

—

C₂³

—

436

—

C₂³

—

436

—

C₂³

—

436

—

100

C₂³

—

218

—

101

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

102

D₄

—

436

—

103

D₄

—

436

—

104

D₄

—

218

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

#13182006

yes

—

109

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

110

C₂×C₄

D₄×D₁₀₉

yes

—

111

C₂×C₄

C₂×D₄₃₆

yes

characteristic

112

C₂×C₄

#13182030

yes

—

113

C₈

—

218

—

114

C₂²

—

872

—

115

C₂²

—

872

—

116

C₂²

—

436

—

117

C₂²

—

436

—

118

C₂²

C₂×#13182006

yes

characteristic

119

C₂²

#16812688

yes

characteristic

120

C₂²

#16812695

yes

characteristic

121

C₂²

#16812724

yes

—

122

C₄

—

123

C₄

—

124

C₄

#16812661

yes

—

125

C₄

#16812674

yes

—

126

C₂

—

872

—

127

C₂

#20294927

yes

characteristic

128

C₂

#20295030

yes

—

129

C₂

#20295392

yes

—

130

C₂

#20295416

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#382×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂×D₄
prime radical	C₂×#16812696	C₂
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#382	D₁₀₉
hypocenter	C₁₀₉	#1961
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#382×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂×D₄

prime radical

C₂×#16812696

C₂

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
2. C₂×#16812619	C₂	semidirect
3. C₂×#16812610	C₂	semidirect
4. #382×C₁₀₉	C₂	semidirect
9. #16812696	C₄	semidirect
24. C₂×#11724784	D₄	semidirect
66. C₁₀₉	#1961	semidirect
70. #382	D₁₀₉	semidirect

H ≅ G/N

type

2. C₂×#16812619

C₂

semidirect

3. C₂×#16812610

C₂

semidirect

4. #382×C₁₀₉

C₂

semidirect

9. #16812696

C₄

semidirect

24. C₂×#11724784

D₄

semidirect

66. C₁₀₉

#1961

semidirect

70. #382

D₁₀₉

semidirect

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

subgroup	index
C₂×#16812696	2
C₂×#16812619	2
C₂×#16812610	2
#382×C₁₀₉	2
#1961	109

subgroup

index

#382×C₁₀₉

109

Subgroups — #28363855

Conjugacy classes of subgroups

Complements

Maximal subgroups