Group #28371495 — subgroups

Subgroups — #28371495

Order 13952.

716 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 249 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×#16812747	C₂	1	1	yes	maximal, characteristic
2	#20295504	C₂	1	1	yes	maximal, characteristic, complemented
3	#20294999	C₂	1	12	yes	maximal, complemented
4	#519×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812747	—	2	4	no	—
6	#16812747	—	2	4	no	—
7	C₂×#13182030	C₄	1	4	yes	complemented
8	C₂×#13182029	C₂²	1	6	yes	—
9	C₂×Q₈×D₁₀₉	C₄	1	4	yes	complemented
10	Q₈×Dic₁₀₉	—	2	4	no	—
11	#16812658	C₂²	1	6	yes	—
12	#16812613	C₂²	1	16	yes	—
13	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₁₀₉	—	2	4	no	—
15	#117×C₁₀₉	C₂²	1	6	yes	—
16	#13182030	—	2	16	no	—
17	#13182029	—	2	12	no	—
18	#13182029	—	2	12	no	—
19	C₂×Dic₂₁₈	C₂×C₄	1	6	yes	—
20	Q₈×D₁₀₉	—	2	16	no	—
21	C₂×D₄₃₆	C₂×C₄	1	6	yes	—
22	C₂×C₂²⋊D₁₀₉	C₂³	1	4	yes	—
23	C₂×C₄×D₁₀₉	C₂×C₄	1	12	yes	—
24	C₄⋊Dic₁₀₉	—	2	6	no	—
25	C₂²⋊Dic₁₀₉	C₂³	1	4	yes	—
26	C₄×Dic₁₀₉	—	2	6	no	—
27	C₂×Q₈×C₁₀₉	C₂×C₄	1	4	yes	—
28	C₂×Q₈×C₁₀₉	D₄	1	4	yes	—
29	C₂×Q₈×C₁₀₉	D₄	1	4	yes	—
30	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
31	C₄⋊C₄×C₁₀₉	—	2	6	no	—
32	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
33	C₄²×C₁₀₉	—	2	6	no	—
34	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
35	C₂²×D₁₀₉	C₂²×C₄	1	4	yes	—
36	Q₈×C₁₀₉	C₂²⋊C₄	1	16	yes	—
37	C₂×C₄×C₁₀₉	—	2	4	no	—
38	C₂×C₄×C₁₀₉	C₂²×C₄	1	6	yes	—
39	C₂×C₄×C₁₀₉	C₂×D₄	1	6	yes	—
40	C₂×C₄×C₁₀₉	C₂×D₄	1	6	yes	—
41	C₂²⋊D₁₀₉	—	2	4	no	—
42	C₂²⋊D₁₀₉	—	2	4	no	—
43	C₂×Dic₁₀₉	—	2	4	no	—
44	C₂×Dic₁₀₉	C₂²×C₄	1	4	yes	—
45	D₄₃₆	—	2	12	no	—
46	C₄×D₁₀₉	—	2	24	no	—
47	Dic₂₁₈	—	2	12	no	—
48	C₂²×C₁₀₉	—	4	1	no	—
49	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
50	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
51	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
52	C₂×D₁₀₉	—	2	4	no	—
53	C₂×D₁₀₉	—	4	4	no	—
54	C₄×C₁₀₉	—	4	4	no	—
55	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	12	yes	—
56	Dic₁₀₉	—	2	4	no	—
57	Dic₁₀₉	—	4	4	no	—
58	C₂×C₁₀₉	—	2	1	no	—
59	C₂×C₁₀₉	—	2	1	no	—
60	C₂×C₁₀₉	—	2	1	no	—
61	C₂×C₁₀₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
62	D₁₀₉	—	4	4	no	—
63	#2884	—	109	1	no	maximal, Sylow
64	C₁₀₉	#2884	1	1	yes	characteristic, complemented, Sylow
65	C₂×2₋¹⁺⁴	—	109	1	no	—
66	#519	—	109	1	no	—
67	#519	D₁₀₉	1	1	yes	characteristic, complemented
68	#410	—	109	12	no	—
69	2₋¹⁺⁴	—	218	4	no	—
70	2₋¹⁺⁴	—	218	4	no	—
71	C₂×#41	—	109	4	no	—
72	C₂×#41	—	109	6	no	—
73	C₂²×Q₈	—	109	4	no	—
74	C₂²×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
75	C₄×Q₈	—	2	4	no	—
76	C₄×Q₈	—	218	4	no	—
77	#117	—	109	6	no	—
78	#117	C₂×D₁₀₉	1	6	yes	—
79	#99	—	109	16	no	—
80	#41	—	218	16	no	—
81	#41	—	218	12	no	—
82	#41	—	218	12	no	—
83	C₂×Q₈	—	109	6	no	—
84	C₂×Q₈	—	218	16	no	—
85	C₂×Q₈	C₄×D₁₀₉	1	4	yes	—
86	C₂×Q₈	D₄₃₆	1	4	yes	—
87	C₂×Q₈	C₂²⋊D₁₀₉	1	4	yes	—
88	C₂×D₄	—	109	4	no	—
89	C₂×D₄	—	109	6	no	—
90	C₂²×C₄	—	109	12	no	—
91	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
92	C₄⋊C₄	—	2	6	no	—
93	C₄⋊C₄	—	218	6	no	—
94	C₂²⋊C₄	—	109	4	no	—
95	C₂²⋊C₄	C₂²×D₁₀₉	1	4	yes	—
96	C₄²	—	2	6	no	—
97	C₄²	—	218	6	no	—
98	C₂³	—	109	4	no	—
99	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
100	Q₈	—	218	12	no	—
101	Q₈	#13182006	1	16	yes	—
102	D₄	—	218	4	no	—
103	D₄	—	218	4	no	—
104	D₄	—	218	12	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	109	4	no	—
107	C₂×C₄	—	218	4	no	—
108	C₂×C₄	—	218	24	no	—
109	C₂×C₄	C₂×C₄×D₁₀₉	1	6	yes	—
110	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	6	yes	—
111	C₂×C₄	C₂×D₄₃₆	1	6	yes	—
112	C₂²	—	4	1	no	—
113	C₂²	—	436	4	no	—
114	C₂²	—	218	4	no	—
115	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
116	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
117	C₂²	C₂²×D₄₃₆	1	1	yes	characteristic
118	C₄	—	4	4	no	—
119	C₄	—	436	4	no	—
120	C₄	—	218	4	no	—
121	C₄	C₂×#13182006	1	12	yes	—
122	C₂	—	2	1	no	—
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	—	436	4	no	—
126	C₂	C₂²×#13182006	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812747

C₂

yes

maximal, characteristic

#20295504

C₂

yes

maximal, characteristic, complemented

#20294999

C₂

yes

maximal, complemented

#519×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#16812747

—

#16812747

—

C₂×#13182030

C₄

yes

complemented

C₂×#13182029

C₂²

yes

—

C₂×Q₈×D₁₀₉

C₄

yes

complemented

Q₈×Dic₁₀₉

—

#16812658

C₂²

yes

—

#16812613

C₂²

yes

—

C₂²×Q₈×C₁₀₉

C₂²

yes

characteristic

C₄×Q₈×C₁₀₉

—

#117×C₁₀₉

C₂²

yes

—

#13182030

—

#13182029

—

#13182029

—

C₂×Dic₂₁₈

C₂×C₄

yes

—

Q₈×D₁₀₉

—

C₂×D₄₃₆

C₂×C₄

yes

—

C₂×C₂²⋊D₁₀₉

C₂³

yes

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

—

C₄⋊Dic₁₀₉

—

C₂²⋊Dic₁₀₉

C₂³

yes

—

C₄×Dic₁₀₉

—

C₂×Q₈×C₁₀₉

C₂×C₄

yes

—

C₂×Q₈×C₁₀₉

D₄

yes

—

C₂×Q₈×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

C₂³

yes

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

C₂⁴

yes

characteristic

C₂²×D₁₀₉

C₂²×C₄

yes

—

Q₈×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂²⋊D₁₀₉

—

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂²×C₄

yes

—

D₄₃₆

—

C₄×D₁₀₉

—

Dic₂₁₈

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂³×C₄

yes

characteristic

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₂²⋊C₄

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂²×C₂²⋊C₄

yes

characteristic

D₁₀₉

—

#2884

—

109

maximal, Sylow

C₁₀₉

#2884

yes

characteristic, complemented, Sylow

C₂×2₋¹⁺⁴

—

109

—

#519

—

109

—

#519

D₁₀₉

yes

characteristic, complemented

#410

—

109

—

2₋¹⁺⁴

—

218

—

2₋¹⁺⁴

—

218

—

C₂×#41

—

109

—

C₂×#41

—

109

—

C₂²×Q₈

—

109

—

C₂²×Q₈

C₂×D₁₀₉

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

218

—

#117

—

109

—

#117

C₂×D₁₀₉

yes

—

#99

—

109

—

#41

—

218

—

#41

—

218

—

#41

—

218

—

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂×Q₈

C₄×D₁₀₉

yes

—

C₂×Q₈

D₄₃₆

yes

—

C₂×Q₈

C₂²⋊D₁₀₉

yes

—

C₂×D₄

—

109

—

C₂×D₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

C₂²×D₁₀₉

yes

—

C₄²

—

C₄²

—

218

—

C₂³

—

109

—

C₂³

C₂³×D₁₀₉

yes

characteristic

100

Q₈

—

218

—

101

Q₈

#13182006

yes

—

102

D₄

—

218

—

103

D₄

—

218

—

104

D₄

—

218

—

105

C₂×C₄

—

106

C₂×C₄

—

109

—

107

C₂×C₄

—

218

—

108

C₂×C₄

—

218

—

109

C₂×C₄

C₂×C₄×D₁₀₉

yes

—

110

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

111

C₂×C₄

C₂×D₄₃₆

yes

—

112

C₂²

—

113

C₂²

—

436

—

114

C₂²

—

218

—

115

C₂²

C₂²×C₄×D₁₀₉

yes

characteristic

116

C₂²

C₂²×C₂²⋊D₁₀₉

yes

characteristic

117

C₂²

C₂²×D₄₃₆

yes

characteristic

118

C₄

—

119

C₄

—

436

—

120

C₄

—

218

—

121

C₄

C₂×#13182006

yes

—

122

C₂

—

123

C₂

—

124

C₂

—

125

C₂

—

436

—

126

C₂

C₂²×#13182006

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×#13182006
commutator	C₂²×C₁₀₉	C₂³×C₄
Frattini	C₂³	C₂³×D₁₀₉
Fitting	#519×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂²×C₂²⋊C₄
prime radical	C₂×#16812747	C₂
cosocle	C₂³×C₁₀₉	C₂⁴
hypercenter	#519	D₁₀₉
hypocenter	C₁₀₉	#2884
LCS	G ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂²×Q₈ ◁ #519

subgroup

G/H

center

C₂

C₂²×#13182006

commutator

C₂²×C₁₀₉

C₂³×C₄

Frattini

C₂³

C₂³×D₁₀₉

Fitting

#519×C₁₀₉

C₂

socle

C₂×C₁₀₉

C₂²×C₂²⋊C₄

prime radical

C₂×#16812747

C₂

cosocle

C₂³×C₁₀₉

C₂⁴

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂ ◁ C₂²×Q₈ ◁ #519

N	H ≅ G/N	type
2. #20295504	C₂	semidirect
3. #20294999	C₂	semidirect
4. #519×C₁₀₉	C₂	semidirect
7. C₂×#13182030	C₄	semidirect
9. C₂×Q₈×D₁₀₉	C₄	semidirect
64. C₁₀₉	#2884	semidirect
67. #519	D₁₀₉	semidirect

H ≅ G/N

type

2. #20295504

C₂

semidirect

3. #20294999

C₂

semidirect

4. #519×C₁₀₉

C₂

semidirect

7. C₂×#13182030

C₄

semidirect

9. C₂×Q₈×D₁₀₉

C₄

semidirect

64. C₁₀₉

#2884

semidirect

67. #519

D₁₀₉

semidirect

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

subgroup	index
C₂×#16812747	2
#20295504	2
#20294999	2
#519×C₁₀₉	2
#2884	109

subgroup

index

#519×C₁₀₉

109

Subgroups — #28371495

Conjugacy classes of subgroups

Complements

Maximal subgroups