Group #28361693 — subgroups

Subgroups — #28361693

Order 13952.

246 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 67 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	#20295577	C₂	1	1	yes	maximal, characteristic
2	#20295078	C₂	1	2	yes	maximal, complemented
3	#20294775	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294769	C₂	1	2	yes	maximal, complemented
5	#333×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13182030	C₄	1	2	yes	—
7	#16812696	C₂²	1	1	yes	characteristic
8	#16812688	C₄	1	2	yes	—
9	#16812674	—	2	2	no	—
10	#16812661	—	2	2	no	—
11	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
12	C₄⋊#11724784	C₂²	1	2	yes	—
13	#16812611	—	2	2	no	—
14	C₈×Dic₁₀₉	—	2	2	no	—
15	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
17	#13182030	—	4	4	no	—
18	C₂×D₄₃₆	—	2	2	no	—
19	C₂×D₄₃₆	—	2	2	no	—
20	C₂×D₄₃₆	C₂×C₄	1	2	yes	—
21	C₂×C₄×D₁₀₉	—	2	2	no	—
22	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₁₀₉	D₄	1	2	yes	—
24	C₂×#11724784	—	2	2	no	—
25	#13182006	—	2	4	no	—
26	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₀₉	D₄	1	2	yes	—
28	C₂×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₁₀₉	—	2	2	no	—
30	C₄⋊C₄×C₁₀₉	—	2	2	no	—
31	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
32	C₂²×D₁₀₉	—	2	2	no	—
33	C₂²×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
34	Q₈×C₁₀₉	—	4	2	no	—
35	C₂×C₄×C₁₀₉	—	2	2	no	—
36	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
38	C₂×Dic₁₀₉	C₂²⋊C₄	1	2	yes	—
39	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
40	D₄₃₆	—	4	2	no	—
41	D₄₃₆	—	4	4	no	—
42	D₄₃₆	—	4	2	no	—
43	D₄₃₆	—	4	2	no	—
44	C₄×D₁₀₉	—	2	2	no	—
45	C₄×D₁₀₉	—	2	2	no	—
46	C₄×D₁₀₉	—	2	2	no	—
47	C₄×D₁₀₉	—	4	4	no	—
48	C₈×C₁₀₉	—	4	2	no	—
49	#11724784	—	4	2	no	—
50	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂×D₁₀₉	—	2	1	no	—
52	C₂×D₁₀₉	—	2	2	no	—
53	C₂×D₁₀₉	—	4	2	no	—
54	C₂×D₁₀₉	—	4	4	no	—
55	C₄×C₁₀₉	—	2	1	no	—
56	C₄×C₁₀₉	—	4	2	no	—
57	C₄×C₁₀₉	#135	1	2	yes	—
58	C₄×C₁₀₉	#135	1	2	yes	—
59	Dic₁₀₉	—	2	2	no	—
60	Dic₁₀₉	#100	1	2	yes	—
61	C₂×C₁₀₉	C₂×#100	1	1	yes	characteristic
62	C₂×C₁₀₉	#416	1	2	yes	—
63	D₁₀₉	—	4	1	no	—
64	D₁₀₉	—	8	2	no	—
65	#1543	—	109	1	no	maximal, Sylow
66	C₁₀₉	#1543	1	1	yes	characteristic, complemented, Sylow
67	#532	—	109	1	no	—
68	#431	—	109	2	no	—
69	#333	—	109	1	no	—
70	#333	D₁₀₉	1	1	yes	characteristic, complemented
71	#331	—	109	2	no	—
72	C₂×#41	—	109	2	no	—
73	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
74	#127	—	109	1	no	—
75	#124	—	109	2	no	—
76	#121	—	218	2	no	—
77	C₄×D₄	—	218	2	no	—
78	C₂×C₄²	—	109	1	no	—
79	C₄⋊C₈	—	109	2	no	—
80	C₄⋊C₈	C₂×D₁₀₉	1	2	yes	—
81	C₂²⋊C₈	—	218	2	no	—
82	C₄×C₈	—	218	2	no	—
83	#41	—	436	4	no	—
84	C₂×Q₈	C₄×D₁₀₉	1	2	yes	—
85	C₂×D₄	—	109	2	no	—
86	C₂×D₄	—	218	2	no	—
87	C₂×D₄	—	218	2	no	—
88	C₂²×C₄	—	109	1	no	—
89	C₂²×C₄	—	109	2	no	—
90	C₂²×C₄	—	218	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	218	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₂²⋊C₄	—	218	4	no	—
95	C₄²	—	109	1	no	—
96	C₄²	—	109	2	no	—
97	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
98	C₂³	—	109	1	no	—
99	C₂³	—	218	2	no	—
100	Q₈	—	4	2	no	—
101	D₄	—	436	2	no	—
102	D₄	—	436	4	no	—
103	D₄	—	436	2	no	—
104	D₄	—	436	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	109	1	no	—
107	C₂×C₄	—	109	2	no	—
108	C₂×C₄	—	436	4	no	—
109	C₂×C₄	—	218	2	no	—
110	C₂×C₄	—	218	2	no	—
111	C₂×C₄	—	218	2	no	—
112	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
113	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
114	C₈	—	4	2	no	—
115	C₈	—	436	2	no	—
116	C₂²	—	436	2	no	—
117	C₂²	—	436	4	no	—
118	C₂²	—	218	1	no	—
119	C₂²	—	218	2	no	—
120	C₂²	C₂²⋊C₄×D₁₀₉	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	4	2	no	—
123	C₄	—	109	2	no	—
124	C₄	—	218	2	no	—
125	C₄	#16812720	1	2	yes	—
126	C₄	#16812722	1	2	yes	—
127	C₂	—	872	2	no	—
128	C₂	—	436	1	no	—
129	C₂	#100×D₁₀₉	1	1	yes	characteristic
130	C₂	#20295021	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295577

C₂

yes

maximal, characteristic

#20295078

C₂

yes

maximal, complemented

#20294775

C₂

yes

maximal, characteristic, complemented

#20294769

C₂

yes

maximal, complemented

#333×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#13182030

C₄

yes

—

#16812696

C₂²

yes

characteristic

#16812688

C₄

yes

—

#16812674

—

#16812661

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄⋊#11724784

C₂²

yes

—

#16812611

—

C₈×Dic₁₀₉

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

C₂²

yes

—

#13182030

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

—

C₂×#11724784

—

#13182006

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

—

C₂×Q₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

C₂²⋊C₄

yes

characteristic

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×Dic₁₀₉

C₂²⋊C₄

yes

—

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#135

yes

—

C₄×C₁₀₉

#135

yes

—

Dic₁₀₉

—

Dic₁₀₉

#100

yes

—

C₂×C₁₀₉

C₂×#100

yes

characteristic

C₂×C₁₀₉

#416

yes

—

D₁₀₉

—

D₁₀₉

—

#1543

—

109

maximal, Sylow

C₁₀₉

#1543

yes

characteristic, complemented, Sylow

#532

—

109

—

#431

—

109

—

#333

—

109

—

#333

D₁₀₉

yes

characteristic, complemented

#331

—

109

—

C₂×#41

—

109

—

C₄⋊Q₈

C₂×D₁₀₉

yes

characteristic

#127

—

109

—

#124

—

109

—

#121

—

218

—

C₄×D₄

—

218

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

C₂×D₁₀₉

yes

—

C₂²⋊C₈

—

218

—

C₄×C₈

—

218

—

#41

—

436

—

C₂×Q₈

C₄×D₁₀₉

yes

—

C₂×D₄

—

109

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂×C₈

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₂²⋊C₄

—

218

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₂²×D₁₀₉

yes

characteristic

C₂³

—

109

—

C₂³

—

218

—

100

Q₈

—

101

D₄

—

436

—

102

D₄

—

436

—

103

D₄

—

436

—

104

D₄

—

436

—

105

C₂×C₄

—

106

C₂×C₄

—

109

—

107

C₂×C₄

—

109

—

108

C₂×C₄

—

436

—

109

C₂×C₄

—

218

—

110

C₂×C₄

—

218

—

111

C₂×C₄

—

218

—

112

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

113

C₂×C₄

D₄×D₁₀₉

yes

—

114

C₈

—

115

C₈

—

436

—

116

C₂²

—

436

—

117

C₂²

—

436

—

118

C₂²

—

218

—

119

C₂²

—

218

—

120

C₂²

C₂²⋊C₄×D₁₀₉

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

109

—

124

C₄

—

218

—

125

C₄

#16812720

yes

—

126

C₄

#16812722

yes

—

127

C₂

—

872

—

128

C₂

—

436

—

129

C₂

#100×D₁₀₉

yes

characteristic

130

C₂

#20295021

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#333×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
prime radical	#20295577	C₂
cosocle	C₄²×C₁₀₉	C₂³
hypercenter	#333	D₁₀₉
hypocenter	C₁₀₉	#1543
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₄²

C₂²×D₁₀₉

Fitting

#333×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

prime radical

#20295577

C₂

cosocle

C₄²×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
2. #20295078	C₂	semidirect
3. #20294775	C₂	semidirect
4. #20294769	C₂	semidirect
5. #333×C₁₀₉	C₂	semidirect
66. C₁₀₉	#1543	semidirect
70. #333	D₁₀₉	semidirect

H ≅ G/N

type

2. #20295078

C₂

semidirect

3. #20294775

C₂

semidirect

4. #20294769

C₂

semidirect

5. #333×C₁₀₉

C₂

semidirect

66. C₁₀₉

#1543

semidirect

70. #333

D₁₀₉

semidirect

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

subgroup	index
#20295577	2
#20295078	2
#20294775	2
#20294769	2
#333×C₁₀₉	2
#1543	109

subgroup

index

#333×C₁₀₉

109

Subgroups — #28361693

Conjugacy classes of subgroups

Complements

Maximal subgroups