Group #20294883 — subgroups

Subgroups — #20294883

Order 6976.

234 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 87 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×C₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13182006	C₂	1	2	yes	maximal
4	#16812606	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₁₀₉	C₂²	1	1	yes	characteristic
7	C₂²×Dic₁₀₉	C₂²	1	2	yes	—
8	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₁₀₉	—	2	2	no	—
10	C₂×C₄×D₁₀₉	—	2	2	no	—
11	C₂×C₄×D₁₀₉	C₄	1	4	yes	complemented
12	C₄⋊Dic₁₀₉	—	2	2	no	—
13	#13182006	—	2	4	no	—
14	C₂²×C₄×C₁₀₉	C₂²	1	2	yes	—
15	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₁₀₉	—	2	2	no	—
17	C₂³×C₁₀₉	C₂³	1	1	yes	characteristic
18	C₂²×D₁₀₉	—	2	1	no	—
19	C₂²×D₁₀₉	—	2	1	no	—
20	C₂²×D₁₀₉	—	2	1	no	—
21	C₂²×D₁₀₉	—	2	1	no	—
22	C₂²×D₁₀₉	C₂×C₄	1	2	yes	—
23	C₂²×D₁₀₉	D₄	1	2	yes	—
24	C₂²×D₁₀₉	Q₈	1	2	yes	—
25	C₂×C₄×C₁₀₉	—	2	2	no	—
26	C₂×C₄×C₁₀₉	—	2	2	no	—
27	C₂×C₄×C₁₀₉	—	2	2	no	—
28	C₂×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
29	C₂×C₄×C₁₀₉	D₄	1	2	yes	—
30	C₂×C₄×C₁₀₉	D₄	1	2	yes	—
31	C₂×Dic₁₀₉	—	2	1	no	—
32	C₂×Dic₁₀₉	—	2	1	no	—
33	C₂×Dic₁₀₉	—	2	2	no	—
34	C₂×Dic₁₀₉	—	2	2	no	—
35	C₂×Dic₁₀₉	—	2	2	no	—
36	C₂×Dic₁₀₉	C₂×C₄	1	2	yes	—
37	C₄×D₁₀₉	—	2	8	no	—
38	C₂²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
43	C₂²×C₁₀₉	#41	1	1	yes	characteristic
44	C₂²×C₁₀₉	#41	1	1	yes	characteristic
45	C₂×D₁₀₉	—	2	2	no	—
46	C₂×D₁₀₉	—	2	2	no	—
47	C₂×D₁₀₉	—	2	2	no	—
48	C₂×D₁₀₉	—	2	2	no	—
49	C₂×D₁₀₉	—	2	2	no	—
50	C₂×D₁₀₉	—	2	2	no	—
51	C₂×D₁₀₉	C₄⋊C₄	1	4	yes	complemented
52	C₄×C₁₀₉	—	4	2	no	—
53	C₄×C₁₀₉	C₂²⋊C₄	1	4	yes	complemented
54	Dic₁₀₉	—	2	2	no	—
55	Dic₁₀₉	—	4	2	no	—
56	C₂×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₁₀₉	#117	1	1	yes	characteristic
59	C₂×C₁₀₉	#121	1	1	yes	characteristic
60	C₂×C₁₀₉	#121	1	1	yes	characteristic
61	C₂×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
63	D₁₀₉	—	2	4	no	—
64	C₁₀₉	#380	1	1	yes	characteristic, complemented, Sylow
65	#380	—	109	1	no	maximal, Sylow
66	C₂³×C₄	—	109	1	no	—
67	C₂×C₄⋊C₄	—	109	1	no	—
68	C₂×C₄⋊C₄	D₁₀₉	1	1	yes	characteristic, complemented
69	C₂×C₂²⋊C₄	—	109	2	no	—
70	#95	—	109	2	no	—
71	C₂⁴	—	109	1	no	—
72	C₂²×C₄	—	109	1	no	—
73	C₂²×C₄	—	109	4	no	—
74	C₂²×C₄	—	109	2	no	—
75	C₂²×C₄	—	218	2	no	—
76	C₂²×C₄	—	218	2	no	—
77	C₂²×C₄	C₂×D₁₀₉	1	2	yes	—
78	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
79	C₄⋊C₄	—	2	2	no	—
80	C₄⋊C₄	—	218	2	no	—
81	C₂²⋊C₄	—	218	4	no	—
82	C₂³	—	109	2	no	—
83	C₂³	—	109	2	no	—
84	C₂³	—	109	2	no	—
85	C₂³	—	218	1	no	—
86	C₂³	—	218	1	no	—
87	C₂³	—	218	1	no	—
88	C₂³	—	218	1	no	—
89	C₂³	C₂²×D₁₀₉	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	109	2	no	—
94	C₂×C₄	—	218	2	no	—
95	C₂×C₄	—	218	2	no	—
96	C₂×C₄	—	218	2	no	—
97	C₂×C₄	—	218	1	no	—
98	C₂×C₄	—	218	1	no	—
99	C₂×C₄	—	218	8	no	—
100	C₂×C₄	C₄×D₁₀₉	1	2	yes	—
101	C₂×C₄	D₄₃₆	1	2	yes	—
102	C₂×C₄	C₂²⋊D₁₀₉	1	2	yes	—
103	C₂²	—	109	4	no	—
104	C₂²	—	218	2	no	—
105	C₂²	—	218	2	no	—
106	C₂²	—	218	2	no	—
107	C₂²	—	218	2	no	—
108	C₂²	—	218	2	no	—
109	C₂²	—	218	2	no	—
110	C₂²	C₂×C₄×D₁₀₉	1	1	yes	characteristic
111	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
112	C₂²	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
113	C₂²	C₂×D₄₃₆	1	1	yes	characteristic
114	C₂²	Q₈×D₁₀₉	1	1	yes	characteristic
115	C₂²	#13182028	1	1	yes	characteristic
116	C₂²	#13182030	1	1	yes	characteristic
117	C₄	—	4	2	no	—
118	C₄	—	436	2	no	—
119	C₄	—	218	2	no	—
120	C₄	#13182006	1	4	yes	complemented
121	C₂	—	218	4	no	—
122	C₂	C₂×#13182006	1	1	yes	characteristic
123	C₂	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
124	C₂	#16812657	1	1	yes	characteristic
125	C₂	#16812670	1	1	yes	characteristic
126	C₂	#16812674	1	1	yes	characteristic
127	C₂	#16812675	1	1	yes	characteristic
128	C₂	#16812679	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×D₁₀₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#13182006

C₂

yes

maximal

#16812606

C₂

yes

maximal, complemented

C₂×C₄⋊C₄×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

—

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

C₄

yes

complemented

C₄⋊Dic₁₀₉

—

#13182006

—

C₂²×C₄×C₁₀₉

C₂²

yes

—

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₂³×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

C₂×C₄

yes

—

C₂²×D₁₀₉

D₄

yes

—

C₂²×D₁₀₉

Q₈

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₄

yes

—

C₂×C₄×C₁₀₉

D₄

yes

—

C₂×C₄×C₁₀₉

D₄

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×C₄

yes

—

C₄×D₁₀₉

—

C₂²×C₁₀₉

C₂²×C₄

yes

characteristic

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×C₁₀₉

C₂×Q₈

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₄⋊C₄

yes

complemented

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²⋊C₄

yes

complemented

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₁₀₉

C₂×C₄⋊C₄

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₂×C₁₀₉

C₂²⋊Q₈

yes

characteristic

D₁₀₉

—

C₁₀₉

#380

yes

characteristic, complemented, Sylow

#380

—

109

maximal, Sylow

C₂³×C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

D₁₀₉

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

109

—

#95

—

109

—

C₂⁴

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂×D₁₀₉

yes

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₂³

—

109

—

C₂³

—

109

—

C₂³

—

109

—

C₂³

—

218

—

C₂³

—

218

—

C₂³

—

218

—

C₂³

—

218

—

C₂³

C₂²×D₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

100

C₂×C₄

C₄×D₁₀₉

yes

—

101

C₂×C₄

D₄₃₆

yes

—

102

C₂×C₄

C₂²⋊D₁₀₉

yes

—

103

C₂²

—

109

—

104

C₂²

—

218

—

105

C₂²

—

218

—

106

C₂²

—

218

—

107

C₂²

—

218

—

108

C₂²

—

218

—

109

C₂²

—

218

—

110

C₂²

C₂×C₄×D₁₀₉

yes

characteristic

111

C₂²

D₄×D₁₀₉

yes

characteristic

112

C₂²

C₂×C₂²⋊D₁₀₉

yes

characteristic

113

C₂²

C₂×D₄₃₆

yes

characteristic

114

C₂²

Q₈×D₁₀₉

yes

characteristic

115

C₂²

#13182028

yes

characteristic

116

C₂²

#13182030

yes

characteristic

117

C₄

—

118

C₄

—

436

—

119

C₄

—

218

—

120

C₄

#13182006

yes

complemented

121

C₂

—

218

—

122

C₂

C₂×#13182006

yes

characteristic

123

C₂

C₄⋊C₄×D₁₀₉

yes

characteristic

124

C₂

#16812657

yes

characteristic

125

C₂

#16812670

yes

characteristic

126

C₂

#16812674

yes

characteristic

127

C₂

#16812675

yes

characteristic

128

C₂

#16812679

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₀₉
commutator	C₂²×C₁₀₉	C₂²×C₄
Fitting	C₂×C₄⋊C₄×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂³
hypercenter	C₂×C₄⋊C₄	D₁₀₉
hypocenter	C₁₀₉	#380
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₀₉

commutator

C₂²×C₁₀₉

C₂²×C₄

Fitting

C₂×C₄⋊C₄×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂³

hypercenter

C₂×C₄⋊C₄

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
2. C₂×C₄⋊Dic₁₀₉	C₂	semidirect
4. #16812606	C₂	semidirect
5. C₂×C₄⋊C₄×C₁₀₉	C₂	semidirect
11. C₂×C₄×D₁₀₉	C₄	semidirect
51. C₂×D₁₀₉	C₄⋊C₄	semidirect
53. C₄×C₁₀₉	C₂²⋊C₄	semidirect
64. C₁₀₉	#380	semidirect
68. C₂×C₄⋊C₄	D₁₀₉	semidirect
120. C₄	#13182006	semidirect

H ≅ G/N

type

2. C₂×C₄⋊Dic₁₀₉

C₂

semidirect

4. #16812606

C₂

semidirect

5. C₂×C₄⋊C₄×C₁₀₉

C₂

semidirect

11. C₂×C₄×D₁₀₉

C₄

semidirect

51. C₂×D₁₀₉

C₄⋊C₄

semidirect

53. C₄×C₁₀₉

C₂²⋊C₄

semidirect

64. C₁₀₉

#380

semidirect

68. C₂×C₄⋊C₄

D₁₀₉

semidirect

120. C₄

#13182006

semidirect

Subgroups — #20294883

Conjugacy classes of subgroups

Complements