Group #20294913 — subgroups

Subgroups — #20294913

Order 6976.

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

normal subgroups only

This group has 154 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	C₂×C₄⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#13182008	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#16812606	C₂	1	1	yes	maximal, characteristic
5	#16812606	C₂	1	1	yes	maximal, characteristic
6	#16812606	C₂	1	1	yes	maximal, characteristic
7	#95×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
10	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
11	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₀₉	—	2	2	no	—
13	#13182008	C₄	1	4	yes	complemented
14	C₄×Dic₁₀₉	—	2	2	no	—
15	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂³×C₁₀₉	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₁₀₉	—	2	1	no	—
20	C₂×C₄×C₁₀₉	—	2	1	no	—
21	C₂×C₄×C₁₀₉	—	2	1	no	—
22	C₂×C₄×C₁₀₉	—	2	1	no	—
23	C₂×C₄×C₁₀₉	—	2	1	no	—
24	C₂×C₄×C₁₀₉	—	2	1	no	—
25	C₂×C₄×C₁₀₉	—	2	1	no	—
26	C₂×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₁₀₉	Q₈	1	2	yes	—
28	C₂×Dic₁₀₉	—	2	1	no	—
29	C₂×Dic₁₀₉	—	2	1	no	—
30	C₂×Dic₁₀₉	—	2	1	no	—
31	C₂×Dic₁₀₉	—	2	1	no	—
32	C₂×Dic₁₀₉	—	2	1	no	—
33	C₂×Dic₁₀₉	—	2	1	no	—
34	C₂×Dic₁₀₉	—	2	1	no	—
35	C₂×Dic₁₀₉	—	2	1	no	—
36	C₂×Dic₁₀₉	—	2	1	no	—
37	C₂×Dic₁₀₉	C₂×C₄	1	2	yes	—
38	C₂×Dic₁₀₉	C₂×C₄	1	2	yes	—
39	C₂×Dic₁₀₉	D₄	1	2	yes	—
40	C₂²×C₁₀₉	C₂²×C₄	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	C₂²×C₁₀₉	#41	1	1	yes	characteristic
46	C₂²×C₁₀₉	#41	1	1	yes	characteristic
47	C₄×C₁₀₉	—	2	2	no	—
48	C₄×C₁₀₉	—	2	2	no	—
49	C₄×C₁₀₉	—	4	1	no	—
50	Dic₁₀₉	—	2	2	no	—
51	Dic₁₀₉	—	2	2	no	—
52	Dic₁₀₉	—	2	2	no	—
53	Dic₁₀₉	—	4	1	no	—
54	C₂×C₁₀₉	#117	1	1	yes	characteristic
55	C₂×C₁₀₉	C₄×D₄	1	1	yes	characteristic
56	C₂×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
57	C₂×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
58	C₂×C₁₀₉	#123	1	1	yes	characteristic
59	C₂×C₁₀₉	#125	1	1	yes	characteristic
60	C₂×C₁₀₉	#126	1	1	yes	characteristic
61	C₁₀₉	#387	1	1	yes	characteristic, complemented, Sylow
62	#387	—	109	1	no	maximal, Sylow
63	C₂×C₄⋊C₄	—	109	1	no	—
64	C₂×C₄⋊C₄	—	109	1	no	—
65	C₂×C₄²	—	109	1	no	—
66	#95	—	109	1	no	—
67	#95	—	109	1	no	—
68	#95	—	109	1	no	—
69	#95	D₁₀₉	1	1	yes	characteristic
70	C₂²×C₄	—	109	1	no	—
71	C₂²×C₄	—	109	1	no	—
72	C₂²×C₄	—	109	1	no	—
73	C₂²×C₄	—	109	1	no	—
74	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
75	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
76	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
77	C₄⋊C₄	—	109	4	no	—
78	C₄⋊C₄	—	218	2	no	—
79	C₄²	—	218	2	no	—
80	C₂³	C₂²×D₁₀₉	1	1	yes	characteristic
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	109	2	no	—
89	C₂×C₄	—	109	2	no	—
90	C₂×C₄	—	109	2	no	—
91	C₂×C₄	—	218	1	no	—
92	C₂×C₄	—	218	1	no	—
93	C₂×C₄	—	218	1	no	—
94	C₂×C₄	—	218	1	no	—
95	C₂×C₄	—	218	1	no	—
96	C₂×C₄	—	218	1	no	—
97	C₂×C₄	—	218	1	no	—
98	C₂×C₄	—	218	1	no	—
99	C₂×C₄	—	218	1	no	—
100	C₂×C₄	Dic₂₁₈	1	2	yes	—
101	C₂×C₄	C₄×D₁₀₉	1	2	yes	—
102	C₂²	C₂×C₄×D₁₀₉	1	1	yes	characteristic
103	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
104	C₂²	C₂×Dic₂₁₈	1	1	yes	characteristic
105	C₂²	#13182028	1	1	yes	characteristic
106	C₂²	#13182028	1	1	yes	characteristic
107	C₂²	#13182029	1	1	yes	characteristic
108	C₂²	#13182030	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	2	2	no	—
111	C₄	—	4	1	no	—
112	C₄	—	436	1	no	—
113	C₄	—	218	2	no	—
114	C₄	—	218	2	no	—
115	C₄	—	218	2	no	—
116	C₂	#16812657	1	1	yes	characteristic
117	C₂	#16812659	1	1	yes	characteristic
118	C₂	C₄×Dic₂₁₈	1	1	yes	characteristic
119	C₂	#16812676	1	1	yes	characteristic
120	C₂	#16812684	1	1	yes	characteristic
121	C₂	#16812690	1	1	yes	characteristic
122	C₂	#16812692	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Dic₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182008

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₀₉

C₂

yes

maximal, characteristic

#16812606

C₂

yes

maximal, characteristic

#16812606

C₂

yes

maximal, characteristic

#16812606

C₂

yes

maximal, characteristic

#95×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

—

#13182008

C₄

yes

complemented

C₄×Dic₁₀₉

—

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₂³×C₁₀₉

C₂³

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₄

yes

—

C₂×C₄×C₁₀₉

Q₈

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×C₄

yes

—

C₂×Dic₁₀₉

C₂×C₄

yes

—

C₂×Dic₁₀₉

D₄

yes

—

C₂²×C₁₀₉

C₂²×C₄

yes

characteristic

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×C₁₀₉

C₂×Q₈

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#117

yes

characteristic

C₂×C₁₀₉

C₄×D₄

yes

characteristic

C₂×C₁₀₉

C₄×Q₈

yes

characteristic

C₂×C₁₀₉

C₂²⋊Q₈

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₁₀₉

yes

characteristic, complemented, Sylow

#387

—

109

maximal, Sylow

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄²

—

109

—

#95

—

109

—

#95

—

109

—

#95

—

109

—

#95

D₁₀₉

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄²

—

218

—

C₂³

C₂²×D₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

100

C₂×C₄

Dic₂₁₈

yes

—

101

C₂×C₄

C₄×D₁₀₉

yes

—

102

C₂²

C₂×C₄×D₁₀₉

yes

characteristic

103

C₂²

D₄×D₁₀₉

yes

characteristic

104

C₂²

C₂×Dic₂₁₈

yes

characteristic

105

C₂²

#13182028

yes

characteristic

106

C₂²

#13182028

yes

characteristic

107

C₂²

#13182029

yes

characteristic

108

C₂²

#13182030

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

436

—

113

C₄

—

218

—

114

C₄

—

218

—

115

C₄

—

218

—

116

C₂

#16812657

yes

characteristic

117

C₂

#16812659

yes

characteristic

118

C₂

C₄×Dic₂₁₈

yes

characteristic

119

C₂

#16812676

yes

characteristic

120

C₂

#16812684

yes

characteristic

121

C₂

#16812690

yes

characteristic

122

C₂

#16812692

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₀₉
commutator	C₂²×C₁₀₉	C₂²×C₄
Fitting	#95×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂³
hypercenter	#95	D₁₀₉
hypocenter	C₁₀₉	#387
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ #95

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₀₉

commutator

C₂²×C₁₀₉

C₂²×C₄

Fitting

#95×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
13. #13182008	C₄	semidirect
61. C₁₀₉	#387	semidirect

H ≅ G/N

type

13. #13182008

C₄

semidirect

61. C₁₀₉

#387

semidirect

Subgroups — #20294913

Conjugacy classes of subgroups

Complements