Group #20295770 — subgroups

Subgroups — #20295770

Order 6976.

220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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	#16812691	C₂	1	1	yes	maximal, characteristic, complemented
2	#16812687	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812683	C₂	1	2	yes	maximal, complemented
4	#16812681	C₂	1	2	yes	maximal, complemented
5	#16812676	C₂	1	2	yes	maximal, complemented
6	#16812675	C₂	1	2	yes	maximal, complemented
7	#16812672	C₂	1	2	yes	maximal, complemented
8	#16812671	C₂	1	2	yes	maximal, complemented
9	#124×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₂₁₈	C₂²	1	1	yes	characteristic
11	C₂×D₄₃₆	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₁₀₉	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₁₀₉	C₂²	1	2	yes	—
14	C₂²×Dic₁₀₉	C₂²	1	2	yes	—
15	C₂×C₄×D₁₀₉	C₂²	1	2	yes	—
16	C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
17	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
19	#13182008	C₂²	1	2	yes	—
20	#13182008	C₂²	1	2	yes	—
21	#13182008	C₂²	1	2	yes	—
22	C₂²⋊Dic₁₀₉	C₂²	1	2	yes	—
23	#13182006	C₂²	1	2	yes	—
24	#13182006	C₂²	1	2	yes	—
25	#13182006	C₂²	1	2	yes	—
26	C₂×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₂×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
30	C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
31	C₂³×C₁₀₉	C₂³	1	2	yes	—
32	C₂²×D₁₀₉	C₂³	1	2	yes	—
33	Q₈×C₁₀₉	—	4	1	no	—
34	D₄×C₁₀₉	—	4	1	no	—
35	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	C₂³	1	2	yes	—
39	C₂²⋊D₁₀₉	—	4	2	no	—
40	C₂²⋊D₁₀₉	—	4	2	no	—
41	C₂×Dic₁₀₉	—	4	2	no	—
42	C₂×Dic₁₀₉	C₂³	1	2	yes	—
43	C₂×Dic₁₀₉	C₂³	1	2	yes	—
44	C₂×Dic₁₀₉	C₂³	1	2	yes	—
45	D₄₃₆	—	4	1	no	—
46	C₄×D₁₀₉	—	4	2	no	—
47	Dic₂₁₈	—	4	1	no	—
48	C₂²×C₁₀₉	—	2	2	no	—
49	C₂²×C₁₀₉	—	2	2	no	—
50	C₂²×C₁₀₉	—	2	2	no	—
51	C₂²×C₁₀₉	C₂⁴	1	1	yes	characteristic
52	C₂×D₁₀₉	—	2	2	no	—
53	C₂×D₁₀₉	—	2	2	no	—
54	C₂×D₁₀₉	—	2	2	no	—
55	C₄×C₁₀₉	—	2	1	no	—
56	C₄×C₁₀₉	—	2	2	no	—
57	C₄×C₁₀₉	—	2	1	no	—
58	C₄×C₁₀₉	—	2	1	no	—
59	Dic₁₀₉	—	2	2	no	—
60	Dic₁₀₉	—	2	2	no	—
61	Dic₁₀₉	—	2	2	no	—
62	C₂×C₁₀₉	—	4	2	no	—
63	C₂×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
66	D₁₀₉	—	4	2	no	—
67	C₁₀₉	#562	1	1	yes	characteristic, complemented, Sylow
68	#562	—	109	1	no	maximal, Sylow
69	#125	—	109	1	no	—
70	#124	—	109	1	no	—
71	#124	D₁₀₉	1	1	yes	characteristic, complemented
72	#123	—	109	2	no	—
73	#123	—	109	2	no	—
74	C₂²⋊Q₈	—	109	2	no	—
75	C₂²⋊Q₈	—	109	2	no	—
76	#121	—	109	2	no	—
77	#121	—	109	2	no	—
78	C₂×Q₈	—	109	1	no	—
79	C₂×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
80	C₂×D₄	—	109	2	no	—
81	C₂×D₄	—	109	2	no	—
82	C₂×D₄	—	109	1	no	—
83	C₂×D₄	C₂×D₁₀₉	1	1	yes	characteristic
84	C₂²×C₄	—	109	2	no	—
85	C₂²×C₄	—	109	2	no	—
86	C₄⋊C₄	—	109	2	no	—
87	C₄⋊C₄	—	109	2	no	—
88	C₄⋊C₄	—	109	2	no	—
89	C₄⋊C₄	—	109	2	no	—
90	C₄⋊C₄	—	109	1	no	—
91	C₄⋊C₄	—	109	1	no	—
92	C₂²⋊C₄	—	109	2	no	—
93	C₂²⋊C₄	—	109	2	no	—
94	C₂²⋊C₄	—	109	2	no	—
95	C₂²⋊C₄	—	109	2	no	—
96	C₂²⋊C₄	C₂×D₁₀₉	1	2	yes	—
97	C₂²⋊C₄	C₂×D₁₀₉	1	2	yes	—
98	C₄²	C₂×D₁₀₉	1	1	yes	characteristic
99	C₂³	—	109	2	no	—
100	C₂³	C₂²×D₁₀₉	1	2	yes	—
101	Q₈	—	4	1	no	—
102	Q₈	—	436	1	no	—
103	D₄	—	4	1	no	—
104	D₄	—	436	1	no	—
105	D₄	—	436	2	no	—
106	D₄	—	436	2	no	—
107	C₂×C₄	—	109	2	no	—
108	C₂×C₄	—	109	2	no	—
109	C₂×C₄	—	109	2	no	—
110	C₂×C₄	—	436	2	no	—
111	C₂×C₄	—	436	2	no	—
112	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
115	C₂×C₄	C₂²×D₁₀₉	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	218	2	no	—
120	C₂²	—	218	2	no	—
121	C₂²	—	218	2	no	—
122	C₂²	C₂³×D₁₀₉	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	218	2	no	—
128	C₄	—	218	2	no	—
129	C₄	—	218	2	no	—
130	C₂	—	4	2	no	—
131	C₂	—	436	2	no	—
132	C₂	#16812744	1	1	yes	characteristic
133	C₂	#16812745	1	1	yes	characteristic
134	C₂	#16812746	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16812691

C₂

yes

maximal, characteristic, complemented

#16812687

C₂

yes

maximal, characteristic, complemented

#16812683

C₂

yes

maximal, complemented

#16812681

C₂

yes

maximal, complemented

#16812676

C₂

yes

maximal, complemented

#16812675

C₂

yes

maximal, complemented

#16812672

C₂

yes

maximal, complemented

#16812671

C₂

yes

maximal, complemented

#124×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₈

C₂²

yes

characteristic

C₂×D₄₃₆

C₂²

yes

characteristic

C₂×C₂²⋊D₁₀₉

C₂²

yes

—

C₂×C₂²⋊D₁₀₉

C₂²

yes

—

C₂²×Dic₁₀₉

C₂²

yes

—

C₂×C₄×D₁₀₉

C₂²

yes

—

C₄⋊Dic₁₀₉

C₂²

yes

—

C₄⋊Dic₁₀₉

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

C₂²

yes

characteristic

#13182008

C₂²

yes

—

#13182008

C₂²

yes

—

#13182008

C₂²

yes

—

C₂²⋊Dic₁₀₉

C₂²

yes

—

#13182006

C₂²

yes

—

#13182006

C₂²

yes

—

#13182006

C₂²

yes

—

C₂×Q₈×C₁₀₉

C₂²

yes

characteristic

C₂×D₄×C₁₀₉

C₂²

yes

characteristic

C₂²⋊C₄×C₁₀₉

C₂²

yes

—

C₂²⋊C₄×C₁₀₉

C₂²

yes

—

C₄²×C₁₀₉

C₂²

yes

characteristic

C₂³×C₁₀₉

C₂³

yes

—

C₂²×D₁₀₉

C₂³

yes

—

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂²⋊D₁₀₉

—

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

D₄₃₆

—

C₄×D₁₀₉

—

Dic₂₁₈

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂⁴

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

2₋¹⁺⁴

yes

characteristic

D₁₀₉

—

C₁₀₉

#562

yes

characteristic, complemented, Sylow

#562

—

109

maximal, Sylow

#125

—

109

—

#124

—

109

—

#124

D₁₀₉

yes

characteristic, complemented

#123

—

109

—

#123

—

109

—

C₂²⋊Q₈

—

109

—

C₂²⋊Q₈

—

109

—

#121

—

109

—

#121

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

C₂×D₁₀₉

yes

characteristic

C₂×D₄

—

109

—

C₂×D₄

—

109

—

C₂×D₄

—

109

—

C₂×D₄

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

C₂×D₁₀₉

yes

—

C₂²⋊C₄

C₂×D₁₀₉

yes

—

C₄²

C₂×D₁₀₉

yes

characteristic

C₂³

—

109

—

100

C₂³

C₂²×D₁₀₉

yes

—

101

Q₈

—

102

Q₈

—

436

—

103

D₄

—

104

D₄

—

436

—

105

D₄

—

436

—

106

D₄

—

436

—

107

C₂×C₄

—

109

—

108

C₂×C₄

—

109

—

109

C₂×C₄

—

109

—

110

C₂×C₄

—

436

—

111

C₂×C₄

—

436

—

112

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

113

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

114

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

115

C₂×C₄

C₂²×D₁₀₉

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

218

—

120

C₂²

—

218

—

121

C₂²

—

218

—

122

C₂²

C₂³×D₁₀₉

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

218

—

128

C₄

—

218

—

129

C₄

—

218

—

130

C₂

—

131

C₂

—

436

—

132

C₂

#16812744

yes

characteristic

133

C₂

#16812745

yes

characteristic

134

C₂

#16812746

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂⁴
Fitting	#124×C₁₀₉	C₂
hypercenter	#124	D₁₀₉
hypocenter	C₁₀₉	#562
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂⁴

Fitting

#124×C₁₀₉

C₂

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #16812691	C₂	semidirect
2. #16812687	C₂	semidirect
3. #16812683	C₂	semidirect
4. #16812681	C₂	semidirect
5. #16812676	C₂	semidirect
6. #16812675	C₂	semidirect
7. #16812672	C₂	semidirect
8. #16812671	C₂	semidirect
9. #124×C₁₀₉	C₂	semidirect
67. C₁₀₉	#562	semidirect
71. #124	D₁₀₉	semidirect

H ≅ G/N

type

1. #16812691

C₂

semidirect

2. #16812687

C₂

semidirect

3. #16812683

C₂

semidirect

4. #16812681

C₂

semidirect

5. #16812676

C₂

semidirect

6. #16812675

C₂

semidirect

7. #16812672

C₂

semidirect

8. #16812671

C₂

semidirect

9. #124×C₁₀₉

C₂

semidirect

67. C₁₀₉

#562

semidirect

71. #124

D₁₀₉

semidirect

Subgroups — #20295770

Conjugacy classes of subgroups

Complements