Group #20295522 — subgroups

Subgroups — C₂×#16812676 #20295522

Order 6976.

322 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 135 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×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#16812676	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13182008	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13182008	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂²⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂²⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₂₁₈	—	2	4	no	—
10	C₂×Dic₂₁₈	C₂²	1	4	yes	complemented
11	C₂²×Dic₁₀₉	—	2	4	no	—
12	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₀₉	C₂²	1	4	yes	—
16	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₁₀₉	C₂²	1	4	yes	complemented
18	#13182008	C₂²	1	4	yes	complemented
19	#13182008	C₂²	1	4	yes	complemented
20	C₂²⋊Dic₁₀₉	C₂²	1	4	yes	—
21	C₂⁴×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	—
25	C₂³×C₁₀₉	—	2	2	no	—
26	C₂³×C₁₀₉	—	2	2	no	—
27	C₂³×C₁₀₉	Q₈	1	4	yes	complemented
28	C₂³×C₁₀₉	C₂³	1	2	yes	—
29	C₂³×C₁₀₉	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	—	2	2	no	—
31	C₂×C₄×C₁₀₉	—	2	2	no	—
32	C₂×C₄×C₁₀₉	C₂³	1	2	yes	—
33	C₂×C₄×C₁₀₉	C₂³	1	2	yes	—
34	C₂×Dic₁₀₉	—	2	2	no	—
35	C₂×Dic₁₀₉	—	2	2	no	—
36	C₂×Dic₁₀₉	—	2	2	no	—
37	C₂×Dic₁₀₉	—	2	8	no	—
38	C₂×Dic₁₀₉	D₄	1	4	yes	—
39	C₂×Dic₁₀₉	C₂³	1	2	yes	—
40	C₂×Dic₁₀₉	C₂³	1	2	yes	—
41	C₂×Dic₁₀₉	C₂³	1	2	yes	—
42	C₂×Dic₁₀₉	C₂³	1	2	yes	—
43	Dic₂₁₈	—	2	8	no	—
44	C₂²×C₁₀₉	—	2	8	no	—
45	C₂²×C₁₀₉	—	2	2	no	—
46	C₂²×C₁₀₉	—	2	2	no	—
47	C₂²×C₁₀₉	C₂×D₄	1	2	yes	—
48	C₂²×C₁₀₉	C₂×Q₈	1	2	yes	—
49	C₂²×C₁₀₉	C₂×Q₈	1	4	yes	complemented
50	C₂²×C₁₀₉	#41	1	2	yes	—
51	C₂²×C₁₀₉	C₂⁴	1	1	yes	characteristic
52	C₄×C₁₀₉	—	2	2	no	—
53	C₄×C₁₀₉	—	2	2	no	—
54	Dic₁₀₉	—	2	2	no	—
55	Dic₁₀₉	—	2	2	no	—
56	Dic₁₀₉	—	2	2	no	—
57	Dic₁₀₉	C₂×D₄	1	4	yes	—
58	C₂×C₁₀₉	—	2	4	no	—
59	C₂×C₁₀₉	C₂²⋊Q₈	1	4	yes	complemented
60	C₂×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
61	C₂×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
62	C₂×C₁₀₉	C₂×#41	1	1	yes	characteristic
63	C₁₀₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₂²⋊Q₈	—	109	1	no	maximal, Sylow
65	C₂²×Q₈	—	109	1	no	—
66	C₂³×C₄	—	109	1	no	—
67	C₂²⋊Q₈	—	109	8	no	—
68	C₂×C₄⋊C₄	—	109	1	no	—
69	C₂×C₄⋊C₄	—	109	1	no	—
70	C₂×C₄⋊C₄	—	109	1	no	—
71	C₂×C₂²⋊C₄	—	109	1	no	—
72	C₂×C₂²⋊C₄	D₁₀₉	1	1	yes	characteristic
73	C₂⁴	C₂×D₁₀₉	1	1	yes	characteristic
74	C₂×Q₈	—	109	4	no	—
75	C₂×Q₈	—	218	4	no	—
76	C₂²×C₄	—	109	1	no	—
77	C₂²×C₄	—	109	4	no	—
78	C₂²×C₄	—	109	1	no	—
79	C₂²×C₄	—	109	1	no	—
80	C₂²×C₄	—	109	1	no	—
81	C₂²×C₄	—	218	4	no	—
82	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
83	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
84	C₄⋊C₄	—	109	4	no	—
85	C₄⋊C₄	—	109	4	no	—
86	C₄⋊C₄	—	109	4	no	—
87	C₂²⋊C₄	—	109	4	no	—
88	C₂²⋊C₄	C₂×D₁₀₉	1	4	yes	—
89	C₂³	—	2	2	no	—
90	C₂³	—	2	2	no	—
91	C₂³	Dic₂₁₈	1	4	yes	complemented
92	C₂³	C₂²×D₁₀₉	1	2	yes	—
93	C₂³	C₂²×D₁₀₉	1	1	yes	characteristic
94	Q₈	—	218	8	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	109	4	no	—
98	C₂×C₄	—	109	2	no	—
99	C₂×C₄	—	109	2	no	—
100	C₂×C₄	—	109	2	no	—
101	C₂×C₄	—	109	2	no	—
102	C₂×C₄	—	218	8	no	—
103	C₂×C₄	—	218	2	no	—
104	C₂×C₄	—	218	2	no	—
105	C₂×C₄	—	218	2	no	—
106	C₂×C₄	C₂²×D₁₀₉	1	2	yes	—
107	C₂×C₄	C₂²×D₁₀₉	1	2	yes	—
108	C₂²	—	2	8	no	—
109	C₂²	—	2	2	no	—
110	C₂²	—	2	2	no	—
111	C₂²	D₄×D₁₀₉	1	2	yes	—
112	C₂²	C₂×Dic₂₁₈	1	2	yes	—
113	C₂²	C₂×Dic₂₁₈	1	4	yes	complemented
114	C₂²	#13182028	1	2	yes	—
115	C₂²	C₂³×D₁₀₉	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	109	4	no	—
119	C₄	—	218	2	no	—
120	C₄	—	218	2	no	—
121	C₄	—	218	2	no	—
122	C₂	—	2	4	no	—
123	C₂	#16812676	1	4	yes	direct product factor
124	C₂	C₂×D₄×D₁₀₉	1	1	yes	characteristic
125	C₂	C₂²×Dic₂₁₈	1	1	yes	characteristic
126	C₂	C₂×#13182028	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₁₈

C₂

yes

maximal, characteristic, complemented

C₂³×Dic₁₀₉

C₂

yes

maximal, characteristic

#16812676

C₂

yes

maximal, direct product factor

C₂×C₄⋊Dic₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#13182008

C₂

yes

maximal, characteristic, complemented

C₂×#13182008

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊Dic₁₀₉

C₂

yes

maximal, characteristic

C₂×C₂²⋊C₄×C₁₀₉

C₂

yes

maximal, characteristic

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂²

yes

complemented

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

—

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

C₂²

yes

complemented

#13182008

C₂²

yes

complemented

#13182008

C₂²

yes

complemented

C₂²⋊Dic₁₀₉

C₂²

yes

—

C₂⁴×C₁₀₉

C₂²

yes

characteristic

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₂²⋊C₄×C₁₀₉

C₂²

yes

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

Q₈

yes

complemented

C₂³×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂³

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₄

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

Dic₂₁₈

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×D₄

yes

—

C₂²×C₁₀₉

C₂×Q₈

yes

—

C₂²×C₁₀₉

C₂×Q₈

yes

complemented

C₂²×C₁₀₉

#41

yes

—

C₂²×C₁₀₉

C₂⁴

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂×D₄

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂²⋊Q₈

yes

complemented

C₂×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×C₁₀₉

C₂²×Q₈

yes

characteristic

C₂×C₁₀₉

C₂×#41

yes

characteristic

C₁₀₉

C₂×C₂²⋊Q₈

yes

characteristic, complemented, Sylow

C₂×C₂²⋊Q₈

—

109

maximal, Sylow

C₂²×Q₈

—

109

—

C₂³×C₄

—

109

—

C₂²⋊Q₈

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₂²⋊C₄

D₁₀₉

yes

characteristic

C₂⁴

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

C₂×D₁₀₉

yes

—

C₂³

—

C₂³

—

C₂³

Dic₂₁₈

yes

complemented

C₂³

C₂²×D₁₀₉

yes

—

C₂³

C₂²×D₁₀₉

yes

characteristic

Q₈

—

218

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

100

C₂×C₄

—

109

—

101

C₂×C₄

—

109

—

102

C₂×C₄

—

218

—

103

C₂×C₄

—

218

—

104

C₂×C₄

—

218

—

105

C₂×C₄

—

218

—

106

C₂×C₄

C₂²×D₁₀₉

yes

—

107

C₂×C₄

C₂²×D₁₀₉

yes

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

D₄×D₁₀₉

yes

—

112

C₂²

C₂×Dic₂₁₈

yes

—

113

C₂²

C₂×Dic₂₁₈

yes

complemented

114

C₂²

#13182028

yes

—

115

C₂²

C₂³×D₁₀₉

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

109

—

119

C₄

—

218

—

120

C₄

—

218

—

121

C₄

—

218

—

122

C₂

—

123

C₂

#16812676

yes

direct product factor

124

C₂

C₂×D₄×D₁₀₉

yes

characteristic

125

C₂

C₂²×Dic₂₁₈

yes

characteristic

126

C₂

C₂×#13182028

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₀₉
commutator	C₂²×C₁₀₉	C₂⁴
Frattini	C₂²	C₂³×D₁₀₉
Fitting	C₂×C₂²⋊C₄×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂³
hypercenter	C₂×C₂²⋊C₄	D₁₀₉
hypocenter	C₁₀₉	C₂×C₂²⋊Q₈
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

subgroup

G/H

center

C₂³

C₂²×D₁₀₉

commutator

C₂²×C₁₀₉

C₂⁴

Frattini

C₂²

C₂³×D₁₀₉

Fitting

C₂×C₂²⋊C₄×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂³

hypercenter

C₂×C₂²⋊C₄

D₁₀₉

hypocenter

C₁₀₉

C₂×C₂²⋊Q₈

LCS

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

N	H ≅ G/N	type
1. C₂²×Dic₂₁₈	C₂	semidirect
3. #16812676	C₂	direct (N × H)
4. C₂×C₄⋊Dic₁₀₉	C₂	semidirect
5. C₂×#13182008	C₂	semidirect
6. C₂×#13182008	C₂	semidirect
10. C₂×Dic₂₁₈	C₂²	semidirect
17. C₄⋊Dic₁₀₉	C₂²	semidirect
18. #13182008	C₂²	semidirect
19. #13182008	C₂²	semidirect
27. C₂³×C₁₀₉	Q₈	semidirect
49. C₂²×C₁₀₉	C₂×Q₈	semidirect
59. C₂×C₁₀₉	C₂²⋊Q₈	semidirect
63. C₁₀₉	C₂×C₂²⋊Q₈	semidirect
91. C₂³	Dic₂₁₈	semidirect
113. C₂²	C₂×Dic₂₁₈	semidirect
123. C₂	#16812676	direct (N × H)

H ≅ G/N

type

1. C₂²×Dic₂₁₈

C₂

semidirect

3. #16812676

C₂

direct (N × H)

4. C₂×C₄⋊Dic₁₀₉

C₂

semidirect

5. C₂×#13182008

C₂

semidirect

6. C₂×#13182008

C₂

semidirect

10. C₂×Dic₂₁₈

C₂²

semidirect

17. C₄⋊Dic₁₀₉

C₂²

semidirect

18. #13182008

C₂²

semidirect

19. #13182008

C₂²

semidirect

27. C₂³×C₁₀₉

Q₈

semidirect

49. C₂²×C₁₀₉

C₂×Q₈

semidirect

59. C₂×C₁₀₉

C₂²⋊Q₈

semidirect

63. C₁₀₉

C₂×C₂²⋊Q₈

semidirect

91. C₂³

Dic₂₁₈

semidirect

113. C₂²

C₂×Dic₂₁₈

semidirect

123. C₂

#16812676

direct (N × H)

Subgroups — C₂×#16812676 #20295522

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16812676 #20295522

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16812676 #20295522