Group #20295601 — subgroups

Subgroups — #20295601

Order 6976.

270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 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×#13182030	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812697	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812688	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812687	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812684	C₂	1	4	yes	maximal, complemented
7	#16812677	C₂	1	4	yes	maximal, complemented
8	#16812656	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#13182030	—	2	4	no	—
11	C₂×Dic₂₁₈	C₂²	1	2	yes	—
12	C₂×Dic₂₁₈	C₂²	1	1	yes	characteristic
13	Q₈×D₁₀₉	—	2	4	no	—
14	C₂×D₄₃₆	C₂²	1	2	yes	—
15	C₂×D₄₃₆	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₁₀₉	C₂²	1	2	yes	—
17	C₂×C₄×D₁₀₉	C₂²	1	2	yes	—
18	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
19	#13182008	C₂²	1	2	yes	—
20	#13182008	C₂²	1	4	yes	—
21	#13182006	C₂²	1	2	yes	—
22	#13182006	C₂²	1	4	yes	—
23	#13182006	C₂²	1	4	yes	—
24	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₂×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
28	C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
29	C₂²×D₁₀₉	C₂³	1	2	yes	—
30	C₂²×D₁₀₉	C₂³	1	1	yes	characteristic
31	Q₈×C₁₀₉	—	2	2	no	—
32	Q₈×C₁₀₉	—	2	2	no	—
33	C₂×C₄×C₁₀₉	C₂³	1	2	yes	—
34	C₂×C₄×C₁₀₉	C₂³	1	2	yes	—
35	C₂×C₄×C₁₀₉	C₂³	1	2	yes	—
36	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₂×Dic₁₀₉	C₂³	1	2	yes	—
38	C₂×Dic₁₀₉	C₂³	1	2	yes	—
39	C₂×Dic₁₀₉	C₂³	1	1	yes	characteristic
40	D₄₃₆	—	2	4	no	—
41	D₄₃₆	—	2	2	no	—
42	C₄×D₁₀₉	—	2	4	no	—
43	C₄×D₁₀₉	—	2	4	no	—
44	C₄×D₁₀₉	D₄	1	4	yes	—
45	Dic₂₁₈	—	2	2	no	—
46	Dic₂₁₈	—	2	4	no	—
47	C₂²×C₁₀₉	C₂⁴	1	1	yes	characteristic
48	C₂×D₁₀₉	—	2	2	no	—
49	C₂×D₁₀₉	—	2	2	no	—
50	C₂×D₁₀₉	—	2	4	no	—
51	C₂×D₁₀₉	C₂×D₄	1	2	yes	—
52	C₄×C₁₀₉	—	2	2	no	—
53	C₄×C₁₀₉	—	2	2	no	—
54	C₄×C₁₀₉	—	2	2	no	—
55	C₄×C₁₀₉	C₂×D₄	1	2	yes	—
56	Dic₁₀₉	—	2	2	no	—
57	Dic₁₀₉	—	2	2	no	—
58	Dic₁₀₉	C₂×D₄	1	2	yes	—
59	C₂×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
60	C₂×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
61	D₁₀₉	—	2	2	no	—
62	D₁₀₉	—	4	2	no	—
63	C₁₀₉	#536	1	1	yes	characteristic, complemented, Sylow
64	#536	—	109	1	no	maximal, Sylow
65	C₂×#41	—	109	1	no	—
66	C₂²×Q₈	—	109	1	no	—
67	C₄⋊Q₈	—	109	1	no	—
68	C₄⋊Q₈	D₁₀₉	1	1	yes	characteristic, complemented
69	#124	—	109	1	no	—
70	#124	—	109	1	no	—
71	#123	—	109	4	no	—
72	C₂²⋊Q₈	—	109	4	no	—
73	#117	—	109	1	no	—
74	#41	—	218	4	no	—
75	C₂×Q₈	—	109	2	no	—
76	C₂×Q₈	—	109	1	no	—
77	C₂×Q₈	—	218	4	no	—
78	C₂×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
79	C₂×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
80	C₂×D₄	—	109	2	no	—
81	C₂×D₄	—	109	1	no	—
82	C₂²×C₄	—	109	1	no	—
83	C₂²×C₄	—	109	2	no	—
84	C₂²×C₄	—	109	2	no	—
85	C₄⋊C₄	—	109	2	no	—
86	C₄⋊C₄	—	109	4	no	—
87	C₄⋊C₄	C₂×D₁₀₉	1	4	yes	—
88	C₂²⋊C₄	—	109	2	no	—
89	C₂²⋊C₄	—	109	4	no	—
90	C₂²⋊C₄	—	109	4	no	—
91	C₄²	—	109	1	no	—
92	C₄²	C₂×D₁₀₉	1	1	yes	characteristic
93	C₂³	—	109	1	no	—
94	C₂³	—	109	2	no	—
95	Q₈	—	2	2	no	—
96	Q₈	—	2	2	no	—
97	Q₈	—	218	4	no	—
98	Q₈	—	218	2	no	—
99	D₄	—	218	4	no	—
100	D₄	—	218	2	no	—
101	C₂×C₄	—	109	1	no	—
102	C₂×C₄	—	109	4	no	—
103	C₂×C₄	—	109	2	no	—
104	C₂×C₄	—	109	2	no	—
105	C₂×C₄	—	218	4	no	—
106	C₂×C₄	—	218	4	no	—
107	C₂×C₄	C₂²×D₁₀₉	1	2	yes	—
108	C₂×C₄	C₂²×D₁₀₉	1	2	yes	—
109	C₂×C₄	C₂²×D₁₀₉	1	2	yes	—
110	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
111	C₂²	—	109	2	no	—
112	C₂²	—	218	2	no	—
113	C₂²	—	218	4	no	—
114	C₂²	—	218	2	no	—
115	C₂²	C₂³×D₁₀₉	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	109	2	no	—
120	C₄	—	218	2	no	—
121	C₄	—	218	2	no	—
122	C₄	D₄×D₁₀₉	1	2	yes	—
123	C₂	—	436	2	no	—
124	C₂	—	218	2	no	—
125	C₂	C₂×D₄×D₁₀₉	1	1	yes	characteristic
126	C₂	#16812747	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182030

C₂

yes

maximal, characteristic

C₂×Q₈×D₁₀₉

C₂

yes

maximal, characteristic, complemented

#16812697

C₂

yes

maximal, characteristic, complemented

#16812688

C₂

yes

maximal, characteristic, complemented

#16812687

C₂

yes

maximal, characteristic, complemented

#16812684

C₂

yes

maximal, complemented

#16812677

C₂

yes

maximal, complemented

#16812656

C₂

yes

maximal, characteristic, complemented

C₄⋊Q₈×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#13182030

—

C₂×Dic₂₁₈

C₂²

yes

—

C₂×Dic₂₁₈

C₂²

yes

characteristic

Q₈×D₁₀₉

—

C₂×D₄₃₆

C₂²

yes

—

C₂×D₄₃₆

C₂²

yes

characteristic

C₂×C₄×D₁₀₉

C₂²

yes

—

C₂×C₄×D₁₀₉

C₂²

yes

—

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

#13182008

C₂²

yes

—

#13182008

C₂²

yes

—

#13182006

C₂²

yes

—

#13182006

C₂²

yes

—

#13182006

C₂²

yes

—

C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×Q₈×C₁₀₉

C₂²

yes

characteristic

C₂×Q₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₄²×C₁₀₉

C₂²

yes

characteristic

C₂²×D₁₀₉

C₂³

yes

—

C₂²×D₁₀₉

C₂³

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

characteristic

D₄₃₆

—

D₄₃₆

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

D₄

yes

—

Dic₂₁₈

—

Dic₂₁₈

—

C₂²×C₁₀₉

C₂⁴

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂×D₄

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×D₄

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂×D₄

yes

—

C₂×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×C₁₀₉

2₋¹⁺⁴

yes

—

D₁₀₉

—

D₁₀₉

—

C₁₀₉

#536

yes

characteristic, complemented, Sylow

#536

—

109

maximal, Sylow

C₂×#41

—

109

—

C₂²×Q₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

D₁₀₉

yes

characteristic, complemented

#124

—

109

—

#124

—

109

—

#123

—

109

—

C₂²⋊Q₈

—

109

—

#117

—

109

—

#41

—

218

—

C₂×Q₈

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂×Q₈

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

C₂×D₁₀₉

yes

characteristic

C₂×D₄

—

109

—

C₂×D₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₄²

—

109

—

C₄²

C₂×D₁₀₉

yes

characteristic

C₂³

—

109

—

C₂³

—

109

—

Q₈

—

Q₈

—

Q₈

—

218

—

Q₈

—

218

—

D₄

—

218

—

100

D₄

—

218

—

101

C₂×C₄

—

109

—

102

C₂×C₄

—

109

—

103

C₂×C₄

—

109

—

104

C₂×C₄

—

109

—

105

C₂×C₄

—

218

—

106

C₂×C₄

—

218

—

107

C₂×C₄

C₂²×D₁₀₉

yes

—

108

C₂×C₄

C₂²×D₁₀₉

yes

—

109

C₂×C₄

C₂²×D₁₀₉

yes

—

110

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

111

C₂²

—

109

—

112

C₂²

—

218

—

113

C₂²

—

218

—

114

C₂²

—

218

—

115

C₂²

C₂³×D₁₀₉

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

109

—

120

C₄

—

218

—

121

C₄

—

218

—

122

C₄

D₄×D₁₀₉

yes

—

123

C₂

—

436

—

124

C₂

—

218

—

125

C₂

C₂×D₄×D₁₀₉

yes

characteristic

126

C₂

#16812747

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂⁴
Fitting	C₄⋊Q₈×C₁₀₉	C₂
hypercenter	C₄⋊Q₈	D₁₀₉
hypocenter	C₁₀₉	#536
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂⁴

Fitting

C₄⋊Q₈×C₁₀₉

C₂

hypercenter

C₄⋊Q₈

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. C₂×Q₈×D₁₀₉	C₂	semidirect
3. #16812697	C₂	semidirect
4. #16812688	C₂	semidirect
5. #16812687	C₂	semidirect
6. #16812684	C₂	semidirect
7. #16812677	C₂	semidirect
8. #16812656	C₂	semidirect
9. C₄⋊Q₈×C₁₀₉	C₂	semidirect
63. C₁₀₉	#536	semidirect
68. C₄⋊Q₈	D₁₀₉	semidirect

H ≅ G/N

type

2. C₂×Q₈×D₁₀₉

C₂

semidirect

3. #16812697

C₂

semidirect

4. #16812688

C₂

semidirect

5. #16812687

C₂

semidirect

6. #16812684

C₂

semidirect

7. #16812677

C₂

semidirect

8. #16812656

C₂

semidirect

9. C₄⋊Q₈×C₁₀₉

C₂

semidirect

63. C₁₀₉

#536

semidirect

68. C₄⋊Q₈

D₁₀₉

semidirect

Subgroups — #20295601

Conjugacy classes of subgroups

Complements