Group #28376434 — subgroups

Subgroups — #28376434

Order 13952.

996 conjugacy classes of subgroups; only the 509 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 509 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	#20295655	C₂	1	12	yes	maximal, complemented
2	#20295652	C₂	1	3	yes	maximal
3	#20295578	C₂	1	12	yes	maximal, complemented
4	C₂×#16812655	C₂	1	3	yes	maximal
5	#543×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₁₀₉	C₂²	1	1	yes	characteristic
7	#16812698	C₂²	1	12	yes	complemented
8	#16812697	C₂²	1	12	yes	complemented
9	#16812689	C₂²	1	24	yes	complemented
10	#16812678	C₂²	1	12	yes	—
11	#16812676	C₂²	1	24	yes	—
12	Q₈×Dic₁₀₉	C₂²	1	4	yes	complemented
13	#16812664	C₂²	1	12	yes	complemented
14	#16812655	C₂²	1	12	yes	—
15	#16812655	C₂²	1	12	yes	—
16	C₂×#13182008	C₂²	1	6	yes	—
17	C₂×C₂²⋊Dic₁₀₉	C₂²	1	3	yes	—
18	C₂×C₄×Dic₁₀₉	C₂²	1	6	yes	—
19	C₂²⋊Q₈×C₁₀₉	C₂²	1	12	yes	—
20	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	3	yes	—
21	C₂×Dic₂₁₈	C₂³	1	12	yes	—
22	C₂²×Dic₁₀₉	Q₈	1	8	yes	complemented
23	C₂²×Dic₁₀₉	C₂³	1	6	yes	—
24	C₂²×Dic₁₀₉	C₂³	1	3	yes	—
25	C₂²×Dic₁₀₉	C₂³	1	3	yes	—
26	C₄⋊Dic₁₀₉	C₂³	1	12	yes	—
27	#13182008	C₂³	1	24	yes	—
28	#13182008	C₂³	1	24	yes	—
29	C₂²⋊Dic₁₀₉	C₂³	1	12	yes	—
30	C₄×Dic₁₀₉	C₂³	1	12	yes	—
31	C₄×Dic₁₀₉	C₂³	1	12	yes	—
32	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₁₀₉	C₂³	1	4	yes	—
34	C₂²×C₄×C₁₀₉	C₂³	1	6	yes	—
35	C₄⋊C₄×C₁₀₉	C₂³	1	12	yes	—
36	C₂²⋊C₄×C₁₀₉	C₂³	1	12	yes	—
37	C₂³×C₁₀₉	—	4	2	no	—
38	C₂³×C₁₀₉	C₂×Q₈	1	4	yes	—
39	C₂³×C₁₀₉	C₂⁴	1	3	yes	—
40	Q₈×C₁₀₉	—	4	4	no	—
41	C₂×C₄×C₁₀₉	—	4	6	no	—
42	C₂×C₄×C₁₀₉	C₂⁴	1	12	yes	—
43	C₂×Dic₁₀₉	—	4	6	no	—
44	C₂×Dic₁₀₉	C₂×Q₈	1	12	yes	—
45	C₂×Dic₁₀₉	C₂×Q₈	1	12	yes	—
46	C₂×Dic₁₀₉	C₂⁴	1	12	yes	—
47	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₁₀₉	C₂⁴	1	3	yes	—
49	Dic₂₁₈	—	4	12	no	—
50	C₂²×C₁₀₉	—	2	6	no	—
51	C₂²×C₁₀₉	—	4	4	no	—
52	C₂²×C₁₀₉	C₂²×Q₈	1	6	yes	—
53	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
54	C₄×C₁₀₉	—	2	12	no	—
55	Dic₁₀₉	—	2	12	no	—
56	Dic₁₀₉	C₂²×Q₈	1	2	yes	—
57	Dic₁₀₉	C₂²×Q₈	1	6	yes	—
58	C₂×C₁₀₉	—	2	6	no	—
59	C₂×C₁₀₉	C₂³×Q₈	1	1	yes	characteristic
60	C₂×C₁₀₉	#585	1	2	yes	—
61	#3458	—	109	1	no	maximal, Sylow
62	C₁₀₉	#3458	1	1	yes	characteristic, complemented, Sylow
63	#544	—	109	12	no	—
64	#543	—	109	3	no	—
65	#543	D₁₀₉	1	1	yes	characteristic
66	#533	—	109	12	no	—
67	C₂×#117	—	109	3	no	—
68	C₂³×C₄	—	109	1	no	—
69	C₄⋊Q₈	—	109	12	no	—
70	C₄⋊Q₈	—	109	12	no	—
71	#125	—	109	24	no	—
72	C₂²⋊Q₈	—	109	12	no	—
73	C₂²⋊Q₈	—	109	24	no	—
74	C₂²⋊Q₈	C₂×D₁₀₉	1	12	yes	—
75	C₄×Q₈	—	109	4	no	—
76	C₄×Q₈	—	109	12	no	—
77	#117	—	109	12	no	—
78	#117	—	109	12	no	—
79	C₂×C₄⋊C₄	—	109	6	no	—
80	C₂×C₂²⋊C₄	—	109	3	no	—
81	C₂×C₂²⋊C₄	C₂×D₁₀₉	1	3	yes	—
82	C₂×C₄²	—	109	6	no	—
83	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
84	C₂×Q₈	—	109	12	no	—
85	C₂×Q₈	C₂²×D₁₀₉	1	4	yes	—
86	C₂²×C₄	—	109	8	no	—
87	C₂²×C₄	—	109	3	no	—
88	C₂²×C₄	—	109	3	no	—
89	C₂²×C₄	—	109	6	no	—
90	C₂²×C₄	C₂²×D₁₀₉	1	6	yes	—
91	C₄⋊C₄	—	109	24	no	—
92	C₄⋊C₄	—	109	24	no	—
93	C₄⋊C₄	—	109	12	no	—
94	C₄⋊C₄	C₂²×D₁₀₉	1	12	yes	—
95	C₂²⋊C₄	—	109	12	no	—
96	C₂²⋊C₄	C₂²×D₁₀₉	1	12	yes	—
97	C₄²	—	109	12	no	—
98	C₄²	—	109	12	no	—
99	C₂³	—	4	2	no	—
100	C₂³	Q₈×D₁₀₉	1	4	yes	—
101	C₂³	C₂³×D₁₀₉	1	3	yes	—
102	Q₈	—	4	4	no	—
103	Q₈	—	436	12	no	—
104	C₂×C₄	—	4	6	no	—
105	C₂×C₄	—	109	12	no	—
106	C₂×C₄	—	109	12	no	—
107	C₂×C₄	—	109	1	no	—
108	C₂×C₄	—	109	3	no	—
109	C₂×C₄	—	109	12	no	—
110	C₂×C₄	—	436	6	no	—
111	C₂×C₄	C₂³×D₁₀₉	1	12	yes	—
112	C₂²	—	2	6	no	—
113	C₂²	—	4	4	no	—
114	C₂²	C₂×Q₈×D₁₀₉	1	6	yes	—
115	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
116	C₄	—	2	12	no	—
117	C₄	—	109	2	no	—
118	C₄	—	109	6	no	—
119	C₄	—	218	12	no	—
120	C₂	—	2	6	no	—
121	C₂	C₂²×Q₈×D₁₀₉	1	1	yes	characteristic
122	C₂	#20295861	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295655

C₂

yes

maximal, complemented

#20295652

C₂

yes

maximal

#20295578

C₂

yes

maximal, complemented

C₂×#16812655

C₂

yes

maximal

#543×C₁₀₉

C₂

yes

maximal, characteristic

C₂³×Dic₁₀₉

C₂²

yes

characteristic

#16812698

C₂²

yes

complemented

#16812697

C₂²

yes

complemented

#16812689

C₂²

yes

complemented

#16812678

C₂²

yes

—

#16812676

C₂²

yes

—

Q₈×Dic₁₀₉

C₂²

yes

complemented

#16812664

C₂²

yes

complemented

#16812655

C₂²

yes

—

#16812655

C₂²

yes

—

C₂×#13182008

C₂²

yes

—

C₂×C₂²⋊Dic₁₀₉

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

C₂²⋊Q₈×C₁₀₉

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₀₉

C₂²

yes

—

C₂×Dic₂₁₈

C₂³

yes

—

C₂²×Dic₁₀₉

Q₈

yes

complemented

C₂²×Dic₁₀₉

C₂³

yes

—

C₂²×Dic₁₀₉

C₂³

yes

—

C₂²×Dic₁₀₉

C₂³

yes

—

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182008

C₂³

yes

—

#13182008

C₂³

yes

—

C₂²⋊Dic₁₀₉

C₂³

yes

—

C₄×Dic₁₀₉

C₂³

yes

—

C₄×Dic₁₀₉

C₂³

yes

—

C₂⁴×C₁₀₉

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₂²⋊C₄×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂×Q₈

yes

—

C₂³×C₁₀₉

C₂⁴

yes

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂⁴

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×Q₈

yes

—

C₂×Dic₁₀₉

C₂×Q₈

yes

—

C₂×Dic₁₀₉

C₂⁴

yes

—

C₂×Dic₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

C₂⁴

yes

—

Dic₂₁₈

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²×Q₈

yes

—

C₂²×C₁₀₉

C₂⁵

yes

characteristic

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂²×Q₈

yes

—

Dic₁₀₉

C₂²×Q₈

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂³×Q₈

yes

characteristic

C₂×C₁₀₉

#585

yes

—

#3458

—

109

maximal, Sylow

C₁₀₉

#3458

yes

characteristic, complemented, Sylow

#544

—

109

—

#543

—

109

—

#543

D₁₀₉

yes

characteristic

#533

—

109

—

C₂×#117

—

109

—

C₂³×C₄

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

109

—

#125

—

109

—

C₂²⋊Q₈

—

109

—

C₂²⋊Q₈

—

109

—

C₂²⋊Q₈

C₂×D₁₀₉

yes

—

C₄×Q₈

—

109

—

C₄×Q₈

—

109

—

#117

—

109

—

#117

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₂²⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄²

—

109

—

C₂⁴

C₂²×D₁₀₉

yes

characteristic

C₂×Q₈

—

109

—

C₂×Q₈

C₂²×D₁₀₉

yes

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

C₂²×D₁₀₉

yes

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

C₂²×D₁₀₉

yes

—

C₄²

—

109

—

C₄²

—

109

—

C₂³

—

100

C₂³

Q₈×D₁₀₉

yes

—

101

C₂³

C₂³×D₁₀₉

yes

—

102

Q₈

—

103

Q₈

—

436

—

104

C₂×C₄

—

105

C₂×C₄

—

109

—

106

C₂×C₄

—

109

—

107

C₂×C₄

—

109

—

108

C₂×C₄

—

109

—

109

C₂×C₄

—

109

—

110

C₂×C₄

—

436

—

111

C₂×C₄

C₂³×D₁₀₉

yes

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×Q₈×D₁₀₉

yes

—

115

C₂²

C₂⁴×D₁₀₉

yes

characteristic

116

C₄

—

117

C₄

—

109

—

118

C₄

—

109

—

119

C₄

—

218

—

120

C₂

—

121

C₂

C₂²×Q₈×D₁₀₉

yes

characteristic

122

C₂

#20295861

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₉
commutator, socle, cosocle	C₂²×C₁₀₉	C₂⁵
Fitting	#543×C₁₀₉	C₂
prime radical	C₂⁴×C₁₀₉	C₂³
hypercenter	#543	D₁₀₉
hypocenter	C₁₀₉	#3458
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #543

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₁₀₉

commutator, socle, cosocle

C₂²×C₁₀₉

C₂⁵

Fitting

#543×C₁₀₉

C₂

prime radical

C₂⁴×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂² ◁ #543

N	H ≅ G/N	type
1. #20295655	C₂	semidirect
3. #20295578	C₂	semidirect
7. #16812698	C₂²	semidirect
8. #16812697	C₂²	semidirect
9. #16812689	C₂²	semidirect
12. Q₈×Dic₁₀₉	C₂²	semidirect
13. #16812664	C₂²	semidirect
22. C₂²×Dic₁₀₉	Q₈	semidirect
62. C₁₀₉	#3458	semidirect

H ≅ G/N

type

1. #20295655

C₂

semidirect

3. #20295578

C₂

semidirect

7. #16812698

C₂²

semidirect

8. #16812697

C₂²

semidirect

9. #16812689

C₂²

semidirect

12. Q₈×Dic₁₀₉

C₂²

semidirect

13. #16812664

C₂²

semidirect

22. C₂²×Dic₁₀₉

Q₈

semidirect

62. C₁₀₉

#3458

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20295655	2
#20295652	2
#20295578	2
C₂×#16812655	2
#543×C₁₀₉	2
#3458	109

subgroup

index

#543×C₁₀₉

109

Subgroups — #28376434

Conjugacy classes of subgroups

Complements

Maximal subgroups