Group #28371390 — subgroups

Subgroups — #28371390

Order 13952.

752 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 197 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×#16812696	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812688	C₂	1	4	yes	maximal, complemented
3	#20294927	C₂	1	8	yes	maximal, complemented
4	C₄²×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄₃₆	C₂²	1	4	yes	—
7	C₂²×D₄₃₆	C₂²	1	2	yes	—
8	#16812696	—	2	4	no	—
9	#16812688	—	2	16	no	—
10	C₂×#13182006	C₂²	1	16	yes	—
11	C₂×C₄×Dic₁₀₉	C₂²	1	4	yes	—
12	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
13	C₂²×Q₈×C₁₀₉	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₁₀₉	—	2	4	no	—
15	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
16	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂³×D₁₀₉	C₂³	1	4	yes	—
18	C₂×D₄₃₆	—	2	8	no	—
19	C₂×D₄₃₆	—	2	8	no	—
20	C₂×D₄₃₆	—	2	4	no	—
21	C₂×D₄₃₆	—	2	16	no	—
22	C₂²×Dic₁₀₉	C₂³	1	4	yes	—
23	#13182006	—	2	32	no	—
24	C₄×Dic₁₀₉	—	2	8	no	—
25	C₄×Dic₁₀₉	D₄	1	8	yes	complemented
26	C₂×Q₈×C₁₀₉	—	2	4	no	—
27	C₂×Q₈×C₁₀₉	—	2	8	no	—
28	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
29	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
31	C₄⋊C₄×C₁₀₉	—	2	8	no	—
32	C₄²×C₁₀₉	D₄	1	4	yes	—
33	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
34	C₂²×D₁₀₉	—	2	8	no	—
35	C₂²×D₁₀₉	—	2	4	no	—
36	C₂²×D₁₀₉	—	2	16	no	—
37	Q₈×C₁₀₉	—	4	8	no	—
38	C₂×C₄×C₁₀₉	—	2	4	no	—
39	C₂×C₄×C₁₀₉	—	2	8	no	—
40	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
41	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
42	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₀₉	#41	1	8	yes	—
44	C₂×Dic₁₀₉	—	2	8	no	—
45	C₂×Dic₁₀₉	C₂×D₄	1	8	yes	—
46	D₄₃₆	—	4	16	no	—
47	D₄₃₆	—	4	8	no	—
48	C₂²×C₁₀₉	C₂²×D₄	1	2	yes	—
49	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
50	C₂²×C₁₀₉	C₂×#41	1	4	yes	—
51	C₂×D₁₀₉	—	4	8	no	—
52	C₂×D₁₀₉	—	4	4	no	—
53	C₂×D₁₀₉	—	4	16	no	—
54	C₄×C₁₀₉	—	4	4	no	—
55	C₄×C₁₀₉	#124	1	8	yes	complemented
56	C₄×C₁₀₉	#127	1	4	yes	—
57	Dic₁₀₉	—	2	8	no	—
58	C₂×C₁₀₉	C₂×#124	1	2	yes	—
59	C₂×C₁₀₉	C₂×#127	1	1	yes	characteristic
60	C₂×C₁₀₉	#532	1	4	yes	—
61	D₁₀₉	—	8	4	no	—
62	#2858	—	109	1	no	maximal, Sylow
63	C₁₀₉	#2858	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊Q₈	D₁₀₉	1	1	yes	characteristic, complemented
65	C₂×#127	—	109	1	no	—
66	C₂×#124	—	109	4	no	—
67	#390	—	109	8	no	—
68	C₄³	—	109	1	no	—
69	C₂²×Q₈	C₂×D₁₀₉	1	2	yes	—
70	C₂²×D₄	—	109	2	no	—
71	C₂²×D₄	—	109	4	no	—
72	C₄⋊Q₈	—	2	4	no	—
73	#127	—	218	4	no	—
74	#124	—	218	16	no	—
75	C₂×C₄⋊C₄	C₂×D₁₀₉	1	4	yes	—
76	C₂×C₂²⋊C₄	—	109	16	no	—
77	C₂×C₄²	—	109	2	no	—
78	C₂×C₄²	—	109	4	no	—
79	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
80	C₂⁴	—	109	4	no	—
81	C₂×Q₈	—	2	8	no	—
82	C₂×Q₈	—	2	4	no	—
83	C₂×D₄	—	218	8	no	—
84	C₂×D₄	—	218	8	no	—
85	C₂×D₄	—	218	4	no	—
86	C₂×D₄	—	218	16	no	—
87	C₂²×C₄	—	109	4	no	—
88	C₂²×C₄	C₂²×D₁₀₉	1	4	yes	—
89	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
91	C₄⋊C₄	—	2	8	no	—
92	C₂²⋊C₄	—	218	32	no	—
93	C₄²	—	109	8	no	—
94	C₄²	—	218	8	no	—
95	C₄²	C₂²⋊D₁₀₉	1	4	yes	—
96	C₂³	—	218	8	no	—
97	C₂³	—	218	4	no	—
98	C₂³	—	218	16	no	—
99	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
100	Q₈	—	4	8	no	—
101	D₄	—	436	16	no	—
102	D₄	—	436	8	no	—
103	C₂×C₄	—	2	8	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	109	8	no	—
106	C₂×C₄	—	218	8	no	—
107	C₂×C₄	D₄×D₁₀₉	1	4	yes	—
108	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
110	C₂×C₄	#13182030	1	8	yes	—
111	C₂²	—	436	8	no	—
112	C₂²	—	436	4	no	—
113	C₂²	—	436	16	no	—
114	C₂²	C₂×D₄×D₁₀₉	1	2	yes	—
115	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
116	C₂²	C₂×#13182030	1	4	yes	—
117	C₄	—	4	4	no	—
118	C₄	—	218	8	no	—
119	C₄	#16812688	1	8	yes	complemented
120	C₄	#16812695	1	4	yes	—
121	C₂	—	872	4	no	—
122	C₂	C₂×#16812688	1	2	yes	—
123	C₂	C₂×#16812695	1	1	yes	characteristic
124	C₂	#20295577	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812696

C₂

yes

maximal, characteristic

C₂×#16812688

C₂

yes

maximal, complemented

#20294927

C₂

yes

maximal, complemented

C₄²×Dic₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊Q₈×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₃₆

C₂²

yes

—

C₂²×D₄₃₆

C₂²

yes

—

#16812696

—

#16812688

—

C₂×#13182006

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

C₂²×Q₈×C₁₀₉

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂³×D₁₀₉

C₂³

yes

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂²×Dic₁₀₉

C₂³

yes

—

#13182006

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

complemented

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

C₂⁴

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×D₄

yes

—

D₄₃₆

—

D₄₃₆

—

C₂²×C₁₀₉

C₂²×D₄

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₂²×C₁₀₉

C₂×#41

yes

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#124

yes

complemented

C₄×C₁₀₉

#127

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

C₂×#124

yes

—

C₂×C₁₀₉

C₂×#127

yes

characteristic

C₂×C₁₀₉

#532

yes

—

D₁₀₉

—

#2858

—

109

maximal, Sylow

C₁₀₉

#2858

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₁₀₉

yes

characteristic, complemented

C₂×#127

—

109

—

C₂×#124

—

109

—

#390

—

109

—

C₄³

—

109

—

C₂²×Q₈

C₂×D₁₀₉

yes

—

C₂²×D₄

—

109

—

C₂²×D₄

—

109

—

C₄⋊Q₈

—

#127

—

218

—

#124

—

218

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₂⁴

—

109

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

218

—

C₄²

—

109

—

C₄²

—

218

—

C₄²

C₂²⋊D₁₀₉

yes

—

C₂³

—

218

—

C₂³

—

218

—

C₂³

—

218

—

C₂³

C₂³×D₁₀₉

yes

characteristic

100

Q₈

—

101

D₄

—

436

—

102

D₄

—

436

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

109

—

106

C₂×C₄

—

218

—

107

C₂×C₄

D₄×D₁₀₉

yes

—

108

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

109

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

110

C₂×C₄

#13182030

yes

—

111

C₂²

—

436

—

112

C₂²

—

436

—

113

C₂²

—

436

—

114

C₂²

C₂×D₄×D₁₀₉

yes

—

115

C₂²

C₂²×C₂²⋊D₁₀₉

yes

characteristic

116

C₂²

C₂×#13182030

yes

—

117

C₄

—

118

C₄

—

218

—

119

C₄

#16812688

yes

complemented

120

C₄

#16812695

yes

—

121

C₂

—

872

—

122

C₂

C₂×#16812688

yes

—

123

C₂

C₂×#16812695

yes

characteristic

124

C₂

#20295577

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₉
commutator, socle, cosocle	C₂³×C₁₀₉	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₁₀₉	C₂
prime radical	C₂×#16812696	C₂
hypercenter	C₂×C₄⋊Q₈	D₁₀₉
hypocenter	C₁₀₉	#2858
LCS	G ▷ C₂³×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₀₉

commutator, socle, cosocle

C₂³×C₁₀₉

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₁₀₉

C₂

prime radical

C₂×#16812696

C₂

hypercenter

C₂×C₄⋊Q₈

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. C₂×#16812688	C₂	semidirect
3. #20294927	C₂	semidirect
4. C₄²×Dic₁₀₉	C₂	semidirect
5. C₂×C₄⋊Q₈×C₁₀₉	C₂	semidirect
25. C₄×Dic₁₀₉	D₄	semidirect
55. C₄×C₁₀₉	#124	semidirect
63. C₁₀₉	#2858	semidirect
64. C₂×C₄⋊Q₈	D₁₀₉	semidirect
119. C₄	#16812688	semidirect

H ≅ G/N

type

2. C₂×#16812688

C₂

semidirect

3. #20294927

C₂

semidirect

4. C₄²×Dic₁₀₉

C₂

semidirect

5. C₂×C₄⋊Q₈×C₁₀₉

C₂

semidirect

25. C₄×Dic₁₀₉

D₄

semidirect

55. C₄×C₁₀₉

#124

semidirect

63. C₁₀₉

#2858

semidirect

64. C₂×C₄⋊Q₈

D₁₀₉

semidirect

119. C₄

#16812688

semidirect

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

subgroup	index
C₂×#16812696	2
C₂×#16812688	2
#20294927	2
C₄²×Dic₁₀₉	2
C₂×C₄⋊Q₈×C₁₀₉	2
#2858	109

subgroup

index

C₂×#16812696

C₂×#16812688

#20294927

C₄²×Dic₁₀₉

C₂×C₄⋊Q₈×C₁₀₉

#2858

109

Subgroups — #28371390

Conjugacy classes of subgroups

Complements

Maximal subgroups