Group #28371408 — subgroups

Subgroups — #28371408

Order 13952.

504 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 181 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×#16812698	C₂	1	2	yes	maximal
2	#20294930	C₂	1	8	yes	maximal
3	#20294892	C₂	1	1	yes	maximal, characteristic
4	#20294891	C₂	1	2	yes	maximal
5	C₄²×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
6	#382×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₁₈	C₂²	1	4	yes	—
8	#16812698	—	2	8	no	—
9	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
10	C₂×#13182008	C₂²	1	4	yes	—
11	C₂×C₄×Dic₁₀₉	C₂²	1	4	yes	—
12	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
14	#16812606	C₂²	1	8	yes	—
15	#16812606	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
18	#95×C₁₀₉	C₂²	1	4	yes	—
19	C₂×Dic₂₁₈	—	2	8	no	—
20	C₂×Dic₂₁₈	—	2	16	no	—
21	C₂²×Dic₁₀₉	C₂³	1	4	yes	—
22	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
23	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
24	C₄⋊Dic₁₀₉	—	2	4	no	—
25	#13182008	—	2	8	no	—
26	C₄×Dic₁₀₉	—	2	8	no	—
27	C₄×Dic₁₀₉	D₄	1	4	yes	—
28	C₄×Dic₁₀₉	Q₈	1	4	yes	complemented
29	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
30	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
31	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₁₀₉	—	2	4	no	—
33	C₄²×C₁₀₉	Q₈	1	4	yes	complemented
34	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	—	2	4	no	—
36	C₂×C₄×C₁₀₉	—	2	4	no	—
37	C₂×C₄×C₁₀₉	—	2	8	no	—
38	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₁₀₉	C₂×Q₈	1	4	yes	—
41	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
42	C₂×Dic₁₀₉	—	2	4	no	—
43	C₂×Dic₁₀₉	—	2	4	no	—
44	C₂×Dic₁₀₉	—	2	8	no	—
45	C₂×Dic₁₀₉	C₂×D₄	1	4	yes	—
46	C₂×Dic₁₀₉	C₂×Q₈	1	4	yes	—
47	C₂×Dic₁₀₉	#41	1	8	yes	—
48	Dic₂₁₈	—	4	16	no	—
49	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
50	C₂²×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
51	C₂²×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
52	C₂²×C₁₀₉	C₂×#41	1	4	yes	—
53	C₄×C₁₀₉	—	2	4	no	—
54	C₄×C₁₀₉	—	4	4	no	—
55	C₄×C₁₀₉	C₄⋊Q₈	1	4	yes	—
56	Dic₁₀₉	—	2	4	no	—
57	Dic₁₀₉	—	4	4	no	—
58	Dic₁₀₉	#124	1	8	yes	—
59	C₂×C₁₀₉	C₂×#124	1	2	yes	—
60	C₂×C₁₀₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₁₀₉	#533	1	2	yes	—
62	C₂×C₁₀₉	#533	1	2	yes	—
63	#2862	—	109	1	no	maximal, Sylow
64	C₁₀₉	#2862	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₄⋊Q₈	—	109	2	no	—
66	#391	—	109	8	no	—
67	#382	—	109	2	no	—
68	#382	—	109	1	no	—
69	#382	D₁₀₉	1	1	yes	characteristic
70	C₄³	—	109	1	no	—
71	C₂²×Q₈	—	109	4	no	—
72	C₄⋊Q₈	—	218	8	no	—
73	C₂×C₄⋊C₄	—	109	4	no	—
74	C₂×C₄⋊C₄	—	109	2	no	—
75	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
76	C₂×C₄²	—	109	1	no	—
77	C₂×C₄²	—	109	1	no	—
78	C₂×C₄²	—	109	4	no	—
79	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
80	#95	—	109	8	no	—
81	#95	—	109	4	no	—
82	#95	C₂×D₁₀₉	1	4	yes	—
83	C₂×Q₈	—	218	8	no	—
84	C₂×Q₈	—	218	16	no	—
85	C₂²×C₄	—	109	2	no	—
86	C₂²×C₄	—	109	2	no	—
87	C₂²×C₄	—	109	4	no	—
88	C₂²×C₄	C₂²×D₁₀₉	1	4	yes	—
89	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	218	8	no	—
93	C₄⋊C₄	—	218	4	no	—
94	C₄²	—	109	4	no	—
95	C₄²	—	109	4	no	—
96	C₄²	—	218	8	no	—
97	C₄²	Dic₂₁₈	1	4	yes	complemented
98	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
99	Q₈	—	436	16	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	8	no	—
103	C₂×C₄	—	109	4	no	—
104	C₂×C₄	—	109	8	no	—
105	C₂×C₄	—	109	4	no	—
106	C₂×C₄	—	218	4	no	—
107	C₂×C₄	—	218	8	no	—
108	C₂×C₄	—	218	4	no	—
109	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
110	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
111	C₂×C₄	C₂×Dic₂₁₈	1	4	yes	—
112	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
113	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
114	C₂²	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
115	C₂²	C₂²×Dic₂₁₈	1	1	yes	characteristic
116	C₂²	#41×D₁₀₉	1	4	yes	—
117	C₄	—	2	4	no	—
118	C₄	—	4	4	no	—
119	C₄	—	109	8	no	—
120	C₄	—	436	4	no	—
121	C₄	—	218	4	no	—
122	C₄	#16812698	1	4	yes	—
123	C₂	#124×D₁₀₉	1	2	yes	—
124	C₂	C₂×#16812698	1	1	yes	characteristic
125	C₂	#20295579	1	2	yes	—
126	C₂	#20295582	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812698

C₂

yes

maximal

#20294930

C₂

yes

maximal

#20294892

C₂

yes

maximal, characteristic

#20294891

C₂

yes

maximal

C₄²×Dic₁₀₉

C₂

yes

maximal, characteristic

#382×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

C₂²

yes

—

#16812698

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×#13182008

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812606

C₂²

yes

—

#16812606

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

#95×C₁₀₉

C₂²

yes

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂²×Dic₁₀₉

C₂³

yes

—

C₂²×Dic₁₀₉

C₂³

yes

—

C₂²×Dic₁₀₉

C₂³

yes

—

C₄⋊Dic₁₀₉

—

#13182008

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

—

C₄×Dic₁₀₉

Q₈

yes

complemented

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

Q₈

yes

complemented

C₂³×C₁₀₉

C₂⁴

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×D₄

yes

—

C₂×Dic₁₀₉

C₂×Q₈

yes

—

C₂×Dic₁₀₉

#41

yes

—

Dic₂₁₈

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₂²×C₁₀₉

C₂²×Q₈

yes

characteristic

C₂²×C₁₀₉

C₂²×Q₈

yes

characteristic

C₂²×C₁₀₉

C₂×#41

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄⋊Q₈

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#124

yes

—

C₂×C₁₀₉

C₂×#124

yes

—

C₂×C₁₀₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₀₉

#533

yes

—

C₂×C₁₀₉

#533

yes

—

#2862

—

109

maximal, Sylow

C₁₀₉

#2862

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

109

—

#391

—

109

—

#382

—

109

—

#382

—

109

—

#382

D₁₀₉

yes

characteristic

C₄³

—

109

—

C₂²×Q₈

—

109

—

C₄⋊Q₈

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#95

—

109

—

#95

—

109

—

#95

C₂×D₁₀₉

yes

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

218

—

C₄²

Dic₂₁₈

yes

complemented

C₂³

C₂³×D₁₀₉

yes

characteristic

Q₈

—

436

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

109

—

104

C₂×C₄

—

109

—

105

C₂×C₄

—

109

—

106

C₂×C₄

—

218

—

107

C₂×C₄

—

218

—

108

C₂×C₄

—

218

—

109

C₂×C₄

D₄×D₁₀₉

yes

—

110

C₂×C₄

Q₈×D₁₀₉

yes

—

111

C₂×C₄

C₂×Dic₂₁₈

yes

—

112

C₂×C₄

C₂×Dic₂₁₈

yes

—

113

C₂²

C₂×D₄×D₁₀₉

yes

characteristic

114

C₂²

C₂×Q₈×D₁₀₉

yes

characteristic

115

C₂²

C₂²×Dic₂₁₈

yes

characteristic

116

C₂²

#41×D₁₀₉

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

109

—

120

C₄

—

436

—

121

C₄

—

218

—

122

C₄

#16812698

yes

—

123

C₂

#124×D₁₀₉

yes

—

124

C₂

C₂×#16812698

yes

characteristic

125

C₂

#20295579

yes

—

126

C₂

#20295582

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₉
commutator, socle, prime radical, cosocle	C₂³×C₁₀₉	C₂⁴
Fitting	#382×C₁₀₉	C₂
hypercenter	#382	D₁₀₉
hypocenter	C₁₀₉	#2862
LCS	G ▷ C₂³×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₀₉

commutator, socle, prime radical, cosocle

C₂³×C₁₀₉

C₂⁴

Fitting

#382×C₁₀₉

C₂

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
28. C₄×Dic₁₀₉	Q₈	semidirect
33. C₄²×C₁₀₉	Q₈	semidirect
64. C₁₀₉	#2862	semidirect
97. C₄²	Dic₂₁₈	semidirect

H ≅ G/N

type

28. C₄×Dic₁₀₉

Q₈

semidirect

33. C₄²×C₁₀₉

Q₈

semidirect

64. C₁₀₉

#2862

semidirect

97. C₄²

Dic₂₁₈

semidirect

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

subgroup	index
C₂×#16812698	2
#20294930	2
#20294892	2
#20294891	2
C₄²×Dic₁₀₉	2
#382×C₁₀₉	2
#2862	109

subgroup

index

C₄²×Dic₁₀₉

#382×C₁₀₉

#2862

109

Subgroups — #28371408

Conjugacy classes of subgroups

Complements

Maximal subgroups