Group #28364004 — subgroups

Subgroups — #28364004

Order 13952.

276 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 105 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×#16812699	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812610	C₂	1	1	yes	maximal, characteristic
3	#20294788	C₂	1	4	yes	maximal
4	#382×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₁₈	—	2	1	no	—
6	C₂²×#11724784	C₂²	1	2	yes	—
7	#16812699	—	2	2	no	—
8	#16812699	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₀₉	—	2	1	no	—
10	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
11	#16812610	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
14	#95×C₁₀₉	—	2	1	no	—
15	#95×C₁₀₉	—	2	1	no	—
16	C₂×Dic₂₁₈	—	2	4	no	—
17	C₂×Dic₂₁₈	—	4	4	no	—
18	C₂²×Dic₁₀₉	—	2	2	no	—
19	C₂×#11724784	—	2	2	no	—
20	C₂×#11724784	—	2	2	no	—
21	C₂×#11724784	Q₈	1	4	yes	complemented
22	C₄⋊Dic₁₀₉	—	2	4	no	—
23	C₄⋊Dic₁₀₉	—	2	2	no	—
24	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₁₀₉	—	2	2	no	—
26	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₀₉	—	2	4	no	—
30	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
31	C₄²×C₁₀₉	D₄	1	2	yes	—
32	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	—	2	2	no	—
34	C₂×C₄×C₁₀₉	—	2	2	no	—
35	C₂×C₄×C₁₀₉	—	4	2	no	—
36	C₂×C₄×C₁₀₉	—	4	2	no	—
37	C₂×C₄×C₁₀₉	—	4	2	no	—
38	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₁₀₉	#41	1	2	yes	—
44	C₂×Dic₁₀₉	—	2	4	no	—
45	C₂×Dic₁₀₉	—	4	4	no	—
46	Dic₂₁₈	—	4	8	no	—
47	#11724784	—	2	4	no	—
48	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₀₉	#124	1	1	yes	characteristic
50	C₂²×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
52	C₂²×C₁₀₉	#137	1	2	yes	—
53	C₄×C₁₀₉	—	2	2	no	—
54	C₄×C₁₀₉	—	2	2	no	—
55	C₄×C₁₀₉	—	8	2	no	—
56	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
57	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
58	Dic₁₀₉	—	4	4	no	—
59	C₂×C₁₀₉	#391	1	1	yes	characteristic
60	C₂×C₁₀₉	#418	1	1	yes	characteristic
61	C₂×C₁₀₉	#419	1	1	yes	characteristic
62	C₂×C₁₀₉	#489	1	2	yes	—
63	C₂×C₁₀₉	#501	1	2	yes	—
64	#1993	—	109	1	no	maximal, Sylow
65	C₁₀₉	#1993	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₄⋊Q₈	—	109	1	no	—
67	C₂×#97	—	109	1	no	—
68	#382	D₁₀₉	1	1	yes	characteristic
69	#340	—	109	4	no	—
70	C₂²×Q₈	—	218	1	no	—
71	C₂²×C₈	—	109	2	no	—
72	C₄⋊Q₈	—	109	4	no	—
73	C₄⋊Q₈	—	218	2	no	—
74	C₂×C₄⋊C₄	—	109	2	no	—
75	C₂×C₄⋊C₄	—	218	1	no	—
76	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
77	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
78	#97	—	218	2	no	—
79	#95	—	2	1	no	—
80	#95	—	2	1	no	—
81	C₂×Q₈	—	436	4	no	—
82	C₂×Q₈	—	218	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	218	2	no	—
85	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
88	C₂×C₈	—	109	4	no	—
89	C₂×C₈	—	218	2	no	—
90	C₂×C₈	—	218	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	109	4	no	—
93	C₄⋊C₄	—	218	4	no	—
94	C₄⋊C₄	—	218	2	no	—
95	C₄²	C₄×D₁₀₉	1	2	yes	—
96	C₄²	D₄₃₆	1	2	yes	—
97	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
98	Q₈	—	436	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	436	4	no	—
105	C₂×C₄	—	218	4	no	—
106	C₂×C₄	#13182006	1	2	yes	—
107	C₂×C₄	#13182006	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
109	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
110	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
111	C₂×C₄	#13182028	1	2	yes	—
112	C₈	—	218	4	no	—
113	C₂²	C₂×#13182006	1	1	yes	characteristic
114	C₂²	#16812686	1	1	yes	characteristic
115	C₂²	#16812697	1	1	yes	characteristic
116	C₂²	#16812724	1	2	yes	—
117	C₂²	#16812729	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	8	2	no	—
121	C₄	—	436	4	no	—
122	C₄	#16812664	1	2	yes	—
123	C₄	#16812679	1	2	yes	—
124	C₂	#20294931	1	1	yes	characteristic
125	C₂	#20295030	1	1	yes	characteristic
126	C₂	#20295034	1	1	yes	characteristic
127	C₂	#20295385	1	2	yes	—
128	C₂	#20295436	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812699

C₂

yes

maximal, characteristic

C₂×#16812610

C₂

yes

maximal, characteristic

#20294788

C₂

yes

maximal

#382×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

—

C₂²×#11724784

C₂²

yes

—

#16812699

—

#16812699

C₄

yes

complemented

C₂×C₄⋊Dic₁₀₉

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

#16812610

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

#95×C₁₀₉

—

#95×C₁₀₉

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂²×Dic₁₀₉

—

C₂×#11724784

—

C₂×#11724784

—

C₂×#11724784

Q₈

yes

complemented

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

#11724784

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

#124

yes

characteristic

C₂²×C₁₀₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₀₉

AGL(1,ℤ/8)

yes

—

C₂²×C₁₀₉

#137

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×Q₈

yes

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

C₂×C₁₀₉

#501

yes

—

#1993

—

109

maximal, Sylow

C₁₀₉

#1993

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

109

—

C₂×#97

—

109

—

#382

D₁₀₉

yes

characteristic

#340

—

109

—

C₂²×Q₈

—

218

—

C₂²×C₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#97

—

218

—

#95

—

#95

—

C₂×Q₈

—

436

—

C₂×Q₈

—

218

—

C₂²×C₄

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄²

C₄×D₁₀₉

yes

—

C₄²

D₄₃₆

yes

—

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

Q₈

—

436

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

436

—

105

C₂×C₄

—

218

—

106

C₂×C₄

#13182006

yes

—

107

C₂×C₄

#13182006

yes

—

108

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

109

C₂×C₄

C₂×D₄₃₆

yes

characteristic

110

C₂×C₄

Q₈×D₁₀₉

yes

—

111

C₂×C₄

#13182028

yes

—

112

C₈

—

218

—

113

C₂²

C₂×#13182006

yes

characteristic

114

C₂²

#16812686

yes

characteristic

115

C₂²

#16812697

yes

characteristic

116

C₂²

#16812724

yes

—

117

C₂²

#16812729

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

436

—

122

C₄

#16812664

yes

—

123

C₄

#16812679

yes

—

124

C₂

#20294931

yes

characteristic

125

C₂

#20295030

yes

characteristic

126

C₂

#20295034

yes

characteristic

127

C₂

#20295385

yes

—

128

C₂

#20295436

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#382×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#382	D₁₀₉
hypocenter	C₁₀₉	#1993
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#382×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
8. #16812699	C₄	semidirect
21. C₂×#11724784	Q₈	semidirect
65. C₁₀₉	#1993	semidirect

H ≅ G/N

type

8. #16812699

C₄

semidirect

21. C₂×#11724784

Q₈

semidirect

65. C₁₀₉

#1993

semidirect

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

subgroup	index
C₂×#16812699	2
C₂×#16812610	2
#20294788	2
#382×C₁₀₉	2
#1993	109

subgroup

index

#382×C₁₀₉

109

Subgroups — #28364004

Conjugacy classes of subgroups

Complements

Maximal subgroups