Group #28373529 — subgroups

Subgroups — C₂×#20295422 #28373529

Order 13952.

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

normal subgroups only

This group has 424 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×#16812699	C₂	1	1	yes	maximal, characteristic
2	#20295422	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16812630	C₂	1	4	yes	maximal
4	C₂×C₄×#11724784	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₈	—	2	1	no	—
7	C₂²×#11724784	C₂²	1	2	yes	—
8	#16812699	—	2	2	no	—
9	#16812699	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₁₀₉	—	2	1	no	—
11	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
12	#16812630	C₂²	1	16	yes	—
13	C₄×#11724784	C₂²	1	4	yes	—
14	C₂²×Q₈×C₁₀₉	—	2	1	no	—
15	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
16	C₄⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
18	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
19	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×Dic₂₁₈	—	2	4	no	—
21	C₂×Dic₂₁₈	—	4	4	no	—
22	C₂²×Dic₁₀₉	—	2	2	no	—
23	C₂×#11724784	Q₈	1	8	yes	complemented
24	C₂×#11724784	C₂³	1	4	yes	—
25	C₄⋊Dic₁₀₉	—	2	4	no	—
26	C₄⋊Dic₁₀₉	—	2	2	no	—
27	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
28	C₂×Q₈×C₁₀₉	—	2	4	no	—
29	C₂×Q₈×C₁₀₉	—	4	4	no	—
30	C₂²×C₄×C₁₀₉	—	2	2	no	—
31	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₀₉	—	2	4	no	—
35	C₄⋊C₄×C₁₀₉	—	2	2	no	—
36	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
37	C₄²×C₁₀₉	D₄	1	2	yes	—
38	C₄²×C₁₀₉	C₂³	1	2	yes	—
39	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
40	Q₈×C₁₀₉	—	4	8	no	—
41	C₂×C₄×C₁₀₉	—	2	4	no	—
42	C₂×C₄×C₁₀₉	—	4	4	no	—
43	C₂×C₄×C₁₀₉	SD₁₆	1	4	yes	—
44	C₂×C₄×C₁₀₉	SD₁₆	1	4	yes	—
45	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₀₉	C₂×Q₈	1	4	yes	—
49	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₁₀₉	—	2	4	no	—
51	C₂×Dic₁₀₉	—	4	4	no	—
52	Dic₂₁₈	—	4	8	no	—
53	#11724784	C₂×Q₈	1	8	yes	complemented
54	C₂²×C₁₀₉	C₄⋊Q₈	1	2	yes	—
55	C₂²×C₁₀₉	C₂×SD₁₆	1	2	yes	—
56	C₂²×C₁₀₉	C₂×SD₁₆	1	2	yes	—
57	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₁₀₉	—	4	4	no	—
59	C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
60	C₄×C₁₀₉	C₂×SD₁₆	1	4	yes	—
61	C₄×C₁₀₉	C₂×SD₁₆	1	4	yes	—
62	C₄×C₁₀₉	C₂²×Q₈	1	2	yes	—
63	Dic₁₀₉	—	4	4	no	—
64	C₂×C₁₀₉	#498	1	4	yes	complemented
65	C₂×C₁₀₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₁₀₉	C₂²×SD₁₆	1	1	yes	characteristic
67	C₂×C₁₀₉	C₂²×SD₁₆	1	1	yes	characteristic
68	C₂×#498	—	109	1	no	maximal, Sylow
69	C₁₀₉	C₂×#498	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₄⋊Q₈	—	109	1	no	—
71	C₂×C₄⋊Q₈	D₁₀₉	1	1	yes	characteristic
72	#498	—	109	8	no	—
73	C₂×#106	—	109	4	no	—
74	C₂×C₄×C₈	—	109	1	no	—
75	C₂²×Q₈	—	2	1	no	—
76	C₂²×Q₈	—	218	1	no	—
77	C₂²×C₈	—	109	2	no	—
78	C₄⋊Q₈	—	2	2	no	—
79	C₄⋊Q₈	—	109	4	no	—
80	C₄⋊Q₈	—	218	2	no	—
81	C₄⋊Q₈	C₂×D₁₀₉	1	4	yes	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	109	2	no	—
84	C₂×C₄⋊C₄	—	218	1	no	—
85	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
86	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
87	#106	—	109	16	no	—
88	C₄×C₈	—	109	4	no	—
89	C₂×Q₈	—	2	4	no	—
90	C₂×Q₈	—	4	4	no	—
91	C₂×Q₈	—	436	4	no	—
92	C₂×Q₈	—	218	4	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	218	2	no	—
95	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
98	C₂×C₈	—	109	8	no	—
99	C₂×C₈	—	109	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	109	4	no	—
103	C₄⋊C₄	—	218	2	no	—
104	C₄⋊C₄	—	218	4	no	—
105	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	—
106	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
107	C₄²	C₂²×D₁₀₉	1	2	yes	—
108	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
109	Q₈	—	4	8	no	—
110	Q₈	—	436	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	4	4	no	—
113	C₂×C₄	—	436	4	no	—
114	C₂×C₄	—	218	4	no	—
115	C₂×C₄	#13182016	1	4	yes	—
116	C₂×C₄	#13182017	1	4	yes	—
117	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
120	C₂×C₄	Q₈×D₁₀₉	1	4	yes	—
121	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
122	C₈	—	109	8	no	—
123	C₂²	#16812697	1	2	yes	—
124	C₂²	C₂×#13182016	1	2	yes	—
125	C₂²	C₂×#13182017	1	2	yes	—
126	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
127	C₄	—	4	4	no	—
128	C₄	—	436	4	no	—
129	C₄	#16812697	1	2	yes	—
130	C₄	C₂×#13182016	1	4	yes	—
131	C₄	C₂×#13182017	1	4	yes	—
132	C₄	C₂×Q₈×D₁₀₉	1	2	yes	—
133	C₂	#20295422	1	4	yes	direct product factor
134	C₂	C₂×#16812697	1	1	yes	characteristic
135	C₂	C₂²×#13182016	1	1	yes	characteristic
136	C₂	C₂²×#13182017	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812699

C₂

yes

maximal, characteristic

#20295422

C₂

yes

maximal, direct product factor

C₂×#16812630

C₂

yes

maximal

C₂×C₄×#11724784

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

—

C₂²×#11724784

C₂²

yes

—

#16812699

—

#16812699

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

#16812630

C₂²

yes

—

C₄×#11724784

C₂²

yes

—

C₂²×Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂²×Dic₁₀₉

—

C₂×#11724784

Q₈

yes

complemented

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₄²×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

SD₁₆

yes

—

C₂×C₄×C₁₀₉

SD₁₆

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

#11724784

C₂×Q₈

yes

complemented

C₂²×C₁₀₉

C₄⋊Q₈

yes

—

C₂²×C₁₀₉

C₂×SD₁₆

yes

—

C₂²×C₁₀₉

C₂×SD₁₆

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄⋊Q₈

yes

—

C₄×C₁₀₉

C₂×SD₁₆

yes

—

C₄×C₁₀₉

C₂×SD₁₆

yes

—

C₄×C₁₀₉

C₂²×Q₈

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

#498

yes

complemented

C₂×C₁₀₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₀₉

C₂²×SD₁₆

yes

characteristic

C₂×C₁₀₉

C₂²×SD₁₆

yes

characteristic

C₂×#498

—

109

maximal, Sylow

C₁₀₉

C₂×#498

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

109

—

C₂×C₄⋊Q₈

D₁₀₉

yes

characteristic

#498

—

109

—

C₂×#106

—

109

—

C₂×C₄×C₈

—

109

—

C₂²×Q₈

—

C₂²×Q₈

—

218

—

C₂²×C₈

—

109

—

C₄⋊Q₈

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

218

—

C₄⋊Q₈

C₂×D₁₀₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#106

—

109

—

C₄×C₈

—

109

—

C₂×Q₈

—

C₂×Q₈

—

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

—

109

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

109

—

103

C₄⋊C₄

—

218

—

104

C₄⋊C₄

—

218

—

105

C₄⋊C₄

C₂²×D₁₀₉

yes

—

106

C₄²

C₂²⋊D₁₀₉

yes

—

107

C₄²

C₂²×D₁₀₉

yes

—

108

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

109

Q₈

—

110

Q₈

—

436

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

436

—

114

C₂×C₄

—

218

—

115

C₂×C₄

#13182016

yes

—

116

C₂×C₄

#13182017

yes

—

117

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

119

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

120

C₂×C₄

Q₈×D₁₀₉

yes

—

121

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

122

C₈

—

109

—

123

C₂²

#16812697

yes

—

124

C₂²

C₂×#13182016

yes

—

125

C₂²

C₂×#13182017

yes

—

126

C₂²

C₂²×C₂²⋊D₁₀₉

yes

characteristic

127

C₄

—

128

C₄

—

436

—

129

C₄

#16812697

yes

—

130

C₄

C₂×#13182016

yes

—

131

C₄

C₂×#13182017

yes

—

132

C₄

C₂×Q₈×D₁₀₉

yes

—

133

C₂

#20295422

yes

direct product factor

134

C₂

C₂×#16812697

yes

characteristic

135

C₂

C₂²×#13182016

yes

characteristic

136

C₂

C₂²×#13182017

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×C₄⋊Q₈×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₀₉
hypocenter	C₁₀₉	C₂×#498
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

C₂×C₄⋊Q₈×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₀₉

hypocenter

C₁₀₉

C₂×#498

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #20295422	C₂	direct (N × H)
23. C₂×#11724784	Q₈	semidirect
53. #11724784	C₂×Q₈	semidirect
64. C₂×C₁₀₉	#498	semidirect
69. C₁₀₉	C₂×#498	semidirect
133. C₂	#20295422	direct (N × H)

H ≅ G/N

type

2. #20295422

C₂

direct (N × H)

23. C₂×#11724784

Q₈

semidirect

53. #11724784

C₂×Q₈

semidirect

64. C₂×C₁₀₉

#498

semidirect

69. C₁₀₉

C₂×#498

semidirect

133. C₂

#20295422

direct (N × H)

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

subgroup	index
C₂×#16812699	2
#20295422	2
C₂×#16812630	2
C₂×C₄×#11724784	2
C₂×C₄⋊Q₈×C₁₀₉	2
C₂×#498	109

subgroup

index

C₂×C₄⋊Q₈×C₁₀₉

109

Subgroups — C₂×#20295422 #28373529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20295422 #28373529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295422 #28373529