Group #28363817 — subgroups

Subgroups — #28363817

Order 13952.

196 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 93 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×#16812635	C₂	1	1	yes	maximal, characteristic
2	#20295069	C₂	1	2	yes	maximal
3	C₂×#16812633	C₂	1	1	yes	maximal, characteristic
4	#20294794	C₂	1	2	yes	maximal
5	#405×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
8	#16812635	C₄	1	4	yes	complemented
9	#16812633	—	2	1	no	—
10	#16812633	—	2	1	no	—
11	#16812632	—	2	2	no	—
12	#16812630	—	2	2	no	—
13	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	#117×C₁₀₉	C₂²	1	1	yes	characteristic
16	#97×C₁₀₉	—	2	1	no	—
17	C₄×C₈×C₁₀₉	—	2	1	no	—
18	C₂²×Dic₁₀₉	—	2	2	no	—
19	#13182013	—	2	2	no	—
20	#13182013	C₂×C₄	1	4	yes	—
21	C₂×#11724784	—	2	2	no	—
22	C₄⋊Dic₁₀₉	—	2	2	no	—
23	C₄⋊Dic₁₀₉	—	2	2	no	—
24	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
25	M₁₆×C₁₀₉	—	2	1	no	—
26	M₁₆×C₁₀₉	Q₈	1	2	yes	—
27	C₂×C₈×C₁₀₉	—	2	1	no	—
28	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
30	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
31	C₂×C₈×C₁₀₉	Q₈	1	2	yes	—
32	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
33	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
34	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
35	C₄²×C₁₀₉	—	2	1	no	—
36	C₂³×C₁₀₉	#41	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	—	2	1	no	—
38	C₂×C₄×C₁₀₉	—	2	1	no	—
39	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
45	C₂×Dic₁₀₉	—	4	2	no	—
46	C₂×Dic₁₀₉	—	8	2	no	—
47	C₈×C₁₀₉	—	2	2	no	—
48	C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
49	C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
50	#11724784	—	2	4	no	—
51	C₂²×C₁₀₉	—	2	1	no	—
52	C₂²×C₁₀₉	—	2	1	no	—
53	C₂²×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
54	C₂²×C₁₀₉	#121	1	1	yes	characteristic
55	C₂²×C₁₀₉	#125	1	1	yes	characteristic
56	C₄×C₁₀₉	—	4	1	no	—
57	C₄×C₁₀₉	—	4	1	no	—
58	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
60	C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
61	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
62	Dic₁₀₉	—	8	2	no	—
63	C₂×C₁₀₉	—	2	1	no	—
64	C₂×C₁₀₉	—	2	1	no	—
65	C₂×C₁₀₉	#389	1	1	yes	characteristic
66	C₂×C₁₀₉	#472	1	1	yes	characteristic
67	C₂×C₁₀₉	#473	1	1	yes	characteristic
68	#1952	—	109	1	no	maximal, Sylow
69	C₁₀₉	#1952	1	1	yes	characteristic, complemented, Sylow
70	C₂×#108	—	109	1	no	—
71	#428	—	109	2	no	—
72	C₂×#107	—	109	1	no	—
73	#405	D₁₀₉	1	1	yes	characteristic
74	#343	—	109	2	no	—
75	C₂×M₁₆	—	109	2	no	—
76	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
77	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
78	#117	C₂×D₁₀₉	1	1	yes	characteristic
79	C₂×C₄⋊C₄	—	109	2	no	—
80	#108	—	109	4	no	—
81	#107	—	218	1	no	—
82	#107	—	218	1	no	—
83	#107	—	218	2	no	—
84	#106	—	218	2	no	—
85	#97	—	2	1	no	—
86	C₄×C₈	—	2	1	no	—
87	C₂²×C₄	—	218	2	no	—
88	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
89	M₁₆	—	2	1	no	—
90	M₁₆	—	109	4	no	—
91	M₁₆	—	218	2	no	—
92	M₁₆	Dic₂₁₈	1	2	yes	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	218	2	no	—
95	C₂×C₈	Dic₂₁₈	1	2	yes	—
96	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
97	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
98	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
99	C₄⋊C₄	—	218	2	no	—
100	C₄⋊C₄	—	218	2	no	—
101	C₄⋊C₄	D₄₃₆	1	1	yes	characteristic
102	C₄⋊C₄	D₄₃₆	1	1	yes	characteristic
103	C₂²⋊C₄	—	2	1	no	—
104	C₄²	—	2	1	no	—
105	C₂³	#13182029	1	1	yes	characteristic
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	872	2	no	—
109	C₂×C₄	—	436	2	no	—
110	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
111	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
112	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
113	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
114	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
115	C₂×C₄	#13182029	1	1	yes	characteristic
116	C₈	—	2	2	no	—
117	C₈	—	218	4	no	—
118	C₈	#13182008	1	2	yes	—
119	C₈	#13182008	1	2	yes	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	C₄×Dic₂₁₈	1	1	yes	characteristic
123	C₂²	#16812673	1	1	yes	characteristic
124	C₂²	#16812691	1	1	yes	characteristic
125	C₄	—	4	1	no	—
126	C₄	—	4	1	no	—
127	C₄	—	872	2	no	—
128	C₄	C₂×#13182008	1	1	yes	characteristic
129	C₄	C₄×D₄₃₆	1	1	yes	characteristic
130	C₄	#16812678	1	1	yes	characteristic
131	C₄	#16812699	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	2	1	no	—
134	C₂	#20294924	1	1	yes	characteristic
135	C₂	#20295296	1	1	yes	characteristic
136	C₂	#20295301	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812635

C₂

yes

maximal, characteristic

#20295069

C₂

yes

maximal

C₂×#16812633

C₂

yes

maximal, characteristic

#20294794

C₂

yes

maximal

#405×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

#16812635

C₄

yes

—

—

—

—

C₂×M₁₆×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

#117×C₁₀₉

C₂²

yes

characteristic

#97×C₁₀₉

—

C₄×C₈×C₁₀₉

—

C₂²×Dic₁₀₉

—

#13182013

—

#13182013

C₂×C₄

yes

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

Q₈

yes

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

Q₈

yes

—

C₄⋊C₄×C₁₀₉

D₄

yes

characteristic

C₄⋊C₄×C₁₀₉

D₄

yes

characteristic

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

#41

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₄⋊C₄

yes

—

C₈×C₁₀₉

C₄⋊C₄

yes

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄×Q₈

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₄×C₁₀₉

C₄⋊Q₈

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1952

yes

characteristic, complemented, Sylow

C₂×#108

—

109

—

#428

—

109

—

C₂×#107

—

109

—

#405

D₁₀₉

yes

characteristic

#343

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

#117

C₂×D₁₀₉

yes

characteristic

C₂×C₄⋊C₄

—

109

—

#108

—

109

—

#107

—

218

—

#107

—

218

—

#107

—

218

—

#106

—

218

—

#97

—

C₄×C₈

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

109

—

M₁₆

—

218

—

M₁₆

Dic₂₁₈

yes

—

C₂×C₈

—

C₂×C₈

—

218

—

C₂×C₈

Dic₂₁₈

yes

—

C₂×C₈

C₄×D₁₀₉

yes

—

C₂×C₈

C₂²⋊D₁₀₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₀₉

yes

characteristic

C₄⋊C₄

—

218

—

100

C₄⋊C₄

—

218

—

101

C₄⋊C₄

D₄₃₆

yes

characteristic

102

C₄⋊C₄

D₄₃₆

yes

characteristic

103

C₂²⋊C₄

—

104

C₄²

—

105

C₂³

#13182029

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

872

—

109

C₂×C₄

—

436

—

110

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

111

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

112

C₂×C₄

C₂×D₄₃₆

yes

characteristic

113

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

114

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

115

C₂×C₄

#13182029

yes

characteristic

116

C₈

—

117

C₈

—

218

—

118

C₈

#13182008

yes

—

119

C₈

#13182008

yes

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₄×Dic₂₁₈

yes

characteristic

123

C₂²

#16812673

yes

characteristic

124

C₂²

#16812691

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

872

—

128

C₄

C₂×#13182008

yes

characteristic

129

C₄

C₄×D₄₃₆

yes

characteristic

130

C₄

#16812678

yes

characteristic

131

C₄

#16812699

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

#20294924

yes

characteristic

135

C₂

#20295296

yes

characteristic

136

C₂

#20295301

yes

characteristic

	subgroup	G/H
center	C₂²	#16812673
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#405×C₁₀₉	C₂
socle	C₂²×C₁₀₉	#121
prime radical	C₂³×C₁₀₉	#41
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#405	D₁₀₉
hypocenter	C₁₀₉	#1952
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

subgroup

G/H

center

C₂²

#16812673

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#405×C₁₀₉

C₂

C₂²×C₁₀₉

C₂³×C₁₀₉

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

N	H ≅ G/N	type
8. #16812635	C₄	semidirect
69. C₁₀₉	#1952	semidirect

H ≅ G/N

type

8. #16812635

C₄

semidirect

69. C₁₀₉

#1952

semidirect

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

subgroup	index
C₂×#16812635	2
#20295069	2
C₂×#16812633	2
#20294794	2
#405×C₁₀₉	2
#1952	109

subgroup

index

#405×C₁₀₉

109

Subgroups — #28363817

Conjugacy classes of subgroups

Complements

Maximal subgroups