Group #28378852 — subgroups

Subgroups — #28378852

Order 13952.

220 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 92 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×#16812770	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812769	C₂	1	1	yes	maximal, characteristic
3	C₄×#13182039	C₂	1	2	yes	maximal
4	#20295902	C₂	1	2	yes	maximal
5	#97×D₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182039	C₂²	1	2	yes	—
7	C₂×#13182039	C₂²	1	2	yes	—
8	#16812770	C₄	1	4	yes	complemented
9	#16812769	C₄	1	4	yes	complemented
10	#16812754	—	2	2	no	—
11	C₄×#11724786	—	2	2	no	—
12	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
14	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
15	#16812610	C₄	1	1	yes	characteristic
16	#16812609	C₄	1	1	yes	characteristic
17	#16812609	C₄	1	1	yes	characteristic
18	#97×C₁₀₉	C₄	1	1	yes	characteristic
19	#13182039	—	2	2	no	—
20	#13182039	—	2	2	no	—
21	#13182039	C₂×C₄	1	4	yes	—
22	#13182039	C₂×C₄	1	4	yes	—
23	C₂×#11724786	—	2	2	no	—
24	C₂×#11724786	—	2	2	no	—
25	C₂×C₄×D₁₀₉	Q₈	1	2	yes	—
26	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
27	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
28	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
29	C₈×D₁₀₉	—	2	1	no	—
30	C₈×D₁₀₉	—	2	1	no	—
31	C₈×D₁₀₉	C₂×C₄	1	2	yes	—
32	C₈×D₁₀₉	C₂×C₄	1	2	yes	—
33	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₁₀₉	D₄	1	2	yes	—
35	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×D₁₀₉	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₁₀₉	C₄⋊C₄	1	2	yes	—
42	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
43	C₄×D₁₀₉	—	4	2	no	—
44	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
45	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
46	C₄×D₁₀₉	#41	1	1	yes	characteristic
47	C₄×D₁₀₉	#41	1	1	yes	characteristic
48	#11724786	—	2	4	no	—
49	#11724786	—	2	4	no	—
50	C₈×C₁₀₉	—	2	1	no	—
51	C₈×C₁₀₉	C₄²	1	2	yes	—
52	#11724784	—	2	1	no	—
53	#11724784	C₄²	1	2	yes	—
54	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×D₁₀₉	—	2	2	no	—
56	C₂×D₁₀₉	C₄×Q₈	1	1	yes	characteristic
57	C₂×D₁₀₉	C₄×Q₈	1	1	yes	characteristic
58	C₄×C₁₀₉	—	2	2	no	—
59	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
60	C₄×C₁₀₉	#117	1	1	yes	characteristic
61	Dic₁₀₉	—	2	2	no	—
62	Dic₁₀₉	C₄×D₄	1	1	yes	characteristic
63	Dic₁₀₉	C₄×D₄	1	1	yes	characteristic
64	C₂×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₁₀₉	#430	1	2	yes	—
66	D₁₀₉	—	2	1	no	—
67	D₁₀₉	—	2	1	no	—
68	#1778	—	109	1	no	maximal, Sylow
69	C₁₀₉	#1778	1	1	yes	characteristic, complemented, Sylow
70	C₂×#108	—	109	1	no	—
71	C₂×#108	—	109	1	no	—
72	C₄×M₁₆	—	109	2	no	—
73	C₂×#97	—	109	1	no	—
74	#341	—	109	2	no	—
75	C₂×M₁₆	—	109	2	no	—
76	C₂×M₁₆	—	109	2	no	—
77	C₂²×C₈	—	109	1	no	—
78	C₂²×C₈	—	109	1	no	—
79	C₂×C₄²	—	109	1	no	—
80	#108	—	109	4	no	—
81	#108	—	109	4	no	—
82	#97	—	109	1	no	—
83	#97	—	109	1	no	—
84	#97	—	109	1	no	—
85	#97	—	218	2	no	—
86	#97	#89137	1	1	yes	characteristic
87	C₄×C₈	—	218	2	no	—
88	C₂²×C₄	—	109	1	no	—
89	C₂²×C₄	—	109	2	no	—
90	M₁₆	—	109	4	no	—
91	M₁₆	—	109	4	no	—
92	M₁₆	—	218	2	no	—
93	M₁₆	—	218	2	no	—
94	C₂×C₈	—	109	1	no	—
95	C₂×C₈	—	109	2	no	—
96	C₂×C₈	—	109	2	no	—
97	C₂×C₈	—	109	1	no	—
98	C₂×C₈	—	218	1	no	—
99	C₂×C₈	—	218	1	no	—
100	C₂×C₈	—	218	2	no	—
101	C₂×C₈	—	218	2	no	—
102	C₂×C₈	C₂×#89137	1	1	yes	characteristic
103	C₂×C₈	C₂×#89137	1	1	yes	characteristic
104	C₄²	—	109	2	no	—
105	C₄²	—	109	1	no	—
106	C₄²	C₂×#89137	1	1	yes	characteristic
107	C₂³	—	109	1	no	—
108	C₂×C₄	—	109	1	no	—
109	C₂×C₄	—	109	1	no	—
110	C₂×C₄	—	109	1	no	—
111	C₂×C₄	—	109	1	no	—
112	C₂×C₄	—	109	1	no	—
113	C₂×C₄	—	109	2	no	—
114	C₂×C₄	—	436	2	no	—
115	C₂×C₄	C₄⋊#89137	1	2	yes	—
116	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
117	C₈	—	2	1	no	—
118	C₈	—	109	2	no	—
119	C₈	—	218	4	no	—
120	C₈	—	218	4	no	—
121	C₈	—	218	1	no	—
122	C₈	C₄×#89137	1	2	yes	—
123	C₂²	—	109	1	no	—
124	C₂²	—	109	1	no	—
125	C₂²	—	218	2	no	—
126	C₂²	C₂×C₄⋊#89137	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	109	1	no	—
129	C₄	—	109	1	no	—
130	C₄	—	218	2	no	—
131	C₄	C₂×C₄×#89137	1	1	yes	characteristic
132	C₄	#16812778	1	1	yes	characteristic
133	C₂	—	218	1	no	—
134	C₂	—	218	1	no	—
135	C₂	C₄×C₄⋊#89137	1	1	yes	characteristic
136	C₂	#20296041	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812770

C₂

yes

maximal, characteristic

C₂×#16812769

C₂

yes

maximal, characteristic

C₄×#13182039

C₂

yes

maximal

#20295902

C₂

yes

maximal

#97×D₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182039

C₂²

yes

—

C₂×#13182039

C₂²

yes

—

#16812770

C₄

yes

complemented

#16812769

C₄

yes

complemented

#16812754

—

C₄×#11724786

—

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

C₄²×D₁₀₉

C₂²

yes

characteristic

#16812610

C₄

yes

characteristic

#16812609

C₄

yes

characteristic

#16812609

C₄

yes

characteristic

#97×C₁₀₉

C₄

yes

—

—

C₂×C₄

yes

—

#13182039

C₂×C₄

yes

—

C₂×#11724786

—

C₂×#11724786

—

C₂×C₄×D₁₀₉

Q₈

yes

—

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₈×D₁₀₉

—

C₈×D₁₀₉

—

C₈×D₁₀₉

C₂×C₄

yes

—

C₈×D₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂²×C₄

yes

characteristic

C₄×D₁₀₉

C₂²×C₄

yes

C₄×D₁₀₉

yes

C₄×D₁₀₉

yes

—

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₄²

yes

—

#11724784

—

#11724784

C₄²

yes

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

C₄×Q₈

yes

characteristic

C₂×D₁₀₉

C₄×Q₈

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄²

yes

characteristic

C₄×C₁₀₉

#117

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

C₄×D₄

yes

characteristic

Dic₁₀₉

C₄×D₄

yes

characteristic

C₂×C₁₀₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₀₉

#430

yes

—

D₁₀₉

—

D₁₀₉

—

#1778

—

109

maximal, Sylow

C₁₀₉

#1778

yes

characteristic, complemented, Sylow

C₂×#108

—

109

—

C₂×#108

—

109

—

C₄×M₁₆

—

109

—

C₂×#97

—

109

—

#341

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

#108

—

109

—

#108

—

109

—

#97

—

109

—

#97

—

109

—

#97

—

109

—

#97

—

218

—

#97

#89137

yes

characteristic

C₄×C₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

M₁₆

—

109

—

M₁₆

—

109

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

100

C₂×C₈

—

218

—

101

C₂×C₈

—

218

—

102

C₂×C₈

C₂×#89137

yes

characteristic

103

C₂×C₈

C₂×#89137

yes

characteristic

104

C₄²

—

109

—

105

C₄²

—

109

—

106

C₄²

C₂×#89137

yes

characteristic

107

C₂³

—

109

—

108

C₂×C₄

—

109

—

109

C₂×C₄

—

109

—

110

C₂×C₄

—

109

—

111

C₂×C₄

—

109

—

112

C₂×C₄

—

109

—

113

C₂×C₄

—

109

—

114

C₂×C₄

—

436

—

115

C₂×C₄

C₄⋊#89137

yes

—

116

C₂×C₄

C₂²×#89137

yes

characteristic

117

C₈

—

118

C₈

—

109

—

119

C₈

—

218

—

120

C₈

—

218

—

121

C₈

—

218

—

122

C₈

C₄×#89137

yes

—

123

C₂²

—

109

—

124

C₂²

—

109

—

125

C₂²

—

218

—

126

C₂²

C₂×C₄⋊#89137

yes

characteristic

127

C₄

—

128

C₄

—

109

—

129

C₄

—

109

—

130

C₄

—

218

—

131

C₄

C₂×C₄×#89137

yes

characteristic

132

C₄

#16812778

yes

characteristic

133

C₂

—

218

—

134

C₂

—

218

—

135

C₂

C₄×C₄⋊#89137

yes

characteristic

136

C₂

#20296041

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#89137
commutator	C₄×C₁₀₉	C₂×C₄²
Frattini	C₂×C₄	C₂²×#89137
Fitting	#97×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂²×D₁₀₉	C₂×Q₈
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	#97	#89137
hypocenter	C₁₀₉	#1778
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊#89137

commutator

C₄×C₁₀₉

C₂×C₄²

Frattini

C₂×C₄

C₂²×#89137

Fitting

#97×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂²×D₁₀₉

C₂×Q₈

cosocle

C₂×C₄×D₁₀₉

C₂³

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
8. #16812770	C₄	semidirect
9. #16812769	C₄	semidirect
69. C₁₀₉	#1778	semidirect

H ≅ G/N

type

8. #16812770

C₄

semidirect

9. #16812769

C₄

semidirect

69. C₁₀₉

#1778

semidirect

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

subgroup	index
C₂×#16812770	2
C₂×#16812769	2
C₄×#13182039	2
#20295902	2
#97×D₁₀₉	2
#1778	109

subgroup

index

#97×D₁₀₉

109

Subgroups — #28378852

Conjugacy classes of subgroups

Complements

Maximal subgroups