Group #28379678 — subgroups

Subgroups — #28379678

Order 13952.

484 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 206 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×#16812787	C₂	1	2	yes	maximal
2	#20296033	C₂	1	6	yes	maximal, complemented
3	#20296030	C₂	1	3	yes	maximal, complemented
4	D₁₀₉⋊(C₄⋊C₈)	C₂	1	3	yes	maximal, complemented
5	C₄×#13182030	C₂	1	1	yes	maximal, characteristic
6	#16812787	—	2	8	no	—
7	C₂×#13182039	C₂²	1	6	yes	—
8	C₂×D₁₀₉⋊C₈	C₂²	1	6	yes	—
9	#16812766	C₂²	1	6	yes	—
10	C₄⋊#11724786	C₂²	1	1	yes	characteristic
11	C₄⋊#11724786	C₂²	1	3	yes	—
12	C₄⋊#11724786	C₂²	1	6	yes	—
13	C₂×#13182030	C₂²	1	1	yes	characteristic
14	Q₈×Dic₁₀₉	C₄	1	1	yes	characteristic
15	C₄×D₄₃₆	C₄	1	3	yes	—
16	#16812661	C₄	1	3	yes	—
17	#16812657	C₂²	1	3	yes	—
18	C₄²×D₁₀₉	C₂²	1	3	yes	—
19	C₄×Q₈×C₁₀₉	C₄	1	1	yes	characteristic
20	#13182039	—	2	12	no	—
21	D₁₀₉⋊C₈	—	2	12	no	—
22	C₂×#11724786	C₂³	1	6	yes	—
23	C₂×#11724786	C₂³	1	2	yes	—
24	#13182030	D₄	1	4	yes	—
25	#13182030	Q₈	1	4	yes	—
26	C₂×D₄₃₆	C₂×C₄	1	3	yes	—
27	C₂×C₄×D₁₀₉	C₂×C₄	1	3	yes	—
28	C₂×C₄×D₁₀₉	C₂×C₄	1	3	yes	—
29	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
30	C₄⋊Dic₁₀₉	C₂×C₄	1	3	yes	—
31	#13182006	C₂×C₄	1	6	yes	—
32	C₄×Dic₁₀₉	C₂×C₄	1	3	yes	—
33	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
34	C₄×Dic₁₀₉	C₂³	1	3	yes	—
35	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₄×C₁₀₉	C₂×C₄	1	3	yes	—
37	C₄²×C₁₀₉	C₂×C₄	1	3	yes	—
38	C₂²×D₁₀₉	C₂²×C₄	1	3	yes	—
39	Q₈×C₁₀₉	C₄⋊C₄	1	4	yes	—
40	C₂×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
41	C₂×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
42	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×Dic₁₀₉	C₂²×C₄	1	3	yes	—
45	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
46	D₄₃₆	C₄⋊C₄	1	12	yes	—
47	C₄×D₁₀₉	—	2	6	no	—
48	C₄×D₁₀₉	—	4	3	no	—
49	C₄×D₁₀₉	C₂×D₄	1	6	yes	—
50	C₄×D₁₀₉	C₂×Q₈	1	6	yes	—
51	#11724786	—	2	2	no	—
52	#11724786	—	2	6	no	—
53	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
54	C₂×D₁₀₉	—	2	3	no	—
55	C₂×D₁₀₉	—	2	3	no	—
56	C₂×D₁₀₉	C₂×C₄⋊C₄	1	6	yes	—
57	C₄×C₁₀₉	—	2	3	no	—
58	C₄×C₁₀₉	C₂×C₄⋊C₄	1	6	yes	—
59	C₄×C₁₀₉	#131	1	2	yes	complemented
60	Dic₁₀₉	—	2	3	no	—
61	Dic₁₀₉	—	2	1	no	—
62	Dic₁₀₉	C₂²×D₄	1	1	yes	characteristic
63	Dic₁₀₉	C₂²×Q₈	1	1	yes	characteristic
64	C₂×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
66	C₂×C₁₀₉	#568	1	1	yes	characteristic
67	D₁₀₉	—	2	6	no	—
68	#2906	—	109	1	no	maximal, Sylow
69	C₁₀₉	#2906	1	1	yes	characteristic, complemented, Sylow
70	C₂×#131	—	109	2	no	—
71	C₄×#41	—	109	1	no	—
72	#424	—	109	6	no	—
73	#423	—	109	3	no	—
74	C₂×C₄⋊C₈	—	109	3	no	—
75	C₂×#41	—	109	1	no	—
76	#131	—	218	8	no	—
77	C₂×M₁₆	—	109	6	no	—
78	C₂²×C₈	—	109	6	no	—
79	C₄×Q₈	—	109	1	no	—
80	C₄×Q₈	#89137	1	1	yes	characteristic
81	C₄×D₄	—	109	3	no	—
82	C₄×D₄	—	109	3	no	—
83	#117	—	109	3	no	—
84	C₂×C₄²	—	109	3	no	—
85	C₄⋊C₈	—	109	1	no	—
86	C₄⋊C₈	—	109	3	no	—
87	C₄⋊C₈	—	109	6	no	—
88	C₄⋊C₈	—	109	6	no	—
89	#41	—	109	4	no	—
90	#41	—	109	4	no	—
91	C₂×Q₈	C₂×#89137	1	1	yes	characteristic
92	C₂×D₄	—	109	3	no	—
93	C₂²×C₄	—	109	3	no	—
94	C₂²×C₄	—	109	3	no	—
95	C₂²×C₄	—	109	3	no	—
96	M₁₆	—	218	12	no	—
97	C₂×C₈	—	109	2	no	—
98	C₂×C₈	—	109	6	no	—
99	C₂×C₈	—	218	12	no	—
100	C₄⋊C₄	—	109	3	no	—
101	C₄⋊C₄	C₂×#89137	1	3	yes	—
102	C₂²⋊C₄	—	109	6	no	—
103	C₄²	—	109	3	no	—
104	C₄²	—	109	1	no	—
105	C₄²	—	109	3	no	—
106	C₄²	C₂×#89137	1	3	yes	—
107	C₂³	—	109	3	no	—
108	Q₈	C₄⋊#89137	1	4	yes	—
109	D₄	—	109	12	no	—
110	C₂×C₄	—	109	1	no	—
111	C₂×C₄	—	109	6	no	—
112	C₂×C₄	—	109	6	no	—
113	C₂×C₄	—	109	1	no	—
114	C₂×C₄	—	109	3	no	—
115	C₂×C₄	—	436	3	no	—
116	C₂×C₄	—	218	6	no	—
117	C₂×C₄	C₂²×#89137	1	3	yes	—
118	C₂×C₄	C₂²×#89137	1	3	yes	—
119	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
120	C₈	—	218	2	no	—
121	C₈	—	218	6	no	—
122	C₂²	—	109	6	no	—
123	C₂²	—	218	3	no	—
124	C₂²	—	218	3	no	—
125	C₂²	C₂³×#89137	1	1	yes	characteristic
126	C₄	—	2	3	no	—
127	C₄	—	109	1	no	—
128	C₄	—	109	1	no	—
129	C₄	—	218	3	no	—
130	C₄	—	218	1	no	—
131	C₄	C₂×C₄⋊#89137	1	6	yes	—
132	C₄	#16812787	1	2	yes	complemented
133	C₂	—	218	6	no	—
134	C₂	C₂²×C₄⋊#89137	1	1	yes	characteristic
135	C₂	C₂×#16812787	1	1	yes	characteristic
136	C₂	#20296105	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812787

C₂

yes

maximal

#20296033

C₂

yes

maximal, complemented

#20296030

C₂

yes

maximal, complemented

D₁₀₉⋊(C₄⋊C₈)

C₂

yes

maximal, complemented

C₄×#13182030

C₂

yes

maximal, characteristic

#16812787

—

C₂×#13182039

C₂²

yes

—

C₂×D₁₀₉⋊C₈

C₂²

yes

—

#16812766

C₂²

yes

—

C₄⋊#11724786

C₂²

yes

characteristic

C₄⋊#11724786

C₂²

yes

—

C₄⋊#11724786

C₂²

yes

—

C₂×#13182030

C₂²

yes

characteristic

Q₈×Dic₁₀₉

C₄

yes

characteristic

C₄×D₄₃₆

C₄

yes

—

#16812661

C₄

yes

—

#16812657

C₂²

yes

—

C₄²×D₁₀₉

C₂²

yes

—

C₄×Q₈×C₁₀₉

C₄

yes

characteristic

#13182039

—

D₁₀₉⋊C₈

—

C₂×#11724786

C₂³

yes

—

C₂×#11724786

C₂³

yes

—

#13182030

D₄

yes

—

#13182030

Q₈

yes

—

C₂×D₄₃₆

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂³

yes

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182006

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₄×Dic₁₀₉

C₂³

yes

—

C₂×Q₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₂²×D₁₀₉

C₂²×C₄

yes

—

Q₈×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²×C₄

yes

—

C₂×Dic₁₀₉

C₂⁴

yes

characteristic

D₄₃₆

C₄⋊C₄

yes

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×D₄

yes

—

C₄×D₁₀₉

C₂×Q₈

yes

—

#11724786

—

#11724786

—

C₂²×C₁₀₉

C₂³×C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂×C₄⋊C₄

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

—

C₄×C₁₀₉

#131

yes

complemented

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂²×D₄

yes

characteristic

Dic₁₀₉

C₂²×Q₈

yes

characteristic

C₂×C₁₀₉

C₂²×C₄⋊C₄

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

#2906

—

109

maximal, Sylow

C₁₀₉

#2906

yes

characteristic, complemented, Sylow

C₂×#131

—

109

—

C₄×#41

—

109

—

#424

—

109

—

#423

—

109

—

C₂×C₄⋊C₈

—

109

—

C₂×#41

—

109

—

#131

—

218

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

C₄×Q₈

—

109

—

C₄×Q₈

#89137

yes

characteristic

C₄×D₄

—

109

—

C₄×D₄

—

109

—

#117

—

109

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

#41

—

109

—

#41

—

109

—

C₂×Q₈

C₂×#89137

yes

characteristic

C₂×D₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

100

C₄⋊C₄

—

109

—

101

C₄⋊C₄

C₂×#89137

yes

—

102

C₂²⋊C₄

—

109

—

103

C₄²

—

109

—

104

C₄²

—

109

—

105

C₄²

—

109

—

106

C₄²

C₂×#89137

yes

—

107

C₂³

—

109

—

108

Q₈

C₄⋊#89137

yes

—

109

D₄

—

109

—

110

C₂×C₄

—

109

—

111

C₂×C₄

—

109

—

112

C₂×C₄

—

109

—

113

C₂×C₄

—

109

—

114

C₂×C₄

—

109

—

115

C₂×C₄

—

436

—

116

C₂×C₄

—

218

—

117

C₂×C₄

C₂²×#89137

yes

—

118

C₂×C₄

C₂²×#89137

yes

—

119

C₂×C₄

C₂²×#89137

yes

characteristic

120

C₈

—

218

—

121

C₈

—

218

—

122

C₂²

—

109

—

123

C₂²

—

218

—

124

C₂²

—

218

—

125

C₂²

C₂³×#89137

yes

characteristic

126

C₄

—

127

C₄

—

109

—

128

C₄

—

109

—

129

C₄

—

218

—

130

C₄

—

218

—

131

C₄

C₂×C₄⋊#89137

yes

—

132

C₄

#16812787

yes

complemented

133

C₂

—

218

—

134

C₂

C₂²×C₄⋊#89137

yes

characteristic

135

C₂

C₂×#16812787

yes

characteristic

136

C₂

#20296105

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#89137
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	C₄×Q₈×C₁₀₉	C₄
prime radical	C₂×#13182030	C₂²
cosocle	C₂×Dic₁₀₉	C₂⁴
hypercenter	C₄×Q₈	#89137
hypocenter	C₁₀₉	#2906
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#89137

commutator, socle

C₂²×C₁₀₉

C₂³×C₄

Fitting

C₄×Q₈×C₁₀₉

C₄

prime radical

C₂×#13182030

C₂²

cosocle

C₂×Dic₁₀₉

C₂⁴

C₄×Q₈

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
2. #20296033	C₂	semidirect
3. #20296030	C₂	semidirect
4. D₁₀₉⋊(C₄⋊C₈)	C₂	semidirect
59. C₄×C₁₀₉	#131	semidirect
69. C₁₀₉	#2906	semidirect
132. C₄	#16812787	semidirect

H ≅ G/N

type

2. #20296033

C₂

semidirect

3. #20296030

C₂

semidirect

4. D₁₀₉⋊(C₄⋊C₈)

C₂

semidirect

59. C₄×C₁₀₉

#131

semidirect

69. C₁₀₉

#2906

semidirect

132. C₄

#16812787

semidirect

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

subgroup	index
C₂×#16812787	2
#20296033	2
#20296030	2
D₁₀₉⋊(C₄⋊C₈)	2
C₄×#13182030	2
#2906	109

subgroup

index

D₁₀₉⋊(C₄⋊C₈)

109

Subgroups — #28379678

Conjugacy classes of subgroups

Complements

Maximal subgroups