Group #28360393 — subgroups

Subgroups — C₁₀₉×#3139 #28360393

Order 13952.

380 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 192 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	#533×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#491×C₁₀₉	C₂	1	4	yes	maximal, complemented
4	#487×C₁₀₉	C₂	1	4	yes	maximal, complemented
5	#419×C₁₀₉	C₂	1	4	yes	maximal
6	C₄×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₁₀₉	C₂²	1	2	yes	—
8	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
10	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
13	#125×C₁₀₉	C₂²	1	2	yes	—
14	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
15	C₄×Q₈×C₁₀₉	—	2	2	no	—
16	#117×C₁₀₉	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
18	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
19	#103×C₁₀₉	C₂²	1	16	yes	—
20	#97×C₁₀₉	C₂²	1	2	yes	—
21	C₄×C₈×C₁₀₉	C₂²	1	2	yes	—
22	C₂×Q₈×C₁₀₉	—	2	2	no	—
23	C₂×Q₈×C₁₀₉	—	2	2	no	—
24	C₂×Q₈×C₁₀₉	—	2	8	no	—
25	C₂×Q₈×C₁₀₉	C₂³	1	4	yes	—
26	C₂²×C₄×C₁₀₉	—	2	2	no	—
27	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
28	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
29	M₁₆×C₁₀₉	—	2	4	no	—
30	C₂×C₈×C₁₀₉	C₂³	1	4	yes	—
31	C₄⋊C₄×C₁₀₉	—	2	4	no	—
32	C₄⋊C₄×C₁₀₉	—	2	4	no	—
33	C₄⋊C₄×C₁₀₉	—	2	4	no	—
34	C₄⋊C₄×C₁₀₉	—	2	2	no	—
35	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
36	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
37	C₄²×C₁₀₉	—	2	2	no	—
38	C₄²×C₁₀₉	D₄	1	2	yes	—
39	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
40	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
41	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₁₀₉	—	2	8	no	—
43	Q₈×C₁₀₉	—	4	4	no	—
44	Q₈×C₁₀₉	—	4	4	no	—
45	C₂×C₄×C₁₀₉	—	2	4	no	—
46	C₂×C₄×C₁₀₉	—	2	4	no	—
47	C₂×C₄×C₁₀₉	—	2	4	no	—
48	C₂×C₄×C₁₀₉	—	4	4	no	—
49	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₀₉	#41	1	4	yes	—
53	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
54	C₈×C₁₀₉	—	2	4	no	—
55	C₂²×C₁₀₉	—	2	2	no	—
56	C₂²×C₁₀₉	#124	1	2	yes	—
57	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₁₀₉	—	2	2	no	—
59	C₄×C₁₀₉	—	4	4	no	—
60	C₄×C₁₀₉	—	4	4	no	—
61	C₄×C₁₀₉	#124	1	2	yes	—
62	C₄×C₁₀₉	#137	1	4	yes	—
63	C₄×C₁₀₉	C₂×#41	1	2	yes	—
64	C₂×C₁₀₉	—	2	2	no	—
65	C₂×C₁₀₉	C₂×#124	1	1	yes	characteristic
66	C₂×C₁₀₉	C₂×#137	1	2	yes	—
67	#3139	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₁₀₉	#3139	1	1	yes	characteristic, direct product factor, Sylow
69	#533	C₂×C₁₀₉	1	1	yes	characteristic
70	C₂×C₄⋊Q₈	C₂×C₁₀₉	1	1	yes	characteristic
71	#491	C₂×C₁₀₉	1	4	yes	complemented
72	#487	C₂×C₁₀₉	1	4	yes	complemented
73	#419	C₂×C₁₀₉	1	4	yes	—
74	C₄×M₁₆	C₂×C₁₀₉	1	1	yes	characteristic
75	C₂²×Q₈	C₂²×C₁₀₉	1	2	yes	—
76	C₂×M₁₆	C₂²×C₁₀₉	1	2	yes	—
77	C₄⋊Q₈	—	2	2	no	—
78	C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
79	C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
80	C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
81	#125	C₂²×C₁₀₉	1	2	yes	—
82	C₂²⋊Q₈	—	2	2	no	—
83	C₄×Q₈	—	2	2	no	—
84	#117	C₂²×C₁₀₉	1	2	yes	—
85	C₂×C₄⋊C₄	—	2	2	no	—
86	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
87	#103	C₂²×C₁₀₉	1	16	yes	—
88	#97	C₂²×C₁₀₉	1	2	yes	—
89	C₄×C₈	C₂²×C₁₀₉	1	2	yes	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	2	2	no	—
92	C₂×Q₈	—	2	8	no	—
93	C₂×Q₈	C₂³×C₁₀₉	1	4	yes	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	D₄×C₁₀₉	1	2	yes	—
96	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
97	M₁₆	—	2	4	no	—
98	C₂×C₈	C₂³×C₁₀₉	1	4	yes	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	2	4	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
104	C₂²⋊C₄	—	2	2	no	—
105	C₄²	—	2	2	no	—
106	C₄²	D₄×C₁₀₉	1	2	yes	—
107	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
108	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
109	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	4	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
118	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
119	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
120	C₂×C₄	#41×C₁₀₉	1	4	yes	—
121	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
122	C₈	—	2	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#124×C₁₀₉	1	2	yes	—
125	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	4	4	no	—
128	C₄	—	4	4	no	—
129	C₄	#124×C₁₀₉	1	2	yes	—
130	C₄	#137×C₁₀₉	1	4	yes	—
131	C₄	C₂×#41×C₁₀₉	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₂×#124×C₁₀₉	1	1	yes	characteristic
134	C₂	C₂×#137×C₁₀₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#533×C₁₀₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₀₉

C₂

yes

maximal, characteristic

#491×C₁₀₉

C₂

yes

maximal, complemented

#487×C₁₀₉

C₂

yes

maximal, complemented

#419×C₁₀₉

C₂

yes

maximal

C₄×M₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₀₉

C₂²

yes

—

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

—

#125×C₁₀₉

C₂²

yes

—

C₂²⋊Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

—

#117×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

#103×C₁₀₉

C₂²

yes

—

#97×C₁₀₉

C₂²

yes

—

C₄×C₈×C₁₀₉

C₂²

yes

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₈×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#124

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#124

yes

—

C₄×C₁₀₉

#137

yes

—

C₄×C₁₀₉

C₂×#41

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×#124

yes

characteristic

C₂×C₁₀₉

C₂×#137

yes

—

#3139

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₁₀₉

#3139

yes

characteristic, direct product factor, Sylow

#533

C₂×C₁₀₉

yes

characteristic

C₂×C₄⋊Q₈

C₂×C₁₀₉

yes

characteristic

#491

C₂×C₁₀₉

yes

complemented

#487

C₂×C₁₀₉

yes

complemented

#419

C₂×C₁₀₉

yes

—

C₄×M₁₆

C₂×C₁₀₉

yes

characteristic

C₂²×Q₈

C₂²×C₁₀₉

yes

—

C₂×M₁₆

C₂²×C₁₀₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²×C₁₀₉

yes

—

C₄⋊Q₈

C₂²×C₁₀₉

yes

—

C₄⋊Q₈

C₂²×C₁₀₉

yes

—

#125

C₂²×C₁₀₉

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂²×C₁₀₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×C₁₀₉

yes

characteristic

#103

C₂²×C₁₀₉

yes

—

#97

C₂²×C₁₀₉

yes

—

C₄×C₈

C₂²×C₁₀₉

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂³×C₁₀₉

yes

—

C₂²×C₄

—

C₂²×C₄

D₄×C₁₀₉

yes

—

C₂²×C₄

C₂³×C₁₀₉

yes

characteristic

M₁₆

—

C₂×C₈

C₂³×C₁₀₉

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

C₂³×C₁₀₉

yes

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

D₄×C₁₀₉

yes

—

107

C₄²

C₂³×C₁₀₉

yes

characteristic

108

C₄²

C₂³×C₁₀₉

yes

characteristic

109

C₂³

C₂×D₄×C₁₀₉

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂×D₄×C₁₀₉

yes

—

118

C₂×C₄

C₂×D₄×C₁₀₉

yes

characteristic

119

C₂×C₄

C₂×D₄×C₁₀₉

yes

—

120

C₂×C₄

#41×C₁₀₉

yes

—

121

C₂×C₄

C₂⁴×C₁₀₉

yes

characteristic

122

C₈

—

123

C₂²

—

124

C₂²

#124×C₁₀₉

yes

—

125

C₂²

C₂²×D₄×C₁₀₉

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

#124×C₁₀₉

yes

—

130

C₄

#137×C₁₀₉

yes

—

131

C₄

C₂×#41×C₁₀₉

yes

—

132

C₂

—

133

C₂

C₂×#124×C₁₀₉

yes

characteristic

134

C₂

C₂×#137×C₁₀₉

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
prime radical	C₂³×C₁₀₉	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₉

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₁₀₉

prime radical

C₂³×C₁₀₉

C₂×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

N	H ≅ G/N	type
3. #491×C₁₀₉	C₂	semidirect
4. #487×C₁₀₉	C₂	semidirect
67. #3139	C₁₀₉	direct (N × H)
68. C₁₀₉	#3139	direct (N × H)
71. #491	C₂×C₁₀₉	semidirect
72. #487	C₂×C₁₀₉	semidirect

H ≅ G/N

type

3. #491×C₁₀₉

C₂

semidirect

4. #487×C₁₀₉

C₂

semidirect

67. #3139

C₁₀₉

direct (N × H)

68. C₁₀₉

#3139

direct (N × H)

71. #491

C₂×C₁₀₉

semidirect

72. #487

C₂×C₁₀₉

semidirect

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

subgroup	index
#533×C₁₀₉	2
C₂×C₄⋊Q₈×C₁₀₉	2
#491×C₁₀₉	2
#487×C₁₀₉	2
#419×C₁₀₉	2
C₄×M₁₆×C₁₀₉	2
#3139	109

subgroup

index

#533×C₁₀₉

C₂×C₄⋊Q₈×C₁₀₉

#491×C₁₀₉

#487×C₁₀₉

#419×C₁₀₉

C₄×M₁₆×C₁₀₉

#3139

109

Subgroups — C₁₀₉×#3139 #28360393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#3139 #28360393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3139 #28360393