Group #41038032 — subgroups

Subgroups — C₁₀₇×#12068 #41038032

Order 27392.

204 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 102 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	C₁₀₇×#2179	C₂	1	1	yes	maximal, characteristic
2	C₁₀₇×#2171	C₂	1	2	yes	maximal, complemented
3	C₁₀₇×#1369	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₇×#1336	C₂	1	2	yes	maximal, complemented
5	C₁₀₇×#1313	C₂	1	1	yes	maximal, characteristic, complemented
6	#508×C₁₀₇	C₄	1	1	yes	characteristic
7	#504×C₁₀₇	—	2	2	no	—
8	#443×C₁₀₇	C₂²	1	2	yes	—
9	#368×C₁₀₇	C₄	1	1	yes	characteristic
10	#364×C₁₀₇	C₂²	1	2	yes	—
11	#359×C₁₀₇	C₄	1	2	yes	—
12	#350×C₁₀₇	—	2	2	no	—
13	#346×C₁₀₇	C₂²	1	1	yes	characteristic
14	#346×C₁₀₇	C₂²	1	1	yes	characteristic
15	#322×C₁₀₇	C₂²	1	1	yes	characteristic
16	#135×C₁₀₇	C₂×C₄	1	2	yes	—
17	#131×C₁₀₇	—	2	2	no	—
18	Q₃₂×C₁₀₇	C₈	1	1	yes	characteristic, complemented
19	SD₃₂×C₁₀₇	C₈	1	2	yes	complemented
20	D₁₆×C₁₀₇	C₈	1	1	yes	characteristic, complemented
21	M₃₂×C₁₀₇	—	2	2	no	—
22	M₃₂×C₁₀₇	—	4	2	no	—
23	C₂×C₁₆×C₁₀₇	—	4	2	no	—
24	C₂×C₁₆×C₁₀₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₁₀₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₁₀₇	D₄	1	2	yes	—
27	#108×C₁₀₇	C₂×C₄	1	2	yes	—
28	#104×C₁₀₇	—	2	2	no	—
29	C₄×C₈×C₁₀₇	—	2	1	no	—
30	C₄×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
31	#41×C₁₀₇	—	2	2	no	—
32	Q₁₆×C₁₀₇	C₂×C₈	1	2	yes	—
33	SD₁₆×C₁₀₇	—	2	2	no	—
34	D₈×C₁₀₇	C₂×C₈	1	2	yes	—
35	M₁₆×C₁₀₇	—	2	2	no	—
36	M₁₆×C₁₀₇	—	4	2	no	—
37	C₂×C₈×C₁₀₇	—	4	2	no	—
38	C₂×C₈×C₁₀₇	—	4	1	no	—
39	C₂×C₈×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₀₇	C₂×D₄	1	1	yes	characteristic
41	C₄²×C₁₀₇	#41	1	1	yes	characteristic
42	C₁₆×C₁₀₇	—	2	1	no	—
43	C₁₆×C₁₀₇	—	2	2	no	—
44	C₁₆×C₁₀₇	—	4	2	no	—
45	C₁₆×C₁₀₇	C₂×C₈	1	1	yes	characteristic
46	C₁₆×C₁₀₇	C₂×C₈	1	1	yes	characteristic
47	Q₈×C₁₀₇	—	2	2	no	—
48	D₄×C₁₀₇	—	2	2	no	—
49	D₄×C₁₀₇	—	4	2	no	—
50	C₂×C₄×C₁₀₇	—	2	1	no	—
51	C₂×C₄×C₁₀₇	—	4	2	no	—
52	C₂×C₄×C₁₀₇	C₄×D₄	1	1	yes	characteristic
53	C₈×C₁₀₇	—	4	2	no	—
54	C₈×C₁₀₇	—	8	1	no	—
55	C₈×C₁₀₇	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₁₀₇	#131	1	1	yes	characteristic
57	C₈×C₁₀₇	AGL(1,ℤ/8)	1	2	yes	—
58	C₂²×C₁₀₇	—	4	2	no	—
59	C₂²×C₁₀₇	#444	1	1	yes	characteristic
60	C₄×C₁₀₇	—	4	1	no	—
61	C₄×C₁₀₇	—	4	2	no	—
62	C₄×C₁₀₇	C₈×D₄	1	1	yes	characteristic
63	C₄×C₁₀₇	#442	1	1	yes	characteristic
64	#12068	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂×C₁₀₇	—	2	1	no	—
66	C₂×C₁₀₇	—	8	2	no	—
67	C₂×C₁₀₇	#1580	1	1	yes	characteristic
68	#2179	C₂×C₁₀₇	1	1	yes	characteristic
69	#2171	C₂×C₁₀₇	1	2	yes	complemented
70	#1369	C₂×C₁₀₇	1	1	yes	characteristic, complemented
71	#1336	C₂×C₁₀₇	1	2	yes	complemented
72	#1313	C₂×C₁₀₇	1	1	yes	characteristic, complemented
73	C₁₀₇	#12068	1	1	yes	characteristic, direct product factor, Sylow
74	#508	C₄×C₁₀₇	1	1	yes	characteristic
75	#504	—	2	2	no	—
76	#443	C₂²×C₁₀₇	1	2	yes	—
77	#368	C₄×C₁₀₇	1	1	yes	characteristic
78	#364	C₂²×C₁₀₇	1	2	yes	—
79	#359	C₄×C₁₀₇	1	2	yes	—
80	#350	—	2	2	no	—
81	#346	C₂²×C₁₀₇	1	1	yes	characteristic
82	#346	C₂²×C₁₀₇	1	1	yes	characteristic
83	#322	C₂²×C₁₀₇	1	1	yes	characteristic
84	#135	C₂×C₄×C₁₀₇	1	2	yes	—
85	#131	—	2	2	no	—
86	Q₃₂	C₈×C₁₀₇	1	1	yes	characteristic, complemented
87	SD₃₂	C₈×C₁₀₇	1	2	yes	complemented
88	D₁₆	C₈×C₁₀₇	1	1	yes	characteristic, complemented
89	M₃₂	—	2	2	no	—
90	M₃₂	—	4	2	no	—
91	C₂×C₁₆	—	4	2	no	—
92	C₂×C₁₆	C₂×C₄×C₁₀₇	1	1	yes	characteristic
93	C₂×C₁₆	C₂×C₄×C₁₀₇	1	1	yes	characteristic
94	C₂×C₁₆	D₄×C₁₀₇	1	2	yes	—
95	#108	C₂×C₄×C₁₀₇	1	2	yes	—
96	#104	—	2	2	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	C₂³×C₁₀₇	1	1	yes	characteristic
99	#41	—	2	2	no	—
100	Q₁₆	C₂×C₈×C₁₀₇	1	2	yes	—
101	SD₁₆	—	2	2	no	—
102	D₈	C₂×C₈×C₁₀₇	1	2	yes	—
103	M₁₆	—	2	2	no	—
104	M₁₆	—	4	2	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	—	4	1	no	—
107	C₂×C₈	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
108	C₂×C₈	C₂×D₄×C₁₀₇	1	1	yes	characteristic
109	C₄²	#41×C₁₀₇	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	2	2	no	—
112	C₁₆	—	4	2	no	—
113	C₁₆	C₂×C₈×C₁₀₇	1	1	yes	characteristic
114	C₁₆	C₂×C₈×C₁₀₇	1	1	yes	characteristic
115	Q₈	—	2	2	no	—
116	D₄	—	2	2	no	—
117	D₄	—	4	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	C₄×D₄×C₁₀₇	1	1	yes	characteristic
121	C₈	—	4	2	no	—
122	C₈	—	8	1	no	—
123	C₈	C₂²×C₈×C₁₀₇	1	1	yes	characteristic
124	C₈	#131×C₁₀₇	1	1	yes	characteristic
125	C₈	AGL(1,ℤ/8)×C₁₀₇	1	2	yes	—
126	C₂²	—	4	2	no	—
127	C₂²	#444×C₁₀₇	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	4	2	no	—
130	C₄	C₈×D₄×C₁₀₇	1	1	yes	characteristic
131	C₄	#442×C₁₀₇	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	8	2	no	—
134	C₂	C₁₀₇×#1580	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₇×#2179

C₂

yes

maximal, characteristic

C₁₀₇×#2171

C₂

yes

maximal, complemented

C₁₀₇×#1369

C₂

yes

maximal, characteristic, complemented

C₁₀₇×#1336

C₂

yes

maximal, complemented

C₁₀₇×#1313

C₂

yes

maximal, characteristic, complemented

#508×C₁₀₇

C₄

yes

characteristic

#504×C₁₀₇

—

#443×C₁₀₇

C₂²

yes

—

#368×C₁₀₇

C₄

yes

characteristic

#364×C₁₀₇

C₂²

yes

—

#359×C₁₀₇

C₄

yes

—

#350×C₁₀₇

—

#346×C₁₀₇

C₂²

yes

characteristic

#346×C₁₀₇

C₂²

yes

characteristic

#322×C₁₀₇

C₂²

yes

characteristic

#135×C₁₀₇

C₂×C₄

yes

—

#131×C₁₀₇

—

Q₃₂×C₁₀₇

C₈

yes

characteristic, complemented

SD₃₂×C₁₀₇

C₈

yes

complemented

D₁₆×C₁₀₇

C₈

yes

characteristic, complemented

M₃₂×C₁₀₇

—

M₃₂×C₁₀₇

—

C₂×C₁₆×C₁₀₇

—

C₂×C₁₆×C₁₀₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₀₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₀₇

D₄

yes

—

#108×C₁₀₇

C₂×C₄

yes

—

#104×C₁₀₇

—

C₄×C₈×C₁₀₇

—

C₄×C₈×C₁₀₇

C₂³

yes

characteristic

#41×C₁₀₇

—

Q₁₆×C₁₀₇

C₂×C₈

yes

—

SD₁₆×C₁₀₇

—

D₈×C₁₀₇

C₂×C₈

yes

—

M₁₆×C₁₀₇

—

M₁₆×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₀₇

C₂×D₄

yes

characteristic

C₄²×C₁₀₇

#41

yes

characteristic

C₁₆×C₁₀₇

—

C₁₆×C₁₀₇

—

C₁₆×C₁₀₇

—

C₁₆×C₁₀₇

C₂×C₈

yes

characteristic

C₁₆×C₁₀₇

C₂×C₈

yes

characteristic

Q₈×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₄×D₄

yes

characteristic

C₈×C₁₀₇

—

C₈×C₁₀₇

—

C₈×C₁₀₇

C₂²×C₈

yes

C₈×C₁₀₇

yes

C₈×C₁₀₇

yes

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#444

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₈×D₄

yes

C₄×C₁₀₇

yes

C₁₀₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

#1580

yes

characteristic

#2179

C₂×C₁₀₇

yes

characteristic

#2171

C₂×C₁₀₇

yes

complemented

#1369

C₂×C₁₀₇

yes

characteristic, complemented

#1336

C₂×C₁₀₇

yes

complemented

#1313

C₂×C₁₀₇

yes

characteristic, complemented

C₁₀₇

#12068

yes

characteristic, direct product factor, Sylow

#508

C₄×C₁₀₇

yes

characteristic

#504

—

#443

C₂²×C₁₀₇

yes

—

#368

C₄×C₁₀₇

yes

characteristic

#364

C₂²×C₁₀₇

yes

—

#359

C₄×C₁₀₇

yes

—

#350

—

#346

C₂²×C₁₀₇

yes

characteristic

#346

C₂²×C₁₀₇

yes

characteristic

#322

C₂²×C₁₀₇

yes

characteristic

#135

C₂×C₄×C₁₀₇

yes

—

#131

—

Q₃₂

C₈×C₁₀₇

yes

characteristic, complemented

SD₃₂

C₈×C₁₀₇

yes

complemented

D₁₆

C₈×C₁₀₇

yes

characteristic, complemented

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄×C₁₀₇

yes

characteristic

C₂×C₁₆

C₂×C₄×C₁₀₇

yes

characteristic

C₂×C₁₆

D₄×C₁₀₇

yes

—

#108

C₂×C₄×C₁₀₇

yes

—

#104

—

C₄×C₈

—

C₄×C₈

C₂³×C₁₀₇

yes

characteristic

#41

—

100

Q₁₆

C₂×C₈×C₁₀₇

yes

—

101

SD₁₆

—

102

D₈

C₂×C₈×C₁₀₇

yes

—

103

M₁₆

—

104

M₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

C₂²×C₄×C₁₀₇

yes

characteristic

108

C₂×C₈

C₂×D₄×C₁₀₇

yes

characteristic

109

C₄²

#41×C₁₀₇

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

113

C₁₆

C₂×C₈×C₁₀₇

yes

characteristic

114

C₁₆

C₂×C₈×C₁₀₇

yes

characteristic

115

Q₈

—

116

D₄

—

117

D₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₄×D₄×C₁₀₇

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

C₂²×C₈×C₁₀₇

yes

characteristic

124

C₈

#131×C₁₀₇

yes

characteristic

125

C₈

AGL(1,ℤ/8)×C₁₀₇

yes

—

126

C₂²

—

127

C₂²

#444×C₁₀₇

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

C₈×D₄×C₁₀₇

yes

characteristic

131

C₄

#442×C₁₀₇

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

C₁₀₇×#1580

yes

characteristic

	subgroup	G/H
center	C₄×C₁₀₇	#442
commutator	C₈	C₂²×C₈×C₁₀₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₀₇
socle	C₂×C₁₀₇	#1580
prime radical	#508×C₁₀₇	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₇ ◁ C₂×C₈×C₁₀₇ ◁ #346×C₁₀₇ ◁ G

subgroup

G/H

center

C₄×C₁₀₇

#442

commutator

C₈

C₂²×C₈×C₁₀₇

Frattini, cosocle

C₄×C₈

C₂³×C₁₀₇

socle

C₂×C₁₀₇

#1580

prime radical

#508×C₁₀₇

C₄

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₁₀₇ ◁ C₂×C₈×C₁₀₇ ◁ #346×C₁₀₇ ◁ G

N	H ≅ G/N	type
2. C₁₀₇×#2171	C₂	semidirect
3. C₁₀₇×#1369	C₂	semidirect
4. C₁₀₇×#1336	C₂	semidirect
5. C₁₀₇×#1313	C₂	semidirect
18. Q₃₂×C₁₀₇	C₈	semidirect
19. SD₃₂×C₁₀₇	C₈	semidirect
20. D₁₆×C₁₀₇	C₈	semidirect
64. #12068	C₁₀₇	direct (N × H)
69. #2171	C₂×C₁₀₇	semidirect
70. #1369	C₂×C₁₀₇	semidirect
71. #1336	C₂×C₁₀₇	semidirect
72. #1313	C₂×C₁₀₇	semidirect
73. C₁₀₇	#12068	direct (N × H)
86. Q₃₂	C₈×C₁₀₇	semidirect
87. SD₃₂	C₈×C₁₀₇	semidirect
88. D₁₆	C₈×C₁₀₇	semidirect

H ≅ G/N

type

2. C₁₀₇×#2171

C₂

semidirect

3. C₁₀₇×#1369

C₂

semidirect

4. C₁₀₇×#1336

C₂

semidirect

5. C₁₀₇×#1313

C₂

semidirect

18. Q₃₂×C₁₀₇

C₈

semidirect

19. SD₃₂×C₁₀₇

C₈

semidirect

20. D₁₆×C₁₀₇

C₈

semidirect

64. #12068

C₁₀₇

direct (N × H)

69. #2171

C₂×C₁₀₇

semidirect

70. #1369

C₂×C₁₀₇

semidirect

71. #1336

C₂×C₁₀₇

semidirect

72. #1313

C₂×C₁₀₇

semidirect

73. C₁₀₇

#12068

direct (N × H)

86. Q₃₂

C₈×C₁₀₇

semidirect

87. SD₃₂

C₈×C₁₀₇

semidirect

88. D₁₆

C₈×C₁₀₇

semidirect

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

subgroup	index
C₁₀₇×#2179	2
C₁₀₇×#2171	2
C₁₀₇×#1369	2
C₁₀₇×#1336	2
C₁₀₇×#1313	2
#12068	107

subgroup

index

C₁₀₇×#2179

C₁₀₇×#2171

C₁₀₇×#1369

C₁₀₇×#1336

C₁₀₇×#1313

#12068

107

Subgroups — C₁₀₇×#12068 #41038032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C107×#12068 #41038032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#12068 #41038032