Group #31778196 — subgroups

Subgroups — #3185×C₁₃₇ #31778196

Order 17536.

358 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 184 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×C₁₃₇	C₂	1	2	yes	maximal
2	#498×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#494×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#487×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#470×C₁₃₇	C₂	1	4	yes	maximal
6	#441×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	C₄×M₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₁₆×C₁₃₇	C₂²	1	4	yes	—
9	C₂×M₁₆×C₁₃₇	C₂²	1	2	yes	—
10	C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
12	#125×C₁₃₇	—	2	2	no	—
13	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
14	C₂²⋊Q₈×C₁₃₇	C₂²	1	4	yes	—
15	C₄×Q₈×C₁₃₇	—	2	2	no	—
16	C₄×Q₈×C₁₃₇	C₂²	1	4	yes	—
17	#117×C₁₃₇	—	2	2	no	—
18	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
19	#106×C₁₃₇	C₂²	1	4	yes	—
20	#103×C₁₃₇	C₂²	1	8	yes	—
21	#97×C₁₃₇	C₂²	1	2	yes	—
22	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
23	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₁₃₇	—	2	2	no	—
25	C₂×Q₈×C₁₃₇	C₂³	1	4	yes	—
26	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
27	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
28	Q₁₆×C₁₃₇	—	2	8	no	—
29	M₁₆×C₁₃₇	—	2	4	no	—
30	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
31	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
32	C₄⋊C₄×C₁₃₇	—	2	2	no	—
33	C₄⋊C₄×C₁₃₇	—	2	2	no	—
34	C₄⋊C₄×C₁₃₇	—	2	2	no	—
35	C₄⋊C₄×C₁₃₇	—	2	4	no	—
36	C₄⋊C₄×C₁₃₇	—	2	4	no	—
37	C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
38	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
39	C₄²×C₁₃₇	—	2	4	no	—
40	C₄²×C₁₃₇	D₄	1	2	yes	—
41	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
42	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
43	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₁₃₇	—	2	8	no	—
45	Q₈×C₁₃₇	—	4	4	no	—
46	C₂×C₄×C₁₃₇	—	2	4	no	—
47	C₂×C₄×C₁₃₇	—	2	2	no	—
48	C₂×C₄×C₁₃₇	—	2	4	no	—
49	C₂×C₄×C₁₃₇	—	2	4	no	—
50	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
54	C₈×C₁₃₇	—	2	2	no	—
55	C₈×C₁₃₇	#41	1	4	yes	—
56	C₂²×C₁₃₇	—	2	1	no	—
57	C₂²×C₁₃₇	—	2	2	no	—
58	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₁₃₇	—	2	1	no	—
60	C₄×C₁₃₇	—	2	2	no	—
61	C₄×C₁₃₇	—	4	4	no	—
62	C₄×C₁₃₇	—	4	4	no	—
63	C₄×C₁₃₇	#137	1	4	yes	—
64	C₄×C₁₃₇	C₂×#41	1	2	yes	—
65	C₂×C₁₃₇	—	4	1	no	—
66	C₂×C₁₃₇	#532	1	1	yes	characteristic
67	C₂×C₁₃₇	C₂×#137	1	2	yes	—
68	C₁₃₇	#3185	1	1	yes	characteristic, direct product factor, Sylow
69	#3185	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	#533	C₂×C₁₃₇	1	2	yes	—
71	#498	C₂×C₁₃₇	1	1	yes	characteristic, complemented
72	#494	C₂×C₁₃₇	1	1	yes	characteristic, complemented
73	#487	C₂×C₁₃₇	1	2	yes	complemented
74	#470	C₂×C₁₃₇	1	4	yes	—
75	#441	C₂×C₁₃₇	1	4	yes	complemented
76	C₄×M₁₆	C₂×C₁₃₇	1	1	yes	characteristic
77	C₂×Q₁₆	C₂²×C₁₃₇	1	4	yes	—
78	C₂×M₁₆	C₂²×C₁₃₇	1	2	yes	—
79	C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
80	C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
81	#125	—	2	2	no	—
82	C₂²⋊Q₈	—	2	2	no	—
83	C₂²⋊Q₈	C₂²×C₁₃₇	1	4	yes	—
84	C₄×Q₈	—	2	2	no	—
85	C₄×Q₈	C₂²×C₁₃₇	1	4	yes	—
86	#117	—	2	2	no	—
87	C₂×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
88	#106	C₂²×C₁₃₇	1	4	yes	—
89	#103	C₂²×C₁₃₇	1	8	yes	—
90	#97	C₂²×C₁₃₇	1	2	yes	—
91	C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
92	C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	C₂³×C₁₃₇	1	4	yes	—
95	C₂²×C₄	D₄×C₁₃₇	1	2	yes	—
96	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
97	Q₁₆	—	2	8	no	—
98	M₁₆	—	2	4	no	—
99	C₂×C₈	C₂³×C₁₃₇	1	2	yes	—
100	C₂×C₈	C₂³×C₁₃₇	1	2	yes	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	4	no	—
105	C₄⋊C₄	—	2	4	no	—
106	C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
107	C₂²⋊C₄	—	2	4	no	—
108	C₄²	—	2	4	no	—
109	C₄²	D₄×C₁₃₇	1	2	yes	—
110	C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
111	C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
112	C₂³	C₂×D₄×C₁₃₇	1	1	yes	characteristic
113	Q₈	—	2	8	no	—
114	Q₈	—	4	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
120	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
121	C₂×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
122	C₂×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	#41×C₁₃₇	1	4	yes	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	2	no	—
127	C₂²	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	2	2	no	—
130	C₄	—	4	4	no	—
131	C₄	—	4	4	no	—
132	C₄	#137×C₁₃₇	1	4	yes	—
133	C₄	C₂×#41×C₁₃₇	1	2	yes	—
134	C₂	—	4	1	no	—
135	C₂	#532×C₁₃₇	1	1	yes	characteristic
136	C₂	C₂×#137×C₁₃₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#533×C₁₃₇

C₂

yes

maximal

#498×C₁₃₇

C₂

yes

maximal, characteristic, complemented

#494×C₁₃₇

C₂

yes

maximal, characteristic, complemented

#487×C₁₃₇

C₂

yes

maximal, complemented

#470×C₁₃₇

C₂

yes

maximal

#441×C₁₃₇

C₂

yes

maximal, complemented

C₄×M₁₆×C₁₃₇

C₂

yes

maximal, characteristic

C₂×Q₁₆×C₁₃₇

C₂²

yes

—

C₂×M₁₆×C₁₃₇

C₂²

yes

—

C₄⋊Q₈×C₁₃₇

C₂²

yes

—

C₄⋊Q₈×C₁₃₇

C₂²

yes

—

#125×C₁₃₇

—

C₂²⋊Q₈×C₁₃₇

—

C₂²⋊Q₈×C₁₃₇

C₂²

yes

—

C₄×Q₈×C₁₃₇

—

C₄×Q₈×C₁₃₇

C₂²

yes

—

#117×C₁₃₇

—

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

#106×C₁₃₇

C₂²

yes

—

#103×C₁₃₇

C₂²

yes

—

#97×C₁₃₇

C₂²

yes

—

C₄×C₈×C₁₃₇

C₂²

yes

characteristic

C₄×C₈×C₁₃₇

C₂²

yes

characteristic

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

C₂³

yes

—

C₂²×C₄×C₁₃₇

D₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

Q₁₆×C₁₃₇

—

M₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂³

yes

—

C₂×C₈×C₁₃₇

C₂³

yes

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

C₂³

yes

—

C₂²⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

—

C₄²×C₁₃₇

D₄

yes

—

C₄²×C₁₃₇

C₂³

yes

characteristic

C₄²×C₁₃₇

C₂³

yes

characteristic

C₂³×C₁₃₇

C₂×D₄

yes

characteristic

Q₈×C₁₃₇

—

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂⁴

yes

characteristic

C₈×C₁₃₇

—

C₈×C₁₃₇

#41

yes

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂²×D₄

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#137

yes

—

C₄×C₁₃₇

C₂×#41

yes

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#532

yes

characteristic

C₂×C₁₃₇

C₂×#137

yes

—

C₁₃₇

#3185

yes

characteristic, direct product factor, Sylow

#3185

C₁₃₇

yes

maximal, characteristic, direct product factor, Sylow

#533

C₂×C₁₃₇

yes

—

#498

C₂×C₁₃₇

yes

characteristic, complemented

#494

C₂×C₁₃₇

yes

characteristic, complemented

#487

C₂×C₁₃₇

yes

complemented

#470

C₂×C₁₃₇

yes

—

#441

C₂×C₁₃₇

yes

complemented

C₄×M₁₆

C₂×C₁₃₇

yes

characteristic

C₂×Q₁₆

C₂²×C₁₃₇

yes

—

C₂×M₁₆

C₂²×C₁₃₇

yes

—

C₄⋊Q₈

C₂²×C₁₃₇

yes

—

C₄⋊Q₈

C₂²×C₁₃₇

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂²×C₁₃₇

yes

—

C₄×Q₈

—

C₄×Q₈

C₂²×C₁₃₇

yes

—

#117

—

C₂×C₄²

C₂²×C₁₃₇

yes

characteristic

#106

C₂²×C₁₃₇

yes

—

#103

C₂²×C₁₃₇

yes

—

#97

C₂²×C₁₃₇

yes

—

C₄×C₈

C₂²×C₁₃₇

yes

characteristic

C₄×C₈

C₂²×C₁₃₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₂³×C₁₃₇

yes

—

C₂²×C₄

D₄×C₁₃₇

yes

—

C₂²×C₄

C₂³×C₁₃₇

yes

characteristic

Q₁₆

—

M₁₆

—

C₂×C₈

C₂³×C₁₃₇

yes

—

100

C₂×C₈

C₂³×C₁₃₇

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

C₂³×C₁₃₇

yes

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₄²

D₄×C₁₃₇

yes

—

110

C₄²

C₂³×C₁₃₇

yes

characteristic

111

C₄²

C₂³×C₁₃₇

yes

characteristic

112

C₂³

C₂×D₄×C₁₃₇

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂×D₄×C₁₃₇

yes

—

120

C₂×C₄

C₂×D₄×C₁₃₇

yes

characteristic

121

C₂×C₄

C₂×D₄×C₁₃₇

yes

—

122

C₂×C₄

C₂⁴×C₁₃₇

yes

characteristic

123

C₈

—

124

C₈

#41×C₁₃₇

yes

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂²×D₄×C₁₃₇

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#137×C₁₃₇

yes

—

133

C₄

C₂×#41×C₁₃₇

yes

—

134

C₂

—

135

C₂

#532×C₁₃₇

yes

characteristic

136

C₂

C₂×#137×C₁₃₇

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃₇
prime radical	C₂³×C₁₃₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₃₇

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₁₃₇

prime radical

C₂³×C₁₃₇

C₂×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

N	H ≅ G/N	type
2. #498×C₁₃₇	C₂	semidirect
3. #494×C₁₃₇	C₂	semidirect
4. #487×C₁₃₇	C₂	semidirect
6. #441×C₁₃₇	C₂	semidirect
68. C₁₃₇	#3185	direct (N × H)
69. #3185	C₁₃₇	direct (N × H)
71. #498	C₂×C₁₃₇	semidirect
72. #494	C₂×C₁₃₇	semidirect
73. #487	C₂×C₁₃₇	semidirect
75. #441	C₂×C₁₃₇	semidirect

H ≅ G/N

type

2. #498×C₁₃₇

C₂

semidirect

3. #494×C₁₃₇

C₂

semidirect

4. #487×C₁₃₇

C₂

semidirect

6. #441×C₁₃₇

C₂

semidirect

68. C₁₃₇

#3185

direct (N × H)

69. #3185

C₁₃₇

direct (N × H)

71. #498

C₂×C₁₃₇

semidirect

72. #494

C₂×C₁₃₇

semidirect

73. #487

C₂×C₁₃₇

semidirect

75. #441

C₂×C₁₃₇

semidirect

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

subgroup	index
#533×C₁₃₇	2
#498×C₁₃₇	2
#494×C₁₃₇	2
#487×C₁₃₇	2
#470×C₁₃₇	2
#441×C₁₃₇	2
C₄×M₁₆×C₁₃₇	2
#3185	137

subgroup

index

#533×C₁₃₇

#498×C₁₃₇

#494×C₁₃₇

#487×C₁₃₇

#470×C₁₃₇

#441×C₁₃₇

C₄×M₁₆×C₁₃₇

#3185

137

Subgroups — #3185×C₁₃₇ #31778196

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3185×C137 #31778196

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3185×C₁₃₇ #31778196