Group #31789575 — subgroups

Subgroups — C₂×#22577509 #31789575

Order 17536.

480 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 197 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	C₂²×Dic₅₄₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984053	C₂	1	2	yes	maximal
3	#22577509	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17984011	C₂	1	2	yes	maximal
5	C₂×#17983998	C₂	1	1	yes	maximal, characteristic
6	C₂×#106×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₇₄	C₂²	1	2	yes	—
8	C₂×Dic₅₄₈	C₄	1	8	yes	complemented
9	C₂×Dic₅₄₈	C₂²	1	4	yes	—
10	C₂²×#11764040	C₂²	1	1	yes	characteristic
11	#17984053	—	2	4	no	—
12	#17984053	C₂²	1	8	yes	—
13	C₂×#14555489	—	2	2	no	—
14	C₂×C₄×Dic₁₃₇	—	2	2	no	—
15	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
16	#17984011	C₂²	1	8	yes	—
17	#17983998	C₂²	1	4	yes	—
18	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
20	#106×C₁₃₇	C₂²	1	4	yes	—
21	C₂×Dic₂₇₄	—	2	2	no	—
22	C₂×Dic₂₇₄	—	2	4	no	—
23	C₂×Dic₂₇₄	C₂×C₄	1	8	yes	—
24	C₂×Dic₂₇₄	C₂³	1	4	yes	—
25	C₂²×Dic₁₃₇	—	2	2	no	—
26	C₂²×Dic₁₃₇	D₄	1	2	yes	—
27	Dic₅₄₈	C₂×C₄	1	16	yes	complemented
28	C₂×#11764040	—	2	2	no	—
29	C₂×#11764040	C₂³	1	2	yes	—
30	#14555489	—	2	8	no	—
31	C₄×Dic₁₃₇	—	2	8	no	—
32	C₄×Dic₁₃₇	D₄	1	2	yes	—
33	C₄×Dic₁₃₇	C₂³	1	2	yes	—
34	C₂²×C₄×C₁₃₇	—	2	2	no	—
35	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
37	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
38	C₄⋊C₄×C₁₃₇	—	2	2	no	—
39	C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
40	Dic₂₇₄	—	2	8	no	—
41	Dic₂₇₄	—	2	4	no	—
42	Dic₂₇₄	C₂²×C₄	1	8	yes	—
43	C₂×Dic₁₃₇	—	2	8	no	—
44	C₂×Dic₁₃₇	—	2	4	no	—
45	C₂×Dic₁₃₇	—	2	4	no	—
46	C₂×Dic₁₃₇	C₂×D₄	1	4	yes	—
47	#11764040	—	2	2	no	—
48	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₃₇	—	2	4	no	—
50	C₂×C₄×C₁₃₇	—	4	4	no	—
51	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₃₇	#41	1	2	yes	—
54	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
55	C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
56	C₂²×C₁₃₇	C₄×D₄	1	2	yes	—
57	C₂²×C₁₃₇	#137	1	4	yes	—
58	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₁₃₇	—	4	4	no	—
60	C₄×C₁₃₇	C₄×D₄	1	2	yes	—
61	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
62	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
63	Dic₁₃₇	—	2	8	no	—
64	Dic₁₃₇	—	2	4	no	—
65	C₂×C₁₃₇	#441	1	4	yes	complemented
66	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
67	C₂×C₁₃₇	C₂×#137	1	2	yes	—
68	C₁₃₇	C₂×#441	1	1	yes	characteristic, complemented, Sylow
69	C₂×#441	—	137	1	no	maximal, Sylow
70	C₂²×Q₁₆	—	137	1	no	—
71	C₂×C₄×Q₈	—	137	2	no	—
72	#441	—	137	8	no	—
73	C₂×#106	D₁₃₇	1	1	yes	characteristic
74	C₂×#103	—	137	2	no	—
75	C₂×#97	—	137	1	no	—
76	C₂²×Q₈	—	137	2	no	—
77	C₂×Q₁₆	—	137	8	no	—
78	C₂×Q₁₆	—	137	4	no	—
79	C₂²×C₈	—	137	1	no	—
80	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
81	C₄×Q₈	—	274	4	no	—
82	C₄×Q₈	—	137	8	no	—
83	C₂×C₄⋊C₄	—	274	2	no	—
84	C₂×C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
85	C₂×C₄²	—	274	2	no	—
86	C₂×C₄²	—	137	1	no	—
87	#106	C₂×D₁₃₇	1	4	yes	—
88	#103	—	137	8	no	—
89	#97	—	137	4	no	—
90	C₂×Q₈	—	274	2	no	—
91	C₂×Q₈	—	274	4	no	—
92	C₂×Q₈	—	137	8	no	—
93	C₂×Q₈	—	137	4	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	274	2	no	—
96	C₂²×C₄	—	137	2	no	—
97	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
98	Q₁₆	—	137	16	no	—
99	C₂×C₈	—	274	2	no	—
100	C₂×C₈	—	137	2	no	—
101	C₂×C₈	C₄×D₁₃₇	1	4	yes	—
102	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	274	8	no	—
105	C₄⋊C₄	C₂²×D₁₃₇	1	4	yes	—
106	C₄²	—	274	8	no	—
107	C₄²	—	137	2	no	—
108	C₄²	—	137	2	no	—
109	C₂³	D₄×D₁₃₇	1	1	yes	characteristic
110	Q₈	—	274	8	no	—
111	Q₈	—	274	4	no	—
112	Q₈	—	137	8	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	4	no	—
115	C₂×C₄	—	274	8	no	—
116	C₂×C₄	—	274	4	no	—
117	C₂×C₄	—	274	4	no	—
118	C₂×C₄	—	137	4	no	—
119	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
120	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
121	C₂×C₄	#14555511	1	2	yes	—
122	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
123	C₈	—	274	2	no	—
124	C₈	C₂×C₄×D₁₃₇	1	4	yes	—
125	C₂²	#17984050	1	2	yes	—
126	C₂²	#17984117	1	4	yes	—
127	C₂²	C₂×D₄×D₁₃₇	1	1	yes	characteristic
128	C₄	—	4	4	no	—
129	C₄	—	274	4	no	—
130	C₄	—	274	8	no	—
131	C₄	#17984050	1	2	yes	—
132	C₄	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
133	C₄	C₂×#14555511	1	1	yes	characteristic
134	C₂	#22577509	1	4	yes	direct product factor
135	C₂	C₂×#17984050	1	1	yes	characteristic
136	C₂	C₂×#17984117	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₄₈

C₂

yes

maximal, characteristic

C₂×#17984053

C₂

yes

maximal

#22577509

C₂

yes

maximal, direct product factor

C₂×#17984011

C₂

yes

maximal

C₂×#17983998

C₂

yes

maximal, characteristic

C₂×#106×C₁₃₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₇₄

C₂²

yes

—

C₂×Dic₅₄₈

C₄

yes

complemented

C₂×Dic₅₄₈

C₂²

yes

—

C₂²×#11764040

C₂²

yes

characteristic

#17984053

—

#17984053

C₂²

yes

—

C₂×#14555489

—

C₂×C₄×Dic₁₃₇

—

C₂×C₄×Dic₁₃₇

C₂²

yes

characteristic

#17984011

C₂²

yes

—

#17983998

C₂²

yes

—

C₂²×C₈×C₁₃₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₃₇

C₂²

yes

—

#106×C₁₃₇

C₂²

yes

—

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

C₂×C₄

yes

—

C₂×Dic₂₇₄

C₂³

yes

—

C₂²×Dic₁₃₇

—

C₂²×Dic₁₃₇

D₄

yes

—

Dic₅₄₈

C₂×C₄

yes

complemented

C₂×#11764040

—

C₂×#11764040

C₂³

yes

—

#14555489

—

C₄×Dic₁₃₇

—

C₄×Dic₁₃₇

D₄

yes

—

C₄×Dic₁₃₇

C₂³

yes

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

—

C₂×C₈×C₁₃₇

C₂³

yes

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

C₂³

yes

—

Dic₂₇₄

—

Dic₂₇₄

—

Dic₂₇₄

C₂²×C₄

yes

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₂×D₄

yes

—

#11764040

—

C₂³×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

#41

yes

—

C₂×C₄×C₁₃₇

C₂⁴

yes

characteristic

C₈×C₁₃₇

C₂²×C₄

yes

—

C₂²×C₁₃₇

C₄×D₄

yes

—

C₂²×C₁₃₇

#137

yes

—

C₂²×C₁₃₇

C₂²×D₄

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

C₄×D₄

yes

—

C₄×C₁₃₇

C₂³×C₄

yes

characteristic

C₄×C₁₃₇

C₂×#41

yes

characteristic

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

#441

yes

complemented

C₂×C₁₃₇

C₂×C₄×D₄

yes

characteristic

C₂×C₁₃₇

C₂×#137

yes

—

C₁₃₇

C₂×#441

yes

characteristic, complemented, Sylow

C₂×#441

—

137

maximal, Sylow

C₂²×Q₁₆

—

137

—

C₂×C₄×Q₈

—

137

—

#441

—

137

—

C₂×#106

D₁₃₇

yes

characteristic

C₂×#103

—

137

—

C₂×#97

—

137

—

C₂²×Q₈

—

137

—

C₂×Q₁₆

—

137

—

C₂×Q₁₆

—

137

—

C₂²×C₈

—

137

—

C₂²×C₈

C₂×D₁₃₇

yes

characteristic

C₄×Q₈

—

274

—

C₄×Q₈

—

137

—

C₂×C₄⋊C₄

—

274

—

C₂×C₄⋊C₄

C₂×D₁₃₇

yes

—

C₂×C₄²

—

274

—

C₂×C₄²

—

137

—

#106

C₂×D₁₃₇

yes

—

#103

—

137

—

#97

—

137

—

C₂×Q₈

—

274

—

C₂×Q₈

—

274

—

C₂×Q₈

—

137

—

C₂×Q₈

—

137

—

C₂²×C₄

—

C₂²×C₄

—

274

—

C₂²×C₄

—

137

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

Q₁₆

—

137

—

C₂×C₈

—

274

—

100

C₂×C₈

—

137

—

101

C₂×C₈

C₄×D₁₃₇

yes

—

102

C₂×C₈

C₂²×D₁₃₇

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

274

—

105

C₄⋊C₄

C₂²×D₁₃₇

yes

—

106

C₄²

—

274

—

107

C₄²

—

137

—

108

C₄²

—

137

—

109

C₂³

D₄×D₁₃₇

yes

characteristic

110

Q₈

—

274

—

111

Q₈

—

274

—

112

Q₈

—

137

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

274

—

116

C₂×C₄

—

274

—

117

C₂×C₄

—

274

—

118

C₂×C₄

—

137

—

119

C₂×C₄

C₂×C₄×D₁₃₇

yes

—

120

C₂×C₄

D₄×D₁₃₇

yes

characteristic

121

C₂×C₄

#14555511

yes

—

122

C₂×C₄

C₂³×D₁₃₇

yes

characteristic

123

C₈

—

274

—

124

C₈

C₂×C₄×D₁₃₇

yes

—

125

C₂²

#17984050

yes

—

126

C₂²

#17984117

yes

—

127

C₂²

C₂×D₄×D₁₃₇

yes

characteristic

128

C₄

—

129

C₄

—

274

—

130

C₄

—

274

—

131

C₄

#17984050

yes

—

132

C₄

C₂²×C₄×D₁₃₇

yes

characteristic

133

C₄

C₂×#14555511

yes

characteristic

134

C₂

#22577509

yes

direct product factor

135

C₂

C₂×#17984050

yes

characteristic

136

C₂

C₂×#17984117

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₃₇
commutator	C₄×C₁₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	C₂×#106×C₁₃₇	C₂
socle, prime radical	C₂³×C₁₃₇	C₂×D₄
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	C₂×#106	D₁₃₇
hypocenter	C₁₃₇	C₂×#441
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₁₃₇

commutator

C₄×C₁₃₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₃₇

Fitting

C₂×#106×C₁₃₇

C₂

socle, prime radical

C₂³×C₁₃₇

C₂×D₄

cosocle

C₂×C₄×C₁₃₇

C₂⁴

D₁₃₇

C₁₃₇

G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
3. #22577509	C₂	direct (N × H)
8. C₂×Dic₅₄₈	C₄	semidirect
27. Dic₅₄₈	C₂×C₄	semidirect
65. C₂×C₁₃₇	#441	semidirect
68. C₁₃₇	C₂×#441	semidirect
134. C₂	#22577509	direct (N × H)

H ≅ G/N

type

3. #22577509

C₂

direct (N × H)

8. C₂×Dic₅₄₈

C₄

semidirect

27. Dic₅₄₈

C₂×C₄

semidirect

65. C₂×C₁₃₇

#441

semidirect

68. C₁₃₇

C₂×#441

semidirect

134. C₂

#22577509

direct (N × H)

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

subgroup	index
C₂²×Dic₅₄₈	2
C₂×#17984053	2
#22577509	2
C₂×#17984011	2
C₂×#17983998	2
C₂×#106×C₁₃₇	2
C₂×#441	137

subgroup

index

C₂²×Dic₅₄₈

C₂×#106×C₁₃₇

137

Subgroups — C₂×#22577509 #31789575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22577509 #31789575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22577509 #31789575