Group #31779932 — subgroups

Subgroups — #31779932

Order 17536.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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	#22578119	C₂	1	1	yes	maximal, characteristic
2	#22577690	C₂	1	2	yes	maximal, complemented
3	#22577152	C₂	1	1	yes	maximal, characteristic, complemented
4	#22577132	C₂	1	2	yes	maximal, complemented
5	#494×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984077	—	2	2	no	—
7	#17984063	—	2	2	no	—
8	C₄×D₅₄₈	C₂²	1	2	yes	—
9	#17984046	—	2	2	no	—
10	#17984019	—	2	2	no	—
11	C₄⋊#11764040	C₂²	1	2	yes	—
12	#17984008	—	2	2	no	—
13	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
14	C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×D₅₄₈	—	2	2	no	—
18	C₂×D₅₄₈	D₄	1	2	yes	—
19	C₂×C₄×D₁₃₇	—	2	2	no	—
20	C₂×#11764040	—	2	2	no	—
21	C₄⋊Dic₁₃₇	D₄	1	2	yes	—
22	#14555487	—	2	4	no	—
23	#14555487	—	2	2	no	—
24	C₄×Dic₁₃₇	—	2	2	no	—
25	C₂×Q₈×C₁₃₇	—	2	1	no	—
26	C₂×Q₈×C₁₃₇	D₄	1	2	yes	—
27	Q₁₆×C₁₃₇	—	4	4	no	—
28	C₂×C₈×C₁₃₇	—	2	1	no	—
29	C₄⋊C₄×C₁₃₇	—	2	2	no	—
30	C₄⋊C₄×C₁₃₇	—	2	2	no	—
31	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
32	C₂²×D₁₃₇	—	2	2	no	—
33	D₅₄₈	—	2	2	no	—
34	D₅₄₈	—	2	2	no	—
35	D₅₄₈	—	8	2	no	—
36	C₂×Dic₁₃₇	—	2	2	no	—
37	C₄×D₁₃₇	—	4	2	no	—
38	C₄×D₁₃₇	—	4	2	no	—
39	#11764040	—	4	2	no	—
40	Q₈×C₁₃₇	—	4	2	no	—
41	Q₈×C₁₃₇	—	4	2	no	—
42	C₂×C₄×C₁₃₇	—	2	2	no	—
43	C₂×C₄×C₁₃₇	—	4	1	no	—
44	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
46	C₈×C₁₃₇	—	2	2	no	—
47	C₂²×C₁₃₇	#120	1	1	yes	characteristic
48	C₂×D₁₃₇	—	4	2	no	—
49	C₂×D₁₃₇	—	4	2	no	—
50	C₂×D₁₃₇	—	4	2	no	—
51	C₄×C₁₃₇	—	2	1	no	—
52	C₄×C₁₃₇	—	4	2	no	—
53	C₄×C₁₃₇	—	8	1	no	—
54	C₄×C₁₃₇	#135	1	2	yes	—
55	C₄×C₁₃₇	#137	1	2	yes	—
56	Dic₁₃₇	—	4	2	no	—
57	C₂×C₁₃₇	#452	1	2	yes	—
58	C₂×C₁₃₇	#453	1	1	yes	characteristic
59	D₁₃₇	—	8	2	no	—
60	C₁₃₇	#1634	1	1	yes	characteristic, complemented, Sylow
61	#1634	—	137	1	no	maximal, Sylow
62	#555	—	137	1	no	—
63	#494	D₁₃₇	1	1	yes	characteristic, complemented
64	#476	—	137	2	no	—
65	#333	—	137	1	no	—
66	D₄⋊C₈	—	137	2	no	—
67	C₂×Q₁₆	—	2	2	no	—
68	C₄⋊Q₈	C₂×D₁₃₇	1	1	yes	characteristic
69	C₄⋊Q₈	C₂×D₁₃₇	1	1	yes	characteristic
70	#124	—	274	2	no	—
71	#121	—	274	2	no	—
72	C₄×D₄	—	137	2	no	—
73	#117	—	274	2	no	—
74	#106	—	274	2	no	—
75	C₄⋊C₈	—	137	2	no	—
76	#102	—	274	2	no	—
77	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	C₂²⋊D₁₃₇	1	2	yes	—
80	C₂×D₄	—	274	2	no	—
81	C₂×D₄	—	137	2	no	—
82	C₂²×C₄	—	274	2	no	—
83	Q₁₆	—	4	4	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	274	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	137	2	no	—
89	C₂²⋊C₄	—	274	4	no	—
90	C₂²⋊C₄	—	274	2	no	—
91	C₄²	—	274	2	no	—
92	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
93	C₂³	—	274	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	4	2	no	—
96	D₄	—	274	2	no	—
97	D₄	—	274	2	no	—
98	D₄	—	1096	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	274	2	no	—
102	C₂×C₄	—	548	2	no	—
103	C₂×C₄	—	548	2	no	—
104	C₂×C₄	D₄×D₁₃₇	1	2	yes	—
105	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
106	C₈	—	2	2	no	—
107	C₈	—	548	2	no	—
108	C₂²	—	548	2	no	—
109	C₂²	—	548	2	no	—
110	C₂²	—	548	2	no	—
111	C₂²	#17984056	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	8	1	no	—
115	C₄	—	548	2	no	—
116	C₄	#17984111	1	2	yes	—
117	C₄	#17984120	1	2	yes	—
118	C₂	—	1096	2	no	—
119	C₂	#22577564	1	2	yes	—
120	C₂	#22577570	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22578119

C₂

yes

maximal, characteristic

#22577690

C₂

yes

maximal, complemented

#22577152

C₂

yes

maximal, characteristic, complemented

#22577132

C₂

yes

maximal, complemented

#494×C₁₃₇

C₂

yes

maximal, characteristic, complemented

#17984077

—

#17984063

—

C₄×D₅₄₈

C₂²

yes

—

#17984046

—

#17984019

—

C₄⋊#11764040

C₂²

yes

—

#17984008

—

C₂×Q₁₆×C₁₃₇

—

C₄⋊Q₈×C₁₃₇

C₂²

yes

characteristic

C₄⋊Q₈×C₁₃₇

C₂²

yes

characteristic

C₄×C₈×C₁₃₇

C₂²

yes

characteristic

C₂×D₅₄₈

—

C₂×D₅₄₈

D₄

yes

—

C₂×C₄×D₁₃₇

—

C₂×#11764040

—

C₄⋊Dic₁₃₇

D₄

yes

—

#14555487

—

#14555487

—

C₄×Dic₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

D₄

yes

—

Q₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

C₂³

yes

characteristic

C₂²×D₁₃₇

—

D₅₄₈

—

D₅₄₈

—

D₅₄₈

—

C₂×Dic₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

#11764040

—

Q₈×C₁₃₇

—

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₈×C₁₃₇

—

C₂²×C₁₃₇

#120

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#135

yes

—

C₄×C₁₃₇

#137

yes

—

Dic₁₃₇

—

C₂×C₁₃₇

#452

yes

—

C₂×C₁₃₇

#453

yes

characteristic

D₁₃₇

—

C₁₃₇

#1634

yes

characteristic, complemented, Sylow

#1634

—

137

maximal, Sylow

#555

—

137

—

#494

D₁₃₇

yes

characteristic, complemented

#476

—

137

—

#333

—

137

—

D₄⋊C₈

—

137

—

C₂×Q₁₆

—

C₄⋊Q₈

C₂×D₁₃₇

yes

characteristic

C₄⋊Q₈

C₂×D₁₃₇

yes

characteristic

#124

—

274

—

#121

—

274

—

C₄×D₄

—

137

—

#117

—

274

—

#106

—

274

—

C₄⋊C₈

—

137

—

#102

—

274

—

C₄×C₈

C₂×D₁₃₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₁₃₇

yes

—

C₂×D₄

—

274

—

C₂×D₄

—

137

—

C₂²×C₄

—

274

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

274

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

137

—

C₂²⋊C₄

—

274

—

C₂²⋊C₄

—

274

—

C₄²

—

274

—

C₄²

C₂²×D₁₃₇

yes

characteristic

C₂³

—

274

—

Q₈

—

Q₈

—

D₄

—

274

—

D₄

—

274

—

D₄

—

1096

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

274

—

102

C₂×C₄

—

548

—

103

C₂×C₄

—

548

—

104

C₂×C₄

D₄×D₁₃₇

yes

—

105

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

106

C₈

—

107

C₈

—

548

—

108

C₂²

—

548

—

109

C₂²

—

548

—

110

C₂²

—

548

—

111

C₂²

#17984056

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

548

—

116

C₄

#17984111

yes

—

117

C₄

#17984120

yes

—

118

C₂

—

1096

—

119

C₂

#22577564

yes

—

120

C₂

#22577570

yes

characteristic

	subgroup	G/H
center	C₂²	#17984056
commutator, cosocle	C₄²×C₁₃₇	C₂³
Frattini	C₄²	C₂²×D₁₃₇
Fitting	#494×C₁₃₇	C₂
socle	C₂²×C₁₃₇	#120
prime radical	#22578119	C₂
hypercenter	#494	D₁₃₇
hypocenter	C₁₃₇	#1634
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#17984056

commutator, cosocle

C₄²×C₁₃₇

C₂³

Frattini

C₄²

C₂²×D₁₃₇

Fitting

#494×C₁₃₇

C₂

C₂²×C₁₃₇

C₂

D₁₃₇

C₁₃₇

G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
2. #22577690	C₂	semidirect
3. #22577152	C₂	semidirect
4. #22577132	C₂	semidirect
5. #494×C₁₃₇	C₂	semidirect
60. C₁₃₇	#1634	semidirect
63. #494	D₁₃₇	semidirect

H ≅ G/N

type

2. #22577690

C₂

semidirect

3. #22577152

C₂

semidirect

4. #22577132

C₂

semidirect

5. #494×C₁₃₇

C₂

semidirect

60. C₁₃₇

#1634

semidirect

63. #494

D₁₃₇

semidirect

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

subgroup	index
#22578119	2
#22577690	2
#22577152	2
#22577132	2
#494×C₁₃₇	2
#1634	137

subgroup

index

#494×C₁₃₇

137

Subgroups — #31779932

Conjugacy classes of subgroups

Complements

Maximal subgroups