Group #31779865 — subgroups

Subgroups — #31779865

Order 17536.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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	#22578133	C₂	1	1	yes	maximal, characteristic, complemented
2	#22577600	C₂	1	2	yes	maximal
3	#22577152	C₂	1	1	yes	maximal, characteristic, complemented
4	#22577135	C₂	1	2	yes	maximal, complemented
5	#492×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555498	—	2	2	no	—
7	#17984087	—	2	2	no	—
8	#17984086	—	2	2	no	—
9	C₄×Dic₂₇₄	C₂²	1	2	yes	—
10	#17984053	—	2	2	no	—
11	C₄⋊#11764040	C₂²	1	2	yes	—
12	#17984009	—	2	2	no	—
13	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
14	C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	#127×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₇₄	—	2	2	no	—
18	C₂×Dic₂₇₄	D₄	1	2	yes	—
19	#14555498	—	4	4	no	—
20	C₂×#11764040	—	2	2	no	—
21	C₄⋊Dic₁₃₇	D₄	1	2	yes	—
22	#14555489	—	2	4	no	—
23	#14555489	—	2	2	no	—
24	C₄×Dic₁₃₇	—	2	2	no	—
25	C₄×Dic₁₃₇	—	2	2	no	—
26	C₂×Q₈×C₁₃₇	D₄	1	2	yes	—
27	C₂×D₄×C₁₃₇	—	2	2	no	—
28	C₂×D₄×C₁₃₇	—	2	1	no	—
29	SD₁₆×C₁₃₇	—	4	4	no	—
30	C₂×C₈×C₁₃₇	—	2	1	no	—
31	C₄⋊C₄×C₁₃₇	—	2	2	no	—
32	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
33	Dic₂₇₄	—	2	2	no	—
34	Dic₂₇₄	—	8	2	no	—
35	Dic₂₇₄	SD₁₆	1	4	yes	complemented
36	C₂×Dic₁₃₇	—	2	2	no	—
37	C₂×Dic₁₃₇	—	2	2	no	—
38	#11764040	—	4	2	no	—
39	C₂³×C₁₃₇	—	4	1	no	—
40	Q₈×C₁₃₇	—	4	2	no	—
41	D₄×C₁₃₇	—	4	2	no	—
42	D₄×C₁₃₇	—	4	2	no	—
43	D₄×C₁₃₇	—	4	2	no	—
44	C₂×C₄×C₁₃₇	—	2	2	no	—
45	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
47	C₈×C₁₃₇	—	2	2	no	—
48	C₂²×C₁₃₇	—	8	2	no	—
49	C₂²×C₁₃₇	—	8	1	no	—
50	C₂²×C₁₃₇	#120	1	1	yes	characteristic
51	C₄×C₁₃₇	—	2	1	no	—
52	C₄×C₁₃₇	—	4	2	no	—
53	C₄×C₁₃₇	C₂×SD₁₆	1	2	yes	—
54	C₄×C₁₃₇	#137	1	2	yes	—
55	Dic₁₃₇	—	2	4	no	—
56	Dic₁₃₇	—	4	2	no	—
57	C₂×C₁₃₇	—	16	1	no	—
58	C₂×C₁₃₇	#448	1	2	yes	—
59	C₂×C₁₃₇	#453	1	1	yes	characteristic
60	C₁₃₇	#1623	1	1	yes	characteristic, complemented, Sylow
61	#1623	—	137	1	no	maximal, Sylow
62	Q₈²	—	137	1	no	—
63	#492	D₁₃₇	1	1	yes	characteristic
64	#460	—	137	2	no	—
65	#333	—	137	1	no	—
66	Q₈⋊C₈	—	137	2	no	—
67	C₂×SD₁₆	—	2	2	no	—
68	C₂×SD₁₆	—	274	2	no	—
69	C₄⋊Q₈	—	274	2	no	—
70	C₄⋊Q₈	—	274	2	no	—
71	C₄⋊Q₈	C₂×D₁₃₇	1	1	yes	characteristic
72	#127	C₂×D₁₃₇	1	1	yes	characteristic
73	C₄×Q₈	—	274	2	no	—
74	C₄×Q₈	—	137	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	—	274	2	no	—
79	C₂×Q₈	—	137	2	no	—
80	C₂×Q₈	C₂²⋊D₁₃₇	1	2	yes	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	2	1	no	—
83	SD₁₆	—	4	4	no	—
84	SD₁₆	—	548	4	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	274	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	274	4	no	—
89	C₄⋊C₄	—	274	2	no	—
90	C₄⋊C₄	—	137	2	no	—
91	C₄²	—	274	2	no	—
92	C₄²	—	274	2	no	—
93	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
94	C₂³	—	4	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	274	2	no	—
97	Q₈	—	1096	2	no	—
98	Q₈	—	137	4	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	D₄	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	274	2	no	—
104	C₂×C₄	—	274	2	no	—
105	C₂×C₄	D₄×D₁₃₇	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	548	2	no	—
109	C₂²	—	8	2	no	—
110	C₂²	—	8	1	no	—
111	C₂²	#17984056	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	274	4	no	—
115	C₄	—	548	2	no	—
116	C₄	SD₁₆×D₁₃₇	1	2	yes	—
117	C₄	#17984117	1	2	yes	—
118	C₂	—	16	1	no	—
119	C₂	#22577540	1	2	yes	—
120	C₂	#22577570	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22578133

C₂

yes

maximal, characteristic, complemented

#22577600

C₂

yes

maximal

#22577152

C₂

yes

maximal, characteristic, complemented

#22577135

C₂

yes

maximal, complemented

#492×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555498

—

#17984087

—

#17984086

—

C₄×Dic₂₇₄

C₂²

yes

—

#17984053

—

C₄⋊#11764040

C₂²

yes

—

#17984009

—

C₂×SD₁₆×C₁₃₇

—

C₄⋊Q₈×C₁₃₇

C₂²

yes

characteristic

#127×C₁₃₇

C₂²

yes

characteristic

C₄×C₈×C₁₃₇

C₂²

yes

characteristic

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

D₄

yes

—

#14555498

—

C₂×#11764040

—

C₄⋊Dic₁₃₇

D₄

yes

—

#14555489

—

#14555489

—

C₄×Dic₁₃₇

—

C₄×Dic₁₃₇

—

C₂×Q₈×C₁₃₇

D₄

yes

—

C₂×D₄×C₁₃₇

—

C₂×D₄×C₁₃₇

—

SD₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

C₂³

yes

characteristic

Dic₂₇₄

—

Dic₂₇₄

—

Dic₂₇₄

SD₁₆

yes

complemented

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

#11764040

—

C₂³×C₁₃₇

—

Q₈×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₈×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

#120

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×SD₁₆

yes

—

C₄×C₁₃₇

#137

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#448

yes

—

C₂×C₁₃₇

#453

yes

characteristic

C₁₃₇

#1623

yes

characteristic, complemented, Sylow

#1623

—

137

maximal, Sylow

Q₈²

—

137

—

#492

D₁₃₇

yes

characteristic

#460

—

137

—

#333

—

137

—

Q₈⋊C₈

—

137

—

C₂×SD₁₆

—

C₂×SD₁₆

—

274

—

C₄⋊Q₈

—

274

—

C₄⋊Q₈

—

274

—

C₄⋊Q₈

C₂×D₁₃₇

yes

characteristic

#127

C₂×D₁₃₇

yes

characteristic

C₄×Q₈

—

274

—

C₄×Q₈

—

137

—

C₄⋊C₈

—

137

—

#102

—

274

—

C₄×C₈

C₂×D₁₃₇

yes

characteristic

C₂×Q₈

—

274

—

C₂×Q₈

—

137

—

C₂×Q₈

C₂²⋊D₁₃₇

yes

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

548

—

C₂×C₈

—

C₂×C₈

—

274

—

C₄⋊C₄

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

137

—

C₄²

—

274

—

C₄²

—

274

—

C₄²

C₂²×D₁₃₇

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

274

—

Q₈

—

1096

—

Q₈

—

137

—

D₄

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

274

—

104

C₂×C₄

—

274

—

105

C₂×C₄

D₄×D₁₃₇

yes

—

106

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

107

C₈

—

108

C₈

—

548

—

109

C₂²

—

110

C₂²

—

111

C₂²

#17984056

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

274

—

115

C₄

—

548

—

116

C₄

SD₁₆×D₁₃₇

yes

—

117

C₄

#17984117

yes

—

118

C₂

—

119

C₂

#22577540

yes

—

120

C₂

#22577570

yes

characteristic

	subgroup	G/H
center	C₂²	#17984056
commutator, cosocle	C₄²×C₁₃₇	C₂³
Frattini	C₄²	C₂²×D₁₃₇
Fitting	#492×C₁₃₇	C₂
socle	C₂²×C₁₃₇	#120
prime radical	#127×C₁₃₇	C₂²
hypercenter	#492	D₁₃₇
hypocenter	C₁₃₇	#1623
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#17984056

commutator, cosocle

C₄²×C₁₃₇

C₂³

Frattini

C₄²

C₂²×D₁₃₇

Fitting

#492×C₁₃₇

C₂

socle

C₂²×C₁₃₇

#120

prime radical

#127×C₁₃₇

C₂²

D₁₃₇

C₁₃₇

G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #22578133	C₂	semidirect
3. #22577152	C₂	semidirect
4. #22577135	C₂	semidirect
35. Dic₂₇₄	SD₁₆	semidirect
60. C₁₃₇	#1623	semidirect

H ≅ G/N

type

1. #22578133

C₂

semidirect

3. #22577152

C₂

semidirect

4. #22577135

C₂

semidirect

35. Dic₂₇₄

SD₁₆

semidirect

60. C₁₃₇

#1623

semidirect

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

subgroup	index
#22578133	2
#22577600	2
#22577152	2
#22577135	2
#492×C₁₃₇	2
#1623	137

subgroup

index

#492×C₁₃₇

137

Subgroups — #31779865

Conjugacy classes of subgroups

Complements

Maximal subgroups