Group #31796715 — subgroups

Subgroups — #31796715

Order 17536.

180 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 66 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	#22578408	C₂	1	1	yes	maximal, characteristic
2	#22578407	C₂	1	1	yes	maximal, characteristic
3	#22578279	C₂	1	4	yes	maximal
4	#97×D₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14555520	C₂²	1	2	yes	—
6	C₂×#14555520	C₂²	1	2	yes	—
7	#17984155	—	2	2	no	—
8	C₄⋊#11764047	—	2	2	no	—
9	C₂×C₈×D₁₃₇	C₂²	1	2	yes	—
10	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
11	#17983999	C₄	1	1	yes	characteristic
12	#17983998	C₄	1	2	yes	—
13	#97×C₁₃₇	C₄	1	1	yes	characteristic
14	#14555520	—	2	4	no	—
15	#14555520	—	2	4	no	—
16	C₂×#11764047	—	2	2	no	—
17	C₂×#11764047	—	2	2	no	—
18	C₂×C₄×D₁₃₇	Q₈	1	2	yes	—
19	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
20	C₂×#11764040	C₂×C₄	1	2	yes	—
21	C₈×D₁₃₇	—	2	2	no	—
22	C₈×D₁₃₇	—	2	2	no	—
23	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₁₃₇	D₄	1	2	yes	—
25	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
26	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
27	#11764047	—	4	2	no	—
28	#11764047	—	4	2	no	—
29	C₂²×D₁₃₇	C₂×Q₈	1	1	yes	characteristic
30	C₂×Dic₁₃₇	C₄⋊C₄	1	2	yes	—
31	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
32	C₄×D₁₃₇	—	4	2	no	—
33	C₄×D₁₃₇	#41	1	2	yes	—
34	C₄×D₁₃₇	#41	1	2	yes	—
35	#11764040	—	2	2	no	—
36	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₁₃₇	—	2	2	no	—
39	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×D₁₃₇	—	2	2	no	—
41	C₂×D₁₃₇	#125	1	2	yes	—
42	C₄×C₁₃₇	—	2	2	no	—
43	C₄×C₁₃₇	#117	1	2	yes	—
44	Dic₁₃₇	—	2	2	no	—
45	Dic₁₃₇	#124	1	1	yes	characteristic
46	Dic₁₃₇	#124	1	1	yes	characteristic
47	C₂×C₁₃₇	#382	1	1	yes	characteristic
48	C₂×C₁₃₇	#430	1	2	yes	—
49	D₁₃₇	—	2	2	no	—
50	C₁₃₇	#1838	1	1	yes	characteristic, complemented, Sylow
51	#1838	—	137	1	no	maximal, Sylow
52	#423	—	137	1	no	—
53	#423	—	137	1	no	—
54	C₂×#97	—	137	1	no	—
55	#341	—	137	4	no	—
56	C₂×M₁₆	—	137	2	no	—
57	C₂×M₁₆	—	137	2	no	—
58	C₂²×C₈	—	137	2	no	—
59	C₂×C₄²	—	137	1	no	—
60	C₄⋊C₈	—	274	2	no	—
61	C₄⋊C₈	—	274	2	no	—
62	#97	—	137	2	no	—
63	#97	—	137	1	no	—
64	#97	#10587739	1	1	yes	characteristic
65	C₂²×C₄	—	137	2	no	—
66	C₂²×C₄	—	137	1	no	—
67	M₁₆	—	274	4	no	—
68	M₁₆	—	274	4	no	—
69	C₂×C₈	—	274	2	no	—
70	C₂×C₈	—	274	2	no	—
71	C₂×C₈	—	274	2	no	—
72	C₂×C₈	—	274	2	no	—
73	C₂×C₈	—	137	2	no	—
74	C₂×C₈	C₂×#10587739	1	2	yes	—
75	C₄²	—	137	2	no	—
76	C₄²	—	137	1	no	—
77	C₄²	C₂×#10587739	1	1	yes	characteristic
78	C₂³	—	137	1	no	—
79	C₂×C₄	—	548	2	no	—
80	C₂×C₄	—	137	2	no	—
81	C₂×C₄	—	137	1	no	—
82	C₂×C₄	—	137	2	no	—
83	C₂×C₄	—	137	2	no	—
84	C₂×C₄	C₄⋊#10587739	1	2	yes	—
85	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
86	C₈	—	2	2	no	—
87	C₈	—	274	2	no	—
88	C₈	—	548	2	no	—
89	C₈	—	548	2	no	—
90	C₂²	—	274	2	no	—
91	C₂²	—	137	2	no	—
92	C₂²	C₂×C₄⋊#10587739	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	274	2	no	—
95	C₄	—	137	1	no	—
96	C₄	—	137	1	no	—
97	C₄	#17984167	1	2	yes	—
98	C₂	—	274	2	no	—
99	C₂	#22578347	1	1	yes	characteristic
100	C₂	#22578418	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22578408

C₂

yes

maximal, characteristic

#22578407

C₂

yes

maximal, characteristic

#22578279

C₂

yes

maximal

#97×D₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555520

C₂²

yes

—

C₂×#14555520

C₂²

yes

—

#17984155

—

C₄⋊#11764047

—

C₂×C₈×D₁₃₇

C₂²

yes

—

C₄²×D₁₃₇

C₂²

yes

characteristic

#17983999

C₄

yes

characteristic

#17983998

C₄

yes

—

#97×C₁₃₇

C₄

yes

—

—

—

—

C₂×C₄×D₁₃₇

Q₈

yes

—

C₂×C₄×D₁₃₇

C₂³

yes

characteristic

C₂×#11764040

C₂×C₄

yes

—

C₈×D₁₃₇

—

C₈×D₁₃₇

—

C₄×Dic₁₃₇

C₂×C₄

yes

characteristic

C₄×Dic₁₃₇

D₄

yes

—

C₂×C₈×C₁₃₇

C₂×C₄

yes

—

C₄²×C₁₃₇

C₂×C₄

yes

characteristic

#11764047

—

#11764047

—

C₂²×D₁₃₇

C₂×Q₈

yes

characteristic

C₂×Dic₁₃₇

C₄⋊C₄

yes

—

C₂×Dic₁₃₇

C₂×D₄

yes

characteristic

C₄×D₁₃₇

—

C₄×D₁₃₇

#41

yes

—

C₄×D₁₃₇

#41

yes

—

#11764040

—

C₂×C₄×C₁₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₈×C₁₃₇

—

C₂²×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

#125

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#117

yes

—

Dic₁₃₇

—

Dic₁₃₇

yes

Dic₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

—

D₁₃₇

—

C₁₃₇

#1838

yes

characteristic, complemented, Sylow

#1838

—

137

maximal, Sylow

#423

—

137

—

#423

—

137

—

C₂×#97

—

137

—

#341

—

137

—

C₂×M₁₆

—

137

—

C₂×M₁₆

—

137

—

C₂²×C₈

—

137

—

C₂×C₄²

—

137

—

C₄⋊C₈

—

274

—

C₄⋊C₈

—

274

—

#97

—

137

—

#97

—

137

—

#97

#10587739

yes

characteristic

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

M₁₆

—

274

—

M₁₆

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

C₂×C₈

C₂×#10587739

yes

—

C₄²

—

137

—

C₄²

—

137

—

C₄²

C₂×#10587739

yes

characteristic

C₂³

—

137

—

C₂×C₄

—

548

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

C₄⋊#10587739

yes

—

C₂×C₄

C₂²×#10587739

yes

characteristic

C₈

—

C₈

—

274

—

C₈

—

548

—

C₈

—

548

—

C₂²

—

274

—

C₂²

—

137

—

C₂²

C₂×C₄⋊#10587739

yes

characteristic

C₄

—

C₄

—

274

—

C₄

—

137

—

C₄

—

137

—

C₄

#17984167

yes

—

C₂

—

274

—

C₂

#22578347

yes

characteristic

100

C₂

#22578418

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#10587739
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10587739
Fitting	#97×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂×C₄⋊C₄
prime radical	C₂²×D₁₃₇	C₂×Q₈
cosocle	C₂×C₄×D₁₃₇	C₂³
hypercenter	#97	#10587739
hypocenter	C₁₃₇	#1838
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊#10587739

commutator

C₂×C₄×C₁₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#10587739

Fitting

#97×C₁₃₇

C₄

socle

C₂²×C₁₃₇

C₂×C₄⋊C₄

prime radical

C₂²×D₁₃₇

C₂×Q₈

cosocle

C₂×C₄×D₁₃₇

C₂³

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
50. C₁₃₇	#1838	semidirect

H ≅ G/N

type

50. C₁₃₇

#1838

semidirect

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

subgroup	index
#22578408	2
#22578407	2
#22578279	2
#97×D₁₃₇	2
#1838	137

subgroup

index

#97×D₁₃₇

137

Subgroups — #31796715

Conjugacy classes of subgroups

Complements

Maximal subgroups