Group #31797000 — subgroups

Subgroups — #31797000

Order 17536.

184 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 114 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	C₂×#17984162	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14555513	C₂	1	1	yes	maximal, characteristic
3	#22578292	C₂	1	2	yes	maximal, complemented
4	#22578289	C₂	1	2	yes	maximal, complemented
5	#22577361	C₂	1	1	yes	maximal, characteristic
6	#17984162	C₄	1	4	yes	complemented
7	C₂×#14555513	C₄	1	4	yes	complemented
8	C₂×#14555513	C₂²	1	1	yes	characteristic
9	C₂×#14555513	C₂²	1	1	yes	characteristic
10	C₂×#14555513	C₂²	1	2	yes	—
11	C₂²×#11764040	C₂²	1	1	yes	characteristic
12	#17984047	C₄	1	1	yes	characteristic
13	#17983998	C₄	1	2	yes	—
14	C₈×Dic₁₃₇	C₂²	1	2	yes	—
15	C₂×M₁₆×C₁₃₇	C₄	1	1	yes	characteristic
16	#14555513	C₂×C₄	1	2	yes	—
17	#14555513	C₂×C₄	1	2	yes	—
18	#14555513	C₂×C₄	1	4	yes	—
19	C₂×#11764040	C₂×C₄	1	2	yes	—
20	C₂×#11764040	C₂×C₄	1	2	yes	—
21	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
22	C₂×#11764040	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₁₃₇	C₈	1	2	yes	—
24	C₂²⋊Dic₁₃₇	C₈	1	2	yes	—
25	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
27	M₁₆×C₁₃₇	C₈	1	4	yes	—
28	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₃₇	C₂×C₈	1	4	yes	—
30	#11764040	C₄²	1	2	yes	—
31	#11764040	C₂²×C₄	1	1	yes	characteristic
32	#11764040	C₂²×C₄	1	1	yes	characteristic
33	C₂³×C₁₃₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₁₃₇	C₄²	1	2	yes	—
35	C₂×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
37	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₁₃₇	C₂×C₈	1	4	yes	—
39	C₂²×C₁₃₇	—	2	2	no	—
40	C₂²×C₁₃₇	C₄×C₈	1	2	yes	—
41	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₁₃₇	C₄×C₈	1	2	yes	—
43	C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
44	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
45	Dic₁₃₇	—	2	4	no	—
46	C₂×C₁₃₇	—	2	2	no	—
47	C₂×C₁₃₇	C₂×C₄×C₈	1	1	yes	characteristic
48	C₂×C₁₃₇	#504	1	2	yes	—
49	C₁₃₇	#2108	1	1	yes	characteristic, complemented, Sylow
50	#2108	—	137	1	no	maximal, Sylow
51	C₂×M₃₂	—	137	1	no	—
52	C₂²×C₁₆	—	137	1	no	—
53	#405	—	137	1	no	—
54	#346	—	137	2	no	—
55	C₄×C₁₆	—	137	2	no	—
56	C₂×M₁₆	#10587739	1	1	yes	characteristic
57	C₂²×C₈	—	137	1	no	—
58	#117	—	137	1	no	—
59	M₃₂	—	137	4	no	—
60	C₂×C₁₆	—	137	1	no	—
61	C₂×C₁₆	—	137	2	no	—
62	C₂×C₁₆	—	137	1	no	—
63	C₂×C₁₆	—	137	4	no	—
64	#97	—	137	2	no	—
65	C₄×C₈	—	137	2	no	—
66	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
67	M₁₆	#11764047	1	4	yes	—
68	C₂×C₈	—	137	2	no	—
69	C₂×C₈	—	137	2	no	—
70	C₂×C₈	—	137	1	no	—
71	C₂×C₈	—	137	1	no	—
72	C₂×C₈	C₂×#10587739	1	2	yes	—
73	C₄⋊C₄	—	137	2	no	—
74	C₂²⋊C₄	—	137	2	no	—
75	C₄²	—	137	2	no	—
76	C₁₆	—	137	2	no	—
77	C₁₆	—	137	2	no	—
78	C₁₆	—	137	4	no	—
79	C₂³	#14555518	1	1	yes	characteristic
80	C₂×C₄	—	137	4	no	—
81	C₂×C₄	C₄×#10587739	1	2	yes	—
82	C₂×C₄	C₂×#11764047	1	2	yes	—
83	C₂×C₄	#14555518	1	1	yes	characteristic
84	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
85	C₈	—	137	1	no	—
86	C₈	—	137	1	no	—
87	C₈	—	137	2	no	—
88	C₈	C₂×#11764047	1	4	yes	—
89	C₂²	—	2	2	no	—
90	C₂²	C₄×#11764047	1	2	yes	—
91	C₂²	C₂×#14555518	1	1	yes	characteristic
92	C₄	—	274	4	no	—
93	C₄	C₄×#11764047	1	2	yes	—
94	C₄	C₂×C₄×#10587739	1	1	yes	characteristic
95	C₄	C₂²×#11764047	1	1	yes	characteristic
96	C₂	—	2	2	no	—
97	C₂	C₂×C₄×#11764047	1	1	yes	characteristic
98	C₂	#22578462	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984162

C₂

yes

maximal, characteristic

C₂²×#14555513

C₂

yes

maximal, characteristic

#22578292

C₂

yes

maximal, complemented

#22578289

C₂

yes

maximal, complemented

#22577361

C₂

yes

maximal, characteristic

#17984162

C₄

yes

complemented

C₂×#14555513

C₄

yes

complemented

C₂×#14555513

C₂²

yes

characteristic

C₂×#14555513

C₂²

yes

characteristic

C₂×#14555513

C₂²

yes

—

C₂²×#11764040

C₂²

yes

characteristic

#17984047

C₄

yes

characteristic

#17983998

C₄

yes

—

C₈×Dic₁₃₇

C₂²

yes

—

C₂×M₁₆×C₁₃₇

C₄

yes

characteristic

#14555513

C₂×C₄

yes

—

#14555513

C₂×C₄

yes

—

#14555513

C₂×C₄

yes

—

C₂×#11764040

C₂×C₄

yes

—

C₂×#11764040

C₂×C₄

yes

—

C₂×#11764040

C₂×C₄

yes

characteristic

C₂×#11764040

C₂³

yes

characteristic

C₄⋊Dic₁₃₇

C₈

yes

—

C₂²⋊Dic₁₃₇

C₈

yes

—

C₄×Dic₁₃₇

C₂×C₄

yes

—

C₂²×C₄×C₁₃₇

C₂×C₄

yes

characteristic

M₁₆×C₁₃₇

C₈

yes

—

C₂×C₈×C₁₃₇

C₂×C₄

yes

—

C₂×Dic₁₃₇

C₂×C₈

yes

—

#11764040

C₄²

yes

—

#11764040

C₂²×C₄

yes

characteristic

#11764040

C₂²×C₄

yes

characteristic

C₂³×C₁₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₃₇

C₄²

yes

—

C₂×C₄×C₁₃₇

C₂×C₈

yes

—

C₂×C₄×C₁₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₈×C₁₃₇

C₂×C₈

yes

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₄×C₈

yes

—

C₂²×C₁₃₇

C₂²×C₈

yes

characteristic

C₄×C₁₃₇

C₄×C₈

yes

—

C₄×C₁₃₇

C₂×C₄²

yes

characteristic

C₄×C₁₃₇

C₂²×C₈

yes

characteristic

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂×C₄×C₈

yes

characteristic

C₂×C₁₃₇

#504

yes

—

C₁₃₇

#2108

yes

characteristic, complemented, Sylow

#2108

—

137

maximal, Sylow

C₂×M₃₂

—

137

—

C₂²×C₁₆

—

137

—

#405

—

137

—

#346

—

137

—

C₄×C₁₆

—

137

—

C₂×M₁₆

#10587739

yes

characteristic

C₂²×C₈

—

137

—

#117

—

137

—

M₃₂

—

137

—

C₂×C₁₆

—

137

—

C₂×C₁₆

—

137

—

C₂×C₁₆

—

137

—

C₂×C₁₆

—

137

—

#97

—

137

—

C₄×C₈

—

137

—

C₂²×C₄

C₂×#10587739

yes

characteristic

M₁₆

#11764047

yes

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

C₂×#10587739

yes

—

C₄⋊C₄

—

137

—

C₂²⋊C₄

—

137

—

C₄²

—

137

—

C₁₆

—

137

—

C₁₆

—

137

—

C₁₆

—

137

—

C₂³

#14555518

yes

characteristic

C₂×C₄

—

137

—

C₂×C₄

C₄×#10587739

yes

—

C₂×C₄

C₂×#11764047

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

137

—

C₈

—

137

—

C₈

—

137

—

C₈

C₂×#11764047

yes

—

C₂²

—

C₂²

C₄×#11764047

yes

—

C₂²

C₂×#14555518

yes

characteristic

C₄

—

274

—

C₄

C₄×#11764047

yes

—

C₄

yes

C₄

yes

C₂

—

C₂

C₂×C₄×#11764047

yes

characteristic

C₂

#22578462

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10587739
commutator	C₂×C₁₃₇	C₂×C₄×C₈
Fitting	C₂×M₁₆×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂²×C₈
prime radical	C₂³×C₁₃₇	C₂×C₈
cosocle	C₂×#11764040	C₂³
hypercenter	C₂×M₁₆	#10587739
hypocenter	C₁₃₇	#2108
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10587739

commutator

C₂×C₁₃₇

C₂×C₄×C₈

Fitting

C₂×M₁₆×C₁₃₇

C₄

socle

C₂²×C₁₃₇

C₂²×C₈

prime radical

C₂³×C₁₃₇

C₂×C₈

C₂³

C₂×M₁₆

C₁₃₇

G ▷ C₂×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
3. #22578292	C₂	semidirect
4. #22578289	C₂	semidirect
6. #17984162	C₄	semidirect
7. C₂×#14555513	C₄	semidirect
49. C₁₃₇	#2108	semidirect

H ≅ G/N

type

3. #22578292

C₂

semidirect

4. #22578289

C₂

semidirect

6. #17984162

C₄

semidirect

7. C₂×#14555513

C₄

semidirect

49. C₁₃₇

#2108

semidirect

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

subgroup	index
C₂×#17984162	2
C₂²×#14555513	2
#22578292	2
#22578289	2
#22577361	2
#2108	137

subgroup

index

137

Subgroups — #31797000

Conjugacy classes of subgroups

Complements

Maximal subgroups