Group #31797120 — subgroups

Subgroups — C₂×#22578324 #31797120

Order 17536.

184 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 106 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×#14555513	C₂	1	2	yes	maximal
2	#22578324	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₈×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555513	—	2	4	no	—
5	C₂×#14555513	C₂²	1	4	yes	—
6	C₂²×#11764040	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₃₇	C₄	1	1	yes	characteristic
8	C₈×Dic₁₃₇	C₄	1	2	yes	—
9	C₈×Dic₁₃₇	C₂²	1	2	yes	—
10	C₂²×C₈×C₁₃₇	C₄	1	1	yes	characteristic
11	#14555513	—	2	4	no	—
12	C₂²×Dic₁₃₇	C₈	1	1	yes	characteristic
13	C₂²×Dic₁₃₇	C₈	1	1	yes	characteristic
14	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
15	C₂×#11764040	D₄	1	2	yes	—
16	C₂×#11764040	Q₈	1	2	yes	—
17	C₂×#11764040	C₂³	1	1	yes	characteristic
18	C₄×Dic₁₃₇	C₈	1	2	yes	—
19	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₃₇	—	2	2	no	—
22	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
23	C₂×Dic₁₃₇	—	2	2	no	—
24	C₂×Dic₁₃₇	C₁₆	1	4	yes	complemented
25	C₂×Dic₁₃₇	C₂×C₈	1	2	yes	—
26	C₂×Dic₁₃₇	C₂×C₈	1	2	yes	—
27	#11764040	C₄⋊C₄	1	2	yes	—
28	#11764040	C₂×D₄	1	1	yes	characteristic
29	#11764040	C₂×Q₈	1	1	yes	characteristic
30	C₂³×C₁₃₇	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
32	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₁₃₇	M₁₆	1	2	yes	—
34	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
35	C₈×C₁₃₇	—	2	2	no	—
36	C₂²×C₁₃₇	C₄⋊C₈	1	2	yes	—
37	C₂²×C₁₃₇	C₂×C₁₆	1	2	yes	—
38	C₂²×C₁₃₇	M₃₂	1	2	yes	—
39	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
40	C₄×C₁₃₇	C₄⋊C₈	1	2	yes	—
41	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
43	Dic₁₃₇	—	2	2	no	—
44	Dic₁₃₇	C₂×C₁₆	1	4	yes	complemented
45	C₂×C₁₃₇	C₄⋊C₁₆	1	4	yes	complemented
46	C₂×C₁₃₇	C₂×C₄⋊C₈	1	1	yes	characteristic
47	C₂×C₁₃₇	C₂²×C₁₆	1	1	yes	characteristic
48	C₂×C₁₃₇	C₂×M₃₂	1	1	yes	characteristic
49	C₁₃₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₄⋊C₁₆	—	137	1	no	maximal, Sylow
51	C₂²×C₁₆	—	137	2	no	—
52	C₂×C₄×C₈	—	137	1	no	—
53	C₄⋊C₁₆	—	137	4	no	—
54	C₂²×C₈	—	137	1	no	—
55	C₂²×C₈	#10587739	1	1	yes	characteristic
56	C₂×C₄²	—	137	1	no	—
57	C₂×C₁₆	—	274	4	no	—
58	C₂×C₁₆	—	137	4	no	—
59	C₄×C₈	—	137	2	no	—
60	C₄×C₈	—	137	2	no	—
61	C₂²×C₄	—	137	1	no	—
62	C₂²×C₄	—	137	1	no	—
63	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	137	2	no	—
66	C₂×C₈	—	137	2	no	—
67	C₂×C₈	—	137	1	no	—
68	C₂×C₈	—	137	1	no	—
69	C₂×C₈	C₂×#10587739	1	2	yes	—
70	C₄²	—	137	2	no	—
71	C₄²	—	137	2	no	—
72	C₁₆	—	274	4	no	—
73	C₂³	#14555518	1	1	yes	characteristic
74	C₂×C₄	—	274	2	no	—
75	C₂×C₄	—	137	2	no	—
76	C₂×C₄	—	137	4	no	—
77	C₂×C₄	—	137	2	no	—
78	C₂×C₄	C₄⋊#10587739	1	2	yes	—
79	C₂×C₄	#14555518	1	1	yes	characteristic
80	C₂×C₄	#14555519	1	2	yes	—
81	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
82	C₈	—	2	2	no	—
83	C₈	—	137	1	no	—
84	C₈	—	137	2	no	—
85	C₈	—	137	1	no	—
86	C₂²	#17984155	1	2	yes	—
87	C₂²	#17984161	1	2	yes	—
88	C₂²	#17984163	1	2	yes	—
89	C₂²	C₂×#14555518	1	1	yes	characteristic
90	C₄	—	274	2	no	—
91	C₄	—	137	4	no	—
92	C₄	#17984155	1	2	yes	—
93	C₄	C₂×C₄⋊#10587739	1	1	yes	characteristic
94	C₄	C₂×#14555519	1	1	yes	characteristic
95	C₂	#22578324	1	4	yes	direct product factor
96	C₂	C₂×#17984155	1	1	yes	characteristic
97	C₂	C₂×#17984161	1	1	yes	characteristic
98	C₂	C₂×#17984163	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#14555513

C₂

yes

maximal

#22578324

C₂

yes

maximal, direct product factor

C₂×C₈×Dic₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555513

—

C₂×#14555513

C₂²

yes

—

C₂²×#11764040

C₂²

yes

characteristic

C₂×C₄×Dic₁₃₇

C₄

yes

characteristic

C₈×Dic₁₃₇

C₄

yes

—

C₈×Dic₁₃₇

C₂²

yes

—

C₂²×C₈×C₁₃₇

C₄

yes

characteristic

#14555513

—

C₂²×Dic₁₃₇

C₈

yes

characteristic

C₂²×Dic₁₃₇

C₈

yes

characteristic

C₂×#11764040

C₂×C₄

yes

characteristic

C₂×#11764040

D₄

yes

—

C₂×#11764040

Q₈

yes

—

C₂×#11764040

C₂³

yes

characteristic

C₄×Dic₁₃₇

C₈

yes

—

C₄×Dic₁₃₇

C₂×C₄

yes

—

C₂²×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂×C₄

yes

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₁₆

yes

complemented

C₂×Dic₁₃₇

C₂×C₈

yes

—

C₂×Dic₁₃₇

C₂×C₈

yes

—

#11764040

C₄⋊C₄

yes

—

#11764040

C₂×D₄

yes

characteristic

#11764040

C₂×Q₈

yes

characteristic

C₂³×C₁₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₃₇

M₁₆

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₈×C₁₃₇

—

C₂²×C₁₃₇

C₄⋊C₈

yes

—

C₂²×C₁₃₇

C₂×C₁₆

yes

—

C₂²×C₁₃₇

M₃₂

yes

—

C₂²×C₁₃₇

C₂²×C₈

yes

characteristic

C₄×C₁₃₇

C₄⋊C₈

yes

—

C₄×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₃₇

C₂×M₁₆

yes

characteristic

Dic₁₃₇

—

Dic₁₃₇

C₂×C₁₆

yes

complemented

C₂×C₁₃₇

C₄⋊C₁₆

yes

complemented

C₂×C₁₃₇

C₂×C₄⋊C₈

yes

characteristic

C₂×C₁₃₇

C₂²×C₁₆

yes

characteristic

C₂×C₁₃₇

C₂×M₃₂

yes

characteristic

C₁₃₇

C₂×C₄⋊C₁₆

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₁₆

—

137

maximal, Sylow

C₂²×C₁₆

—

137

—

C₂×C₄×C₈

—

137

—

C₄⋊C₁₆

—

137

—

C₂²×C₈

—

137

—

C₂²×C₈

#10587739

yes

characteristic

C₂×C₄²

—

137

—

C₂×C₁₆

—

274

—

C₂×C₁₆

—

137

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

C₂×#10587739

yes

characteristic

C₂×C₈

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

C₂×#10587739

yes

—

C₄²

—

137

—

C₄²

—

137

—

C₁₆

—

274

—

C₂³

#14555518

yes

characteristic

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

C₄⋊#10587739

yes

—

C₂×C₄

#14555518

yes

characteristic

C₂×C₄

#14555519

yes

—

C₂×C₄

C₂²×#10587739

yes

characteristic

C₈

—

C₈

—

137

—

C₈

—

137

—

C₈

—

137

—

C₂²

#17984155

yes

—

C₂²

#17984161

yes

—

C₂²

#17984163

yes

—

C₂²

C₂×#14555518

yes

characteristic

C₄

—

274

—

C₄

—

137

—

C₄

#17984155

yes

—

C₄

yes

C₄

yes

C₂

yes

direct product factor

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²×C₄	C₂×#10587739
commutator	C₂×C₁₃₇	C₂²×C₁₆
Frattini	C₂×C₄	C₂²×#10587739
Fitting	C₂²×C₈×C₁₃₇	C₄
socle, prime radical	C₂³×C₁₃₇	C₂×C₈
cosocle	C₂×#11764040	C₂³
hypercenter	C₂²×C₈	#10587739
hypocenter	C₁₃₇	C₂×C₄⋊C₁₆
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

subgroup

G/H

center

C₂²×C₄

C₂×#10587739

commutator

C₂×C₁₃₇

C₂²×C₁₆

Frattini

C₂×C₄

C₂²×#10587739

Fitting

C₂²×C₈×C₁₃₇

C₄

socle, prime radical

C₂³×C₁₃₇

C₂×C₈

C₂³

C₂²×C₈

C₁₃₇

C₂×C₄⋊C₁₆

LCS

G ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

N	H ≅ G/N	type
2. #22578324	C₂	direct (N × H)
24. C₂×Dic₁₃₇	C₁₆	semidirect
44. Dic₁₃₇	C₂×C₁₆	semidirect
45. C₂×C₁₃₇	C₄⋊C₁₆	semidirect
49. C₁₃₇	C₂×C₄⋊C₁₆	semidirect
95. C₂	#22578324	direct (N × H)

H ≅ G/N

type

2. #22578324

C₂

direct (N × H)

24. C₂×Dic₁₃₇

C₁₆

semidirect

44. Dic₁₃₇

C₂×C₁₆

semidirect

45. C₂×C₁₃₇

C₄⋊C₁₆

semidirect

49. C₁₃₇

C₂×C₄⋊C₁₆

semidirect

95. C₂

#22578324

direct (N × H)

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

subgroup	index
C₂²×#14555513	2
#22578324	2
C₂×C₈×Dic₁₃₇	2
C₂×C₄⋊C₁₆	137

subgroup

index

C₂²×#14555513

#22578324

C₂×C₈×Dic₁₃₇

C₂×C₄⋊C₁₆

137

Subgroups — C₂×#22578324 #31797120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22578324 #31797120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22578324 #31797120