Group #31783998 — subgroups

Subgroups — C₂×#22577322 #31783998

Order 17536.

908 conjugacy classes of subgroups; only the 285 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 285 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×#14555489	C₂	1	3	yes	maximal
2	#22577322	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17983995	C₂	1	3	yes	maximal
4	C₂³×Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₃₇	C₂²	1	1	yes	characteristic
6	C₂³×Dic₁₃₇	C₂²	1	3	yes	—
7	C₂×#14555489	—	2	12	no	—
8	C₂×#14555489	C₂²	1	12	yes	—
9	#17983995	C₂²	1	12	yes	—
10	C₂²×Q₈×C₁₃₇	—	2	12	no	—
11	C₂²×Q₈×C₁₃₇	C₂²	1	4	yes	—
12	C₂³×C₄×C₁₃₇	C₂²	1	3	yes	—
13	C₂²×Dic₁₃₇	—	2	6	no	—
14	C₂²×Dic₁₃₇	—	2	6	no	—
15	C₂²×Dic₁₃₇	—	2	6	no	—
16	C₂²×Dic₁₃₇	Q₈	1	12	yes	complemented
17	C₂²×Dic₁₃₇	C₂³	1	2	yes	—
18	C₂²×Dic₁₃₇	C₂³	1	6	yes	—
19	#14555489	—	2	24	no	—
20	C₂⁴×C₁₃₇	C₂³	1	1	yes	characteristic
21	C₂×Q₈×C₁₃₇	—	2	32	no	—
22	C₂×Q₈×C₁₃₇	—	2	24	no	—
23	C₂²×C₄×C₁₃₇	—	2	12	no	—
24	C₂²×C₄×C₁₃₇	D₄	1	12	yes	—
25	C₂²×C₄×C₁₃₇	C₂³	1	6	yes	—
26	C₂×Dic₁₃₇	—	2	24	no	—
27	C₂×Dic₁₃₇	—	2	6	no	—
28	C₂×Dic₁₃₇	—	2	6	no	—
29	C₂×Dic₁₃₇	—	2	6	no	—
30	C₂×Dic₁₃₇	—	4	4	no	—
31	C₂×Dic₁₃₇	C₂×Q₈	1	12	yes	complemented
32	C₂³×C₁₃₇	C₂×D₄	1	6	yes	—
33	C₂³×C₁₃₇	C₂×Q₈	1	6	yes	—
34	C₂³×C₁₃₇	#41	1	2	yes	—
35	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
36	Q₈×C₁₃₇	—	2	32	no	—
37	C₂×C₄×C₁₃₇	—	2	24	no	—
38	C₂×C₄×C₁₃₇	—	2	12	no	—
39	C₂×C₄×C₁₃₇	C₂×D₄	1	12	yes	—
40	C₂²×C₁₃₇	#120	1	4	yes	—
41	C₂²×C₁₃₇	C₂²⋊Q₈	1	12	yes	—
42	C₂²×C₁₃₇	C₄⋊Q₈	1	12	yes	—
43	C₂²×C₁₃₇	C₂²×D₄	1	3	yes	—
44	C₂²×C₁₃₇	C₂²×Q₈	1	3	yes	—
45	C₂²×C₁₃₇	C₂×#41	1	1	yes	characteristic
46	C₄×C₁₃₇	—	2	12	no	—
47	Dic₁₃₇	—	2	12	no	—
48	Dic₁₃₇	—	4	2	no	—
49	C₂×C₁₃₇	#395	1	8	yes	complemented
50	C₂×C₁₃₇	C₂×#120	1	1	yes	characteristic
51	C₂×C₁₃₇	C₂×C₂²⋊Q₈	1	3	yes	—
52	C₂×C₁₃₇	C₂×C₄⋊Q₈	1	3	yes	—
53	C₁₃₇	C₂×#395	1	1	yes	characteristic, complemented, Sylow
54	C₂×#395	—	137	1	no	maximal, Sylow
55	C₂³×Q₈	D₁₃₇	1	1	yes	characteristic
56	C₂²×C₄⋊C₄	—	137	3	no	—
57	#395	—	137	8	no	—
58	C₂×#95	—	137	3	no	—
59	C₂²×Q₈	—	2	12	no	—
60	C₂²×Q₈	C₂×D₁₃₇	1	4	yes	—
61	C₂³×C₄	—	137	1	no	—
62	C₂³×C₄	—	137	3	no	—
63	C₂³×C₄	C₂×D₁₃₇	1	3	yes	—
64	C₂×C₄⋊C₄	—	274	12	no	—
65	C₂×C₄⋊C₄	—	137	12	no	—
66	#95	—	137	12	no	—
67	C₂⁴	C₂²×D₁₃₇	1	1	yes	characteristic
68	C₂×Q₈	—	2	32	no	—
69	C₂×Q₈	—	2	24	no	—
70	C₂²×C₄	—	2	12	no	—
71	C₂²×C₄	—	274	6	no	—
72	C₂²×C₄	—	274	6	no	—
73	C₂²×C₄	—	274	6	no	—
74	C₂²×C₄	—	137	2	no	—
75	C₂²×C₄	—	137	12	no	—
76	C₂²×C₄	—	137	6	no	—
77	C₂²×C₄	C₂²⋊D₁₃₇	1	12	yes	—
78	C₂²×C₄	C₂²×D₁₃₇	1	6	yes	—
79	C₄⋊C₄	—	274	24	no	—
80	C₂³	C₂×C₂²⋊D₁₃₇	1	6	yes	—
81	C₂³	Q₈×D₁₃₇	1	6	yes	—
82	C₂³	#14555511	1	2	yes	—
83	C₂³	C₂³×D₁₃₇	1	1	yes	characteristic
84	Q₈	—	2	32	no	—
85	C₂×C₄	—	2	24	no	—
86	C₂×C₄	—	2	12	no	—
87	C₂×C₄	—	274	24	no	—
88	C₂×C₄	—	274	6	no	—
89	C₂×C₄	—	274	6	no	—
90	C₂×C₄	—	274	6	no	—
91	C₂×C₄	—	548	4	no	—
92	C₂×C₄	—	137	12	no	—
93	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	12	yes	—
94	C₂²	#17984057	1	4	yes	—
95	C₂²	#17984064	1	12	yes	—
96	C₂²	#17984086	1	12	yes	—
97	C₂²	C₂²×C₂²⋊D₁₃₇	1	3	yes	—
98	C₂²	C₂×Q₈×D₁₃₇	1	3	yes	—
99	C₂²	C₂×#14555511	1	1	yes	characteristic
100	C₄	—	2	12	no	—
101	C₄	—	274	12	no	—
102	C₄	—	548	2	no	—
103	C₂	#22577322	1	8	yes	direct product factor
104	C₂	C₂×#17984057	1	1	yes	characteristic
105	C₂	C₂×#17984064	1	3	yes	—
106	C₂	C₂×#17984086	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#14555489

C₂

yes

maximal

#22577322

C₂

yes

maximal, direct product factor

C₂×#17983995

C₂

yes

maximal

C₂³×Q₈×C₁₃₇

C₂

yes

maximal, characteristic

C₂³×Dic₁₃₇

C₂²

yes

characteristic

C₂³×Dic₁₃₇

C₂²

yes

—

C₂×#14555489

—

C₂×#14555489

C₂²

yes

—

#17983995

C₂²

yes

—

C₂²×Q₈×C₁₃₇

—

C₂²×Q₈×C₁₃₇

C₂²

yes

—

C₂³×C₄×C₁₃₇

C₂²

yes

—

C₂²×Dic₁₃₇

—

C₂²×Dic₁₃₇

—

C₂²×Dic₁₃₇

—

C₂²×Dic₁₃₇

Q₈

yes

complemented

C₂²×Dic₁₃₇

C₂³

yes

—

C₂²×Dic₁₃₇

C₂³

yes

—

#14555489

—

C₂⁴×C₁₃₇

C₂³

yes

characteristic

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

D₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₂×Q₈

yes

complemented

C₂³×C₁₃₇

C₂×D₄

yes

—

C₂³×C₁₃₇

C₂×Q₈

yes

—

C₂³×C₁₃₇

#41

yes

—

C₂³×C₁₃₇

C₂⁴

yes

characteristic

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂²×C₁₃₇

#120

yes

—

C₂²×C₁₃₇

C₂²⋊Q₈

yes

—

C₂²×C₁₃₇

C₄⋊Q₈

yes

—

C₂²×C₁₃₇

C₂²×D₄

yes

—

C₂²×C₁₃₇

C₂²×Q₈

yes

—

C₂²×C₁₃₇

C₂×#41

yes

characteristic

C₄×C₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

C₂×C₂²⋊Q₈

yes

—

C₂×C₁₃₇

C₂×C₄⋊Q₈

yes

—

C₁₃₇

C₂×#395

yes

characteristic, complemented, Sylow

C₂×#395

—

137

maximal, Sylow

C₂³×Q₈

D₁₃₇

yes

characteristic

C₂²×C₄⋊C₄

—

137

—

#395

—

137

—

C₂×#95

—

137

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₁₃₇

yes

—

C₂³×C₄

—

137

—

C₂³×C₄

—

137

—

C₂³×C₄

C₂×D₁₃₇

yes

—

C₂×C₄⋊C₄

—

274

—

C₂×C₄⋊C₄

—

137

—

#95

—

137

—

C₂⁴

C₂²×D₁₃₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

274

—

C₂²×C₄

—

274

—

C₂²×C₄

—

274

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

C₂²⋊D₁₃₇

yes

—

C₂²×C₄

C₂²×D₁₃₇

yes

—

C₄⋊C₄

—

274

—

C₂³

C₂×C₂²⋊D₁₃₇

yes

—

C₂³

Q₈×D₁₃₇

yes

—

C₂³

#14555511

yes

—

C₂³

C₂³×D₁₃₇

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

548

—

C₂×C₄

—

137

—

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

—

C₂²

#17984057

yes

—

C₂²

#17984064

yes

—

C₂²

#17984086

yes

—

C₂²

C₂²×C₂²⋊D₁₃₇

yes

—

C₂²

C₂×Q₈×D₁₃₇

yes

—

C₂²

C₂×#14555511

yes

characteristic

100

C₄

—

101

C₄

—

274

—

102

C₄

—

548

—

103

C₂

#22577322

yes

direct product factor

104

C₂

C₂×#17984057

yes

characteristic

105

C₂

C₂×#17984064

yes

—

106

C₂

C₂×#17984086

yes

—

	subgroup	G/H
center	C₂⁴	C₂²×D₁₃₇
commutator, cosocle	C₂³×C₁₃₇	C₂⁴
Frattini	C₂³	C₂³×D₁₃₇
Fitting	C₂³×Q₈×C₁₃₇	C₂
socle, prime radical	C₂⁴×C₁₃₇	C₂³
hypercenter	C₂³×Q₈	D₁₃₇
hypocenter	C₁₃₇	C₂×#395
LCS	G ▷ C₂³×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×Q₈

subgroup

G/H

center

C₂⁴

C₂²×D₁₃₇

commutator, cosocle

C₂³×C₁₃₇

C₂⁴

Frattini

C₂³

C₂³×D₁₃₇

Fitting

C₂³×Q₈×C₁₃₇

C₂

socle, prime radical

C₂⁴×C₁₃₇

C₂³

hypercenter

C₂³×Q₈

D₁₃₇

C₁₃₇

G ▷ C₂³×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂⁴ ◁ C₂³×Q₈

N	H ≅ G/N	type
2. #22577322	C₂	direct (N × H)
16. C₂²×Dic₁₃₇	Q₈	semidirect
31. C₂×Dic₁₃₇	C₂×Q₈	semidirect
49. C₂×C₁₃₇	#395	semidirect
53. C₁₃₇	C₂×#395	semidirect
103. C₂	#22577322	direct (N × H)

H ≅ G/N

type

2. #22577322

C₂

direct (N × H)

16. C₂²×Dic₁₃₇

Q₈

semidirect

31. C₂×Dic₁₃₇

C₂×Q₈

semidirect

49. C₂×C₁₃₇

#395

semidirect

53. C₁₃₇

C₂×#395

semidirect

103. C₂

#22577322

direct (N × H)

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

subgroup	index
C₂²×#14555489	2
#22577322	2
C₂×#17983995	2
C₂³×Q₈×C₁₃₇	2
C₂×#395	137

subgroup

index

C₂³×Q₈×C₁₃₇

137

Subgroups — C₂×#22577322 #31783998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22577322 #31783998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22577322 #31783998