Group #31777113 — subgroups

Subgroups — #2102×C₁₃₇ #31777113

Order 17536.

158 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 72 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	#430×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#423×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#344×C₁₃₇	C₂	1	2	yes	maximal
4	#339×C₁₃₇	C₂	1	2	yes	maximal
5	H(3,ℤ/4)×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₁₃₇	C₂²	1	2	yes	—
8	C₂×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
9	#117×C₁₃₇	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
11	#108×C₁₃₇	—	2	2	no	—
12	C₄⋊C₈×C₁₃₇	—	2	2	no	—
13	C₂²×C₄×C₁₃₇	—	2	1	no	—
14	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
15	M₁₆×C₁₃₇	—	2	2	no	—
16	M₁₆×C₁₃₇	—	2	2	no	—
17	M₁₆×C₁₃₇	—	2	2	no	—
18	M₁₆×C₁₃₇	—	2	2	no	—
19	C₂×C₈×C₁₃₇	—	2	1	no	—
20	C₂×C₈×C₁₃₇	—	2	2	no	—
21	C₂×C₈×C₁₃₇	D₄	1	2	yes	—
22	C₄⋊C₄×C₁₃₇	—	2	2	no	—
23	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
24	C₄²×C₁₃₇	—	2	1	no	—
25	C₄²×C₁₃₇	—	2	2	no	—
26	C₄²×C₁₃₇	Q₈	1	1	yes	characteristic
27	C₄²×C₁₃₇	Q₈	1	1	yes	characteristic
28	C₂³×C₁₃₇	#41	1	1	yes	characteristic
29	C₂×C₄×C₁₃₇	—	2	1	no	—
30	C₂×C₄×C₁₃₇	—	2	1	no	—
31	C₂×C₄×C₁₃₇	—	4	2	no	—
32	C₂×C₄×C₁₃₇	—	4	2	no	—
33	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₁₃₇	#41	1	2	yes	—
36	C₂×C₄×C₁₃₇	#41	1	2	yes	—
37	C₈×C₁₃₇	—	2	2	no	—
38	C₈×C₁₃₇	—	4	1	no	—
39	C₈×C₁₃₇	—	4	2	no	—
40	C₂²×C₁₃₇	—	4	1	no	—
41	C₂²×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
42	C₂²×C₁₃₇	#126	1	2	yes	—
43	C₄×C₁₃₇	—	2	2	no	—
44	C₄×C₁₃₇	—	4	1	no	—
45	C₄×C₁₃₇	—	8	2	no	—
46	C₄×C₁₃₇	#123	1	1	yes	characteristic
47	C₄×C₁₃₇	#124	1	1	yes	characteristic
48	C₄×C₁₃₇	#125	1	2	yes	—
49	C₂×C₁₃₇	—	2	1	no	—
50	C₂×C₁₃₇	—	2	2	no	—
51	C₂×C₁₃₇	#400	1	1	yes	characteristic
52	C₁₃₇	#2102	1	1	yes	characteristic, direct product factor, Sylow
53	#2102	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	#430	C₂×C₁₃₇	1	1	yes	characteristic
55	#423	C₂×C₁₃₇	1	1	yes	characteristic
56	#344	C₂×C₁₃₇	1	2	yes	—
57	#339	C₂×C₁₃₇	1	2	yes	—
58	H(3,ℤ/4)	C₂×C₁₃₇	1	1	yes	characteristic
59	C₂×M₁₆	C₂²×C₁₃₇	1	2	yes	—
60	C₂×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
61	C₂×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
62	#117	C₂²×C₁₃₇	1	2	yes	—
63	C₂×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
64	#108	—	2	2	no	—
65	C₄⋊C₈	—	2	2	no	—
66	C₂²×C₄	—	2	1	no	—
67	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	2	2	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	D₄×C₁₃₇	1	2	yes	—
75	C₄⋊C₄	—	2	2	no	—
76	C₂²⋊C₄	—	2	2	no	—
77	C₄²	—	2	1	no	—
78	C₄²	—	2	2	no	—
79	C₄²	Q₈×C₁₃₇	1	1	yes	characteristic
80	C₄²	Q₈×C₁₃₇	1	1	yes	characteristic
81	C₂³	#41×C₁₃₇	1	1	yes	characteristic
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	—	4	2	no	—
86	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
87	C₂×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
88	C₂×C₄	#41×C₁₃₇	1	2	yes	—
89	C₂×C₄	#41×C₁₃₇	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	4	1	no	—
92	C₈	—	4	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
95	C₂²	#126×C₁₃₇	1	2	yes	—
96	C₄	—	2	2	no	—
97	C₄	—	4	1	no	—
98	C₄	—	8	2	no	—
99	C₄	#123×C₁₃₇	1	1	yes	characteristic
100	C₄	#124×C₁₃₇	1	1	yes	characteristic
101	C₄	#125×C₁₃₇	1	2	yes	—
102	C₂	—	2	1	no	—
103	C₂	—	2	2	no	—
104	C₂	#400×C₁₃₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#430×C₁₃₇

C₂

yes

maximal, characteristic

#423×C₁₃₇

C₂

yes

maximal, characteristic

#344×C₁₃₇

C₂

yes

maximal

#339×C₁₃₇

C₂

yes

maximal

H(3,ℤ/4)×C₁₃₇

C₂

yes

maximal, characteristic

C₂×M₁₆×C₁₃₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₃₇

C₂²

yes

—

C₂×M₁₆×C₁₃₇

C₂²

yes

characteristic

#117×C₁₃₇

C₂²

yes

—

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

#108×C₁₃₇

—

C₄⋊C₈×C₁₃₇

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

M₁₆×C₁₃₇

—

M₁₆×C₁₃₇

—

M₁₆×C₁₃₇

—

M₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

D₄

yes

—

C₄⋊C₄×C₁₃₇

—

C₂²⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

—

C₄²×C₁₃₇

—

C₄²×C₁₃₇

Q₈

yes

characteristic

C₄²×C₁₃₇

Q₈

yes

characteristic

C₂³×C₁₃₇

#41

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₇

#41

yes

—

C₂×C₄×C₁₃₇

#41

yes

—

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₃₇

#126

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

—

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#400

yes

characteristic

C₁₃₇

#2102

yes

characteristic, direct product factor, Sylow

#2102

C₁₃₇

yes

maximal, characteristic, direct product factor, Sylow

#430

C₂×C₁₃₇

yes

characteristic

#423

C₂×C₁₃₇

yes

characteristic

#344

C₂×C₁₃₇

yes

—

#339

C₂×C₁₃₇

yes

—

H(3,ℤ/4)

C₂×C₁₃₇

yes

characteristic

C₂×M₁₆

C₂²×C₁₃₇

yes

—

C₂×M₁₆

C₂²×C₁₃₇

yes

characteristic

C₂×M₁₆

C₂²×C₁₃₇

yes

characteristic

#117

C₂²×C₁₃₇

yes

—

C₂×C₄²

C₂²×C₁₃₇

yes

characteristic

#108

—

C₄⋊C₈

—

C₂²×C₄

—

C₂²×C₄

C₂³×C₁₃₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₁₃₇

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

Q₈×C₁₃₇

yes

characteristic

C₄²

Q₈×C₁₃₇

yes

characteristic

C₂³

#41×C₁₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₁₃₇

yes

characteristic

C₂×C₄

C₂×Q₈×C₁₃₇

yes

characteristic

C₂×C₄

#41×C₁₃₇

yes

—

C₂×C₄

#41×C₁₃₇

yes

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

C₂²⋊Q₈×C₁₃₇

yes

characteristic

C₂²

#126×C₁₃₇

yes

—

C₄

—

C₄

—

C₄

—

C₄

#123×C₁₃₇

yes

characteristic

100

C₄

#124×C₁₃₇

yes

characteristic

101

C₄

#125×C₁₃₇

yes

—

102

C₂

—

103

C₂

—

104

C₂

#400×C₁₃₇

yes

characteristic

	subgroup	G/H
center	C₄×C₁₃₇	#124
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₃₇
socle	C₂×C₁₃₇	#400
prime radical	C₂³×C₁₃₇	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ G

subgroup

G/H

center

C₄×C₁₃₇

#124

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₁₃₇

C₂×C₁₃₇

C₂³×C₁₃₇

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₄×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ G

N	H ≅ G/N	type
52. C₁₃₇	#2102	direct (N × H)
53. #2102	C₁₃₇	direct (N × H)

H ≅ G/N

type

52. C₁₃₇

#2102

direct (N × H)

53. #2102

C₁₃₇

direct (N × H)

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

subgroup	index
#430×C₁₃₇	2
#423×C₁₃₇	2
#344×C₁₃₇	2
#339×C₁₃₇	2
H(3,ℤ/4)×C₁₃₇	2
#2102	137

subgroup

index

#430×C₁₃₇

#423×C₁₃₇

#344×C₁₃₇

#339×C₁₃₇

H(3,ℤ/4)×C₁₃₇

#2102

137

Subgroups — #2102×C₁₃₇ #31777113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2102×C137 #31777113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2102×C₁₃₇ #31777113