Group #30662765 — subgroups

Subgroups — #2582×C₁₃₁ #30662765

Order 16768.

484 conjugacy classes of subgroups; only the 248 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 248 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×C₄×Q₈×C₁₃₁	C₂	1	2	yes	maximal
2	#398×C₁₃₁	C₂	1	4	yes	maximal
3	#391×C₁₃₁	C₂	1	4	yes	maximal
4	#389×C₁₃₁	C₂	1	4	yes	maximal
5	#384×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₃₁	C₂²	1	2	yes	—
7	C₄×Q₈×C₁₃₁	—	2	8	no	—
8	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
9	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	4	yes	—
10	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	4	yes	—
11	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	4	yes	—
12	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₃₁	C₂²	1	4	yes	—
14	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
15	#95×C₁₃₁	C₂²	1	8	yes	—
16	#95×C₁₃₁	C₂²	1	4	yes	—
17	C₂×Q₈×C₁₃₁	—	2	4	no	—
18	C₂×Q₈×C₁₃₁	—	2	4	no	—
19	C₂×Q₈×C₁₃₁	Q₈	1	8	yes	complemented
20	C₂²×C₄×C₁₃₁	C₂³	1	4	yes	—
21	C₂²×C₄×C₁₃₁	C₂³	1	4	yes	—
22	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₁₃₁	C₂³	1	4	yes	—
24	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
25	C₄⋊C₄×C₁₃₁	—	2	8	no	—
26	C₄⋊C₄×C₁₃₁	—	2	4	no	—
27	C₄⋊C₄×C₁₃₁	—	2	8	no	—
28	C₄⋊C₄×C₁₃₁	—	4	2	no	—
29	C₄⋊C₄×C₁₃₁	—	4	4	no	—
30	C₄²×C₁₃₁	—	2	8	no	—
31	C₄²×C₁₃₁	D₄	1	4	yes	—
32	C₂³×C₁₃₁	C₂⁴	1	1	yes	characteristic
33	Q₈×C₁₃₁	—	2	16	no	—
34	C₂×C₄×C₁₃₁	—	2	4	no	—
35	C₂×C₄×C₁₃₁	—	2	4	no	—
36	C₂×C₄×C₁₃₁	—	2	1	no	—
37	C₂×C₄×C₁₃₁	—	2	1	no	—
38	C₂×C₄×C₁₃₁	—	2	4	no	—
39	C₂×C₄×C₁₃₁	—	2	8	no	—
40	C₂×C₄×C₁₃₁	—	2	4	no	—
41	C₂×C₄×C₁₃₁	—	2	4	no	—
42	C₂×C₄×C₁₃₁	—	2	4	no	—
43	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
45	C₂×C₄×C₁₃₁	C₂×Q₈	1	8	yes	—
46	C₂×C₄×C₁₃₁	C₂×Q₈	1	4	yes	—
47	C₂×C₄×C₁₃₁	#41	1	4	yes	—
48	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
49	C₂²×C₁₃₁	C₂²×Q₈	1	2	yes	—
50	C₂²×C₁₃₁	C₂×#41	1	2	yes	—
51	C₂²×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂²×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
53	C₄×C₁₃₁	—	2	8	no	—
54	C₄×C₁₃₁	—	2	2	no	—
55	C₄×C₁₃₁	—	4	4	no	—
56	C₄×C₁₃₁	—	4	4	no	—
57	C₄×C₁₃₁	C₂²⋊Q₈	1	8	yes	—
58	C₂×C₁₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
59	C₂×C₁₃₁	#537	1	1	yes	characteristic
60	C₂×C₁₃₁	#557	1	2	yes	—
61	C₂×C₁₃₁	Q₈²	1	1	yes	characteristic
62	C₂×C₁₃₁	#559	1	1	yes	characteristic
63	C₁₃₁	#2582	1	1	yes	characteristic, direct product factor, Sylow
64	#2582	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂×C₄×Q₈	C₂×C₁₃₁	1	2	yes	—
66	#398	C₂×C₁₃₁	1	4	yes	—
67	#391	C₂×C₁₃₁	1	4	yes	—
68	#389	C₂×C₁₃₁	1	4	yes	—
69	#384	C₂×C₁₃₁	1	1	yes	characteristic
70	C₂²×Q₈	C₂²×C₁₃₁	1	2	yes	—
71	C₄×Q₈	—	2	8	no	—
72	C₂×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
73	C₂×C₄⋊C₄	C₂²×C₁₃₁	1	4	yes	—
74	C₂×C₄⋊C₄	C₂²×C₁₃₁	1	4	yes	—
75	C₂×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
76	C₂×C₄⋊C₄	C₂²×C₁₃₁	1	4	yes	—
77	C₂×C₄²	C₂²×C₁₃₁	1	4	yes	—
78	C₂×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
79	#95	C₂²×C₁₃₁	1	8	yes	—
80	#95	C₂²×C₁₃₁	1	4	yes	—
81	C₂×Q₈	—	2	4	no	—
82	C₂×Q₈	—	2	4	no	—
83	C₂×Q₈	Q₈×C₁₃₁	1	8	yes	complemented
84	C₂²×C₄	C₂³×C₁₃₁	1	4	yes	—
85	C₂²×C₄	C₂³×C₁₃₁	1	4	yes	—
86	C₂²×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
87	C₂²×C₄	C₂³×C₁₃₁	1	4	yes	—
88	C₂²×C₄	C₂³×C₁₃₁	1	2	yes	—
89	C₄⋊C₄	—	2	8	no	—
90	C₄⋊C₄	—	2	4	no	—
91	C₄⋊C₄	—	2	8	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	4	no	—
94	C₄²	—	2	8	no	—
95	C₄²	D₄×C₁₃₁	1	4	yes	—
96	C₂³	C₂⁴×C₁₃₁	1	1	yes	characteristic
97	Q₈	—	2	16	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	2	8	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
108	C₂×C₄	C₂×D₄×C₁₃₁	1	4	yes	—
109	C₂×C₄	C₂×Q₈×C₁₃₁	1	8	yes	—
110	C₂×C₄	C₂×Q₈×C₁₃₁	1	4	yes	—
111	C₂×C₄	#41×C₁₃₁	1	4	yes	—
112	C₂²	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
113	C₂²	C₂²×Q₈×C₁₃₁	1	2	yes	—
114	C₂²	C₂×#41×C₁₃₁	1	2	yes	—
115	C₂²	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
116	C₂²	2₋¹⁺⁴×C₁₃₁	1	1	yes	characteristic
117	C₄	—	2	8	no	—
118	C₄	—	2	2	no	—
119	C₄	—	4	4	no	—
120	C₄	—	4	4	no	—
121	C₄	C₂²⋊Q₈×C₁₃₁	1	8	yes	—
122	C₂	C₂×C₂²⋊Q₈×C₁₃₁	1	2	yes	—
123	C₂	#537×C₁₃₁	1	1	yes	characteristic
124	C₂	#557×C₁₃₁	1	2	yes	—
125	C₂	Q₈²×C₁₃₁	1	1	yes	characteristic
126	C₂	#559×C₁₃₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Q₈×C₁₃₁

C₂

yes

maximal

#398×C₁₃₁

C₂

yes

maximal

#391×C₁₃₁

C₂

yes

maximal

#389×C₁₃₁

C₂

yes

maximal

#384×C₁₃₁

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₃₁

C₂²

yes

—

C₄×Q₈×C₁₃₁

—

C₂×C₄⋊C₄×C₁₃₁

C₂²

yes

—

C₂×C₄⋊C₄×C₁₃₁

C₂²

yes

—

C₂×C₄⋊C₄×C₁₃₁

C₂²

yes

—

C₂×C₄⋊C₄×C₁₃₁

C₂²

yes

—

C₂×C₄⋊C₄×C₁₃₁

C₂²

yes

—

C₂×C₄²×C₁₃₁

C₂²

yes

—

C₂×C₄²×C₁₃₁

C₂²

yes

characteristic

#95×C₁₃₁

C₂²

yes

—

#95×C₁₃₁

C₂²

yes

—

C₂×Q₈×C₁₃₁

—

C₂×Q₈×C₁₃₁

—

C₂×Q₈×C₁₃₁

Q₈

yes

complemented

C₂²×C₄×C₁₃₁

C₂³

yes

—

C₂²×C₄×C₁₃₁

C₂³

yes

—

C₂²×C₄×C₁₃₁

C₂³

yes

characteristic

C₂²×C₄×C₁₃₁

C₂³

yes

—

C₂²×C₄×C₁₃₁

C₂³

yes

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄²×C₁₃₁

—

C₄²×C₁₃₁

D₄

yes

—

C₂³×C₁₃₁

C₂⁴

yes

characteristic

Q₈×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

—

C₂×C₄×C₁₃₁

C₂×Q₈

yes

—

C₂×C₄×C₁₃₁

C₂×Q₈

yes

—

C₂×C₄×C₁₃₁

#41

yes

—

C₂²×C₁₃₁

C₂²×D₄

yes

characteristic

C₂²×C₁₃₁

C₂²×Q₈

yes

—

C₂²×C₁₃₁

C₂×#41

yes

—

C₂²×C₁₃₁

H(5,𝔽₂)

yes

characteristic

C₂²×C₁₃₁

2₋¹⁺⁴

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂²⋊Q₈

yes

—

C₂×C₁₃₁

C₂×C₂²⋊Q₈

yes

—

C₂×C₁₃₁

#537

yes

characteristic

C₂×C₁₃₁

#557

yes

—

C₂×C₁₃₁

Q₈²

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

characteristic, direct product factor, Sylow

#2582

C₁₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄×Q₈

C₂×C₁₃₁

yes

—

#398

C₂×C₁₃₁

yes

—

#391

C₂×C₁₃₁

yes

—

#389

C₂×C₁₃₁

yes

—

#384

C₂×C₁₃₁

yes

characteristic

C₂²×Q₈

C₂²×C₁₃₁

yes

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₂²×C₁₃₁

yes

—

C₂×C₄⋊C₄

C₂²×C₁₃₁

yes

—

C₂×C₄⋊C₄

C₂²×C₁₃₁

yes

—

C₂×C₄⋊C₄

C₂²×C₁₃₁

yes

—

C₂×C₄⋊C₄

C₂²×C₁₃₁

yes

—

C₂×C₄²

C₂²×C₁₃₁

yes

—

C₂×C₄²

C₂²×C₁₃₁

yes

characteristic

#95

C₂²×C₁₃₁

yes

—

#95

C₂²×C₁₃₁

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Q₈×C₁₃₁

yes

complemented

C₂²×C₄

C₂³×C₁₃₁

yes

—

C₂²×C₄

C₂³×C₁₃₁

yes

—

C₂²×C₄

C₂³×C₁₃₁

yes

characteristic

C₂²×C₄

C₂³×C₁₃₁

yes

—

C₂²×C₄

C₂³×C₁₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

D₄×C₁₃₁

yes

—

C₂³

C₂⁴×C₁₃₁

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂×D₄×C₁₃₁

yes

—

108

C₂×C₄

C₂×D₄×C₁₃₁

yes

—

109

C₂×C₄

C₂×Q₈×C₁₃₁

yes

—

110

C₂×C₄

C₂×Q₈×C₁₃₁

yes

—

111

C₂×C₄

#41×C₁₃₁

yes

—

112

C₂²

C₂²×D₄×C₁₃₁

yes

characteristic

113

C₂²

C₂²×Q₈×C₁₃₁

yes

—

114

C₂²

C₂×#41×C₁₃₁

yes

—

115

C₂²

H(5,𝔽₂)×C₁₃₁

yes

characteristic

116

C₂²

2₋¹⁺⁴×C₁₃₁

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₂²⋊Q₈×C₁₃₁

yes

—

122

C₂

C₂×C₂²⋊Q₈×C₁₃₁

yes

—

123

C₂

#537×C₁₃₁

yes

characteristic

124

C₂

#557×C₁₃₁

yes

—

125

C₂

Q₈²×C₁₃₁

yes

characteristic

126

C₂

#559×C₁₃₁

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₃₁

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₁₃₁

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₃₁ ◁ G

N	H ≅ G/N	type
19. C₂×Q₈×C₁₃₁	Q₈	semidirect
63. C₁₃₁	#2582	direct (N × H)
64. #2582	C₁₃₁	direct (N × H)
83. C₂×Q₈	Q₈×C₁₃₁	semidirect

H ≅ G/N

type

19. C₂×Q₈×C₁₃₁

Q₈

semidirect

63. C₁₃₁

#2582

direct (N × H)

64. #2582

C₁₃₁

direct (N × H)

83. C₂×Q₈

Q₈×C₁₃₁

semidirect

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

subgroup	index
C₂×C₄×Q₈×C₁₃₁	2
#398×C₁₃₁	2
#391×C₁₃₁	2
#389×C₁₃₁	2
#384×C₁₃₁	2
#2582	131

subgroup

index

C₂×C₄×Q₈×C₁₃₁

#398×C₁₃₁

#391×C₁₃₁

#389×C₁₃₁

#384×C₁₃₁

#2582

131

Subgroups — #2582×C₁₃₁ #30662765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2582×C131 #30662765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2582×C₁₃₁ #30662765