Group #135807 — subgroups

Subgroups — #135807

Order 512.

224 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 118 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	#12023	C₂	1	2	yes	maximal
2	#12022	C₂	1	2	yes	maximal
3	C₂×#1312	C₂	1	1	yes	maximal, characteristic
4	#8824	C₂	1	1	yes	maximal, characteristic
5	#8823	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₁₆	C₂²	1	2	yes	—
7	#1848	—	2	1	no	—
8	#1837	—	2	1	no	—
9	#1567	—	2	1	no	—
10	#1567	C₂²	1	2	yes	—
11	#1567	C₂²	1	1	yes	characteristic
12	#1567	C₂²	1	1	yes	characteristic
13	#1566	—	2	1	no	—
14	C₂×C₈²	C₂²	1	1	yes	characteristic
15	#1372	—	2	4	no	—
16	#1371	—	2	4	no	—
17	#1312	C₄	1	2	yes	—
18	#1312	C₄	1	2	yes	—
19	C₂²×C₁₆	Q₈	1	4	yes	—
20	C₂×#108	—	4	2	no	—
21	#423	—	2	1	no	—
22	#423	—	2	1	no	—
23	#423	—	2	2	no	—
24	C₂×C₄×C₈	D₄	1	1	yes	characteristic
25	C₂×C₄×C₈	D₄	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
27	C₄×C₁₆	C₂×C₄	1	4	yes	—
28	C₄×C₁₆	D₄	1	4	yes	—
29	#341	—	4	2	no	—
30	#335	—	2	2	no	—
31	#335	—	2	2	no	—
32	#335	—	2	2	no	—
33	#335	—	2	2	no	—
34	#335	—	4	2	no	—
35	#334	—	4	2	no	—
36	C₈²	C₂×C₄	1	1	yes	characteristic
37	C₈²	C₂×C₄	1	1	yes	characteristic
38	C₈²	Q₈	1	1	yes	characteristic
39	C₈²	Q₈	1	1	yes	characteristic
40	C₂×M₁₆	—	4	1	no	—
41	C₂×M₁₆	—	4	1	no	—
42	C₂×M₁₆	—	4	2	no	—
43	C₂²×C₈	D₈	1	2	yes	—
44	C₂²×C₈	Q₁₆	1	2	yes	—
45	C₂²×C₈	C₂×Q₈	1	2	yes	—
46	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
47	C₂×C₁₆	—	4	4	no	—
48	C₂×C₁₆	C₄⋊C₄	1	8	yes	—
49	#108	—	8	4	no	—
50	C₄⋊C₈	—	4	1	no	—
51	C₄⋊C₈	—	4	2	no	—
52	C₄⋊C₈	—	4	1	no	—
53	C₄⋊C₈	—	4	1	no	—
54	C₄⋊C₈	—	4	1	no	—
55	C₄⋊C₈	—	4	2	no	—
56	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
58	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₈	D₈	1	2	yes	—
61	C₄×C₈	Q₁₆	1	2	yes	—
62	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₄×C₈	C₂×D₄	1	2	yes	—
64	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
66	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
67	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
68	M₁₆	—	8	4	no	—
69	M₁₆	—	8	2	no	—
70	M₁₆	—	8	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	4	no	—
75	C₂×C₈	—	8	1	no	—
76	C₂×C₈	—	8	2	no	—
77	C₂×C₈	—	8	1	no	—
78	C₂×C₈	#107	1	2	yes	—
79	C₂×C₈	#107	1	2	yes	—
80	C₂×C₈	#107	1	2	yes	—
81	C₂×C₈	#107	1	2	yes	—
82	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
83	C₂×C₈	C₄⋊Q₈	1	2	yes	—
84	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₄²	#127	1	1	yes	characteristic
86	C₄²	C₂×D₈	1	1	yes	characteristic
87	C₄²	C₂×Q₁₆	1	1	yes	characteristic
88	C₁₆	—	2	8	no	—
89	C₂³	#500	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	#384	1	1	yes	characteristic
93	C₂×C₄	C₂×#107	1	1	yes	characteristic
94	C₂×C₄	C₂×#107	1	1	yes	characteristic
95	C₂×C₄	#493	1	1	yes	characteristic
96	C₂×C₄	#494	1	1	yes	characteristic
97	C₂×C₄	#500	1	1	yes	characteristic
98	C₂×C₄	#509	1	2	yes	—
99	C₂×C₄	#510	1	2	yes	—
100	C₈	—	2	2	no	—
101	C₈	—	2	2	no	—
102	C₈	—	2	2	no	—
103	C₈	—	2	2	no	—
104	C₈	—	16	2	no	—
105	C₈	—	16	1	no	—
106	C₈	—	16	1	no	—
107	C₈	#368	1	4	yes	—
108	C₈	#430	1	4	yes	—
109	C₂²	—	2	1	no	—
110	C₂²	—	2	1	no	—
111	C₂²	#1845	1	1	yes	characteristic
112	C₂²	#2255	1	2	yes	—
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	#1849	1	2	yes	—
116	C₄	#2159	1	2	yes	—
117	C₄	C₂×#368	1	2	yes	—
118	C₄	#2250	1	1	yes	characteristic
119	C₄	#2251	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	#8826	1	1	yes	characteristic
122	C₂	#12908	1	1	yes	characteristic
123	C₂	#12909	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12023

C₂

yes

maximal

#12022

C₂

yes

maximal

C₂×#1312

C₂

yes

maximal, characteristic

#8824

C₂

yes

maximal, characteristic

#8823

C₂

yes

maximal, characteristic

C₂×C₄×C₁₆

C₂²

yes

—

—

—

—

C₂²

yes

—

#1567

C₂²

yes

characteristic

#1567

C₂²

yes

characteristic

#1566

—

C₂×C₈²

C₂²

yes

—

—

C₄

yes

—

#1312

C₄

yes

—

C₂²×C₁₆

Q₈

yes

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

D₄

yes

—

—

—

—

—

—

—

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

Q₈

yes

characteristic

C₈²

Q₈

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

Q₁₆

yes

—

C₂²×C₈

C₂×Q₈

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₄⋊C₄

yes

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

D₈

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

C₂×Q₈

yes

characteristic

C₂²×C₄

C₄⋊Q₈

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

C₂×Q₁₆

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#107

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

C₂×C₄⋊C₄

yes

—

C₂×C₈

C₄⋊Q₈

yes

—

C₄²

C₂×C₄⋊C₄

yes

characteristic

C₄²

#127

yes

characteristic

C₄²

C₂×D₈

yes

characteristic

C₄²

C₂×Q₁₆

yes

characteristic

C₁₆

—

C₂³

#500

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#510

yes

—

100

C₈

—

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

#368

yes

—

108

C₈

#430

yes

—

109

C₂²

—

110

C₂²

—

111

C₂²

#1845

yes

characteristic

112

C₂²

#2255

yes

—

113

C₄

—

114

C₄

—

115

C₄

#1849

yes

—

116

C₄

#2159

yes

—

117

C₄

C₂×#368

yes

—

118

C₄

#2250

yes

characteristic

119

C₄

#2251

yes

characteristic

120

C₂

—

121

C₂

#8826

yes

characteristic

122

C₂

#12908

yes

characteristic

123

C₂

#12909

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#493
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1845
prime radical	C₂³	#500
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#493

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#12023	2
#12022	2
C₂×#1312	2
#8824	2
#8823	2

subgroup

index

Subgroups — #135807

Conjugacy classes of subgroups

Maximal subgroups