Group #13136461 — subgroups

Subgroups — #13136461

Order 1728.

424 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 163 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	C₂₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#11720193	C₂	1	4	yes	maximal
3	#11720137	C₂	1	4	yes	maximal
4	#11720136	C₂	1	4	yes	maximal
5	C₄×Dic₅₄	C₂	1	2	yes	maximal
6	#10589824	—	3	1	no	maximal
7	C₂×Q₈×C₂₇	C₂²	1	2	yes	—
8	C₂×Dic₅₄	C₂²	1	4	yes	—
9	C₂×Dic₅₄	C₂²	1	2	yes	—
10	C₄⋊C₄×C₂₇	C₂²	1	4	yes	—
11	C₄²×C₂₇	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₇	C₂²	1	2	yes	—
13	Dic₂₇⋊C₄	C₂²	1	4	yes	—
14	Dic₂₇⋊C₄	C₂²	1	8	yes	—
15	C₄×Dic₂₇	C₂²	1	4	yes	—
16	C₄×Dic₂₇	C₂²	1	4	yes	—
17	C₉×C₄⋊Q₈	D₃	1	1	yes	characteristic
18	#63506	—	3	4	no	—
19	#63450	—	3	4	no	—
20	#63449	—	3	4	no	—
21	C₄×Dic₁₈	—	3	2	no	—
22	Q₈×C₂₇	—	4	2	no	—
23	C₂×C₄×C₂₇	C₂³	1	4	yes	—
24	C₂×C₄×C₂₇	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₂₇	C₂³	1	2	yes	—
26	C₂×Dic₂₇	C₂³	1	4	yes	—
27	C₂×Dic₂₇	C₂³	1	4	yes	—
28	Dic₅₄	—	4	4	no	—
29	Dic₅₄	Q₈	1	8	yes	complemented
30	#5722	—	9	1	no	—
31	C₂×Q₈×C₉	C₂×D₃	1	2	yes	—
32	C₂×Dic₁₈	—	3	2	no	—
33	C₂×Dic₁₈	—	3	4	no	—
34	C₉×C₄⋊C₄	C₂×D₃	1	4	yes	—
35	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
36	C₄⋊Dic₉	—	3	2	no	—
37	Dic₉⋊C₄	—	3	4	no	—
38	Dic₉⋊C₄	—	3	8	no	—
39	C₄×Dic₉	—	3	4	no	—
40	C₄×Dic₉	—	3	4	no	—
41	C₂²×C₂₇	C₂⁴	1	1	yes	characteristic
42	C₄×C₂₇	—	2	4	no	—
43	C₄×C₂₇	—	2	1	no	—
44	C₄×C₂₇	C₂×Q₈	1	4	yes	—
45	Dic₂₇	—	2	4	no	—
46	Dic₂₇	C₂×Q₈	1	8	yes	—
47	C₃×C₄⋊Q₈	D₉	1	1	yes	characteristic
48	#944	—	9	4	no	—
49	#888	—	9	4	no	—
50	#887	—	9	4	no	—
51	C₄×Dic₆	—	9	2	no	—
52	Q₈×C₉	—	4	2	no	—
53	C₂×C₄×C₉	C₂²×D₃	1	4	yes	—
54	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
55	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
56	C₂×Dic₉	—	3	4	no	—
57	C₂×Dic₉	—	3	4	no	—
58	Dic₁₈	—	3	8	no	—
59	Dic₁₈	—	12	4	no	—
60	Q₈²	—	27	1	no	Sylow
61	C₂×C₂₇	C₂²×Q₈	1	2	yes	—
62	C₂×C₂₇	H(5,𝔽₂)	1	1	yes	characteristic
63	C₂×C₃×Q₈	C₂×D₉	1	2	yes	—
64	C₂×Dic₆	—	9	2	no	—
65	C₂×Dic₆	—	9	4	no	—
66	C₃×C₄⋊C₄	C₂×D₉	1	4	yes	—
67	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
68	C₄⋊Dic₃	—	9	2	no	—
69	Dic₃⋊C₄	—	9	4	no	—
70	Dic₃⋊C₄	—	9	8	no	—
71	C₄×Dic₃	—	9	4	no	—
72	C₄×Dic₃	—	9	4	no	—
73	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
74	C₄×C₉	—	2	4	no	—
75	C₄×C₉	—	2	1	no	—
76	C₄×C₉	D₃×Q₈	1	4	yes	—
77	Dic₉	—	3	8	no	—
78	Dic₉	—	6	4	no	—
79	C₄⋊Q₈	—	27	4	no	—
80	C₄⋊Q₈	—	27	4	no	—
81	C₄⋊Q₈	D₂₇	1	1	yes	characteristic
82	C₄×Q₈	—	27	4	no	—
83	C₄×Q₈	—	27	2	no	—
84	C₂₇	Q₈²	1	1	yes	characteristic, complemented, Sylow
85	C₃×Q₈	—	4	2	no	—
86	C₂×C₃×C₄	C₂²×D₉	1	4	yes	—
87	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
88	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
89	C₂×Dic₃	—	9	4	no	—
90	C₂×Dic₃	—	9	4	no	—
91	Dic₆	—	9	8	no	—
92	Dic₆	—	36	4	no	—
93	C₂×C₉	#1004	1	1	yes	characteristic
94	C₂×C₉	C₂×D₃×Q₈	1	2	yes	—
95	C₂×Q₈	—	27	2	no	—
96	C₂×Q₈	—	27	4	no	—
97	C₂×Q₈	C₂×D₂₇	1	2	yes	—
98	C₄⋊C₄	—	27	8	no	—
99	C₄⋊C₄	—	27	4	no	—
100	C₄⋊C₄	—	27	2	no	—
101	C₄⋊C₄	C₂×D₂₇	1	4	yes	—
102	C₄²	—	27	4	no	—
103	C₄²	—	27	4	no	—
104	C₄²	C₂×D₂₇	1	1	yes	characteristic
105	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
106	C₃×C₄	—	2	4	no	—
107	C₃×C₄	—	2	1	no	—
108	C₃×C₄	Q₈×D₉	1	4	yes	—
109	Dic₃	—	9	8	no	—
110	Dic₃	—	18	4	no	—
111	C₉	#5722	1	1	yes	characteristic
112	Q₈	—	4	2	no	—
113	Q₈	—	27	8	no	—
114	Q₈	—	108	4	no	—
115	C₂×C₄	—	27	4	no	—
116	C₂×C₄	—	27	4	no	—
117	C₂×C₄	C₂²×D₂₇	1	4	yes	—
118	C₂×C₄	C₂²×D₂₇	1	1	yes	characteristic
119	C₂×C₄	C₂²×D₂₇	1	2	yes	—
120	C₂×C₃	#63710	1	1	yes	characteristic
121	C₂×C₃	C₂×Q₈×D₉	1	2	yes	—
122	C₂²	C₂³×D₂₇	1	1	yes	characteristic
123	C₄	—	2	1	no	—
124	C₄	—	2	4	no	—
125	C₄	—	27	8	no	—
126	C₄	—	54	4	no	—
127	C₄	Q₈×D₂₇	1	4	yes	—
128	C₃	#10589824	1	1	yes	characteristic
129	C₂	#11720670	1	1	yes	characteristic
130	C₂	C₂×Q₈×D₂₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₇×C₄⋊Q₈

C₂

yes

maximal, characteristic

#11720193

C₂

yes

maximal

#11720137

C₂

yes

maximal

#11720136

C₂

yes

maximal

C₄×Dic₅₄

C₂

yes

maximal

#10589824

—

maximal

C₂×Q₈×C₂₇

C₂²

yes

—

C₂×Dic₅₄

C₂²

yes

—

C₂×Dic₅₄

C₂²

yes

—

C₄⋊C₄×C₂₇

C₂²

yes

—

C₄²×C₂₇

C₂²

yes

characteristic

C₄⋊Dic₂₇

C₂²

yes

—

Dic₂₇⋊C₄

C₂²

yes

—

Dic₂₇⋊C₄

C₂²

yes

—

C₄×Dic₂₇

C₂²

yes

—

C₄×Dic₂₇

C₂²

yes

—

C₉×C₄⋊Q₈

D₃

yes

—

—

—

C₄×Dic₁₈

—

Q₈×C₂₇

—

C₂×C₄×C₂₇

C₂³

yes

—

C₂×C₄×C₂₇

C₂³

yes

characteristic

C₂×C₄×C₂₇

C₂³

yes

—

C₂×Dic₂₇

C₂³

yes

—

C₂×Dic₂₇

C₂³

yes

—

Dic₅₄

—

Dic₅₄

Q₈

yes

complemented

#5722

—

C₂×Q₈×C₉

C₂×D₃

yes

—

C₂×Dic₁₈

—

C₂×Dic₁₈

—

C₉×C₄⋊C₄

C₂×D₃

yes

—

C₄²×C₉

C₂×D₃

yes

characteristic

C₄⋊Dic₉

—

Dic₉⋊C₄

—

Dic₉⋊C₄

—

C₄×Dic₉

—

C₄×Dic₉

—

C₂²×C₂₇

C₂⁴

yes

characteristic

C₄×C₂₇

—

C₄×C₂₇

—

C₄×C₂₇

C₂×Q₈

yes

—

Dic₂₇

—

Dic₂₇

C₂×Q₈

yes

—

C₃×C₄⋊Q₈

D₉

yes

—

—

—

C₄×Dic₆

—

Q₈×C₉

—

C₂×C₄×C₉

C₂²×D₃

yes

—

C₂×C₄×C₉

C₂²×D₃

yes

characteristic

C₂×C₄×C₉

C₂²×D₃

yes

—

C₂×Dic₉

—

C₂×Dic₉

—

Dic₁₈

—

Dic₁₈

—

Q₈²

—

Sylow

C₂×C₂₇

C₂²×Q₈

yes

—

C₂×C₂₇

H(5,𝔽₂)

yes

characteristic

C₂×C₃×Q₈

C₂×D₉

yes

—

C₂×Dic₆

—

C₂×Dic₆

—

C₃×C₄⋊C₄

C₂×D₉

yes

—

C₃×C₄²

C₂×D₉

yes

characteristic

C₄⋊Dic₃

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂²×C₉

C₂³×D₃

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

D₃×Q₈

yes

—

Dic₉

—

Dic₉

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₂₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂₇

Q₈²

yes

characteristic, complemented, Sylow

C₃×Q₈

—

C₂×C₃×C₄

C₂²×D₉

yes

—

C₂×C₃×C₄

C₂²×D₉

yes

characteristic

C₂×C₃×C₄

C₂²×D₉

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₂×C₉

#1004

yes

characteristic

C₂×C₉

C₂×D₃×Q₈

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₂₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

C₂×D₂₇

yes

—

102

C₄²

—

103

C₄²

—

104

C₄²

C₂×D₂₇

yes

characteristic

105

C₂²×C₃

C₂³×D₉

yes

characteristic

106

C₃×C₄

—

107

C₃×C₄

—

108

C₃×C₄

Q₈×D₉

yes

—

109

Dic₃

—

110

Dic₃

—

111

C₉

#5722

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

108

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂²×D₂₇

yes

—

118

C₂×C₄

C₂²×D₂₇

yes

characteristic

119

C₂×C₄

C₂²×D₂₇

yes

—

120

C₂×C₃

#63710

yes

characteristic

121

C₂×C₃

C₂×Q₈×D₉

yes

—

122

C₂²

C₂³×D₂₇

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

Q₈×D₂₇

yes

—

128

C₃

#10589824

yes

characteristic

129

C₂

#11720670

yes

characteristic

130

C₂

C₂×Q₈×D₂₇

yes

—

	subgroup	G/H
center	C₂²	C₂³×D₂₇
commutator, cosocle	C₂²×C₂₇	C₂⁴
Frattini	C₂²×C₉	C₂³×D₃
Fitting	C₂₇×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₃	C₂³×D₉
hypercenter	C₄⋊Q₈	D₂₇
hypocenter	C₂₇	Q₈²
LCS	G ▷ C₂²×C₂₇ ▷ C₂₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂³×D₂₇

commutator, cosocle

C₂²×C₂₇

C₂⁴

Frattini

C₂²×C₉

C₂³×D₃

Fitting

C₂₇×C₄⋊Q₈

C₂

socle, prime radical

C₂²×C₃

C₂³×D₉

hypercenter

C₄⋊Q₈

D₂₇

hypocenter

C₂₇

Q₈²

LCS

G ▷ C₂²×C₂₇ ▷ C₂₇

UCS

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
29. Dic₅₄	Q₈	semidirect
84. C₂₇	Q₈²	semidirect

H ≅ G/N

type

29. Dic₅₄

Q₈

semidirect

84. C₂₇

Q₈²

semidirect

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

subgroup	index
C₂₇×C₄⋊Q₈	2
#11720193	2
#11720137	2
#11720136	2
C₄×Dic₅₄	2
#10589824	3

subgroup

index

C₂₇×C₄⋊Q₈

C₄×Dic₅₄

Subgroups — #13136461

Conjugacy classes of subgroups

Complements

Maximal subgroups