Group #20242466 — subgroups

Subgroups — #20242466

Order 6016.

228 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 65 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	#15689497	C₂	1	1	yes	maximal, characteristic
2	#15689323	C₂	1	2	yes	maximal
3	#15688872	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal, complemented
5	Q₈²×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10625336	—	2	2	no	—
7	C₂×#10625336	—	2	2	no	—
8	#13094021	C₂²	1	1	yes	characteristic
9	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	#13093944	—	2	2	no	—
11	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₄₇	—	2	2	no	—
13	C₄⋊Q₈×C₄₇	—	2	2	no	—
14	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₄₇	—	2	2	no	—
16	C₄×Q₈×C₄₇	C₂²	1	2	yes	—
17	C₂×Q₈×C₄₇	—	2	2	no	—
18	C₂×Q₈×C₄₇	D₄	1	2	yes	—
19	C₂×Q₈×C₄₇	D₄	1	2	yes	—
20	C₂×D₁₈₈	—	2	2	no	—
21	C₂×D₁₈₈	—	2	1	no	—
22	C₄⋊C₄×C₄₇	—	2	4	no	—
23	C₄⋊C₄×C₄₇	—	2	2	no	—
24	C₄⋊C₄×C₄₇	—	2	2	no	—
25	C₄⋊C₄×C₄₇	D₄	1	2	yes	—
26	C₄²×C₄₇	—	2	2	no	—
27	C₄²×C₄₇	—	2	2	no	—
28	C₄²×C₄₇	C₂³	1	1	yes	characteristic
29	#10625336	—	4	4	no	—
30	#10625336	—	4	4	no	—
31	C₂×C₄₇⋊C₈	—	2	1	no	—
32	C₂×C₄₇⋊C₈	—	2	2	no	—
33	C₂²×D₄₇	—	4	1	no	—
34	Q₈×C₄₇	—	2	2	no	—
35	Q₈×C₄₇	—	4	2	no	—
36	Q₈×C₄₇	—	8	2	no	—
37	Q₈×C₄₇	SD₁₆	1	4	yes	complemented
38	C₂×C₄×C₄₇	—	2	2	no	—
39	C₂×C₄×C₄₇	—	2	2	no	—
40	C₂×C₄×C₄₇	—	2	2	no	—
41	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
43	D₁₈₈	—	4	2	no	—
44	D₁₈₈	—	4	2	no	—
45	D₁₈₈	—	4	2	no	—
46	C₄₇⋊C₈	—	2	2	no	—
47	C₄₇⋊C₈	—	4	2	no	—
48	C₂²×C₄₇	#120	1	1	yes	characteristic
49	C₂×D₄₇	—	8	2	no	—
50	C₂×D₄₇	—	8	1	no	—
51	C₄×C₄₇	—	2	4	no	—
52	C₄×C₄₇	—	2	1	no	—
53	C₄×C₄₇	—	4	2	no	—
54	C₄×C₄₇	—	4	2	no	—
55	C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
56	C₄×C₄₇	#137	1	2	yes	—
57	#1623	—	47	1	no	maximal, Sylow
58	C₂×C₄₇	#448	1	2	yes	—
59	C₂×C₄₇	#453	1	1	yes	characteristic
60	D₄₇	—	16	1	no	—
61	Q₈²	D₄₇	1	1	yes	characteristic, complemented
62	#492	—	47	1	no	—
63	#460	—	47	2	no	—
64	#333	—	47	1	no	—
65	Q₈⋊C₈	—	47	2	no	—
66	C₄₇	#1623	1	1	yes	characteristic, complemented, Sylow
67	C₂×SD₁₆	—	94	2	no	—
68	C₂×SD₁₆	—	94	2	no	—
69	C₄⋊Q₈	—	2	2	no	—
70	C₄⋊Q₈	—	2	2	no	—
71	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
72	#127	—	47	1	no	—
73	C₄×Q₈	—	2	2	no	—
74	C₄×Q₈	C₂×D₄₇	1	2	yes	—
75	C₄⋊C₈	—	47	2	no	—
76	#102	—	94	2	no	—
77	C₄×C₈	—	47	1	no	—
78	C₂×Q₈	—	2	2	no	—
79	C₂×Q₈	C₂²⋊D₄₇	1	2	yes	—
80	C₂×Q₈	C₂²⋊D₄₇	1	2	yes	—
81	C₂×D₄	—	94	2	no	—
82	C₂×D₄	—	94	1	no	—
83	SD₁₆	—	188	4	no	—
84	SD₁₆	—	188	4	no	—
85	C₂×C₈	—	94	1	no	—
86	C₂×C₈	—	94	2	no	—
87	C₄⋊C₄	—	2	4	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
91	C₄²	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	C₂²×D₄₇	1	1	yes	characteristic
94	C₂³	—	188	1	no	—
95	Q₈	—	2	2	no	—
96	Q₈	—	4	2	no	—
97	Q₈	—	8	2	no	—
98	Q₈	#10625336	1	4	yes	complemented
99	D₄	—	188	2	no	—
100	D₄	—	188	2	no	—
101	D₄	—	188	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
107	C₈	—	94	2	no	—
108	C₈	—	188	2	no	—
109	C₂²	—	376	2	no	—
110	C₂²	—	376	1	no	—
111	C₂²	#13093993	1	1	yes	characteristic
112	C₄	—	2	4	no	—
113	C₄	—	2	1	no	—
114	C₄	—	4	2	no	—
115	C₄	—	4	2	no	—
116	C₄	C₂×#10625336	1	2	yes	—
117	C₄	#13094055	1	2	yes	—
118	C₂	—	752	1	no	—
119	C₂	#15689262	1	2	yes	—
120	C₂	#15689291	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689497

C₂

yes

maximal, characteristic

#15689323

C₂

yes

maximal

#15688872

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₄₇⋊C₈)

C₂

yes

maximal, complemented

Q₈²×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×#10625336

—

C₂×#10625336

—

#13094021

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093944

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

C₂²

yes

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

D₄

yes

—

C₂×Q₈×C₄₇

D₄

yes

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625336

—

#10625336

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

SD₁₆

yes

complemented

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

#120

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×SD₁₆

yes

—

C₄×C₄₇

#137

yes

—

#1623

—

maximal, Sylow

C₂×C₄₇

#448

yes

—

C₂×C₄₇

#453

yes

characteristic

D₄₇

—

Q₈²

D₄₇

yes

characteristic, complemented

#492

—

#460

—

#333

—

Q₈⋊C₈

—

C₄₇

#1623

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

#127

—

C₄×Q₈

—

C₄×Q₈

C₂×D₄₇

yes

—

C₄⋊C₈

—

#102

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₄₇

yes

—

C₂×Q₈

C₂²⋊D₄₇

yes

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

188

—

SD₁₆

—

188

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₄₇

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₄₇

yes

characteristic

C₂³

—

188

—

Q₈

—

Q₈

—

Q₈

—

Q₈

#10625336

yes

complemented

D₄

—

188

—

100

D₄

—

188

—

101

D₄

—

188

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

107

C₈

—

108

C₈

—

188

—

109

C₂²

—

376

—

110

C₂²

—

376

—

111

C₂²

#13093993

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

C₂×#10625336

yes

—

117

C₄

#13094055

yes

—

118

C₂

—

752

—

119

C₂

#15689262

yes

—

120

C₂

#15689291

yes

characteristic

	subgroup	G/H
center	C₂²	#13093993
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	Q₈²×C₄₇	C₂
socle	C₂²×C₄₇	#120
hypercenter	Q₈²	D₄₇
hypocenter	C₄₇	#1623
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#13093993

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

Q₈²×C₄₇

C₂

socle

C₂²×C₄₇

#120

hypercenter

Q₈²

D₄₇

hypocenter

C₄₇

#1623

LCS

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
3. #15688872	C₂	semidirect
4. Q₈⋊(C₄₇⋊C₈)	C₂	semidirect
5. Q₈²×C₄₇	C₂	semidirect
37. Q₈×C₄₇	SD₁₆	semidirect
61. Q₈²	D₄₇	semidirect
66. C₄₇	#1623	semidirect
98. Q₈	#10625336	semidirect

H ≅ G/N

type

3. #15688872

C₂

semidirect

4. Q₈⋊(C₄₇⋊C₈)

C₂

semidirect

5. Q₈²×C₄₇

C₂

semidirect

37. Q₈×C₄₇

SD₁₆

semidirect

61. Q₈²

D₄₇

semidirect

66. C₄₇

#1623

semidirect

98. Q₈

#10625336

semidirect

Subgroups — #20242466

Conjugacy classes of subgroups

Complements