Group #71742 — subgroups

Subgroups — #71742

Order 384.

228 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 97 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×C₃×#125	C₂	1	1	yes	maximal, characteristic
2	#4924	C₂	1	1	yes	maximal, characteristic
3	C₄×#208	C₂	1	1	yes	maximal, characteristic
4	#4530	C₂	1	4	yes	maximal
5	#1998	—	3	1	no	maximal, Sylow
6	C₃×#125	—	2	1	no	—
7	C₃×#125	—	2	1	no	—
8	C₃×#125	—	2	2	no	—
9	C₂×C₃×C₄⋊C₄	C₄	1	4	yes	—
10	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
12	C₂×#208	C₂²	1	2	yes	—
13	C₂×#208	C₂²	1	2	yes	—
14	C₄⋊(C₃⋊C₈)	—	2	2	no	—
15	#803	—	2	1	no	—
16	C₄×C₃⋊C₈	—	2	1	no	—
17	C₂×#125	D₃	1	1	yes	characteristic
18	#423	—	3	1	no	—
19	C₄×M₁₆	—	3	1	no	—
20	#343	—	3	4	no	—
21	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
22	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
23	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
24	C₃×C₄⋊C₄	—	2	4	no	—
25	C₃×C₄⋊C₄	—	2	4	no	—
26	C₃×C₄⋊C₄	—	2	2	no	—
27	C₃×C₄⋊C₄	—	2	2	no	—
28	C₃×C₄²	D₄	1	1	yes	characteristic
29	C₃×C₄²	D₄	1	1	yes	characteristic
30	C₃×C₄²	D₄	1	2	yes	—
31	#208	—	2	2	no	—
32	#208	—	2	4	no	—
33	#208	Q₈	1	4	yes	—
34	C₂×C₃⋊C₈	—	2	2	no	—
35	C₂×C₃⋊C₈	—	2	2	no	—
36	C₂×M₁₆	—	3	2	no	—
37	C₂×M₁₆	—	3	2	no	—
38	#125	—	2	1	no	—
39	#125	—	2	1	no	—
40	#125	—	2	2	no	—
41	C₂×C₄⋊C₄	Dic₃	1	4	yes	—
42	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
43	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
44	C₄⋊C₈	—	6	2	no	—
45	#97	—	6	1	no	—
46	C₄×C₈	—	6	1	no	—
47	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₃×C₄	—	2	4	no	—
49	C₂×C₃×C₄	—	4	2	no	—
50	C₂×C₃×C₄	—	4	4	no	—
51	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
52	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
53	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
54	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
56	C₂×C₃×C₄	#41	1	2	yes	—
57	C₃⋊C₈	—	2	4	no	—
58	C₃⋊C₈	—	4	2	no	—
59	C₂²×C₄	C₂×Dic₃	1	4	yes	—
60	C₂²×C₄	C₂×Dic₃	1	2	yes	—
61	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
62	M₁₆	—	3	4	no	—
63	M₁₆	—	6	2	no	—
64	M₁₆	—	6	4	no	—
65	C₂×C₈	—	6	2	no	—
66	C₂×C₈	—	6	2	no	—
67	C₄⋊C₄	—	2	4	no	—
68	C₄⋊C₄	—	2	4	no	—
69	C₄⋊C₄	—	2	2	no	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄²	#67	1	1	yes	characteristic
72	C₄²	#67	1	1	yes	characteristic
73	C₄²	#67	1	2	yes	—
74	C₂²×C₃	—	2	2	no	—
75	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₃	C₄×Q₈	1	2	yes	—
77	C₃×C₄	—	2	2	no	—
78	C₃×C₄	—	2	2	no	—
79	C₃×C₄	—	4	4	no	—
80	C₃×C₄	C₂²⋊Q₈	1	2	yes	—
81	C₃×C₄	#124	1	1	yes	characteristic
82	C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
83	C₂³	C₂²×Dic₃	1	1	yes	characteristic
84	C₂×C₄	—	2	4	no	—
85	C₂×C₄	—	4	2	no	—
86	C₂×C₄	—	4	4	no	—
87	C₂×C₄	#217	1	2	yes	—
88	C₂×C₄	#217	1	2	yes	—
89	C₂×C₄	D₃×Q₈	1	2	yes	—
90	C₂×C₄	#239	1	2	yes	—
91	C₂×C₄	C₂×#67	1	1	yes	characteristic
92	C₂×C₄	C₂×#67	1	1	yes	characteristic
93	C₈	—	6	4	no	—
94	C₈	—	12	2	no	—
95	C₂×C₃	—	2	2	no	—
96	C₂×C₃	#391	1	1	yes	characteristic
97	C₂×C₃	#413	1	2	yes	—
98	C₂²	—	2	2	no	—
99	C₂²	Q₈×Dic₃	1	2	yes	—
100	C₂²	C₂×#217	1	1	yes	characteristic
101	C₄	—	2	2	no	—
102	C₄	—	2	2	no	—
103	C₄	—	4	4	no	—
104	C₄	#944	1	1	yes	characteristic
105	C₄	#946	1	2	yes	—
106	C₄	#947	1	1	yes	characteristic
107	C₃	#1998	1	1	yes	characteristic, complemented, Sylow
108	C₂	—	2	2	no	—
109	C₂	#5229	1	1	yes	characteristic
110	C₂	#5237	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×#125

C₂

yes

maximal, characteristic

#4924

C₂

yes

maximal, characteristic

C₄×#208

C₂

yes

maximal, characteristic

#4530

C₂

yes

—

—

—

—

C₂×C₃×C₄⋊C₄

C₄

yes

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×#208

C₂²

yes

—

C₂×#208

C₂²

yes

—

C₄⋊(C₃⋊C₈)

—

#803

—

C₄×C₃⋊C₈

—

C₂×#125

D₃

yes

characteristic

#423

—

C₄×M₁₆

—

#343

—

C₂²×C₃×C₄

C₂×C₄

yes

—

C₂²×C₃×C₄

C₂×C₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

D₄

yes

characteristic

C₃×C₄²

D₄

yes

characteristic

C₃×C₄²

D₄

yes

—

#208

—

#208

—

#208

Q₈

yes

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×M₁₆

—

C₂×M₁₆

—

#125

—

#125

—

#125

—

C₂×C₄⋊C₄

Dic₃

yes

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₂³×C₃

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

#41

yes

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂²×C₄

C₂×Dic₃

yes

—

C₂²×C₄

C₂×Dic₃

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

yes

C₄²

yes

C₄²

yes

—

C₂²×C₃

—

C₂²×C₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃

C₄×Q₈

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂²⋊Q₈

yes

—

C₃×C₄

#124

yes

characteristic

C₃×C₄

C₄⋊Q₈

yes

characteristic

C₂³

C₂²×Dic₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#217

yes

—

C₂×C₄

#217

yes

—

C₂×C₄

D₃×Q₈

yes

—

C₂×C₄

#239

yes

—

C₂×C₄

C₂×#67

yes

characteristic

C₂×C₄

C₂×#67

yes

characteristic

C₈

—

C₈

—

C₂×C₃

—

C₂×C₃

#391

yes

characteristic

C₂×C₃

#413

yes

—

C₂²

—

C₂²

Q₈×Dic₃

yes

—

100

C₂²

C₂×#217

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

#944

yes

characteristic

105

C₄

#946

yes

—

106

C₄

#947

yes

characteristic

107

C₃

#1998

yes

characteristic, complemented, Sylow

108

C₂

—

109

C₂

#5229

yes

characteristic

110

C₂

#5237

yes

—

	subgroup	G/H
center	C₂²	C₂×#217
commutator, prime radical	C₂³×C₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂×C₃×#125	C₂
socle	C₂²×C₃	C₂×C₂²⋊C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×#125	D₃
hypocenter	C₃	#1998
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂×#217

commutator, prime radical

C₂³×C₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₂×C₃×#125

C₂

socle

C₂²×C₃

C₂×C₂²⋊C₄

cosocle

C₂²×C₃×C₄

C₂³

D₃

C₃

G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
107. C₃	#1998	semidirect

H ≅ G/N

type

107. C₃

#1998

semidirect

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

subgroup	index
C₂×C₃×#125	2
#4924	2
C₄×#208	2
#4530	2
#1998	3

subgroup

index

C₂×C₃×#125

Subgroups — #71742

Conjugacy classes of subgroups

Complements

Maximal subgroups