Group #73407 — subgroups

Subgroups — #73407

Order 384.

416 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 149 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×#870	C₂	1	1	yes	maximal, characteristic
2	C₂×#869	C₂	1	1	yes	maximal, characteristic
3	C₃×#376	C₂	1	1	yes	maximal, characteristic
4	#4929	C₂	1	4	yes	maximal
5	#4659	C₂	1	8	yes	maximal
6	#2642	—	3	1	no	maximal, Sylow
7	C₂²×Dic₆	C₂²	1	2	yes	—
8	C₂×C₃×C₄²	C₂²	1	2	yes	—
9	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×Dic₃⋊C₄	C₂²	1	4	yes	—
13	#870	—	2	4	no	—
14	#869	—	2	4	no	—
15	C₃×#95	C₂²	1	4	yes	—
16	C₃⋊#95	C₂²	1	16	yes	—
17	C₂×C₄⋊Q₈	—	3	1	no	—
18	C₂×#125	—	3	1	no	—
19	#400	—	3	8	no	—
20	#391	—	3	4	no	—
21	#376	D₃	1	1	yes	characteristic
22	C₂²×C₃×C₄	C₂³	1	4	yes	—
23	C₂²×C₃×C₄	C₂³	1	2	yes	—
24	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
25	C₂²×Dic₃	C₂³	1	4	yes	—
26	C₂²×Dic₃	C₂³	1	4	yes	—
27	C₂×Dic₆	—	2	4	no	—
28	C₂×Dic₆	—	2	8	no	—
29	C₃×C₄²	—	2	4	no	—
30	C₃×C₄²	D₄	1	4	yes	—
31	C₄⋊Dic₃	—	2	8	no	—
32	C₄⋊Dic₃	—	2	4	no	—
33	Dic₃⋊C₄	—	2	8	no	—
34	C₂²×Q₈	—	3	2	no	—
35	C₄⋊Q₈	—	6	4	no	—
36	#125	—	6	4	no	—
37	C₂×C₄⋊C₄	—	3	2	no	—
38	C₂×C₄⋊C₄	—	3	4	no	—
39	C₂×C₄⋊C₄	—	3	4	no	—
40	C₂×C₄²	C₂×D₃	1	2	yes	—
41	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
42	#95	—	3	16	no	—
43	#95	C₂×D₃	1	4	yes	—
44	C₂³×C₃	C₂⁴	1	1	yes	characteristic
45	C₂×C₃×C₄	—	2	4	no	—
46	C₂×C₃×C₄	—	2	4	no	—
47	C₂×C₃×C₄	—	2	8	no	—
48	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
49	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
50	C₂×C₃×C₄	#41	1	4	yes	—
51	C₂×Dic₃	—	2	4	no	—
52	C₂×Dic₃	—	2	4	no	—
53	C₂×Dic₃	—	2	8	no	—
54	C₂×Dic₃	—	2	8	no	—
55	Dic₆	—	4	8	no	—
56	C₂×Q₈	—	6	4	no	—
57	C₂×Q₈	—	6	8	no	—
58	C₂²×C₄	—	3	4	no	—
59	C₂²×C₄	—	3	4	no	—
60	C₂²×C₄	C₂²×D₃	1	4	yes	—
61	C₂²×C₄	C₂²×D₃	1	2	yes	—
62	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
63	C₄⋊C₄	—	6	8	no	—
64	C₄⋊C₄	—	6	4	no	—
65	C₄⋊C₄	—	6	8	no	—
66	C₄²	—	2	4	no	—
67	C₄²	D₁₂	1	4	yes	—
68	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
69	C₂²×C₃	C₂×#41	1	2	yes	—
70	C₂²×C₃	2₋¹⁺⁴	1	4	yes	—
71	C₃×C₄	—	2	4	no	—
72	C₃×C₄	—	4	4	no	—
73	C₃×C₄	#124	1	4	yes	—
74	Dic₃	—	4	4	no	—
75	Dic₃	—	4	4	no	—
76	C₂³	C₂³×D₃	1	1	yes	characteristic
77	Q₈	—	12	8	no	—
78	C₂×C₄	—	2	4	no	—
79	C₂×C₄	—	2	4	no	—
80	C₂×C₄	—	2	8	no	—
81	C₂×C₄	—	6	4	no	—
82	C₂×C₄	—	6	4	no	—
83	C₂×C₄	—	6	8	no	—
84	C₂×C₄	—	6	8	no	—
85	C₂×C₄	C₂×D₁₂	1	4	yes	—
86	C₂×C₄	C₂×D₁₂	1	2	yes	—
87	C₂×C₄	#235	1	4	yes	—
88	C₂×C₃	C₂×#124	1	1	yes	characteristic
89	C₂×C₃	#536	1	2	yes	—
90	C₂×C₃	#541	1	4	yes	—
91	C₂²	C₂²×D₁₂	1	1	yes	characteristic
92	C₂²	C₂×#235	1	2	yes	—
93	C₂²	#1010	1	4	yes	—
94	C₄	—	2	4	no	—
95	C₄	—	4	4	no	—
96	C₄	—	12	4	no	—
97	C₄	—	12	4	no	—
98	C₄	#875	1	4	yes	—
99	C₃	#2642	1	1	yes	characteristic, complemented, Sylow
100	C₂	C₂×#875	1	1	yes	characteristic
101	C₂	#5518	1	4	yes	—
102	C₂	#5526	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#870

C₂

yes

maximal, characteristic

C₂×#869

C₂

yes

maximal, characteristic

C₃×#376

C₂

yes

maximal, characteristic

#4929

C₂

yes

maximal

#4659

C₂

yes

maximal

#2642

—

maximal, Sylow

C₂²×Dic₆

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×Dic₃⋊C₄

C₂²

yes

—

#870

—

#869

—

C₃×#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₂×C₄⋊Q₈

—

—

—

—

D₃

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₂×Dic₆

—

C₂×Dic₆

—

C₃×C₄²

—

C₃×C₄²

D₄

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₂²×Q₈

—

C₄⋊Q₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃

yes

—

C₂×C₄²

C₂×D₃

yes

characteristic

#95

—

#95

C₂×D₃

yes

—

C₂³×C₃

C₂⁴

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

D₁₂

yes

—

C₂²×C₃

C₂²×D₄

yes

characteristic

C₂²×C₃

C₂×#41

yes

—

C₂²×C₃

2₋¹⁺⁴

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#124

yes

—

Dic₃

—

Dic₃

—

C₂³

C₂³×D₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₁₂

yes

—

C₂×C₄

C₂×D₁₂

yes

—

C₂×C₄

#235

yes

—

C₂×C₃

C₂×#124

yes

characteristic

C₂×C₃

#536

yes

—

C₂×C₃

#541

yes

—

C₂²

C₂²×D₁₂

yes

characteristic

C₂²

C₂×#235

yes

—

C₂²

#1010

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

#875

yes

—

C₃

#2642

yes

characteristic, complemented, Sylow

100

C₂

C₂×#875

yes

characteristic

101

C₂

#5518

yes

—

102

C₂

#5526

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator, socle, prime radical, cosocle	C₂³×C₃	C₂⁴
Fitting	C₃×#376	C₂
hypercenter	#376	D₃
hypocenter	C₃	#2642
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ #376

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃

commutator, socle, prime radical, cosocle

C₂³×C₃

C₂⁴

C₂

D₃

C₃

G ▷ C₂³×C₃ ▷ C₃

C₁ ◁ C₂³ ◁ #376

N	H ≅ G/N	type
99. C₃	#2642	semidirect

H ≅ G/N

type

99. C₃

#2642

semidirect

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

subgroup	index
C₂×#870	2
C₂×#869	2
C₃×#376	2
#4929	2
#4659	2
#2642	3

subgroup

index

Subgroups — #73407

Conjugacy classes of subgroups

Complements

Maximal subgroups