Group #68924 — subgroups

Subgroups — #68924

Order 384.

176 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 71 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	#5121	C₂	1	2	yes	maximal
2	C₂×#817	C₂	1	1	yes	maximal, characteristic
3	#5102	C₂	1	1	yes	maximal, characteristic
4	C₃×#341	C₂	1	1	yes	maximal, characteristic
5	#4554	C₂	1	2	yes	maximal
6	#2090	—	3	1	no	maximal, Sylow
7	C₂×C₃×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
9	#926	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₂×#208	C₂²	1	1	yes	characteristic
12	#818	—	2	2	no	—
13	#817	—	2	2	no	—
14	#817	—	2	2	no	—
15	Dic₃⋊C₈	—	2	2	no	—
16	#428	—	3	2	no	—
17	C₂×#106	—	3	1	no	—
18	#424	—	3	1	no	—
19	#343	—	3	2	no	—
20	#341	D₃	1	1	yes	characteristic
21	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
22	C₂²×Dic₃	—	2	2	no	—
23	C₃×M₁₆	—	2	2	no	—
24	C₃×M₁₆	Q₈	1	4	yes	—
25	C₂×C₃×C₈	—	2	2	no	—
26	C₂×C₃×C₈	—	2	2	no	—
27	C₂×C₃×C₈	D₄	1	2	yes	—
28	#217	—	2	1	no	—
29	C₄⋊Dic₃	—	2	2	no	—
30	C₄⋊Dic₃	—	2	2	no	—
31	C₄⋊Dic₃	D₄	1	2	yes	—
32	C₄×Dic₃	—	2	1	no	—
33	#208	—	2	2	no	—
34	C₂×C₃⋊C₈	—	2	1	no	—
35	C₂×M₁₆	—	3	1	no	—
36	C₂×M₁₆	C₂×D₃	1	2	yes	—
37	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
38	#117	—	3	1	no	—
39	C₂×C₄⋊C₄	—	3	2	no	—
40	#107	—	6	2	no	—
41	#106	—	6	2	no	—
42	#106	—	6	2	no	—
43	C₄⋊C₈	—	6	2	no	—
44	C₂³×C₃	#41	1	1	yes	characteristic
45	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
46	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
47	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
48	C₂×C₃×C₄	#41	1	2	yes	—
49	C₂×Dic₃	—	2	2	no	—
50	C₂×Dic₃	—	4	2	no	—
51	C₂×Dic₃	—	8	2	no	—
52	C₃×C₈	—	2	4	no	—
53	C₃×C₈	—	2	2	no	—
54	C₃⋊C₈	—	4	1	no	—
55	C₂²×C₄	—	6	2	no	—
56	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
57	M₁₆	—	2	2	no	—
58	M₁₆	—	6	2	no	—
59	M₁₆	Dic₆	1	4	yes	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	6	1	no	—
63	C₂×C₈	D₁₂	1	2	yes	—
64	C₄⋊C₄	—	3	2	no	—
65	C₄⋊C₄	—	6	2	no	—
66	C₄⋊C₄	—	6	2	no	—
67	C₂²⋊C₄	—	6	1	no	—
68	C₄²	—	6	1	no	—
69	C₂²×C₃	—	2	2	no	—
70	C₂²×C₃	#121	1	1	yes	characteristic
71	C₂²×C₃	#125	1	2	yes	—
72	C₃×C₄	C₂²⋊Q₈	1	2	yes	—
73	C₃×C₄	#123	1	1	yes	characteristic
74	C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
75	Dic₃	—	4	2	no	—
76	Dic₃	—	8	2	no	—
77	C₂³	#239	1	1	yes	characteristic
78	C₂×C₄	—	6	2	no	—
79	C₂×C₄	—	12	2	no	—
80	C₂×C₄	—	24	2	no	—
81	C₂×C₄	C₂×Dic₆	1	2	yes	—
82	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
83	C₂×C₄	D₃×D₄	1	1	yes	characteristic
84	C₂×C₄	#237	1	2	yes	—
85	C₈	—	2	4	no	—
86	C₈	—	2	2	no	—
87	C₈	—	12	1	no	—
88	C₂×C₃	—	2	2	no	—
89	C₂×C₃	#398	1	1	yes	characteristic
90	C₂×C₃	#471	1	2	yes	—
91	C₂²	—	2	2	no	—
92	C₂²	#890	1	2	yes	—
93	C₂²	#895	1	1	yes	characteristic
94	C₄	—	12	2	no	—
95	C₄	—	24	2	no	—
96	C₄	#869	1	1	yes	characteristic
97	C₄	#878	1	2	yes	—
98	C₄	#886	1	1	yes	characteristic
99	C₃	#2090	1	1	yes	characteristic, complemented, Sylow
100	C₂	—	2	2	no	—
101	C₂	#4661	1	1	yes	characteristic
102	C₂	#4898	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#5121

C₂

yes

maximal

C₂×#817

C₂

yes

maximal, characteristic

#5102

C₂

yes

maximal, characteristic

C₃×#341

C₂

yes

maximal, characteristic

#4554

C₂

yes

maximal

#2090

—

maximal, Sylow

C₂×C₃×M₁₆

C₂²

yes

—

C₂²×C₃×C₈

C₂²

yes

characteristic

#926

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×#208

C₂²

yes

—

—

—

Dic₃⋊C₈

—

—

—

—

—

D₃

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

—

C₃×M₁₆

—

C₃×M₁₆

Q₈

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

D₄

yes

—

#217

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

D₄

yes

—

C₄×Dic₃

—

#208

—

C₂×C₃⋊C₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃

yes

—

C₂²×C₈

C₂×D₃

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

#107

—

#106

—

#106

—

C₄⋊C₈

—

C₂³×C₃

#41

yes

characteristic

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

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

Dic₆

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₂

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂²×C₃

—

C₂²×C₃

#121

yes

characteristic

C₂²×C₃

#125

yes

—

C₃×C₄

C₂²⋊Q₈

yes

—

C₃×C₄

#123

yes

characteristic

C₃×C₄

C₄⋊Q₈

yes

characteristic

Dic₃

—

Dic₃

—

C₂³

#239

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×Dic₆

yes

—

C₂×C₄

C₂×D₁₂

yes

characteristic

C₂×C₄

D₃×D₄

yes

characteristic

C₂×C₄

#237

yes

—

C₈

—

C₈

—

C₈

—

C₂×C₃

—

C₂×C₃

#398

yes

characteristic

C₂×C₃

#471

yes

—

C₂²

—

C₂²

#890

yes

—

C₂²

#895

yes

characteristic

C₄

—

C₄

—

C₄

#869

yes

characteristic

C₄

#878

yes

—

C₄

#886

yes

characteristic

C₃

#2090

yes

characteristic, complemented, Sylow

100

C₂

—

101

C₂

#4661

yes

characteristic

102

C₂

#4898

yes

—

	subgroup	G/H
center	C₂²	#895
commutator, cosocle	C₂²×C₃×C₄	C₂³
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#341	C₂
socle	C₂²×C₃	#121
prime radical	C₂³×C₃	#41
hypercenter	#341	D₃
hypocenter	C₃	#2090
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#895

commutator, cosocle

C₂²×C₃×C₄

C₂³

Frattini

C₂²×C₄

C₂²×D₃

C₂

C₂²×C₃

C₂³×C₃

D₃

C₃

G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
99. C₃	#2090	semidirect

H ≅ G/N

type

99. C₃

#2090

semidirect

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

subgroup	index
#5121	2
C₂×#817	2
#5102	2
C₃×#341	2
#4554	2
#2090	3

subgroup

index

Subgroups — #68924

Conjugacy classes of subgroups

Complements

Maximal subgroups