Group #11794609 — subgroups

Subgroups — #11794609

Order 1152.

648 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 145 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×#63650	C₂	1	3	yes	maximal
2	#10592823	C₂	1	3	yes	maximal, complemented
3	C₃²×#338	C₂	1	1	yes	maximal, characteristic, complemented
4	#68722	—	3	4	no	maximal
5	C₂²×C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
6	C₂×C₃²×M₁₆	C₂²	1	3	yes	—
7	C₂×#3748	C₂²	1	3	yes	—
8	#63650	—	2	6	no	—
9	#63649	—	2	6	no	—
10	C₂×#823	—	3	12	no	—
11	#5124	—	3	12	no	—
12	C₃×#338	D₃	1	4	yes	complemented
13	C₂³×C₃⋊D₃	Q₈	1	2	yes	—
14	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
15	C₂×C₄⋊(C₃⋊D₃)	—	2	3	no	—
16	C₂×C₄⋊(C₃⋊D₃)	—	2	3	no	—
17	C₃²×M₁₆	—	2	3	no	—
18	C₃²×M₁₆	—	2	3	no	—
19	C₂×C₃²×C₈	—	2	3	no	—
20	#3748	—	2	3	no	—
21	#3748	D₄	1	6	yes	—
22	C₂×C₃⋊(C₃⋊C₈)	—	2	3	no	—
23	#2069	—	9	1	no	Sylow
24	C₂²×D₁₂	—	3	4	no	—
25	C₂×C₃×M₁₆	C₂×D₃	1	12	yes	—
26	C₂×#208	—	3	12	no	—
27	#823	—	6	24	no	—
28	#822	—	6	24	no	—
29	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
30	C₂²×C₃⋊D₃	—	2	6	no	—
31	C₂²×C₃⋊D₃	—	8	2	no	—
32	C₂×C₃²×C₄	C₂×D₄	1	3	yes	—
33	C₂×C₃²×C₄	#41	1	3	yes	—
34	C₄⋊(C₃⋊D₃)	—	4	3	no	—
35	C₄⋊(C₃⋊D₃)	—	4	1	no	—
36	C₃²×C₈	—	4	3	no	—
37	C₃⋊(C₃⋊C₈)	—	2	6	no	—
38	#430	—	9	3	no	—
39	C₂×#100	—	9	3	no	—
40	#338	C₃⋊D₃	1	1	yes	characteristic, complemented
41	C₂³×D₃	—	3	8	no	—
42	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
43	C₂×D₁₂	—	6	12	no	—
44	C₂×D₁₂	—	6	12	no	—
45	C₃×M₁₆	—	2	12	no	—
46	C₃×M₁₆	—	2	12	no	—
47	C₂×C₃×C₈	—	2	12	no	—
48	#208	—	3	24	no	—
49	#208	—	6	12	no	—
50	C₂×C₃⋊C₈	—	6	12	no	—
51	C₂²×C₃²	—	4	1	no	—
52	C₂²×C₃²	C₂²⋊Q₈	1	3	yes	—
53	C₂×C₃⋊D₃	—	4	2	no	—
54	C₂×C₃⋊D₃	—	8	6	no	—
55	C₃²×C₄	#121	1	3	yes	—
56	C₃²×C₄	#126	1	1	yes	characteristic
57	C₂²×D₄	—	9	1	no	—
58	C₂×M₁₆	—	9	3	no	—
59	C₂×M₁₆	C₂×C₃⋊D₃	1	3	yes	—
60	#108	—	18	6	no	—
61	#100	—	18	6	no	—
62	C₂³×C₃	D₃×Q₈	1	4	yes	—
63	C₂²×D₃	—	6	24	no	—
64	C₂²×D₃	—	24	8	no	—
65	C₂×C₃×C₄	#235	1	12	yes	—
66	C₂×C₃×C₄	D₃×D₄	1	12	yes	—
67	D₁₂	—	12	4	no	—
68	D₁₂	—	12	12	no	—
69	C₃×C₈	—	4	12	no	—
70	C₃⋊C₈	—	6	24	no	—
71	C₂×C₃²	—	2	3	no	—
72	C₂×C₃²	#396	1	1	yes	characteristic
73	C₃⋊D₃	—	8	2	no	—
74	C₂⁴	—	9	2	no	—
75	C₂×D₄	—	18	3	no	—
76	C₂×D₄	—	18	3	no	—
77	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
78	M₁₆	—	2	3	no	—
79	M₁₆	—	2	3	no	—
80	M₁₆	—	9	6	no	—
81	M₁₆	—	18	3	no	—
82	C₂×C₈	—	2	3	no	—
83	C₂×C₈	—	18	3	no	—
84	C₂²×C₃	—	4	4	no	—
85	C₂²×C₃	#896	1	12	yes	—
86	C₂×D₃	—	12	8	no	—
87	C₂×D₃	—	24	24	no	—
88	C₃×C₄	#876	1	4	yes	—
89	C₃×C₄	#884	1	12	yes	—
90	C₃²	#2069	1	1	yes	characteristic, complemented, Sylow
91	C₂³	—	18	6	no	—
92	C₂³	—	72	2	no	—
93	C₂³	Q₈×C₃⋊D₃	1	1	yes	characteristic
94	D₄	—	36	1	no	—
95	D₄	—	36	3	no	—
96	C₂×C₄	#3828	1	3	yes	—
97	C₂×C₄	D₄×C₃⋊D₃	1	3	yes	—
98	C₈	—	4	3	no	—
99	C₈	—	18	6	no	—
100	C₂×C₃	—	2	12	no	—
101	C₂×C₃	#4676	1	4	yes	—
102	D₃	—	24	8	no	—
103	C₂²	—	4	1	no	—
104	C₂²	—	36	2	no	—
105	C₂²	—	72	6	no	—
106	C₂²	#64105	1	3	yes	—
107	C₄	#64085	1	1	yes	characteristic
108	C₄	#64093	1	3	yes	—
109	C₃	#68722	1	4	yes	complemented
110	C₂	—	2	3	no	—
111	C₂	—	72	2	no	—
112	C₂	#10592375	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#63650

C₂

yes

maximal

#10592823

C₂

yes

maximal, complemented

C₃²×#338

C₂

yes

maximal, characteristic, complemented

#68722

—

maximal

C₂²×C₄⋊(C₃⋊D₃)

C₂²

yes

characteristic

C₂×C₃²×M₁₆

C₂²

yes

—

C₂×#3748

C₂²

yes

—

—

—

—

—

D₃

yes

complemented

C₂³×C₃⋊D₃

Q₈

yes

—

C₂²×C₃²×C₄

C₂³

yes

characteristic

C₂×C₄⋊(C₃⋊D₃)

—

C₂×C₄⋊(C₃⋊D₃)

—

C₃²×M₁₆

—

C₃²×M₁₆

—

C₂×C₃²×C₈

—

#3748

—

#3748

D₄

yes

—

C₂×C₃⋊(C₃⋊C₈)

—

#2069

—

Sylow

C₂²×D₁₂

—

C₂×C₃×M₁₆

C₂×D₃

yes

—

C₂×#208

—

#823

—

#822

—

C₂³×C₃²

C₂×Q₈

yes

characteristic

C₂²×C₃⋊D₃

—

C₂²×C₃⋊D₃

—

C₂×C₃²×C₄

C₂×D₄

yes

—

C₂×C₃²×C₄

#41

yes

—

C₄⋊(C₃⋊D₃)

—

C₄⋊(C₃⋊D₃)

—

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

#430

—

C₂×#100

—

#338

C₃⋊D₃

yes

characteristic, complemented

C₂³×D₃

—

C₂²×C₃×C₄

C₂²×D₃

yes

—

C₂×D₁₂

—

C₂×D₁₂

—

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

#208

—

#208

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

C₂²⋊Q₈

yes

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₃²×C₄

#121

yes

—

C₃²×C₄

#126

yes

characteristic

C₂²×D₄

—

C₂×M₁₆

—

C₂×M₁₆

C₂×C₃⋊D₃

yes

—

#108

—

#100

—

C₂³×C₃

D₃×Q₈

yes

—

C₂²×D₃

—

C₂²×D₃

—

C₂×C₃×C₄

#235

yes

—

C₂×C₃×C₄

D₃×D₄

yes

—

D₁₂

—

D₁₂

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

#396

yes

characteristic

C₃⋊D₃

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂²×C₃⋊D₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂²×C₃

—

C₂²×C₃

#896

yes

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

#876

yes

—

C₃×C₄

#884

yes

—

C₃²

#2069

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

C₂³

Q₈×C₃⋊D₃

yes

characteristic

D₄

—

D₄

—

C₂×C₄

#3828

yes

—

C₂×C₄

D₄×C₃⋊D₃

yes

—

C₈

—

C₈

—

100

C₂×C₃

—

101

C₂×C₃

#4676

yes

—

102

D₃

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

#64105

yes

—

107

C₄

#64085

yes

characteristic

108

C₄

#64093

yes

—

109

C₃

#68722

yes

complemented

110

C₂

—

111

C₂

—

112

C₂

#10592375

yes

characteristic

	subgroup	G/H
center	C₂	#10592375
commutator, cosocle	C₂²×C₃²×C₄	C₂³
Frattini	C₂²×C₄	C₂²×C₃⋊D₃
Fitting	C₃²×#338	C₂
socle	C₂×C₃²	#396
prime radical	C₂²×C₄⋊(C₃⋊D₃)	C₂²
hypercenter	#338	C₃⋊D₃
hypocenter	C₃²	#2069
LCS	G ▷ C₂²×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #338

subgroup

G/H

center

C₂

#10592375

commutator, cosocle

C₂²×C₃²×C₄

C₂³

Frattini

C₂²×C₄

C₂²×C₃⋊D₃

Fitting

C₃²×#338

C₂

socle

C₂×C₃²

#396

prime radical

C₂²×C₄⋊(C₃⋊D₃)

C₂²

C₃⋊D₃

C₃²

G ▷ C₂²×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #338

N	H ≅ G/N	type
2. #10592823	C₂	semidirect
3. C₃²×#338	C₂	semidirect
12. C₃×#338	D₃	semidirect
40. #338	C₃⋊D₃	semidirect
90. C₃²	#2069	semidirect
109. C₃	#68722	semidirect

H ≅ G/N

type

2. #10592823

C₂

semidirect

3. C₃²×#338

C₂

semidirect

12. C₃×#338

D₃

semidirect

40. #338

C₃⋊D₃

semidirect

90. C₃²

#2069

semidirect

109. C₃

#68722

semidirect

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

subgroup	index
C₂×#63650	2
#10592823	2
C₃²×#338	2
#68722	3

subgroup

index

C₂×#63650

#10592823

C₃²×#338

#68722

Subgroups — #11794609

Conjugacy classes of subgroups

Complements

Maximal subgroups