Group #11729031 — subgroups

Subgroups — #11729031

Order 896.

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

normal subgroups only

This group has 220 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	C₇×#493	C₂	1	1	yes	maximal, characteristic
2	#89848	C₂	1	2	yes	maximal, complemented
3	#89842	C₂	1	2	yes	maximal, complemented
4	#89452	C₂	1	1	yes	maximal, characteristic, complemented
5	#89250	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×D₈	—	2	2	no	—
7	C₇×#127	C₂²	1	1	yes	characteristic
8	C₇×#127	C₂²	1	1	yes	characteristic
9	C₂×#1142	—	2	4	no	—
10	#6432	C₂²	1	2	yes	—
11	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	#6406	—	2	4	no	—
13	#6391	—	2	2	no	—
14	C₄×C₇⋊C₈	C₂²	1	2	yes	—
15	#1715	—	7	1	no	maximal, Sylow
16	C₂×C₇×D₄	—	2	2	no	—
17	C₂×C₇×D₄	—	2	2	no	—
18	C₂×C₇×D₄	—	2	1	no	—
19	C₂×C₇×D₄	—	2	1	no	—
20	C₂×Dic₁₄	—	2	2	no	—
21	C₇×D₈	—	4	4	no	—
22	C₂×C₇×C₈	D₄	1	2	yes	—
23	C₄²×C₇	C₂³	1	1	yes	characteristic
24	#1142	—	4	8	no	—
25	C₄⋊Dic₇	—	2	2	no	—
26	C₄⋊Dic₇	—	2	2	no	—
27	C₂×C₇⋊C₈	D₄	1	4	yes	—
28	#498	—	7	1	no	—
29	#493	D₇	1	1	yes	characteristic
30	#492	—	7	2	no	—
31	#486	—	7	2	no	—
32	#322	—	7	1	no	—
33	C₂³×C₇	—	4	1	no	—
34	C₂³×C₇	—	4	1	no	—
35	C₇×D₄	—	4	2	no	—
36	C₇×D₄	—	4	2	no	—
37	C₇×D₄	—	4	4	no	—
38	C₇×D₄	—	4	4	no	—
39	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₇	—	4	2	no	—
42	Dic₁₄	—	4	4	no	—
43	C₇×C₈	SD₁₆	1	4	yes	complemented
44	C₇⋊C₈	—	2	4	no	—
45	C₂×SD₁₆	—	14	4	no	—
46	C₂×D₈	—	2	2	no	—
47	C₄⋊Q₈	—	7	2	no	—
48	#127	C₂×D₇	1	1	yes	characteristic
49	#127	C₂×D₇	1	1	yes	characteristic
50	#106	—	14	2	no	—
51	#102	—	14	4	no	—
52	C₄×C₈	—	7	2	no	—
53	C₄×C₈	C₂×D₇	1	1	yes	characteristic
54	C₂²×C₇	—	8	1	no	—
55	C₂²×C₇	—	8	1	no	—
56	C₂²×C₇	—	8	2	no	—
57	C₂²×C₇	—	8	2	no	—
58	C₂²×C₇	#127	1	1	yes	characteristic
59	C₄×C₇	C₂×SD₁₆	1	2	yes	—
60	C₄×C₇	AGL(1,ℤ/8)	1	4	yes	—
61	Dic₇	—	8	2	no	—
62	C₂×Q₈	—	14	2	no	—
63	C₂×D₄	—	2	2	no	—
64	C₂×D₄	—	2	2	no	—
65	C₂×D₄	—	2	1	no	—
66	C₂×D₄	—	2	1	no	—
67	SD₁₆	—	28	8	no	—
68	D₈	—	4	4	no	—
69	C₂×C₈	—	7	4	no	—
70	C₂×C₈	C₂²⋊D₇	1	2	yes	—
71	C₄⋊C₄	—	14	2	no	—
72	C₄⋊C₄	—	14	2	no	—
73	C₄²	C₂²×D₇	1	1	yes	characteristic
74	C₂×C₇	—	16	1	no	—
75	C₂×C₇	—	16	1	no	—
76	C₂×C₇	#492	1	1	yes	characteristic
77	C₂×C₇	#496	1	2	yes	—
78	C₂³	—	4	1	no	—
79	C₂³	—	4	1	no	—
80	Q₈	—	28	4	no	—
81	D₄	—	4	4	no	—
82	D₄	—	4	4	no	—
83	D₄	—	4	2	no	—
84	D₄	—	4	2	no	—
85	C₂×C₄	—	28	2	no	—
86	C₂×C₄	D₄×D₇	1	2	yes	—
87	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
88	C₈	—	14	4	no	—
89	C₈	#1142	1	4	yes	complemented
90	C₇	#1715	1	1	yes	characteristic, complemented, Sylow
91	C₂²	—	8	2	no	—
92	C₂²	—	8	2	no	—
93	C₂²	—	8	1	no	—
94	C₂²	—	8	1	no	—
95	C₂²	#6503	1	1	yes	characteristic
96	C₄	—	56	2	no	—
97	C₄	#6474	1	4	yes	—
98	C₄	C₂×#1142	1	2	yes	—
99	C₂	—	16	1	no	—
100	C₂	—	16	1	no	—
101	C₂	#89848	1	1	yes	characteristic
102	C₂	#89926	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#493

C₂

yes

maximal, characteristic

#89848

C₂

yes

maximal, complemented

#89842

C₂

yes

maximal, complemented

#89452

C₂

yes

maximal, characteristic, complemented

#89250

C₂

yes

maximal, characteristic, complemented

C₂×C₇×D₈

—

C₇×#127

C₂²

yes

characteristic

C₇×#127

C₂²

yes

characteristic

C₂×#1142

—

#6432

C₂²

yes

—

C₄×C₇×C₈

C₂²

yes

characteristic

#6406

—

#6391

—

C₄×C₇⋊C₈

C₂²

yes

—

#1715

—

maximal, Sylow

C₂×C₇×D₄

—

C₂×C₇×D₄

—

C₂×C₇×D₄

—

C₂×C₇×D₄

—

C₂×Dic₁₄

—

C₇×D₈

—

C₂×C₇×C₈

D₄

yes

—

C₄²×C₇

C₂³

yes

characteristic

#1142

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₂×C₇⋊C₈

D₄

yes

—

#498

—

#493

D₇

yes

—

—

—

C₂³×C₇

—

C₂³×C₇

—

C₇×D₄

—

C₇×D₄

—

C₇×D₄

—

C₇×D₄

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×Dic₇

—

Dic₁₄

—

C₇×C₈

SD₁₆

yes

complemented

C₇⋊C₈

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

#127

C₂×D₇

yes

characteristic

#127

C₂×D₇

yes

characteristic

#106

—

#102

—

C₄×C₈

—

C₄×C₈

C₂×D₇

yes

characteristic

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

#127

yes

characteristic

C₄×C₇

C₂×SD₁₆

yes

—

C₄×C₇

AGL(1,ℤ/8)

yes

—

Dic₇

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

D₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₇

yes

characteristic

C₂×C₇

—

C₂×C₇

—

C₂×C₇

#492

yes

characteristic

C₂×C₇

#496

yes

—

C₂³

—

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

D₄×D₇

yes

—

C₂×C₄

C₂×C₂²⋊D₇

yes

characteristic

C₈

—

C₈

#1142

yes

complemented

C₇

#1715

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#6503

yes

characteristic

C₄

—

C₄

#6474

yes

—

C₄

C₂×#1142

yes

—

C₂

—

100

C₂

—

101

C₂

#89848

yes

characteristic

102

C₂

#89926

yes

—

	subgroup	G/H
center	C₂²	#6503
commutator, cosocle	C₄²×C₇	C₂³
Frattini	C₄²	C₂²×D₇
Fitting, prime radical	C₇×#493	C₂
socle	C₂²×C₇	#127
hypercenter	#493	D₇
hypocenter	C₇	#1715
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#6503

commutator, cosocle

C₄²×C₇

C₂³

Frattini

C₄²

C₂²×D₇

Fitting, prime radical

C₂

C₂²×C₇

D₇

C₇

G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
2. #89848	C₂	semidirect
3. #89842	C₂	semidirect
4. #89452	C₂	semidirect
5. #89250	C₂	semidirect
43. C₇×C₈	SD₁₆	semidirect
89. C₈	#1142	semidirect
90. C₇	#1715	semidirect

H ≅ G/N

type

2. #89848

C₂

semidirect

3. #89842

C₂

semidirect

4. #89452

C₂

semidirect

5. #89250

C₂

semidirect

43. C₇×C₈

SD₁₆

semidirect

89. C₈

#1142

semidirect

90. C₇

#1715

semidirect

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

subgroup	index
C₇×#493	2
#89848	2
#89842	2
#89452	2
#89250	2
#1715	7

subgroup

index

Subgroups — #11729031

Conjugacy classes of subgroups

Complements

Maximal subgroups