Group #22596736 — subgroups

Subgroups — #22596736

Order 9088.

648 conjugacy classes of subgroups; only the 245 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 245 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×#14556128	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556126	C₂	1	4	yes	maximal
3	#17985868	C₂	1	8	yes	maximal
4	C₄²×Dic₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₄₂	C₂²	1	2	yes	—
7	#14556128	—	2	4	no	—
8	#14556126	—	2	16	no	—
9	C₂×C₄⋊Dic₇₁	C₂²	1	4	yes	—
10	C₂×Dic₇₁⋊C₄	C₂²	1	16	yes	—
11	C₂×C₄×Dic₇₁	C₂²	1	2	yes	—
12	C₂×C₄×Dic₇₁	C₂²	1	4	yes	—
13	C₂²×Q₈×C₇₁	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₇₁	—	2	4	no	—
15	C₂×C₄⋊C₄×C₇₁	C₂²	1	4	yes	—
16	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₄₂	—	2	8	no	—
18	C₂×Dic₁₄₂	—	2	4	no	—
19	C₂²×Dic₇₁	C₂³	1	4	yes	—
20	C₂²×Dic₇₁	C₂³	1	4	yes	—
21	C₄⋊Dic₇₁	—	2	8	no	—
22	Dic₇₁⋊C₄	—	2	32	no	—
23	C₄×Dic₇₁	D₄	1	8	yes	—
24	C₄×Dic₇₁	Q₈	1	16	yes	complemented
25	C₂×Q₈×C₇₁	—	2	4	no	—
26	C₂×Q₈×C₇₁	—	2	8	no	—
27	C₂²×C₄×C₇₁	C₂³	1	4	yes	—
28	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₇₁	C₂³	1	2	yes	—
30	C₄⋊C₄×C₇₁	—	2	8	no	—
31	C₄²×C₇₁	D₄	1	4	yes	—
32	C₂³×C₇₁	C₂⁴	1	1	yes	characteristic
33	Q₈×C₇₁	—	4	8	no	—
34	C₂×C₄×C₇₁	—	2	4	no	—
35	C₂×C₄×C₇₁	—	2	8	no	—
36	C₂×C₄×C₇₁	C₂×D₄	1	4	yes	—
37	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₇₁	C₂×D₄	1	4	yes	—
39	C₂×C₄×C₇₁	C₂×Q₈	1	8	yes	—
40	C₂×Dic₇₁	—	2	8	no	—
41	C₂×Dic₇₁	—	2	4	no	—
42	C₂×Dic₇₁	C₂×D₄	1	8	yes	—
43	C₂×Dic₇₁	C₂×Q₈	1	16	yes	—
44	Dic₁₄₂	—	4	8	no	—
45	C₂²×C₇₁	C₂²×D₄	1	2	yes	—
46	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
47	C₂²×C₇₁	C₂²×Q₈	1	4	yes	—
48	C₄×C₇₁	—	4	4	no	—
49	C₄×C₇₁	#127	1	4	yes	—
50	C₄×C₇₁	C₄⋊Q₈	1	8	yes	complemented
51	Dic₇₁	—	4	4	no	—
52	Dic₇₁	C₄⋊Q₈	1	16	yes	complemented
53	C₂×C₇₁	C₂×#127	1	1	yes	characteristic
54	C₂×C₇₁	C₂×C₄⋊Q₈	1	4	yes	—
55	C₂×C₇₁	C₂×C₄⋊Q₈	1	2	yes	—
56	#2868	—	71	1	no	maximal, Sylow
57	C₇₁	#2868	1	1	yes	characteristic, complemented, Sylow
58	C₂×C₄⋊Q₈	—	71	4	no	—
59	C₂×C₄⋊Q₈	—	71	1	no	—
60	C₂×C₄⋊Q₈	D₇₁	1	1	yes	characteristic
61	#384	—	71	8	no	—
62	C₄³	—	71	1	no	—
63	C₂²×Q₈	—	71	2	no	—
64	C₂²×Q₈	C₂×D₇₁	1	2	yes	—
65	C₄⋊Q₈	—	2	4	no	—
66	C₄⋊Q₈	—	142	16	no	—
67	C₄⋊Q₈	—	142	4	no	—
68	C₂×C₄⋊C₄	—	71	16	no	—
69	C₂×C₄⋊C₄	—	71	4	no	—
70	C₂×C₄⋊C₄	C₂×D₇₁	1	4	yes	—
71	C₂×C₄²	—	71	2	no	—
72	C₂×C₄²	—	71	4	no	—
73	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
74	C₂×Q₈	—	2	8	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	142	8	no	—
77	C₂×Q₈	—	142	4	no	—
78	C₂²×C₄	—	71	4	no	—
79	C₂²×C₄	—	71	4	no	—
80	C₂²×C₄	C₂²×D₇₁	1	4	yes	—
81	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
82	C₂²×C₄	C₂²×D₇₁	1	2	yes	—
83	C₄⋊C₄	—	2	8	no	—
84	C₄⋊C₄	—	142	32	no	—
85	C₄⋊C₄	—	142	8	no	—
86	C₄²	—	71	8	no	—
87	C₄²	—	71	16	no	—
88	C₄²	C₂²⋊D₇₁	1	4	yes	—
89	C₂³	C₂³×D₇₁	1	1	yes	characteristic
90	Q₈	—	4	8	no	—
91	Q₈	—	284	8	no	—
92	C₂×C₄	—	2	8	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	71	8	no	—
95	C₂×C₄	—	71	16	no	—
96	C₂×C₄	—	142	8	no	—
97	C₂×C₄	—	142	4	no	—
98	C₂×C₄	D₄×D₇₁	1	4	yes	—
99	C₂×C₄	C₂×C₂²⋊D₇₁	1	2	yes	—
100	C₂×C₄	C₂×C₂²⋊D₇₁	1	4	yes	—
101	C₂×C₄	Q₈×D₇₁	1	8	yes	—
102	C₂²	C₂×D₄×D₇₁	1	2	yes	—
103	C₂²	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
104	C₂²	C₂×Q₈×D₇₁	1	4	yes	—
105	C₄	—	4	4	no	—
106	C₄	—	284	4	no	—
107	C₄	—	71	16	no	—
108	C₄	#14556124	1	4	yes	—
109	C₄	#14556126	1	8	yes	complemented
110	C₂	C₂×#14556124	1	1	yes	characteristic
111	C₂	C₄⋊Q₈×D₇₁	1	4	yes	—
112	C₂	C₂×#14556126	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556128

C₂

yes

maximal, characteristic

C₂×#14556126

C₂

yes

maximal

#17985868

C₂

yes

maximal

C₄²×Dic₇₁

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₇₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₂

C₂²

yes

—

#14556128

—

#14556126

—

C₂×C₄⋊Dic₇₁

C₂²

yes

—

C₂×Dic₇₁⋊C₄

C₂²

yes

—

C₂×C₄×Dic₇₁

C₂²

yes

—

C₂×C₄×Dic₇₁

C₂²

yes

—

C₂²×Q₈×C₇₁

C₂²

yes

—

C₄⋊Q₈×C₇₁

—

C₂×C₄⋊C₄×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

C₂³

yes

—

C₂²×Dic₇₁

C₂³

yes

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

D₄

yes

—

C₄×Dic₇₁

Q₈

yes

complemented

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

C₂³

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

D₄

yes

—

C₂³×C₇₁

C₂⁴

yes

characteristic

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

C₂×D₄

yes

—

C₂×Dic₇₁

C₂×Q₈

yes

—

Dic₁₄₂

—

C₂²×C₇₁

C₂²×D₄

yes

—

C₂²×C₇₁

C₂²×D₄

yes

characteristic

C₂²×C₇₁

C₂²×Q₈

yes

—

C₄×C₇₁

—

C₄×C₇₁

#127

yes

—

C₄×C₇₁

C₄⋊Q₈

yes

complemented

Dic₇₁

—

Dic₇₁

C₄⋊Q₈

yes

complemented

C₂×C₇₁

C₂×#127

yes

characteristic

C₂×C₇₁

C₂×C₄⋊Q₈

yes

—

C₂×C₇₁

C₂×C₄⋊Q₈

yes

—

#2868

—

maximal, Sylow

C₇₁

#2868

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₇₁

yes

characteristic

#384

—

C₄³

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₇₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

142

—

C₄⋊Q₈

—

142

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

142

—

C₂×Q₈

—

142

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇₁

yes

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂²×C₄

C₂²×D₇₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₇₁

yes

—

C₂³

C₂³×D₇₁

yes

characteristic

Q₈

—

Q₈

—

284

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

142

—

C₂×C₄

—

142

—

C₂×C₄

D₄×D₇₁

yes

—

C₂×C₄

C₂×C₂²⋊D₇₁

yes

—

100

C₂×C₄

C₂×C₂²⋊D₇₁

yes

—

101

C₂×C₄

Q₈×D₇₁

yes

—

102

C₂²

C₂×D₄×D₇₁

yes

—

103

C₂²

C₂²×C₂²⋊D₇₁

yes

characteristic

104

C₂²

C₂×Q₈×D₇₁

yes

—

105

C₄

—

106

C₄

—

284

—

107

C₄

—

108

C₄

#14556124

yes

—

109

C₄

#14556126

yes

complemented

110

C₂

C₂×#14556124

yes

characteristic

111

C₂

C₄⋊Q₈×D₇₁

yes

—

112

C₂

C₂×#14556126

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₁
commutator, socle	C₂³×C₇₁	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₇₁	C₂
hypercenter	C₂×C₄⋊Q₈	D₇₁
hypocenter	C₇₁	#2868
LCS	G ▷ C₂³×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₁

commutator, socle

C₂³×C₇₁

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₇₁

C₂

hypercenter

C₂×C₄⋊Q₈

D₇₁

C₇₁

G ▷ C₂³×C₇₁ ▷ C₇₁

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
24. C₄×Dic₇₁	Q₈	semidirect
50. C₄×C₇₁	C₄⋊Q₈	semidirect
52. Dic₇₁	C₄⋊Q₈	semidirect
57. C₇₁	#2868	semidirect
109. C₄	#14556126	semidirect

H ≅ G/N

type

24. C₄×Dic₇₁

Q₈

semidirect

50. C₄×C₇₁

C₄⋊Q₈

semidirect

52. Dic₇₁

C₄⋊Q₈

semidirect

57. C₇₁

#2868

semidirect

109. C₄

#14556126

semidirect

Subgroups — #22596736

Conjugacy classes of subgroups

Complements