Group #19147449 — subgroups

Subgroups — Q₈×D₃₄₄ #19147449

Order 5504.

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

normal subgroups only

This group has 404 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	Q₈×D₁₇₂	C₂	1	2	yes	maximal, complemented
2	#14583304	C₂	1	3	yes	maximal, complemented
3	#14583180	C₂	1	6	yes	maximal, complemented
4	C₄×D₃₄₄	C₂	1	3	yes	maximal
5	C₈×Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×D₄₃	—	2	2	no	—
7	C₂×D₃₄₄	C₂²	1	1	yes	characteristic
8	#13063325	C₂²	1	6	yes	—
9	#13063305	—	2	6	no	—
10	C₄×D₁₇₂	C₂²	1	6	yes	—
11	#13063259	C₂²	1	3	yes	—
12	#13063259	C₂²	1	6	yes	—
13	#13063247	C₂²	1	6	yes	—
14	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₄₃	C₂²	1	3	yes	—
16	C₄×C₈×C₄₃	C₂²	1	3	yes	—
17	C₂×Q₈×C₄₃	D₄	1	1	yes	characteristic
18	Q₈×D₄₃	—	4	8	no	—
19	C₂×D₁₇₂	C₂³	1	2	yes	—
20	C₂×C₄×D₄₃	—	2	6	no	—
21	C₂×Dic₈₆	—	2	6	no	—
22	C₂×C₈×C₄₃	C₂³	1	3	yes	—
23	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₄₃	D₄	1	3	yes	—
25	C₄²×C₄₃	C₂³	1	3	yes	—
26	(C₂×D₄₃)⋊C₄	—	2	6	no	—
27	C₄⋊Dic₄₃	—	2	6	no	—
28	C₄⋊Dic₄₃	C₂³	1	6	yes	—
29	D₃₄₄	Q₈	1	4	yes	direct product factor
30	C₂²×D₄₃	—	2	2	no	—
31	Q₈×C₄₃	D₈	1	4	yes	complemented
32	C₂×C₄×C₄₃	C₂×D₄	1	3	yes	—
33	C₂×C₄×C₄₃	C₂×D₄	1	3	yes	—
34	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
35	C₂×Dic₄₃	—	2	6	no	—
36	D₁₇₂	—	2	2	no	—
37	D₁₇₂	C₂×Q₈	1	4	yes	complemented
38	C₄×D₄₃	—	4	12	no	—
39	Dic₈₆	—	4	12	no	—
40	C₈×C₄₃	—	2	3	no	—
41	C₈×C₄₃	C₂×Q₈	1	2	yes	complemented
42	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
43	C₂×D₄₃	—	2	4	no	—
44	C₂×D₄₃	—	4	2	no	—
45	C₂×D₄₃	—	4	2	no	—
46	C₄×C₄₃	—	2	3	no	—
47	C₄×C₄₃	C₂×D₈	1	6	yes	—
48	C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
49	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
50	Dic₄₃	—	4	6	no	—
51	Q₈×D₈	—	43	1	no	maximal, Sylow
52	C₂×C₄₃	D₄×Q₈	1	1	yes	characteristic
53	C₂×C₄₃	C₂²×D₈	1	1	yes	characteristic
54	C₂×C₄₃	#578	1	1	yes	characteristic
55	D₄₃	—	4	4	no	—
56	D₄×Q₈	—	43	2	no	—
57	#500	—	43	3	no	—
58	#474	—	43	6	no	—
59	C₈×Q₈	D₄₃	1	1	yes	characteristic, complemented
60	C₄×D₈	—	43	3	no	—
61	C₄₃	Q₈×D₈	1	1	yes	characteristic, complemented, Sylow
62	C₂²×Q₈	—	86	2	no	—
63	C₂×D₈	—	43	1	no	—
64	C₄⋊Q₈	—	43	6	no	—
65	C₂²⋊Q₈	—	86	6	no	—
66	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
67	C₄×D₄	—	43	6	no	—
68	#107	—	43	3	no	—
69	#107	—	43	6	no	—
70	C₄⋊C₈	C₂×D₄₃	1	3	yes	—
71	#102	—	43	6	no	—
72	C₄×C₈	C₂×D₄₃	1	3	yes	—
73	C₂×Q₈	—	86	6	no	—
74	C₂×Q₈	—	172	8	no	—
75	C₂×Q₈	D₁₇₂	1	1	yes	characteristic
76	C₂×D₄	—	43	2	no	—
77	C₂²×C₄	—	86	6	no	—
78	D₈	—	43	4	no	—
79	C₂×C₈	C₂²×D₄₃	1	3	yes	—
80	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
81	C₄⋊C₄	—	43	6	no	—
82	C₄⋊C₄	—	86	6	no	—
83	C₄⋊C₄	D₁₇₂	1	3	yes	—
84	C₂²⋊C₄	—	86	6	no	—
85	C₄²	C₂²×D₄₃	1	3	yes	—
86	C₂³	—	86	2	no	—
87	Q₈	—	172	12	no	—
88	Q₈	D₃₄₄	1	4	yes	direct product factor
89	D₄	—	43	4	no	—
90	D₄	—	86	2	no	—
91	C₂×C₄	—	86	6	no	—
92	C₂×C₄	—	172	12	no	—
93	C₂×C₄	C₂×D₁₇₂	1	3	yes	—
94	C₂×C₄	C₂×D₁₇₂	1	3	yes	—
95	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
96	C₈	—	2	3	no	—
97	C₈	Q₈×D₄₃	1	2	yes	complemented
98	C₂²	—	86	4	no	—
99	C₂²	—	172	2	no	—
100	C₂²	—	172	2	no	—
101	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
102	C₄	—	2	3	no	—
103	C₄	—	172	6	no	—
104	C₄	C₂×D₃₄₄	1	6	yes	—
105	C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
106	C₄	#13063372	1	1	yes	characteristic
107	C₂	—	172	4	no	—
108	C₂	Q₈×D₁₇₂	1	1	yes	characteristic
109	C₂	C₂²×D₃₄₄	1	1	yes	characteristic
110	C₂	#14583716	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₁₇₂

C₂

yes

maximal, complemented

#14583304

C₂

yes

maximal, complemented

#14583180

C₂

yes

maximal, complemented

C₄×D₃₄₄

C₂

yes

maximal

C₈×Q₈×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₄₃

—

C₂×D₃₄₄

C₂²

yes

characteristic

#13063325

C₂²

yes

—

#13063305

—

C₄×D₁₇₂

C₂²

yes

—

#13063259

C₂²

yes

—

#13063259

C₂²

yes

—

#13063247

C₂²

yes

—

C₄×Q₈×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

—

C₄×C₈×C₄₃

C₂²

yes

—

C₂×Q₈×C₄₃

D₄

yes

characteristic

Q₈×D₄₃

—

C₂×D₁₇₂

C₂³

yes

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

C₂³

yes

—

C₂×C₈×C₄₃

C₂³

yes

characteristic

C₄⋊C₄×C₄₃

D₄

yes

—

C₄²×C₄₃

C₂³

yes

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂³

yes

—

D₃₄₄

Q₈

yes

direct product factor

C₂²×D₄₃

—

Q₈×C₄₃

D₈

yes

complemented

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

C₂×Q₈

yes

complemented

C₄×D₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

C₂×Q₈

yes

complemented

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×D₈

yes

—

C₄×C₄₃

C₂²×Q₈

yes

characteristic

C₄×C₄₃

2₋¹⁺⁴

yes

characteristic

Dic₄₃

—

Q₈×D₈

—

maximal, Sylow

C₂×C₄₃

D₄×Q₈

yes

characteristic

C₂×C₄₃

C₂²×D₈

yes

characteristic

C₂×C₄₃

#578

yes

characteristic

D₄₃

—

D₄×Q₈

—

#500

—

#474

—

C₈×Q₈

D₄₃

yes

characteristic, complemented

C₄×D₈

—

C₄₃

Q₈×D₈

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×D₈

—

C₄⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

C₂×D₄₃

yes

characteristic

C₄×D₄

—

#107

—

#107

—

C₄⋊C₈

C₂×D₄₃

yes

—

#102

—

C₄×C₈

C₂×D₄₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

172

—

C₂×Q₈

D₁₇₂

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

D₈

—

C₂×C₈

C₂²×D₄₃

yes

—

C₂×C₈

C₂²×D₄₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₁₇₂

yes

—

C₂²⋊C₄

—

C₄²

C₂²×D₄₃

yes

—

C₂³

—

Q₈

—

172

—

Q₈

D₃₄₄

yes

direct product factor

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

172

—

C₂×C₄

C₂×D₁₇₂

yes

—

C₂×C₄

C₂×D₁₇₂

yes

—

C₂×C₄

C₂³×D₄₃

yes

characteristic

C₈

—

C₈

Q₈×D₄₃

yes

complemented

C₂²

—

C₂²

—

172

—

100

C₂²

—

172

—

101

C₂²

C₂²×D₁₇₂

yes

characteristic

102

C₄

—

103

C₄

—

172

—

104

C₄

C₂×D₃₄₄

yes

—

105

C₄

C₂×Q₈×D₄₃

yes

characteristic

106

C₄

#13063372

yes

characteristic

107

C₂

—

172

—

108

C₂

Q₈×D₁₇₂

yes

characteristic

109

C₂

C₂²×D₃₄₄

yes

characteristic

110

C₂

#14583716

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₈×Q₈×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	C₈×Q₈	D₄₃
hypocenter	C₄₃	Q₈×D₈
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₈×Q₈×C₄₃

C₂

socle

C₂²×C₄₃

C₂²×D₄

hypercenter

C₈×Q₈

D₄₃

hypocenter

C₄₃

Q₈×D₈

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. Q₈×D₁₇₂	C₂	semidirect
2. #14583304	C₂	semidirect
3. #14583180	C₂	semidirect
5. C₈×Q₈×C₄₃	C₂	semidirect
29. D₃₄₄	Q₈	direct (N × H)
31. Q₈×C₄₃	D₈	semidirect
37. D₁₇₂	C₂×Q₈	semidirect
41. C₈×C₄₃	C₂×Q₈	semidirect
59. C₈×Q₈	D₄₃	semidirect
61. C₄₃	Q₈×D₈	semidirect
88. Q₈	D₃₄₄	direct (N × H)
97. C₈	Q₈×D₄₃	semidirect

H ≅ G/N

type

1. Q₈×D₁₇₂

C₂

semidirect

2. #14583304

C₂

semidirect

3. #14583180

C₂

semidirect

5. C₈×Q₈×C₄₃

C₂

semidirect

29. D₃₄₄

Q₈

direct (N × H)

31. Q₈×C₄₃

D₈

semidirect

37. D₁₇₂

C₂×Q₈

semidirect

41. C₈×C₄₃

C₂×Q₈

semidirect

59. C₈×Q₈

D₄₃

semidirect

61. C₄₃

Q₈×D₈

semidirect

88. Q₈

D₃₄₄

direct (N × H)

97. C₈

Q₈×D₄₃

semidirect

Subgroups — Q₈×D₃₄₄ #19147449

Conjugacy classes of subgroups

Complements

Subgroups — Q8×D344 #19147449

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×D₃₄₄ #19147449