Group #36351194 — subgroups

Subgroups — C₂²×SD₃₂×C₁₇₃ #36351194

Order 22144.

400 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 200 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	C₂²×Q₁₆×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₈×C₁₇₃	C₂	1	1	yes	maximal, characteristic
3	C₂×SD₃₂×C₁₇₃	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₁₆×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×Q₈×C₁₇₃	—	2	1	no	—
6	C₂²×D₄×C₁₇₃	—	2	1	no	—
7	C₂×Q₁₆×C₁₇₃	—	2	3	no	—
8	C₂×Q₁₆×C₁₇₃	C₂²	1	6	yes	complemented
9	C₂×D₈×C₁₇₃	—	2	3	no	—
10	C₂×D₈×C₁₇₃	C₂²	1	6	yes	—
11	C₂²×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
12	SD₃₂×C₁₇₃	C₂²	1	16	yes	direct product factor
13	C₂×C₁₆×C₁₇₃	C₂²	1	6	yes	complemented
14	C₂⁴×C₁₇₃	—	4	1	no	—
15	C₂×Q₈×C₁₇₃	—	2	6	no	—
16	C₂×Q₈×C₁₇₃	—	4	3	no	—
17	C₂×D₄×C₁₇₃	—	2	6	no	—
18	C₂×D₄×C₁₇₃	—	4	3	no	—
19	C₂²×C₄×C₁₇₃	—	4	1	no	—
20	C₂²×C₄×C₁₇₃	D₄	1	1	yes	characteristic
21	Q₁₆×C₁₇₃	—	2	6	no	—
22	Q₁₆×C₁₇₃	C₂³	1	4	yes	complemented
23	D₈×C₁₇₃	—	2	6	no	—
24	D₈×C₁₇₃	C₂³	1	4	yes	—
25	C₂×C₈×C₁₇₃	D₄	1	3	yes	—
26	C₂×C₈×C₁₇₃	C₂³	1	3	yes	—
27	C₁₆×C₁₇₃	C₂³	1	4	yes	complemented
28	C₂³×C₁₇₃	—	4	6	no	—
29	C₂³×C₁₇₃	—	8	4	no	—
30	C₂³×C₁₇₃	D₈	1	1	yes	characteristic
31	Q₈×C₁₇₃	—	2	4	no	—
32	Q₈×C₁₇₃	—	4	6	no	—
33	D₄×C₁₇₃	—	2	4	no	—
34	D₄×C₁₇₃	—	4	6	no	—
35	C₂×C₄×C₁₇₃	—	4	6	no	—
36	C₂×C₄×C₁₇₃	D₈	1	3	yes	—
37	C₂×C₄×C₁₇₃	C₂×D₄	1	3	yes	—
38	C₈×C₁₇₃	C₂×D₄	1	3	yes	—
39	C₈×C₁₇₃	C₂⁴	1	1	yes	characteristic
40	C₂²×C₁₇₃	—	4	4	no	—
41	C₂²×C₁₇₃	—	8	12	no	—
42	C₂²×C₁₇₃	SD₃₂	1	4	yes	direct product factor
43	C₂²×C₁₇₃	C₂×D₈	1	3	yes	—
44	C₄×C₁₇₃	—	4	4	no	—
45	C₄×C₁₇₃	C₂×D₈	1	3	yes	—
46	C₄×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
47	C₂×C₁₇₃	—	8	4	no	—
48	C₂×C₁₇₃	C₂×SD₃₂	1	6	yes	direct product factor
49	C₂×C₁₇₃	C₂²×D₈	1	1	yes	characteristic
50	C₁₇₃	C₂²×SD₃₂	1	1	yes	characteristic, direct product factor, Sylow
51	C₂²×SD₃₂	C₁₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂²×Q₁₆	C₂×C₁₇₃	1	1	yes	characteristic, complemented
53	C₂²×D₈	C₂×C₁₇₃	1	1	yes	characteristic
54	C₂×SD₃₂	C₂×C₁₇₃	1	12	yes	direct product factor
55	C₂²×C₁₆	C₂×C₁₇₃	1	1	yes	characteristic, complemented
56	C₂²×Q₈	—	2	1	no	—
57	C₂²×D₄	—	2	1	no	—
58	C₂×Q₁₆	—	2	3	no	—
59	C₂×Q₁₆	C₂²×C₁₇₃	1	6	yes	complemented
60	C₂×D₈	—	2	3	no	—
61	C₂×D₈	C₂²×C₁₇₃	1	6	yes	—
62	C₂²×C₈	C₂²×C₁₇₃	1	1	yes	characteristic
63	SD₃₂	C₂²×C₁₇₃	1	16	yes	direct product factor
64	C₂×C₁₆	C₂²×C₁₇₃	1	6	yes	complemented
65	C₂⁴	—	4	1	no	—
66	C₂×Q₈	—	2	6	no	—
67	C₂×Q₈	—	4	3	no	—
68	C₂×D₄	—	2	6	no	—
69	C₂×D₄	—	4	3	no	—
70	C₂²×C₄	—	4	1	no	—
71	C₂²×C₄	D₄×C₁₇₃	1	1	yes	characteristic
72	Q₁₆	—	2	6	no	—
73	Q₁₆	C₂³×C₁₇₃	1	4	yes	complemented
74	D₈	—	2	6	no	—
75	D₈	C₂³×C₁₇₃	1	4	yes	—
76	C₂×C₈	D₄×C₁₇₃	1	3	yes	—
77	C₂×C₈	C₂³×C₁₇₃	1	3	yes	—
78	C₁₆	C₂³×C₁₇₃	1	4	yes	complemented
79	C₂³	—	4	6	no	—
80	C₂³	—	8	4	no	—
81	C₂³	D₈×C₁₇₃	1	1	yes	characteristic
82	Q₈	—	2	4	no	—
83	Q₈	—	4	6	no	—
84	D₄	—	2	4	no	—
85	D₄	—	4	6	no	—
86	C₂×C₄	—	4	6	no	—
87	C₂×C₄	D₈×C₁₇₃	1	3	yes	—
88	C₂×C₄	C₂×D₄×C₁₇₃	1	3	yes	—
89	C₈	C₂×D₄×C₁₇₃	1	3	yes	—
90	C₈	C₂⁴×C₁₇₃	1	1	yes	characteristic
91	C₂²	—	4	4	no	—
92	C₂²	—	8	12	no	—
93	C₂²	SD₃₂×C₁₇₃	1	4	yes	direct product factor
94	C₂²	C₂×D₈×C₁₇₃	1	3	yes	—
95	C₄	—	4	4	no	—
96	C₄	C₂×D₈×C₁₇₃	1	3	yes	—
97	C₄	C₂²×D₄×C₁₇₃	1	1	yes	characteristic
98	C₂	—	8	4	no	—
99	C₂	C₂×SD₃₂×C₁₇₃	1	6	yes	direct product factor
100	C₂	C₂²×D₈×C₁₇₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₁₆×C₁₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₈×C₁₇₃

C₂

yes

maximal, characteristic

C₂×SD₃₂×C₁₇₃

C₂

yes

maximal, direct product factor

C₂²×C₁₆×C₁₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₇₃

—

C₂²×D₄×C₁₇₃

—

C₂×Q₁₆×C₁₇₃

—

C₂×Q₁₆×C₁₇₃

C₂²

yes

complemented

C₂×D₈×C₁₇₃

—

C₂×D₈×C₁₇₃

C₂²

yes

—

C₂²×C₈×C₁₇₃

C₂²

yes

characteristic

SD₃₂×C₁₇₃

C₂²

yes

direct product factor

C₂×C₁₆×C₁₇₃

C₂²

yes

complemented

C₂⁴×C₁₇₃

—

C₂×Q₈×C₁₇₃

—

C₂×Q₈×C₁₇₃

—

C₂×D₄×C₁₇₃

—

C₂×D₄×C₁₇₃

—

C₂²×C₄×C₁₇₃

—

C₂²×C₄×C₁₇₃

D₄

yes

characteristic

Q₁₆×C₁₇₃

—

Q₁₆×C₁₇₃

C₂³

yes

complemented

D₈×C₁₇₃

—

D₈×C₁₇₃

C₂³

yes

—

C₂×C₈×C₁₇₃

D₄

yes

—

C₂×C₈×C₁₇₃

C₂³

yes

—

C₁₆×C₁₇₃

C₂³

yes

complemented

C₂³×C₁₇₃

—

C₂³×C₁₇₃

—

C₂³×C₁₇₃

D₈

yes

characteristic

Q₈×C₁₇₃

—

Q₈×C₁₇₃

—

D₄×C₁₇₃

—

D₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

D₈

yes

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₈×C₁₇₃

C₂×D₄

yes

—

C₈×C₁₇₃

C₂⁴

yes

characteristic

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

SD₃₂

yes

direct product factor

C₂²×C₁₇₃

C₂×D₈

yes

—

C₄×C₁₇₃

—

C₄×C₁₇₃

C₂×D₈

yes

—

C₄×C₁₇₃

C₂²×D₄

yes

characteristic

C₂×C₁₇₃

—

C₂×C₁₇₃

C₂×SD₃₂

yes

direct product factor

C₂×C₁₇₃

C₂²×D₈

yes

characteristic

C₁₇₃

C₂²×SD₃₂

yes

characteristic, direct product factor, Sylow

C₂²×SD₃₂

C₁₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×Q₁₆

C₂×C₁₇₃

yes

characteristic, complemented

C₂²×D₈

C₂×C₁₇₃

yes

characteristic

C₂×SD₃₂

C₂×C₁₇₃

yes

direct product factor

C₂²×C₁₆

C₂×C₁₇₃

yes

characteristic, complemented

C₂²×Q₈

—

C₂²×D₄

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂²×C₁₇₃

yes

complemented

C₂×D₈

—

C₂×D₈

C₂²×C₁₇₃

yes

—

C₂²×C₈

C₂²×C₁₇₃

yes

characteristic

SD₃₂

C₂²×C₁₇₃

yes

direct product factor

C₂×C₁₆

C₂²×C₁₇₃

yes

complemented

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₁₇₃

yes

characteristic

Q₁₆

—

Q₁₆

C₂³×C₁₇₃

yes

complemented

D₈

—

D₈

C₂³×C₁₇₃

yes

—

C₂×C₈

D₄×C₁₇₃

yes

—

C₂×C₈

C₂³×C₁₇₃

yes

—

C₁₆

C₂³×C₁₇₃

yes

complemented

C₂³

—

C₂³

—

C₂³

D₈×C₁₇₃

yes

characteristic

Q₈

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

D₈×C₁₇₃

yes

—

C₂×C₄

C₂×D₄×C₁₇₃

yes

—

C₈

C₂×D₄×C₁₇₃

yes

—

C₈

C₂⁴×C₁₇₃

yes

characteristic

C₂²

—

C₂²

—

C₂²

SD₃₂×C₁₇₃

yes

direct product factor

C₂²

C₂×D₈×C₁₇₃

yes

—

C₄

—

C₄

C₂×D₈×C₁₇₃

yes

—

C₄

C₂²×D₄×C₁₇₃

yes

characteristic

C₂

—

C₂

C₂×SD₃₂×C₁₇₃

yes

direct product factor

100

C₂

C₂²×D₈×C₁₇₃

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₁₇₃	D₈
commutator, Frattini, cosocle	C₈	C₂⁴×C₁₇₃
prime radical	C₂²×D₈×C₁₇₃	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇₃ ◁ C₂²×C₄×C₁₇₃ ◁ C₂²×C₈×C₁₇₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₇₃

D₈

commutator, Frattini, cosocle

C₈

C₂⁴×C₁₇₃

prime radical

C₂²×D₈×C₁₇₃

C₂

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₇₃ ◁ C₂²×C₄×C₁₇₃ ◁ C₂²×C₈×C₁₇₃ ◁ G

N	H ≅ G/N	type
1. C₂²×Q₁₆×C₁₇₃	C₂	semidirect
3. C₂×SD₃₂×C₁₇₃	C₂	direct (N × H)
4. C₂²×C₁₆×C₁₇₃	C₂	semidirect
8. C₂×Q₁₆×C₁₇₃	C₂²	semidirect
12. SD₃₂×C₁₇₃	C₂²	direct (N × H)
13. C₂×C₁₆×C₁₇₃	C₂²	semidirect
22. Q₁₆×C₁₇₃	C₂³	semidirect
27. C₁₆×C₁₇₃	C₂³	semidirect
42. C₂²×C₁₇₃	SD₃₂	direct (N × H)
48. C₂×C₁₇₃	C₂×SD₃₂	direct (N × H)
50. C₁₇₃	C₂²×SD₃₂	direct (N × H)
51. C₂²×SD₃₂	C₁₇₃	direct (N × H)
52. C₂²×Q₁₆	C₂×C₁₇₃	semidirect
54. C₂×SD₃₂	C₂×C₁₇₃	direct (N × H)
55. C₂²×C₁₆	C₂×C₁₇₃	semidirect
59. C₂×Q₁₆	C₂²×C₁₇₃	semidirect
63. SD₃₂	C₂²×C₁₇₃	direct (N × H)
64. C₂×C₁₆	C₂²×C₁₇₃	semidirect
73. Q₁₆	C₂³×C₁₇₃	semidirect
78. C₁₆	C₂³×C₁₇₃	semidirect
93. C₂²	SD₃₂×C₁₇₃	direct (N × H)
99. C₂	C₂×SD₃₂×C₁₇₃	direct (N × H)

H ≅ G/N

type

1. C₂²×Q₁₆×C₁₇₃

C₂

semidirect

3. C₂×SD₃₂×C₁₇₃

C₂

direct (N × H)

4. C₂²×C₁₆×C₁₇₃

C₂

semidirect

8. C₂×Q₁₆×C₁₇₃

C₂²

semidirect

12. SD₃₂×C₁₇₃

C₂²

direct (N × H)

13. C₂×C₁₆×C₁₇₃

C₂²

semidirect

22. Q₁₆×C₁₇₃

C₂³

semidirect

27. C₁₆×C₁₇₃

C₂³

semidirect

42. C₂²×C₁₇₃

SD₃₂

direct (N × H)

48. C₂×C₁₇₃

C₂×SD₃₂

direct (N × H)

50. C₁₇₃

C₂²×SD₃₂

direct (N × H)

51. C₂²×SD₃₂

C₁₇₃

direct (N × H)

52. C₂²×Q₁₆

C₂×C₁₇₃

semidirect

54. C₂×SD₃₂

C₂×C₁₇₃

direct (N × H)

55. C₂²×C₁₆

C₂×C₁₇₃

semidirect

59. C₂×Q₁₆

C₂²×C₁₇₃

semidirect

63. SD₃₂

C₂²×C₁₇₃

direct (N × H)

64. C₂×C₁₆

C₂²×C₁₇₃

semidirect

73. Q₁₆

C₂³×C₁₇₃

semidirect

78. C₁₆

C₂³×C₁₇₃

semidirect

93. C₂²

SD₃₂×C₁₇₃

direct (N × H)

99. C₂

C₂×SD₃₂×C₁₇₃

direct (N × H)

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

subgroup	index
C₂²×Q₁₆×C₁₇₃	2
C₂²×D₈×C₁₇₃	2
C₂×SD₃₂×C₁₇₃	2
C₂²×C₁₆×C₁₇₃	2
C₂²×SD₃₂	173

subgroup

index

C₂²×Q₁₆×C₁₇₃

C₂²×D₈×C₁₇₃

C₂×SD₃₂×C₁₇₃

C₂²×C₁₆×C₁₇₃

C₂²×SD₃₂

173

Subgroups — C₂²×SD₃₂×C₁₇₃ #36351194

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×SD32×C173 #36351194

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×SD₃₂×C₁₇₃ #36351194