Group #37591724 — subgroups

Subgroups — C₄×M₃₂×C₁₉₃ #37591724

Order 24704.

196 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 176 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×M₃₂×C₁₉₃	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₁₉₃	C₂	1	1	yes	maximal, characteristic
3	#346×C₁₉₃	C₂	1	2	yes	maximal, complemented
4	C₄×C₁₆×C₁₉₃	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×C₁₉₃	C₄	1	1	yes	characteristic
6	C₂²×C₈×C₁₉₃	C₂²	1	1	yes	characteristic
7	C₂×C₄²×C₁₉₃	C₄	1	1	yes	characteristic
8	M₃₂×C₁₉₃	C₄	1	8	yes	direct product factor
9	C₂×C₁₆×C₁₉₃	C₂²	1	4	yes	—
10	C₄×C₈×C₁₉₃	C₄	1	1	yes	characteristic
11	C₄×C₈×C₁₉₃	C₄	1	1	yes	characteristic
12	C₄×C₈×C₁₉₃	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₁₉₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₁₉₃	C₈	1	2	yes	—
15	C₂²×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₁₉₃	C₈	1	4	yes	—
17	C₂×C₈×C₁₉₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₁₉₃	C₂×C₄	1	4	yes	—
19	C₂×C₈×C₁₉₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₁₉₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₉₃	C₂³	1	1	yes	characteristic
22	C₄²×C₁₉₃	C₈	1	2	yes	—
23	C₄²×C₁₉₃	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₉₃	C₂×C₄	1	1	yes	characteristic
25	C₁₆×C₁₉₃	C₂×C₄	1	8	yes	complemented
26	C₂³×C₁₉₃	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₉₃	—	2	4	no	—
28	C₂×C₄×C₁₉₃	C₄²	1	2	yes	—
29	C₂×C₄×C₁₉₃	C₂×C₈	1	2	yes	—
30	C₂×C₄×C₁₉₃	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₁₉₃	C₂×C₈	1	2	yes	—
32	C₂×C₄×C₁₉₃	C₂×C₈	1	2	yes	—
33	C₂×C₄×C₁₉₃	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₁₉₃	C₄²	1	2	yes	—
35	C₈×C₁₉₃	C₂×C₈	1	2	yes	—
36	C₈×C₁₉₃	C₂×C₈	1	2	yes	—
37	C₈×C₁₉₃	C₂²×C₄	1	2	yes	—
38	C₂²×C₁₉₃	—	2	2	no	—
39	C₂²×C₁₉₃	C₄×C₈	1	2	yes	—
40	C₂²×C₁₉₃	C₂²×C₈	1	1	yes	characteristic
41	C₄×C₁₉₃	—	2	2	no	—
42	C₄×C₁₉₃	C₄×C₈	1	2	yes	—
43	C₄×C₁₉₃	M₃₂	1	4	yes	direct product factor
44	C₄×C₁₉₃	C₂×C₄²	1	1	yes	characteristic
45	C₄×C₁₉₃	C₂²×C₈	1	1	yes	characteristic
46	C₂×C₁₉₃	—	2	2	no	—
47	C₂×C₁₉₃	C₂×C₄×C₈	1	1	yes	characteristic
48	C₂×C₁₉₃	C₂×M₃₂	1	2	yes	—
49	C₁₉₃	C₄×M₃₂	1	1	yes	characteristic, direct product factor, Sylow
50	C₄×M₃₂	C₁₉₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂×M₃₂	C₂×C₁₉₃	1	2	yes	—
52	C₂×C₄×C₈	C₂×C₁₉₃	1	1	yes	characteristic
53	#346	C₂×C₁₉₃	1	2	yes	complemented
54	C₄×C₁₆	C₂×C₁₉₃	1	2	yes	complemented
55	C₂²×C₈	C₄×C₁₉₃	1	1	yes	characteristic
56	C₂²×C₈	C₂²×C₁₉₃	1	1	yes	characteristic
57	C₂×C₄²	C₄×C₁₉₃	1	1	yes	characteristic
58	M₃₂	C₄×C₁₉₃	1	8	yes	direct product factor
59	C₂×C₁₆	C₂²×C₁₉₃	1	4	yes	—
60	C₄×C₈	C₄×C₁₉₃	1	1	yes	characteristic
61	C₄×C₈	C₄×C₁₉₃	1	1	yes	characteristic
62	C₄×C₈	C₂²×C₁₉₃	1	1	yes	characteristic
63	C₄×C₈	C₂²×C₁₉₃	1	1	yes	characteristic
64	C₂²×C₄	C₈×C₁₉₃	1	2	yes	—
65	C₂²×C₄	C₂×C₄×C₁₉₃	1	1	yes	characteristic
66	C₂×C₈	C₈×C₁₉₃	1	4	yes	—
67	C₂×C₈	C₂×C₄×C₁₉₃	1	1	yes	characteristic
68	C₂×C₈	C₂×C₄×C₁₉₃	1	4	yes	—
69	C₂×C₈	C₂×C₄×C₁₉₃	1	1	yes	characteristic
70	C₂×C₈	C₂×C₄×C₁₉₃	1	1	yes	characteristic
71	C₂×C₈	C₂³×C₁₉₃	1	1	yes	characteristic
72	C₄²	C₈×C₁₉₃	1	2	yes	—
73	C₄²	C₂×C₄×C₁₉₃	1	1	yes	characteristic
74	C₄²	C₂×C₄×C₁₉₃	1	1	yes	characteristic
75	C₁₆	C₂×C₄×C₁₉₃	1	8	yes	complemented
76	C₂³	C₂×C₈×C₁₉₃	1	1	yes	characteristic
77	C₂×C₄	—	2	4	no	—
78	C₂×C₄	C₄²×C₁₉₃	1	2	yes	—
79	C₂×C₄	C₂×C₈×C₁₉₃	1	2	yes	—
80	C₂×C₄	C₂×C₈×C₁₉₃	1	1	yes	characteristic
81	C₂×C₄	C₂×C₈×C₁₉₃	1	2	yes	—
82	C₂×C₄	C₂×C₈×C₁₉₃	1	2	yes	—
83	C₂×C₄	C₂²×C₄×C₁₉₃	1	1	yes	characteristic
84	C₈	C₄²×C₁₉₃	1	2	yes	—
85	C₈	C₂×C₈×C₁₉₃	1	2	yes	—
86	C₈	C₂×C₈×C₁₉₃	1	2	yes	—
87	C₈	C₂²×C₄×C₁₉₃	1	2	yes	—
88	C₂²	—	2	2	no	—
89	C₂²	C₄×C₈×C₁₉₃	1	2	yes	—
90	C₂²	C₂²×C₈×C₁₉₃	1	1	yes	characteristic
91	C₄	—	2	2	no	—
92	C₄	C₄×C₈×C₁₉₃	1	2	yes	—
93	C₄	M₃₂×C₁₉₃	1	4	yes	direct product factor
94	C₄	C₂×C₄²×C₁₉₃	1	1	yes	characteristic
95	C₄	C₂²×C₈×C₁₉₃	1	1	yes	characteristic
96	C₂	—	2	2	no	—
97	C₂	C₂×C₄×C₈×C₁₉₃	1	1	yes	characteristic
98	C₂	C₂×M₃₂×C₁₉₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×M₃₂×C₁₉₃

C₂

yes

maximal

C₂×C₄×C₈×C₁₉₃

C₂

yes

maximal, characteristic

#346×C₁₉₃

C₂

yes

maximal, complemented

C₄×C₁₆×C₁₉₃

C₂

yes

maximal, complemented

C₂²×C₈×C₁₉₃

C₄

yes

characteristic

C₂²×C₈×C₁₉₃

C₂²

yes

characteristic

C₂×C₄²×C₁₉₃

C₄

yes

characteristic

M₃₂×C₁₉₃

C₄

yes

direct product factor

C₂×C₁₆×C₁₉₃

C₂²

yes

—

C₄×C₈×C₁₉₃

C₄

yes

characteristic

C₄×C₈×C₁₉₃

C₄

yes

characteristic

C₄×C₈×C₁₉₃

C₂²

yes

characteristic

C₄×C₈×C₁₉₃

C₂²

yes

characteristic

C₂²×C₄×C₁₉₃

C₈

yes

—

C₂²×C₄×C₁₉₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₃

C₈

yes

—

C₂×C₈×C₁₉₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₃

C₂×C₄

yes

—

C₂×C₈×C₁₉₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₃

C₂³

yes

characteristic

C₄²×C₁₉₃

C₈

yes

—

C₄²×C₁₉₃

C₂×C₄

yes

characteristic

C₄²×C₁₉₃

C₂×C₄

yes

characteristic

C₁₆×C₁₉₃

C₂×C₄

yes

complemented

C₂³×C₁₉₃

C₂×C₈

yes

characteristic

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

C₄²

yes

—

C₂×C₄×C₁₉₃

C₂×C₈

yes

—

C₂×C₄×C₁₉₃

C₂×C₈

yes

characteristic

C₂×C₄×C₁₉₃

C₂×C₈

yes

—

C₂×C₄×C₁₉₃

C₂×C₈

yes

—

C₂×C₄×C₁₉₃

C₂²×C₄

yes

characteristic

C₈×C₁₉₃

C₄²

yes

—

C₈×C₁₉₃

C₂×C₈

yes

—

C₈×C₁₉₃

C₂×C₈

yes

—

C₈×C₁₉₃

C₂²×C₄

yes

—

C₂²×C₁₉₃

—

C₂²×C₁₉₃

C₄×C₈

yes

—

C₂²×C₁₉₃

C₂²×C₈

yes

characteristic

C₄×C₁₉₃

—

C₄×C₁₉₃

C₄×C₈

yes

—

C₄×C₁₉₃

M₃₂

yes

direct product factor

C₄×C₁₉₃

C₂×C₄²

yes

characteristic

C₄×C₁₉₃

C₂²×C₈

yes

characteristic

C₂×C₁₉₃

—

C₂×C₁₉₃

C₂×C₄×C₈

yes

characteristic

C₂×C₁₉₃

C₂×M₃₂

yes

—

C₁₉₃

C₄×M₃₂

yes

characteristic, direct product factor, Sylow

C₄×M₃₂

C₁₉₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×M₃₂

C₂×C₁₉₃

yes

—

C₂×C₄×C₈

C₂×C₁₉₃

yes

characteristic

#346

C₂×C₁₉₃

yes

complemented

C₄×C₁₆

C₂×C₁₉₃

yes

complemented

C₂²×C₈

C₄×C₁₉₃

yes

characteristic

C₂²×C₈

C₂²×C₁₉₃

yes

characteristic

C₂×C₄²

C₄×C₁₉₃

yes

characteristic

M₃₂

C₄×C₁₉₃

yes

direct product factor

C₂×C₁₆

C₂²×C₁₉₃

yes

—

C₄×C₈

C₄×C₁₉₃

yes

characteristic

C₄×C₈

C₄×C₁₉₃

yes

characteristic

C₄×C₈

C₂²×C₁₉₃

yes

characteristic

C₄×C₈

C₂²×C₁₉₃

yes

characteristic

C₂²×C₄

C₈×C₁₉₃

yes

—

C₂²×C₄

C₂×C₄×C₁₉₃

yes

characteristic

C₂×C₈

C₈×C₁₉₃

yes

—

C₂×C₈

C₂×C₄×C₁₉₃

yes

characteristic

C₂×C₈

C₂×C₄×C₁₉₃

yes

—

C₂×C₈

C₂×C₄×C₁₉₃

yes

characteristic

C₂×C₈

C₂×C₄×C₁₉₃

yes

characteristic

C₂×C₈

C₂³×C₁₉₃

yes

characteristic

C₄²

C₈×C₁₉₃

yes

—

C₄²

C₂×C₄×C₁₉₃

yes

characteristic

C₄²

C₂×C₄×C₁₉₃

yes

characteristic

C₁₆

C₂×C₄×C₁₉₃

yes

complemented

C₂³

C₂×C₈×C₁₉₃

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄²×C₁₉₃

yes

—

C₂×C₄

C₂×C₈×C₁₉₃

yes

—

C₂×C₄

C₂×C₈×C₁₉₃

yes

characteristic

C₂×C₄

C₂×C₈×C₁₉₃

yes

—

C₂×C₄

C₂×C₈×C₁₉₃

yes

—

C₂×C₄

C₂²×C₄×C₁₉₃

yes

characteristic

C₈

C₄²×C₁₉₃

yes

—

C₈

C₂×C₈×C₁₉₃

yes

—

C₈

C₂×C₈×C₁₉₃

yes

—

C₈

C₂²×C₄×C₁₉₃

yes

—

C₂²

—

C₂²

C₄×C₈×C₁₉₃

yes

—

C₂²

C₂²×C₈×C₁₉₃

yes

characteristic

C₄

—

C₄

C₄×C₈×C₁₉₃

yes

—

C₄

M₃₂×C₁₉₃

yes

direct product factor

C₄

C₂×C₄²×C₁₉₃

yes

characteristic

C₄

C₂²×C₈×C₁₉₃

yes

characteristic

C₂

—

C₂

C₂×C₄×C₈×C₁₉₃

yes

characteristic

C₂

C₂×M₃₂×C₁₉₃

yes

—

	subgroup	G/H
center	C₄×C₈×C₁₉₃	C₂²
commutator	C₂	C₂×C₄×C₈×C₁₉₃
Frattini, cosocle	C₂×C₈	C₂³×C₁₉₃
socle	C₂²×C₁₉₃	C₂²×C₈
prime radical	C₂³×C₁₉₃	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₁₉₃ ◁ G

subgroup

G/H

center

C₄×C₈×C₁₉₃

C₂²

commutator

C₂

C₂×C₄×C₈×C₁₉₃

Frattini, cosocle

C₂×C₈

C₂³×C₁₉₃

socle

C₂²×C₁₉₃

C₂²×C₈

prime radical

C₂³×C₁₉₃

C₂×C₈

LCS

G ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₈×C₁₉₃ ◁ G

N	H ≅ G/N	type
3. #346×C₁₉₃	C₂	semidirect
4. C₄×C₁₆×C₁₉₃	C₂	semidirect
8. M₃₂×C₁₉₃	C₄	direct (N × H)
25. C₁₆×C₁₉₃	C₂×C₄	semidirect
43. C₄×C₁₉₃	M₃₂	direct (N × H)
49. C₁₉₃	C₄×M₃₂	direct (N × H)
50. C₄×M₃₂	C₁₉₃	direct (N × H)
53. #346	C₂×C₁₉₃	semidirect
54. C₄×C₁₆	C₂×C₁₉₃	semidirect
58. M₃₂	C₄×C₁₉₃	direct (N × H)
75. C₁₆	C₂×C₄×C₁₉₃	semidirect
93. C₄	M₃₂×C₁₉₃	direct (N × H)

H ≅ G/N

type

3. #346×C₁₉₃

C₂

semidirect

4. C₄×C₁₆×C₁₉₃

C₂

semidirect

8. M₃₂×C₁₉₃

C₄

direct (N × H)

25. C₁₆×C₁₉₃

C₂×C₄

semidirect

43. C₄×C₁₉₃

M₃₂

direct (N × H)

49. C₁₉₃

C₄×M₃₂

direct (N × H)

50. C₄×M₃₂

C₁₉₃

direct (N × H)

53. #346

C₂×C₁₉₃

semidirect

54. C₄×C₁₆

C₂×C₁₉₃

semidirect

58. M₃₂

C₄×C₁₉₃

direct (N × H)

75. C₁₆

C₂×C₄×C₁₉₃

semidirect

93. C₄

M₃₂×C₁₉₃

direct (N × H)

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

subgroup	index
C₂×M₃₂×C₁₉₃	2
C₂×C₄×C₈×C₁₉₃	2
#346×C₁₉₃	2
C₄×C₁₆×C₁₉₃	2
C₄×M₃₂	193

subgroup

index

C₂×M₃₂×C₁₉₃

C₂×C₄×C₈×C₁₉₃

#346×C₁₉₃

C₄×C₁₆×C₁₉₃

C₄×M₃₂

193

Subgroups — C₄×M₃₂×C₁₉₃ #37591724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×M32×C193 #37591724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×M₃₂×C₁₉₃ #37591724