Group #37591769 — subgroups

Subgroups — C₂×#364×C₁₉₃ #37591769

Order 24704.

168 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 120 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	#364×C₁₉₃	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₈×C₁₉₃	C₄	1	1	yes	characteristic
5	C₂²×C₈×C₁₉₃	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₁₉₃	C₄	1	1	yes	characteristic
7	M₃₂×C₁₉₃	—	2	2	no	—
8	M₃₂×C₁₉₃	C₂²	1	4	yes	—
9	C₂×C₁₆×C₁₉₃	—	2	2	no	—
10	C₄×C₈×C₁₉₃	C₄	1	2	yes	—
11	C₄×C₈×C₁₉₃	C₂²	1	2	yes	—
12	C₂²×C₄×C₁₉₃	—	2	1	no	—
13	C₂²×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₁₉₃	C₈	1	2	yes	—
15	C₂×C₈×C₁₉₃	C₈	1	2	yes	—
16	C₂×C₈×C₁₉₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₁₉₃	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₁₉₃	D₄	1	1	yes	characteristic
19	C₂×C₈×C₁₉₃	D₄	1	1	yes	characteristic
20	C₂×C₈×C₁₉₃	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₁₉₃	Q₈	1	1	yes	characteristic
22	C₂×C₈×C₁₉₃	C₂³	1	1	yes	characteristic
23	C₄²×C₁₉₃	—	2	1	no	—
24	C₄²×C₁₉₃	C₂×C₄	1	2	yes	—
25	C₁₆×C₁₉₃	—	2	4	no	—
26	C₂³×C₁₉₃	M₁₆	1	1	yes	characteristic
27	C₂×C₄×C₁₉₃	—	2	4	no	—
28	C₂×C₄×C₁₉₃	—	2	2	no	—
29	C₂×C₄×C₁₉₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₉₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₉₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₁₉₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₁₉₃	M₁₆	1	1	yes	characteristic
34	C₂×C₄×C₁₉₃	C₂²×C₄	1	1	yes	characteristic
35	C₈×C₁₉₃	C₄⋊C₄	1	1	yes	characteristic
36	C₈×C₁₉₃	C₄⋊C₄	1	1	yes	characteristic
37	C₈×C₁₉₃	C₂×C₈	1	4	yes	—
38	C₈×C₁₉₃	C₂×D₄	1	1	yes	characteristic
39	C₈×C₁₉₃	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₁₉₃	—	2	2	no	—
41	C₂²×C₁₉₃	C₄⋊C₈	1	1	yes	characteristic
42	C₂²×C₁₉₃	C₄⋊C₈	1	1	yes	characteristic
43	C₂²×C₁₉₃	C₂×M₁₆	1	1	yes	characteristic
44	C₄×C₁₉₃	—	2	4	no	—
45	C₄×C₁₉₃	C₄⋊C₈	1	1	yes	characteristic
46	C₄×C₁₉₃	C₄⋊C₈	1	1	yes	characteristic
47	C₄×C₁₉₃	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₄×C₁₉₃	C₂²×C₈	1	1	yes	characteristic
49	C₂×C₁₉₃	—	2	1	no	—
50	C₂×C₁₉₃	—	2	1	no	—
51	C₂×C₁₉₃	#364	1	2	yes	direct product factor
52	C₂×C₁₉₃	C₂×C₄⋊C₈	1	1	yes	characteristic
53	C₁₉₃	C₂×#364	1	1	yes	characteristic, direct product factor, Sylow
54	C₂×#364	C₁₉₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂×M₃₂	C₂×C₁₉₃	1	2	yes	—
56	C₂×C₄×C₈	C₂×C₁₉₃	1	1	yes	characteristic
57	#364	C₂×C₁₉₃	1	4	yes	direct product factor
58	C₂²×C₈	C₄×C₁₉₃	1	1	yes	characteristic
59	C₂²×C₈	C₂²×C₁₉₃	1	1	yes	characteristic
60	C₂×C₄²	C₄×C₁₉₃	1	1	yes	characteristic
61	M₃₂	—	2	2	no	—
62	M₃₂	C₂²×C₁₉₃	1	4	yes	—
63	C₂×C₁₆	—	2	2	no	—
64	C₄×C₈	C₄×C₁₉₃	1	2	yes	—
65	C₄×C₈	C₂²×C₁₉₃	1	2	yes	—
66	C₂²×C₄	—	2	1	no	—
67	C₂²×C₄	C₂×C₄×C₁₉₃	1	1	yes	characteristic
68	C₂×C₈	C₈×C₁₉₃	1	2	yes	—
69	C₂×C₈	C₈×C₁₉₃	1	2	yes	—
70	C₂×C₈	C₂×C₄×C₁₉₃	1	1	yes	characteristic
71	C₂×C₈	C₂×C₄×C₁₉₃	1	2	yes	—
72	C₂×C₈	D₄×C₁₉₃	1	1	yes	characteristic
73	C₂×C₈	D₄×C₁₉₃	1	1	yes	characteristic
74	C₂×C₈	Q₈×C₁₉₃	1	1	yes	characteristic
75	C₂×C₈	Q₈×C₁₉₃	1	1	yes	characteristic
76	C₂×C₈	C₂³×C₁₉₃	1	1	yes	characteristic
77	C₄²	—	2	1	no	—
78	C₄²	C₂×C₄×C₁₉₃	1	2	yes	—
79	C₁₆	—	2	4	no	—
80	C₂³	M₁₆×C₁₉₃	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	C₄⋊C₄×C₁₉₃	1	1	yes	characteristic
84	C₂×C₄	C₄⋊C₄×C₁₉₃	1	1	yes	characteristic
85	C₂×C₄	C₂×C₈×C₁₉₃	1	1	yes	characteristic
86	C₂×C₄	C₂×C₈×C₁₉₃	1	1	yes	characteristic
87	C₂×C₄	M₁₆×C₁₉₃	1	1	yes	characteristic
88	C₂×C₄	C₂²×C₄×C₁₉₃	1	1	yes	characteristic
89	C₈	C₄⋊C₄×C₁₉₃	1	1	yes	characteristic
90	C₈	C₄⋊C₄×C₁₉₃	1	1	yes	characteristic
91	C₈	C₂×C₈×C₁₉₃	1	4	yes	—
92	C₈	C₂×D₄×C₁₉₃	1	1	yes	characteristic
93	C₈	C₂×Q₈×C₁₉₃	1	1	yes	characteristic
94	C₂²	—	2	2	no	—
95	C₂²	C₄⋊C₈×C₁₉₃	1	1	yes	characteristic
96	C₂²	C₄⋊C₈×C₁₉₃	1	1	yes	characteristic
97	C₂²	C₂×M₁₆×C₁₉₃	1	1	yes	characteristic
98	C₄	—	2	4	no	—
99	C₄	C₄⋊C₈×C₁₉₃	1	1	yes	characteristic
100	C₄	C₄⋊C₈×C₁₉₃	1	1	yes	characteristic
101	C₄	C₂×C₄⋊C₄×C₁₉₃	1	1	yes	characteristic
102	C₄	C₂²×C₈×C₁₉₃	1	1	yes	characteristic
103	C₂	—	2	1	no	—
104	C₂	—	2	1	no	—
105	C₂	#364×C₁₉₃	1	2	yes	direct product factor
106	C₂	C₂×C₄⋊C₈×C₁₉₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×M₃₂×C₁₉₃

C₂

yes

maximal

C₂×C₄×C₈×C₁₉₃

C₂

yes

maximal, characteristic

#364×C₁₉₃

C₂

yes

maximal, direct product factor

C₂²×C₈×C₁₉₃

C₄

yes

characteristic

C₂²×C₈×C₁₉₃

C₂²

yes

characteristic

C₂×C₄²×C₁₉₃

C₄

yes

characteristic

M₃₂×C₁₉₃

—

M₃₂×C₁₉₃

C₂²

yes

—

C₂×C₁₆×C₁₉₃

—

C₄×C₈×C₁₉₃

C₄

yes

—

C₄×C₈×C₁₉₃

C₂²

yes

—

C₂²×C₄×C₁₉₃

—

C₂²×C₄×C₁₉₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₃

C₈

yes

—

C₂×C₈×C₁₉₃

C₈

yes

—

C₂×C₈×C₁₉₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₃

C₂×C₄

yes

—

C₂×C₈×C₁₉₃

D₄

yes

characteristic

C₂×C₈×C₁₉₃

D₄

yes

characteristic

C₂×C₈×C₁₉₃

Q₈

yes

characteristic

C₂×C₈×C₁₉₃

Q₈

yes

characteristic

C₂×C₈×C₁₉₃

C₂³

yes

characteristic

C₄²×C₁₉₃

—

C₄²×C₁₉₃

C₂×C₄

yes

—

C₁₆×C₁₉₃

—

C₂³×C₁₉₃

M₁₆

yes

characteristic

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉₃

C₂×C₈

yes

characteristic

C₂×C₄×C₁₉₃

C₂×C₈

yes

characteristic

C₂×C₄×C₁₉₃

M₁₆

yes

characteristic

C₂×C₄×C₁₉₃

C₂²×C₄

yes

characteristic

C₈×C₁₉₃

C₄⋊C₄

yes

characteristic

C₈×C₁₉₃

C₄⋊C₄

yes

characteristic

C₈×C₁₉₃

C₂×C₈

yes

—

C₈×C₁₉₃

C₂×D₄

yes

characteristic

C₈×C₁₉₃

C₂×Q₈

yes

characteristic

C₂²×C₁₉₃

—

C₂²×C₁₉₃

C₄⋊C₈

yes

characteristic

C₂²×C₁₉₃

C₄⋊C₈

yes

characteristic

C₂²×C₁₉₃

C₂×M₁₆

yes

characteristic

C₄×C₁₉₃

—

C₄×C₁₉₃

C₄⋊C₈

yes

characteristic

C₄×C₁₉₃

C₄⋊C₈

yes

characteristic

C₄×C₁₉₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₉₃

C₂²×C₈

yes

characteristic

C₂×C₁₉₃

—

C₂×C₁₉₃

—

C₂×C₁₉₃

#364

yes

direct product factor

C₂×C₁₉₃

C₂×C₄⋊C₈

yes

characteristic

C₁₉₃

C₂×#364

yes

characteristic, direct product factor, Sylow

C₂×#364

C₁₉₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×M₃₂

C₂×C₁₉₃

yes

—

C₂×C₄×C₈

C₂×C₁₉₃

yes

characteristic

#364

C₂×C₁₉₃

yes

direct product factor

C₂²×C₈

C₄×C₁₉₃

yes

characteristic

C₂²×C₈

C₂²×C₁₉₃

yes

characteristic

C₂×C₄²

C₄×C₁₉₃

yes

characteristic

M₃₂

—

M₃₂

C₂²×C₁₉₃

yes

—

C₂×C₁₆

—

C₄×C₈

C₄×C₁₉₃

yes

—

C₄×C₈

C₂²×C₁₉₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₁₉₃

yes

characteristic

C₂×C₈

C₈×C₁₉₃

yes

—

C₂×C₈

C₈×C₁₉₃

yes

—

C₂×C₈

C₂×C₄×C₁₉₃

yes

characteristic

C₂×C₈

C₂×C₄×C₁₉₃

yes

—

C₂×C₈

D₄×C₁₉₃

yes

characteristic

C₂×C₈

D₄×C₁₉₃

yes

characteristic

C₂×C₈

Q₈×C₁₉₃

yes

characteristic

C₂×C₈

Q₈×C₁₉₃

yes

characteristic

C₂×C₈

C₂³×C₁₉₃

yes

characteristic

C₄²

—

C₄²

C₂×C₄×C₁₉₃

yes

—

C₁₆

—

C₂³

M₁₆×C₁₉₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊C₄×C₁₉₃

yes

characteristic

C₂×C₄

C₄⋊C₄×C₁₉₃

yes

characteristic

C₂×C₄

C₂×C₈×C₁₉₃

yes

characteristic

C₂×C₄

C₂×C₈×C₁₉₃

yes

characteristic

C₂×C₄

M₁₆×C₁₉₃

yes

characteristic

C₂×C₄

C₂²×C₄×C₁₉₃

yes

characteristic

C₈

C₄⋊C₄×C₁₉₃

yes

characteristic

C₈

C₄⋊C₄×C₁₉₃

yes

characteristic

C₈

C₂×C₈×C₁₉₃

yes

—

C₈

C₂×D₄×C₁₉₃

yes

characteristic

C₈

C₂×Q₈×C₁₉₃

yes

characteristic

C₂²

—

C₂²

C₄⋊C₈×C₁₉₃

yes

characteristic

C₂²

C₄⋊C₈×C₁₉₃

yes

characteristic

C₂²

C₂×M₁₆×C₁₉₃

yes

characteristic

C₄

—

C₄

C₄⋊C₈×C₁₉₃

yes

characteristic

100

C₄

C₄⋊C₈×C₁₉₃

yes

characteristic

101

C₄

C₂×C₄⋊C₄×C₁₉₃

yes

characteristic

102

C₄

C₂²×C₈×C₁₉₃

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

#364×C₁₉₃

yes

direct product factor

106

C₂

C₂×C₄⋊C₈×C₁₉₃

yes

characteristic

	subgroup	G/H
center	C₂×C₈×C₁₉₃	D₄
commutator	C₄	C₂²×C₈×C₁₉₃
Frattini, cosocle	C₂×C₈	C₂³×C₁₉₃
socle	C₂²×C₁₉₃	C₄⋊C₈
prime radical	C₂³×C₁₉₃	M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₉₃ ◁ C₂²×C₈×C₁₉₃ ◁ G

subgroup

G/H

center

C₂×C₈×C₁₉₃

D₄

commutator

C₄

C₂²×C₈×C₁₉₃

Frattini, cosocle

C₂×C₈

C₂³×C₁₉₃

socle

C₂²×C₁₉₃

C₄⋊C₈

prime radical

C₂³×C₁₉₃

M₁₆

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈×C₁₉₃ ◁ C₂²×C₈×C₁₉₃ ◁ G

N	H ≅ G/N	type
3. #364×C₁₉₃	C₂	direct (N × H)
51. C₂×C₁₉₃	#364	direct (N × H)
53. C₁₉₃	C₂×#364	direct (N × H)
54. C₂×#364	C₁₉₃	direct (N × H)
57. #364	C₂×C₁₉₃	direct (N × H)
105. C₂	#364×C₁₉₃	direct (N × H)

H ≅ G/N

type

3. #364×C₁₉₃

C₂

direct (N × H)

51. C₂×C₁₉₃

#364

direct (N × H)

53. C₁₉₃

C₂×#364

direct (N × H)

54. C₂×#364

C₁₉₃

direct (N × H)

57. #364

C₂×C₁₉₃

direct (N × H)

105. C₂

#364×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
#364×C₁₉₃	2
C₂×#364	193

subgroup

index

C₂×M₃₂×C₁₉₃

C₂×C₄×C₈×C₁₉₃

#364×C₁₉₃

C₂×#364

193

Subgroups — C₂×#364×C₁₉₃ #37591769

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#364×C193 #37591769

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#364×C₁₉₃ #37591769