Group #19154764 — subgroups

Subgroups — C₂₃×#7138 #19154764

Order 5888.

278 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 166 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	C₄×C₂₃×M₃₂	C₂	1	2	yes	maximal
2	C₂×C₈²×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×M₃₂	C₄	1	4	yes	—
4	C₂×C₄×C₈×C₂₃	C₄	1	2	yes	—
5	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
6	C₂₃×#346	—	2	2	no	—
7	C₄×C₁₆×C₂₃	—	2	2	no	—
8	C₈²×C₂₃	—	2	2	no	—
9	C₂²×C₈×C₂₃	—	2	2	no	—
10	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
11	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂₃×M₃₂	—	2	4	no	—
13	C₂₃×M₃₂	C₈	1	8	yes	complemented
14	C₂×C₁₆×C₂₃	—	2	4	no	—
15	C₄×C₈×C₂₃	—	2	2	no	—
16	C₄×C₈×C₂₃	C₈	1	4	yes	—
17	C₄×C₈×C₂₃	D₄	1	2	yes	—
18	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
19	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
20	C₂²×C₄×C₂₃	C₄²	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	M₁₆	1	2	yes	—
22	C₂×C₈×C₂₃	—	2	8	no	—
23	C₂×C₈×C₂₃	—	2	4	no	—
24	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
25	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
26	C₂×C₈×C₂₃	C₂×C₈	1	4	yes	—
27	C₂×C₈×C₂₃	M₁₆	1	4	yes	—
28	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
30	C₄²×C₂₃	C₂×C₈	1	2	yes	—
31	C₁₆×C₂₃	—	2	8	no	—
32	#7138	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂³×C₂₃	#97	1	1	yes	characteristic
34	C₂×C₄×C₂₃	—	2	4	no	—
35	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
36	C₂×C₄×C₂₃	C₄×C₈	1	1	yes	characteristic
37	C₂×C₄×C₂₃	C₄×C₈	1	2	yes	—
38	C₂×C₄×C₂₃	#97	1	2	yes	—
39	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
40	C₂×C₄×C₂₃	C₄⋊C₈	1	2	yes	—
41	C₈×C₂₃	—	2	8	no	—
42	C₈×C₂₃	C₂²⋊C₈	1	4	yes	—
43	C₈×C₂₃	C₄⋊C₈	1	4	yes	—
44	C₄×M₃₂	C₂×C₂₃	1	2	yes	—
45	C₂×C₈²	C₂×C₂₃	1	1	yes	characteristic
46	C₂²×C₂₃	—	2	2	no	—
47	C₂²×C₂₃	#322	1	2	yes	—
48	C₂²×C₂₃	#336	1	1	yes	characteristic
49	C₄×C₂₃	—	2	2	no	—
50	C₄×C₂₃	C₈²	1	1	yes	characteristic
51	C₄×C₂₃	#322	1	1	yes	characteristic
52	C₄×C₂₃	#336	1	2	yes	—
53	C₄×C₂₃	#364	1	4	yes	—
54	C₂×M₃₂	C₄×C₂₃	1	4	yes	—
55	C₂×C₄×C₈	C₄×C₂₃	1	2	yes	—
56	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
57	#346	—	2	2	no	—
58	C₄×C₁₆	—	2	2	no	—
59	C₈²	—	2	2	no	—
60	C₂×C₂₃	—	2	2	no	—
61	C₂×C₂₃	#1273	1	1	yes	characteristic
62	C₂×C₂₃	#1376	1	2	yes	—
63	C₂²×C₈	—	2	2	no	—
64	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
65	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
66	M₃₂	—	2	4	no	—
67	M₃₂	C₈×C₂₃	1	8	yes	complemented
68	C₂×C₁₆	—	2	4	no	—
69	C₄×C₈	—	2	2	no	—
70	C₄×C₈	C₈×C₂₃	1	4	yes	—
71	C₄×C₈	D₄×C₂₃	1	2	yes	—
72	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
73	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
74	C₂₃	#7138	1	1	yes	characteristic, direct product factor, Sylow
75	C₂²×C₄	C₄²×C₂₃	1	1	yes	characteristic
76	C₂²×C₄	M₁₆×C₂₃	1	2	yes	—
77	C₂×C₈	—	2	8	no	—
78	C₂×C₈	—	2	4	no	—
79	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
80	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
81	C₂×C₈	C₂×C₈×C₂₃	1	4	yes	—
82	C₂×C₈	M₁₆×C₂₃	1	4	yes	—
83	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
84	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
85	C₄²	C₂×C₈×C₂₃	1	2	yes	—
86	C₁₆	—	2	8	no	—
87	C₂³	C₂₃×#97	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
90	C₂×C₄	C₄×C₈×C₂₃	1	1	yes	characteristic
91	C₂×C₄	C₄×C₈×C₂₃	1	2	yes	—
92	C₂×C₄	C₂₃×#97	1	2	yes	—
93	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
94	C₂×C₄	C₂₃×C₄⋊C₈	1	2	yes	—
95	C₈	—	2	8	no	—
96	C₈	C₂₃×C₂²⋊C₈	1	4	yes	—
97	C₈	C₂₃×C₄⋊C₈	1	4	yes	—
98	C₂²	—	2	2	no	—
99	C₂²	C₂₃×#322	1	2	yes	—
100	C₂²	C₂₃×#336	1	1	yes	characteristic
101	C₄	—	2	2	no	—
102	C₄	C₈²×C₂₃	1	1	yes	characteristic
103	C₄	C₂₃×#322	1	1	yes	characteristic
104	C₄	C₂₃×#336	1	2	yes	—
105	C₄	C₂₃×#364	1	4	yes	—
106	C₂	—	2	2	no	—
107	C₂	C₂₃×#1273	1	1	yes	characteristic
108	C₂	C₂₃×#1376	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₂₃×M₃₂

C₂

yes

maximal

C₂×C₈²×C₂₃

C₂

yes

maximal, characteristic

C₂×C₂₃×M₃₂

C₄

yes

—

C₂×C₄×C₈×C₂₃

C₄

yes

—

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#346

—

C₄×C₁₆×C₂₃

—

C₈²×C₂₃

—

C₂²×C₈×C₂₃

—

C₂²×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₄²×C₂₃

C₂×C₄

yes

characteristic

C₂₃×M₃₂

—

C₂₃×M₃₂

C₈

yes

complemented

C₂×C₁₆×C₂₃

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

C₈

yes

—

C₄×C₈×C₂₃

D₄

yes

—

C₄×C₈×C₂₃

D₄

yes

characteristic

C₄×C₈×C₂₃

Q₈

yes

characteristic

C₂²×C₄×C₂₃

C₄²

yes

characteristic

C₂²×C₄×C₂₃

M₁₆

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂²⋊C₄

yes

—

C₂×C₈×C₂₃

C₄⋊C₄

yes

—

C₂×C₈×C₂₃

C₂×C₈

yes

—

C₂×C₈×C₂₃

M₁₆

yes

—

C₄²×C₂₃

C₂²⋊C₄

yes

characteristic

C₄²×C₂₃

C₄⋊C₄

yes

characteristic

C₄²×C₂₃

C₂×C₈

yes

—

C₁₆×C₂₃

—

#7138

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

#97

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#95

yes

characteristic

C₂×C₄×C₂₃

C₄×C₈

yes

characteristic

C₂×C₄×C₂₃

C₄×C₈

yes

—

C₂×C₄×C₂₃

#97

yes

—

C₂×C₄×C₂₃

C₂²⋊C₈

yes

—

C₂×C₄×C₂₃

C₄⋊C₈

yes

—

C₈×C₂₃

—

C₈×C₂₃

C₂²⋊C₈

yes

—

C₈×C₂₃

C₄⋊C₈

yes

—

C₄×M₃₂

C₂×C₂₃

yes

—

C₂×C₈²

C₂×C₂₃

yes

characteristic

C₂²×C₂₃

—

C₂²×C₂₃

#322

yes

—

C₂²×C₂₃

#336

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

C₈²

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

C₄×C₂₃

#364

yes

—

C₂×M₃₂

C₄×C₂₃

yes

—

C₂×C₄×C₈

C₄×C₂₃

yes

—

C₂×C₄×C₈

C₂²×C₂₃

yes

characteristic

#346

—

C₄×C₁₆

—

C₈²

—

C₂×C₂₃

—

C₂×C₂₃

#1273

yes

characteristic

C₂×C₂₃

#1376

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄×C₂₃

yes

—

C₂×C₄²

C₂×C₄×C₂₃

yes

characteristic

M₃₂

—

M₃₂

C₈×C₂₃

yes

complemented

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

C₈×C₂₃

yes

—

C₄×C₈

D₄×C₂₃

yes

—

C₄×C₈

D₄×C₂₃

yes

characteristic

C₄×C₈

Q₈×C₂₃

yes

characteristic

C₂₃

#7138

yes

characteristic, direct product factor, Sylow

C₂²×C₄

C₄²×C₂₃

yes

characteristic

C₂²×C₄

M₁₆×C₂₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄×C₂₃

yes

—

C₂×C₈

C₄⋊C₄×C₂₃

yes

—

C₂×C₈

C₂×C₈×C₂₃

yes

—

C₂×C₈

M₁₆×C₂₃

yes

—

C₄²

C₂²⋊C₄×C₂₃

yes

characteristic

C₄²

C₄⋊C₄×C₂₃

yes

characteristic

C₄²

C₂×C₈×C₂₃

yes

—

C₁₆

—

C₂³

C₂₃×#97

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂₃×#95

yes

characteristic

C₂×C₄

C₄×C₈×C₂₃

yes

characteristic

C₂×C₄

C₄×C₈×C₂₃

yes

—

C₂×C₄

C₂₃×#97

yes

—

C₂×C₄

C₂₃×C₂²⋊C₈

yes

—

C₂×C₄

C₂₃×C₄⋊C₈

yes

—

C₈

—

C₈

C₂₃×C₂²⋊C₈

yes

—

C₈

C₂₃×C₄⋊C₈

yes

—

C₂²

—

C₂²

C₂₃×#322

yes

—

100

C₂²

C₂₃×#336

yes

characteristic

101

C₄

—

102

C₄

C₈²×C₂₃

yes

characteristic

103

C₄

C₂₃×#322

yes

characteristic

104

C₄

C₂₃×#336

yes

—

105

C₄

C₂₃×#364

yes

—

106

C₂

—

107

C₂

C₂₃×#1273

yes

characteristic

108

C₂

C₂₃×#1376

yes

—

	subgroup	G/H
center	C₄×C₈×C₂₃	D₄
commutator	C₄	C₈²×C₂₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₂₃
socle	C₂²×C₂₃	#336
prime radical	C₂³×C₂₃	#97
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

subgroup

G/H

center

C₄×C₈×C₂₃

D₄

commutator

C₄

C₈²×C₂₃

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₂₃

C₂²×C₂₃

C₂³×C₂₃

G ▷ C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₄×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

N	H ≅ G/N	type
13. C₂₃×M₃₂	C₈	semidirect
32. #7138	C₂₃	direct (N × H)
67. M₃₂	C₈×C₂₃	semidirect
74. C₂₃	#7138	direct (N × H)

H ≅ G/N

type

13. C₂₃×M₃₂

C₈

semidirect

32. #7138

C₂₃

direct (N × H)

67. M₃₂

C₈×C₂₃

semidirect

74. C₂₃

#7138

direct (N × H)

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

subgroup	index
C₄×C₂₃×M₃₂	2
C₂×C₈²×C₂₃	2
#7138	23

subgroup

index

C₄×C₂₃×M₃₂

C₂×C₈²×C₂₃

#7138

Subgroups — C₂₃×#7138 #19154764

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#7138 #19154764

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7138 #19154764