Group #43251832 — subgroups

Subgroups — #2817×C₂₂₃ #43251832

Order 28544.

444 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 184 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×C₂²⋊Q₈×C₂₂₃	C₂	1	2	yes	maximal
2	#400×C₂₂₃	C₂	1	4	yes	maximal, complemented
3	#397×C₂₂₃	C₂	1	4	yes	maximal
4	#395×C₂₂₃	C₂	1	4	yes	maximal, complemented
5	#381×C₂₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₂₂₃	C₂²	1	2	yes	—
7	C₂³×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
8	C₂²⋊Q₈×C₂₂₃	—	2	8	no	—
9	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	4	yes	—
10	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	4	yes	—
12	C₂×C₂²⋊C₄×C₂₂₃	C₂²	1	4	yes	—
13	C₂×C₂²⋊C₄×C₂₂₃	C₂²	1	2	yes	—
14	#95×C₂₂₃	C₂²	1	4	yes	—
15	#95×C₂₂₃	C₂²	1	8	yes	—
16	#95×C₂₂₃	C₂²	1	4	yes	—
17	C₂⁴×C₂₂₃	C₂³	1	1	yes	characteristic
18	C₂×Q₈×C₂₂₃	—	2	4	no	—
19	C₂×Q₈×C₂₂₃	—	4	4	no	—
20	C₂²×C₄×C₂₂₃	—	4	2	no	—
21	C₂²×C₄×C₂₂₃	D₄	1	4	yes	—
22	C₂²×C₄×C₂₂₃	C₂³	1	4	yes	—
23	C₂²×C₄×C₂₂₃	C₂³	1	4	yes	—
24	C₂²×C₄×C₂₂₃	C₂³	1	4	yes	—
25	C₂²×C₄×C₂₂₃	C₂³	1	2	yes	—
26	C₄⋊C₄×C₂₂₃	—	2	4	no	—
27	C₄⋊C₄×C₂₂₃	—	2	8	no	—
28	C₄⋊C₄×C₂₂₃	—	4	4	no	—
29	C₂²⋊C₄×C₂₂₃	—	2	8	no	—
30	C₂²⋊C₄×C₂₂₃	—	4	2	no	—
31	C₂³×C₂₂₃	—	2	2	no	—
32	C₂³×C₂₂₃	—	2	4	no	—
33	C₂³×C₂₂₃	C₂×D₄	1	2	yes	—
34	C₂³×C₂₂₃	C₂⁴	1	1	yes	characteristic
35	Q₈×C₂₂₃	—	4	8	no	—
36	C₂×C₄×C₂₂₃	—	2	4	no	—
37	C₂×C₄×C₂₂₃	—	2	4	no	—
38	C₂×C₄×C₂₂₃	—	2	4	no	—
39	C₂×C₄×C₂₂₃	—	2	4	no	—
40	C₂×C₄×C₂₂₃	—	2	8	no	—
41	C₂×C₄×C₂₂₃	—	2	8	no	—
42	C₂×C₄×C₂₂₃	—	2	8	no	—
43	C₂×C₄×C₂₂₃	—	4	4	no	—
44	C₂×C₄×C₂₂₃	C₂×D₄	1	4	yes	—
45	C₂²×C₂₂₃	—	2	2	no	—
46	C₂²×C₂₂₃	—	2	4	no	—
47	C₂²×C₂₂₃	—	4	4	no	—
48	C₂²×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
49	C₂²×C₂₂₃	H(5,𝔽₂)	1	2	yes	—
50	C₂²×C₂₂₃	2₋¹⁺⁴	1	4	yes	—
51	C₄×C₂₂₃	—	2	4	no	—
52	C₄×C₂₂₃	—	4	4	no	—
53	C₄×C₂₂₃	—	4	4	no	—
54	C₄×C₂₂₃	—	4	4	no	—
55	C₂×C₂₂₃	—	4	2	no	—
56	C₂×C₂₂₃	#534	1	1	yes	characteristic
57	C₂×C₂₂₃	#536	1	2	yes	—
58	C₂×C₂₂₃	#563	1	4	yes	—
59	C₂₂₃	#2817	1	1	yes	characteristic, direct product factor, Sylow
60	#2817	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂×C₂²⋊Q₈	C₂×C₂₂₃	1	2	yes	—
62	#400	C₂×C₂₂₃	1	4	yes	complemented
63	#397	C₂×C₂₂₃	1	4	yes	—
64	#395	C₂×C₂₂₃	1	4	yes	complemented
65	#381	C₂×C₂₂₃	1	1	yes	characteristic
66	C₂²×Q₈	C₂²×C₂₂₃	1	2	yes	—
67	C₂³×C₄	C₂²×C₂₂₃	1	1	yes	characteristic
68	C₂²⋊Q₈	—	2	8	no	—
69	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	4	yes	—
70	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	—
71	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	4	yes	—
72	C₂×C₂²⋊C₄	C₂²×C₂₂₃	1	4	yes	—
73	C₂×C₂²⋊C₄	C₂²×C₂₂₃	1	2	yes	—
74	#95	C₂²×C₂₂₃	1	4	yes	—
75	#95	C₂²×C₂₂₃	1	8	yes	—
76	#95	C₂²×C₂₂₃	1	4	yes	—
77	C₂⁴	C₂³×C₂₂₃	1	1	yes	characteristic
78	C₂×Q₈	—	2	4	no	—
79	C₂×Q₈	—	4	4	no	—
80	C₂²×C₄	—	4	2	no	—
81	C₂²×C₄	D₄×C₂₂₃	1	4	yes	—
82	C₂²×C₄	C₂³×C₂₂₃	1	4	yes	—
83	C₂²×C₄	C₂³×C₂₂₃	1	4	yes	—
84	C₂²×C₄	C₂³×C₂₂₃	1	4	yes	—
85	C₂²×C₄	C₂³×C₂₂₃	1	2	yes	—
86	C₄⋊C₄	—	2	4	no	—
87	C₄⋊C₄	—	2	8	no	—
88	C₄⋊C₄	—	4	4	no	—
89	C₂²⋊C₄	—	2	8	no	—
90	C₂²⋊C₄	—	4	2	no	—
91	C₂³	—	2	2	no	—
92	C₂³	—	2	4	no	—
93	C₂³	C₂×D₄×C₂₂₃	1	2	yes	—
94	C₂³	C₂⁴×C₂₂₃	1	1	yes	characteristic
95	Q₈	—	4	8	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	8	no	—
101	C₂×C₄	—	2	8	no	—
102	C₂×C₄	—	2	8	no	—
103	C₂×C₄	—	4	4	no	—
104	C₂×C₄	C₂×D₄×C₂₂₃	1	4	yes	—
105	C₂²	—	2	2	no	—
106	C₂²	—	2	4	no	—
107	C₂²	—	4	4	no	—
108	C₂²	C₂²×D₄×C₂₂₃	1	1	yes	characteristic
109	C₂²	H(5,𝔽₂)×C₂₂₃	1	2	yes	—
110	C₂²	2₋¹⁺⁴×C₂₂₃	1	4	yes	—
111	C₄	—	2	4	no	—
112	C₄	—	4	4	no	—
113	C₄	—	4	4	no	—
114	C₄	—	4	4	no	—
115	C₂	—	4	2	no	—
116	C₂	#534×C₂₂₃	1	1	yes	characteristic
117	C₂	#536×C₂₂₃	1	2	yes	—
118	C₂	#563×C₂₂₃	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊Q₈×C₂₂₃

C₂

yes

maximal

#400×C₂₂₃

C₂

yes

maximal, complemented

#397×C₂₂₃

C₂

yes

maximal

#395×C₂₂₃

C₂

yes

maximal, complemented

#381×C₂₂₃

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₂₃

C₂²

yes

—

C₂³×C₄×C₂₂₃

C₂²

yes

characteristic

C₂²⋊Q₈×C₂₂₃

—

C₂×C₄⋊C₄×C₂₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₂₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₂₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₂₃

C₂²

yes

—

#95×C₂₂₃

C₂²

yes

—

#95×C₂₂₃

C₂²

yes

—

#95×C₂₂₃

C₂²

yes

—

C₂⁴×C₂₂₃

C₂³

yes

characteristic

C₂×Q₈×C₂₂₃

—

C₂×Q₈×C₂₂₃

—

C₂²×C₄×C₂₂₃

—

C₂²×C₄×C₂₂₃

D₄

yes

—

C₂²×C₄×C₂₂₃

C₂³

yes

—

C₂²×C₄×C₂₂₃

C₂³

yes

—

C₂²×C₄×C₂₂₃

C₂³

yes

—

C₂²×C₄×C₂₂₃

C₂³

yes

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

—

C₂²⋊C₄×C₂₂₃

—

C₂²⋊C₄×C₂₂₃

—

C₂³×C₂₂₃

—

C₂³×C₂₂₃

—

C₂³×C₂₂₃

C₂×D₄

yes

—

C₂³×C₂₂₃

C₂⁴

yes

characteristic

Q₈×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

C₂×D₄

yes

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

C₂²×D₄

yes

characteristic

C₂²×C₂₂₃

H(5,𝔽₂)

yes

—

C₂²×C₂₂₃

2₋¹⁺⁴

yes

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₂×C₂₂₃

—

C₂×C₂₂₃

#534

yes

characteristic

C₂×C₂₂₃

#536

yes

—

C₂×C₂₂₃

#563

yes

—

C₂₂₃

#2817

yes

characteristic, direct product factor, Sylow

#2817

C₂₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₂²⋊Q₈

C₂×C₂₂₃

yes

—

#400

C₂×C₂₂₃

yes

complemented

#397

C₂×C₂₂₃

yes

—

#395

C₂×C₂₂₃

yes

complemented

#381

C₂×C₂₂₃

yes

characteristic

C₂²×Q₈

C₂²×C₂₂₃

yes

—

C₂³×C₄

C₂²×C₂₂₃

yes

characteristic

C₂²⋊Q₈

—

C₂×C₄⋊C₄

C₂²×C₂₂₃

yes

—

C₂×C₄⋊C₄

C₂²×C₂₂₃

yes

—

C₂×C₄⋊C₄

C₂²×C₂₂₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₂₂₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₂₂₃

yes

—

#95

C₂²×C₂₂₃

yes

—

#95

C₂²×C₂₂₃

yes

—

#95

C₂²×C₂₂₃

yes

—

C₂⁴

C₂³×C₂₂₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

D₄×C₂₂₃

yes

—

C₂²×C₄

C₂³×C₂₂₃

yes

—

C₂²×C₄

C₂³×C₂₂₃

yes

—

C₂²×C₄

C₂³×C₂₂₃

yes

—

C₂²×C₄

C₂³×C₂₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂³

—

C₂³

—

C₂³

C₂×D₄×C₂₂₃

yes

—

C₂³

C₂⁴×C₂₂₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×D₄×C₂₂₃

yes

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

C₂²×D₄×C₂₂₃

yes

characteristic

109

C₂²

H(5,𝔽₂)×C₂₂₃

yes

—

110

C₂²

2₋¹⁺⁴×C₂₂₃

yes

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₂

—

116

C₂

#534×C₂₂₃

yes

characteristic

117

C₂

#536×C₂₂₃

yes

—

118

C₂

#563×C₂₂₃

yes

—

	subgroup	G/H
center, socle	C₂³×C₂₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₂₃
prime radical	C₂⁴×C₂₂₃	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₂₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₂₂₃

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₂₂₃

prime radical

C₂⁴×C₂₂₃

C₂³

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₂₃ ◁ G

N	H ≅ G/N	type
2. #400×C₂₂₃	C₂	semidirect
4. #395×C₂₂₃	C₂	semidirect
59. C₂₂₃	#2817	direct (N × H)
60. #2817	C₂₂₃	direct (N × H)
62. #400	C₂×C₂₂₃	semidirect
64. #395	C₂×C₂₂₃	semidirect

H ≅ G/N

type

2. #400×C₂₂₃

C₂

semidirect

4. #395×C₂₂₃

C₂

semidirect

59. C₂₂₃

#2817

direct (N × H)

60. #2817

C₂₂₃

direct (N × H)

62. #400

C₂×C₂₂₃

semidirect

64. #395

C₂×C₂₂₃

semidirect

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

subgroup	index
C₂×C₂²⋊Q₈×C₂₂₃	2
#400×C₂₂₃	2
#397×C₂₂₃	2
#395×C₂₂₃	2
#381×C₂₂₃	2
#2817	223

subgroup

index

C₂×C₂²⋊Q₈×C₂₂₃

#400×C₂₂₃

#397×C₂₂₃

#395×C₂₂₃

#381×C₂₂₃

#2817

223

Subgroups — #2817×C₂₂₃ #43251832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2817×C223 #43251832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2817×C₂₂₃ #43251832