Group #22608209 — subgroups

Subgroups — M₁₆×#10596831 #22608209

Order 9344.

224 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 153 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	#17988640	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#10596831	C₂	1	1	yes	maximal, characteristic
3	#17988308	C₂	1	2	yes	maximal, complemented
4	C₈×#10596831	C₂	1	2	yes	maximal, complemented
5	C₄×M₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10596831	C₄	1	2	yes	—
7	C₄⋊#10596831	C₄	1	2	yes	—
8	C₂²⋊#10596831	C₄	1	2	yes	—
9	C₄×#10596831	C₄	1	2	yes	—
10	C₄×#10596831	C₂²	1	2	yes	—
11	C₄×#10596831	C₂²	1	1	yes	characteristic
12	C₄×#10596831	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₇₃	C₄	1	2	yes	—
14	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
15	#97×C₇₃	C₄	1	2	yes	—
16	C₄×C₈×C₇₃	C₂²	1	2	yes	—
17	C₂×#10596831	—	2	4	no	—
18	C₂×#10596831	C₂×C₄	1	2	yes	—
19	C₂×#10596831	C₂×C₄	1	4	yes	—
20	C₂×#10596831	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
24	M₁₆×C₇₃	C₈	1	8	yes	complemented
25	C₂×C₈×C₇₃	C₂×C₄	1	4	yes	—
26	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₇₃	C₂³	1	1	yes	characteristic
30	C₂³×C₇₃	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₇₃	—	2	2	no	—
32	C₂×C₄×C₇₃	—	2	2	no	—
33	C₂×C₄×C₇₃	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₇₃	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₇₃	C₄²	1	1	yes	characteristic
36	C₂×C₄×C₇₃	C₂×C₈	1	4	yes	—
37	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₈×C₇₃	C₂×C₈	1	8	yes	complemented
41	#10596831	—	2	2	no	—
42	#10596831	—	2	4	no	—
43	#10596831	M₁₆	1	4	yes	direct product factor
44	C₂²×C₇₃	—	2	1	no	—
45	C₂²×C₇₃	—	2	1	no	—
46	C₂²×C₇₃	C₄×C₈	1	2	yes	—
47	C₂²×C₇₃	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₇₃	—	2	1	no	—
49	C₄×C₇₃	—	2	1	no	—
50	C₄×C₇₃	C₄×C₈	1	2	yes	—
51	C₄×C₇₃	C₂²×C₈	1	2	yes	—
52	C₄×C₇₃	C₂×M₁₆	1	2	yes	—
53	C₄×C₇₃	#131	1	2	yes	—
54	C₂×C₇₃	—	2	2	no	—
55	C₂×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
56	C₂×C₇₃	C₄×M₁₆	1	1	yes	characteristic
57	C₂×C₇₃	#405	1	1	yes	characteristic
58	C₈×M₁₆	—	73	1	no	maximal, Sylow
59	C₇₃	C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
60	#431	—	73	1	no	—
61	C₄×M₁₆	D₇₃	1	1	yes	characteristic
62	C₂×C₄×C₈	—	73	1	no	—
63	#322	—	73	2	no	—
64	C₈²	—	73	2	no	—
65	C₂×M₁₆	Dic₇₃	1	2	yes	—
66	C₂²×C₈	—	73	2	no	—
67	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
68	C₄⋊C₈	—	73	2	no	—
69	C₂²⋊C₈	—	73	2	no	—
70	#97	Dic₇₃	1	2	yes	—
71	C₄×C₈	—	73	2	no	—
72	C₄×C₈	—	73	1	no	—
73	C₄×C₈	—	73	2	no	—
74	C₄×C₈	—	73	1	no	—
75	C₄×C₈	C₂×D₇₃	1	2	yes	—
76	C₂²×C₄	C₄×D₇₃	1	1	yes	characteristic
77	C₂²×C₄	C₄×D₇₃	1	1	yes	characteristic
78	C₂²×C₄	C₂×Dic₇₃	1	1	yes	characteristic
79	M₁₆	#10596831	1	8	yes	direct product factor
80	C₂×C₈	—	73	2	no	—
81	C₂×C₈	—	73	2	no	—
82	C₂×C₈	—	73	4	no	—
83	C₂×C₈	—	146	4	no	—
84	C₂×C₈	C₂×Dic₇₃	1	4	yes	—
85	C₄²	C₄×D₇₃	1	1	yes	characteristic
86	C₄²	C₄×D₇₃	1	1	yes	characteristic
87	C₄²	C₂×Dic₇₃	1	1	yes	characteristic
88	C₄²	C₂²×D₇₃	1	1	yes	characteristic
89	C₂³	C₄×Dic₇₃	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	C₄×Dic₇₃	1	1	yes	characteristic
93	C₂×C₄	C₄×Dic₇₃	1	1	yes	characteristic
94	C₂×C₄	C₄×Dic₇₃	1	1	yes	characteristic
95	C₂×C₄	C₂×#10596831	1	4	yes	—
96	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
97	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
98	C₂×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
99	C₈	—	73	4	no	—
100	C₈	—	146	2	no	—
101	C₈	—	146	4	no	—
102	C₈	C₂×#10596831	1	8	yes	complemented
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	C₄×#10596831	1	2	yes	—
106	C₂²	C₂×C₄×Dic₇₃	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	C₄×#10596831	1	2	yes	—
110	C₄	C₂²×#10596831	1	2	yes	—
111	C₄	M₁₆×D₇₃	1	2	yes	—
112	C₄	#14556599	1	2	yes	—
113	C₂	—	2	2	no	—
114	C₂	C₂×C₄×#10596831	1	1	yes	characteristic
115	C₂	M₁₆×Dic₇₃	1	1	yes	characteristic
116	C₂	#17988545	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988640

C₂

yes

maximal, characteristic

C₂×C₄×#10596831

C₂

yes

maximal, characteristic

#17988308

C₂

yes

maximal, complemented

C₈×#10596831

C₂

yes

maximal, complemented

C₄×M₁₆×C₇₃

C₂

yes

maximal, characteristic

C₂²×#10596831

C₄

yes

—

C₄⋊#10596831

C₄

yes

—

C₂²⋊#10596831

C₄

yes

—

C₄×#10596831

C₄

yes

—

C₄×#10596831

C₂²

yes

—

C₄×#10596831

C₂²

yes

characteristic

C₄×#10596831

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

C₄

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

#97×C₇₃

C₄

yes

—

C₄×C₈×C₇₃

C₂²

yes

—

C₂×#10596831

—

C₂×#10596831

C₂×C₄

yes

—

C₂×#10596831

C₂×C₄

yes

—

C₂×#10596831

C₂×C₄

yes

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

M₁₆×C₇₃

C₈

yes

complemented

C₂×C₈×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

C₄²

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄²

yes

characteristic

C₂×C₄×C₇₃

C₄²

yes

characteristic

C₂×C₄×C₇₃

C₄²

yes

characteristic

C₂×C₄×C₇₃

C₂×C₈

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₈×C₇₃

C₂×C₈

yes

—

—

M₁₆

yes

direct product factor

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₄×C₈

yes

—

C₂²×C₇₃

C₂×C₄²

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄×C₈

yes

—

C₄×C₇₃

C₂²×C₈

yes

—

C₄×C₇₃

C₂×M₁₆

yes

—

C₄×C₇₃

#131

yes

—

C₂×C₇₃

—

C₂×C₇₃

C₂×C₄×C₈

yes

characteristic

C₂×C₇₃

C₄×M₁₆

yes

characteristic

C₂×C₇₃

#405

yes

characteristic

C₈×M₁₆

—

maximal, Sylow

C₇₃

C₈×M₁₆

yes

characteristic, complemented, Sylow

#431

—

C₄×M₁₆

D₇₃

yes

characteristic

C₂×C₄×C₈

—

#322

—

C₈²

—

C₂×M₁₆

Dic₇₃

yes

—

C₂²×C₈

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

Dic₇₃

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₇₃

yes

—

C₂²×C₄

C₄×D₇₃

yes

characteristic

C₂²×C₄

C₄×D₇₃

yes

characteristic

C₂²×C₄

C₂×Dic₇₃

yes

characteristic

M₁₆

#10596831

yes

direct product factor

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

C₂×Dic₇₃

yes

—

C₄²

C₄×D₇₃

yes

characteristic

C₄²

C₄×D₇₃

yes

characteristic

C₄²

C₂×Dic₇₃

yes

characteristic

C₄²

C₂²×D₇₃

yes

characteristic

C₂³

C₄×Dic₇₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₇₃

yes

characteristic

C₂×C₄

C₄×Dic₇₃

yes

characteristic

C₂×C₄

C₄×Dic₇₃

yes

characteristic

C₂×C₄

C₂×#10596831

yes

—

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

C₂×C₄

C₂²×Dic₇₃

yes

characteristic

C₈

—

100

C₈

—

146

—

101

C₈

—

146

—

102

C₈

C₂×#10596831

yes

complemented

103

C₂²

—

104

C₂²

—

105

C₂²

C₄×#10596831

yes

—

106

C₂²

C₂×C₄×Dic₇₃

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

C₄×#10596831

yes

—

110

C₄

C₂²×#10596831

yes

—

111

C₄

M₁₆×D₇₃

yes

—

112

C₄

#14556599

yes

—

113

C₂

—

114

C₂

C₂×C₄×#10596831

yes

characteristic

115

C₂

M₁₆×Dic₇₃

yes

characteristic

116

C₂

#17988545

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₇₃
commutator	C₂×C₇₃	C₂×C₄×C₈
Fitting	C₄×M₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₄²
hypercenter	C₄×M₁₆	D₇₃
hypocenter	C₇₃	C₈×M₁₆
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

subgroup

G/H

center, Frattini

C₄²

C₂²×D₇₃

commutator

C₂×C₇₃

C₂×C₄×C₈

Fitting

C₄×M₁₆×C₇₃

C₂

socle

C₂²×C₇₃

C₂×C₄²

hypercenter

C₄×M₁₆

D₇₃

hypocenter

C₇₃

C₈×M₁₆

LCS

G ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
3. #17988308	C₂	semidirect
4. C₈×#10596831	C₂	semidirect
24. M₁₆×C₇₃	C₈	semidirect
40. C₈×C₇₃	C₂×C₈	semidirect
43. #10596831	M₁₆	direct (N × H)
59. C₇₃	C₈×M₁₆	semidirect
79. M₁₆	#10596831	direct (N × H)
102. C₈	C₂×#10596831	semidirect

H ≅ G/N

type

3. #17988308

C₂

semidirect

4. C₈×#10596831

C₂

semidirect

24. M₁₆×C₇₃

C₈

semidirect

40. C₈×C₇₃

C₂×C₈

semidirect

43. #10596831

M₁₆

direct (N × H)

59. C₇₃

C₈×M₁₆

semidirect

79. M₁₆

#10596831

direct (N × H)

102. C₈

C₂×#10596831

semidirect

Subgroups — M₁₆×#10596831 #22608209

Conjugacy classes of subgroups

Complements

Subgroups — M16×#10596831 #22608209

Conjugacy classes of subgroups

Complements

Subgroups — M₁₆×#10596831 #22608209