Group #22618196 — subgroups

Subgroups — #22618196

Order 9344.

548 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 125 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	#17988523	C₂	1	3	yes	maximal, complemented
2	#17988511	C₂	1	2	yes	maximal, complemented
3	#17988509	C₂	1	6	yes	maximal, complemented
4	#17988442	C₂	1	3	yes	maximal
5	#399×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×D₇₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₇₃	C₂²	1	3	yes	complemented
8	C₂×#11924007	C₂²	1	6	yes	complemented
9	C₂×C₂²⋊Dic₇₃	C₂²	1	3	yes	—
10	C₂×#11924005	C₂²	1	6	yes	—
11	C₂×#11924005	C₂²	1	6	yes	—
12	#14556498	C₂²	1	3	yes	complemented
13	C₂×C₄⋊C₄×C₇₃	C₂²	1	3	yes	complemented
14	C₂×C₂²⋊C₄×C₇₃	C₂²	1	3	yes	—
15	#95×C₇₃	C₂²	1	1	yes	characteristic, complemented
16	C₂³×D₇₃	—	4	3	no	—
17	C₂³×D₇₃	—	4	3	no	—
18	C₂³×D₇₃	Q₈	1	4	yes	—
19	C₂³×D₇₃	C₂³	1	2	yes	—
20	C₂²×Dic₇₃	C₂³	1	6	yes	—
21	C₄⋊Dic₇₃	—	4	3	no	—
22	#11924007	—	4	6	no	—
23	C₂²⋊Dic₇₃	—	4	3	no	—
24	#11924005	—	4	6	no	—
25	#11924005	—	4	6	no	—
26	C₂⁴×C₇₃	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₇₃	C₂³	1	3	yes	—
28	C₂²×C₄×C₇₃	C₂³	1	3	yes	—
29	C₄⋊C₄×C₇₃	—	4	3	no	—
30	C₂²⋊C₄×C₇₃	—	4	3	no	—
31	C₂³×C₇₃	—	2	3	no	—
32	C₂³×C₇₃	—	2	3	no	—
33	C₂³×C₇₃	C₂×Q₈	1	2	yes	—
34	C₂³×C₇₃	C₂⁴	1	1	yes	characteristic
35	C₂²×D₇₃	—	2	6	no	—
36	C₂²×D₇₃	—	2	6	no	—
37	C₂²×D₇₃	—	4	12	no	—
38	C₂²×D₇₃	—	4	4	no	—
39	C₂²×D₇₃	—	4	12	no	—
40	C₂²×D₇₃	C₂×Q₈	1	4	yes	—
41	C₂×C₄×C₇₃	—	2	3	no	—
42	C₂×C₄×C₇₃	—	2	3	no	—
43	C₂×C₄×C₇₃	—	2	6	no	—
44	C₂×C₄×C₇₃	—	2	6	no	—
45	C₂×Dic₇₃	—	2	6	no	—
46	C₂×Dic₇₃	—	2	6	no	—
47	C₂×Dic₇₃	—	2	6	no	—
48	C₂²×C₇₃	—	2	6	no	—
49	C₂²×C₇₃	—	4	1	no	—
50	C₂²×C₇₃	—	4	3	no	—
51	C₂²×C₇₃	C₂²×Q₈	1	1	yes	characteristic
52	C₂²×C₇₃	H(5,𝔽₂)	1	3	yes	—
53	C₂²×C₇₃	H(5,𝔽₂)	1	3	yes	—
54	C₂×D₇₃	—	2	12	no	—
55	C₂×D₇₃	—	4	16	no	—
56	C₂×D₇₃	—	4	2	no	—
57	C₂×D₇₃	—	4	6	no	—
58	C₄×C₇₃	—	4	3	no	—
59	C₄×C₇₃	—	4	3	no	—
60	Dic₇₃	—	4	6	no	—
61	C₂×C₇₃	—	4	2	no	—
62	C₂×C₇₃	#543	1	3	yes	—
63	C₂×C₇₃	#560	1	1	yes	characteristic
64	C₂×C₇₃	#560	1	3	yes	—
65	D₇₃	—	4	4	no	—
66	#2842	—	73	1	no	maximal, Sylow
67	C₇₃	#2842	1	1	yes	characteristic, complemented, Sylow
68	#399	—	73	3	no	—
69	#399	D₇₃	1	1	yes	characteristic, complemented
70	#396	—	73	6	no	—
71	#396	—	73	2	no	—
72	#379	—	73	3	no	—
73	C₂⁵	—	73	1	no	—
74	C₂×C₄⋊C₄	—	73	6	no	—
75	C₂×C₄⋊C₄	—	73	3	no	—
76	C₂×C₄⋊C₄	C₂×D₇₃	1	3	yes	complemented
77	C₂×C₂²⋊C₄	—	73	3	no	—
78	C₂×C₂²⋊C₄	—	73	6	no	—
79	C₂×C₂²⋊C₄	—	73	6	no	—
80	C₂×C₂²⋊C₄	C₂×D₇₃	1	3	yes	—
81	#95	—	73	3	no	—
82	#95	C₂×D₇₃	1	1	yes	characteristic, complemented
83	C₂⁴	—	292	3	no	—
84	C₂⁴	—	292	3	no	—
85	C₂⁴	—	73	4	no	—
86	C₂⁴	—	73	2	no	—
87	C₂⁴	C₂²×D₇₃	1	1	yes	characteristic
88	C₂²×C₄	—	73	6	no	—
89	C₂²×C₄	C₂²×D₇₃	1	3	yes	—
90	C₂²×C₄	C₂²×D₇₃	1	3	yes	—
91	C₄⋊C₄	—	4	3	no	—
92	C₄⋊C₄	—	292	6	no	—
93	C₄⋊C₄	—	292	3	no	—
94	C₂²⋊C₄	—	4	3	no	—
95	C₂²⋊C₄	—	292	6	no	—
96	C₂²⋊C₄	—	292	6	no	—
97	C₂²⋊C₄	—	292	3	no	—
98	C₂³	—	2	3	no	—
99	C₂³	—	2	3	no	—
100	C₂³	—	292	12	no	—
101	C₂³	—	292	4	no	—
102	C₂³	—	292	12	no	—
103	C₂³	—	73	4	no	—
104	C₂³	—	146	6	no	—
105	C₂³	—	146	6	no	—
106	C₂³	Q₈×D₇₃	1	2	yes	—
107	C₂³	C₂³×D₇₃	1	1	yes	characteristic
108	C₂×C₄	—	2	3	no	—
109	C₂×C₄	—	2	3	no	—
110	C₂×C₄	—	2	6	no	—
111	C₂×C₄	—	2	6	no	—
112	C₂×C₄	—	146	6	no	—
113	C₂×C₄	—	146	6	no	—
114	C₂×C₄	—	146	6	no	—
115	C₂²	—	2	6	no	—
116	C₂²	—	4	1	no	—
117	C₂²	—	4	3	no	—
118	C₂²	—	292	16	no	—
119	C₂²	—	292	2	no	—
120	C₂²	—	292	6	no	—
121	C₂²	—	146	12	no	—
122	C₂²	C₂×Q₈×D₇₃	1	1	yes	characteristic
123	C₂²	#14556636	1	3	yes	—
124	C₂²	#14556637	1	3	yes	—
125	C₄	—	4	3	no	—
126	C₄	—	4	3	no	—
127	C₄	—	292	6	no	—
128	C₂	—	4	2	no	—
129	C₂	—	292	4	no	—
130	C₂	#17989212	1	3	yes	—
131	C₂	#17989324	1	1	yes	characteristic
132	C₂	#17989327	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17988523

C₂

yes

maximal, complemented

#17988511

C₂

yes

maximal, complemented

#17988509

C₂

yes

maximal, complemented

#17988442

C₂

yes

maximal

#399×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂⁴×D₇₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

complemented

C₂×#11924007

C₂²

yes

complemented

C₂×C₂²⋊Dic₇₃

C₂²

yes

—

C₂×#11924005

C₂²

yes

—

C₂×#11924005

C₂²

yes

—

#14556498

C₂²

yes

complemented

C₂×C₄⋊C₄×C₇₃

C₂²

yes

complemented

C₂×C₂²⋊C₄×C₇₃

C₂²

yes

—

#95×C₇₃

C₂²

yes

characteristic, complemented

C₂³×D₇₃

—

C₂³×D₇₃

—

C₂³×D₇₃

Q₈

yes

—

C₂³×D₇₃

C₂³

yes

—

C₂²×Dic₇₃

C₂³

yes

—

C₄⋊Dic₇₃

—

#11924007

—

C₂²⋊Dic₇₃

—

#11924005

—

#11924005

—

C₂⁴×C₇₃

C₂³

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

C₂³

yes

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₂×Q₈

yes

—

C₂³×C₇₃

C₂⁴

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

C₂×Q₈

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²×Q₈

yes

characteristic

C₂²×C₇₃

H(5,𝔽₂)

yes

—

C₂²×C₇₃

H(5,𝔽₂)

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#543

yes

—

C₂×C₇₃

#560

yes

characteristic

C₂×C₇₃

#560

yes

—

D₇₃

—

#2842

—

maximal, Sylow

C₇₃

#2842

yes

characteristic, complemented, Sylow

#399

—

#399

D₇₃

yes

characteristic, complemented

#396

—

#396

—

#379

—

C₂⁵

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₃

yes

complemented

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₇₃

yes

—

#95

—

#95

C₂×D₇₃

yes

characteristic, complemented

C₂⁴

—

292

—

C₂⁴

—

292

—

C₂⁴

—

C₂⁴

—

C₂⁴

C₂²×D₇₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂²×D₇₃

yes

—

C₂²×C₄

C₂²×D₇₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

292

—

C₂²⋊C₄

—

C₂²⋊C₄

—

292

—

C₂²⋊C₄

—

292

—

C₂²⋊C₄

—

292

—

C₂³

—

C₂³

—

100

C₂³

—

292

—

101

C₂³

—

292

—

102

C₂³

—

292

—

103

C₂³

—

104

C₂³

—

146

—

105

C₂³

—

146

—

106

C₂³

Q₈×D₇₃

yes

—

107

C₂³

C₂³×D₇₃

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

146

—

113

C₂×C₄

—

146

—

114

C₂×C₄

—

146

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

292

—

119

C₂²

—

292

—

120

C₂²

—

292

—

121

C₂²

—

146

—

122

C₂²

C₂×Q₈×D₇₃

yes

characteristic

123

C₂²

#14556636

yes

—

124

C₂²

#14556637

yes

—

125

C₄

—

126

C₄

—

127

C₄

—

292

—

128

C₂

—

129

C₂

—

292

—

130

C₂

#17989212

yes

—

131

C₂

#17989324

yes

characteristic

132

C₂

#17989327

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₃
commutator, socle	C₂³×C₇₃	C₂⁴
Fitting	#399×C₇₃	C₂
hypercenter	#399	D₇₃
hypocenter	C₇₃	#2842
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ #399

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₃

commutator, socle

C₂³×C₇₃

C₂⁴

Fitting

#399×C₇₃

C₂

D₇₃

C₇₃

G ▷ C₂³×C₇₃ ▷ C₇₃

C₁ ◁ C₂³ ◁ #399

N	H ≅ G/N	type
1. #17988523	C₂	semidirect
2. #17988511	C₂	semidirect
3. #17988509	C₂	semidirect
5. #399×C₇₃	C₂	semidirect
7. C₂×C₄⋊Dic₇₃	C₂²	semidirect
8. C₂×#11924007	C₂²	semidirect
12. #14556498	C₂²	semidirect
13. C₂×C₄⋊C₄×C₇₃	C₂²	semidirect
15. #95×C₇₃	C₂²	semidirect
67. C₇₃	#2842	semidirect
69. #399	D₇₃	semidirect
76. C₂×C₄⋊C₄	C₂×D₇₃	semidirect
82. #95	C₂×D₇₃	semidirect

H ≅ G/N

type

1. #17988523

C₂

semidirect

2. #17988511

C₂

semidirect

3. #17988509

C₂

semidirect

5. #399×C₇₃

C₂

semidirect

7. C₂×C₄⋊Dic₇₃

C₂²

semidirect

8. C₂×#11924007

C₂²

semidirect

12. #14556498

C₂²

semidirect

13. C₂×C₄⋊C₄×C₇₃

C₂²

semidirect

15. #95×C₇₃

C₂²

semidirect

67. C₇₃

#2842

semidirect

69. #399

D₇₃

semidirect

76. C₂×C₄⋊C₄

C₂×D₇₃

semidirect

82. #95

C₂×D₇₃

semidirect

Subgroups — #22618196

Conjugacy classes of subgroups

Complements