Group #17989331 — subgroups

Subgroups — #17989331

Order 4672.

220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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	#14556583	C₂	1	1	yes	maximal, characteristic, complemented
2	#14556579	C₂	1	1	yes	maximal, characteristic, complemented
3	#14556575	C₂	1	2	yes	maximal, complemented
4	#14556573	C₂	1	2	yes	maximal, complemented
5	#14556568	C₂	1	2	yes	maximal, complemented
6	#14556567	C₂	1	2	yes	maximal, complemented
7	#14556564	C₂	1	2	yes	maximal, complemented
8	#14556563	C₂	1	2	yes	maximal, complemented
9	#124×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₁₄₆	C₂²	1	1	yes	characteristic
11	C₂×D₂₉₂	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₇₃	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₇₃	C₂²	1	2	yes	—
14	C₂²×Dic₇₃	C₂²	1	2	yes	—
15	C₂×C₄×D₇₃	C₂²	1	2	yes	—
16	C₄⋊Dic₇₃	C₂²	1	2	yes	—
17	C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
19	#11924007	C₂²	1	2	yes	—
20	#11924007	C₂²	1	2	yes	—
21	#11924007	C₂²	1	2	yes	—
22	C₂²⋊Dic₇₃	C₂²	1	2	yes	—
23	#11924005	C₂²	1	2	yes	—
24	#11924005	C₂²	1	2	yes	—
25	#11924005	C₂²	1	2	yes	—
26	C₂×Q₈×C₇₃	C₂²	1	1	yes	characteristic
27	C₂×D₄×C₇₃	C₂²	1	1	yes	characteristic
28	C₂²⋊C₄×C₇₃	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₇₃	C₂²	1	2	yes	—
30	C₄²×C₇₃	C₂²	1	1	yes	characteristic
31	C₂³×C₇₃	C₂³	1	2	yes	—
32	C₂²×D₇₃	C₂³	1	2	yes	—
33	Q₈×C₇₃	—	4	1	no	—
34	D₄×C₇₃	—	4	1	no	—
35	C₂×C₄×C₇₃	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₇₃	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₇₃	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₇₃	C₂³	1	2	yes	—
39	C₂²⋊D₇₃	—	4	2	no	—
40	C₂²⋊D₇₃	—	4	2	no	—
41	C₂×Dic₇₃	—	4	2	no	—
42	C₂×Dic₇₃	C₂³	1	2	yes	—
43	C₂×Dic₇₃	C₂³	1	2	yes	—
44	C₂×Dic₇₃	C₂³	1	2	yes	—
45	D₂₉₂	—	4	1	no	—
46	C₄×D₇₃	—	4	2	no	—
47	Dic₁₄₆	—	4	1	no	—
48	C₂²×C₇₃	—	2	2	no	—
49	C₂²×C₇₃	—	2	2	no	—
50	C₂²×C₇₃	—	2	2	no	—
51	C₂²×C₇₃	C₂⁴	1	1	yes	characteristic
52	C₂×D₇₃	—	2	2	no	—
53	C₂×D₇₃	—	2	2	no	—
54	C₂×D₇₃	—	2	2	no	—
55	C₄×C₇₃	—	2	1	no	—
56	C₄×C₇₃	—	2	2	no	—
57	C₄×C₇₃	—	2	1	no	—
58	C₄×C₇₃	—	2	1	no	—
59	Dic₇₃	—	2	2	no	—
60	Dic₇₃	—	2	2	no	—
61	Dic₇₃	—	2	2	no	—
62	C₂×C₇₃	—	4	2	no	—
63	C₂×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
66	D₇₃	—	4	2	no	—
67	C₇₃	#562	1	1	yes	characteristic, complemented, Sylow
68	#562	—	73	1	no	maximal, Sylow
69	#125	—	73	1	no	—
70	#124	—	73	1	no	—
71	#124	D₇₃	1	1	yes	characteristic, complemented
72	#123	—	73	2	no	—
73	#123	—	73	2	no	—
74	C₂²⋊Q₈	—	73	2	no	—
75	C₂²⋊Q₈	—	73	2	no	—
76	#121	—	73	2	no	—
77	#121	—	73	2	no	—
78	C₂×Q₈	—	73	1	no	—
79	C₂×Q₈	C₂×D₇₃	1	1	yes	characteristic
80	C₂×D₄	—	73	2	no	—
81	C₂×D₄	—	73	2	no	—
82	C₂×D₄	—	73	1	no	—
83	C₂×D₄	C₂×D₇₃	1	1	yes	characteristic
84	C₂²×C₄	—	73	2	no	—
85	C₂²×C₄	—	73	2	no	—
86	C₄⋊C₄	—	73	2	no	—
87	C₄⋊C₄	—	73	2	no	—
88	C₄⋊C₄	—	73	2	no	—
89	C₄⋊C₄	—	73	2	no	—
90	C₄⋊C₄	—	73	1	no	—
91	C₄⋊C₄	—	73	1	no	—
92	C₂²⋊C₄	—	73	2	no	—
93	C₂²⋊C₄	—	73	2	no	—
94	C₂²⋊C₄	—	73	2	no	—
95	C₂²⋊C₄	—	73	2	no	—
96	C₂²⋊C₄	C₂×D₇₃	1	2	yes	—
97	C₂²⋊C₄	C₂×D₇₃	1	2	yes	—
98	C₄²	C₂×D₇₃	1	1	yes	characteristic
99	C₂³	—	73	2	no	—
100	C₂³	C₂²×D₇₃	1	2	yes	—
101	Q₈	—	4	1	no	—
102	Q₈	—	292	1	no	—
103	D₄	—	4	1	no	—
104	D₄	—	292	1	no	—
105	D₄	—	292	2	no	—
106	D₄	—	292	2	no	—
107	C₂×C₄	—	292	2	no	—
108	C₂×C₄	—	292	2	no	—
109	C₂×C₄	—	73	2	no	—
110	C₂×C₄	—	73	2	no	—
111	C₂×C₄	—	73	2	no	—
112	C₂×C₄	C₂²×D₇₃	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₇₃	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₇₃	1	1	yes	characteristic
115	C₂×C₄	C₂²×D₇₃	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	146	2	no	—
120	C₂²	—	146	2	no	—
121	C₂²	—	146	2	no	—
122	C₂²	C₂³×D₇₃	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	146	2	no	—
128	C₄	—	146	2	no	—
129	C₄	—	146	2	no	—
130	C₂	—	4	2	no	—
131	C₂	—	292	2	no	—
132	C₂	#14556636	1	1	yes	characteristic
133	C₂	#14556637	1	1	yes	characteristic
134	C₂	#14556638	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14556583

C₂

yes

maximal, characteristic, complemented

#14556579

C₂

yes

maximal, characteristic, complemented

#14556575

C₂

yes

maximal, complemented

#14556573

C₂

yes

maximal, complemented

#14556568

C₂

yes

maximal, complemented

#14556567

C₂

yes

maximal, complemented

#14556564

C₂

yes

maximal, complemented

#14556563

C₂

yes

maximal, complemented

#124×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₄₆

C₂²

yes

characteristic

C₂×D₂₉₂

C₂²

yes

characteristic

C₂×C₂²⋊D₇₃

C₂²

yes

—

C₂×C₂²⋊D₇₃

C₂²

yes

—

C₂²×Dic₇₃

C₂²

yes

—

C₂×C₄×D₇₃

C₂²

yes

—

C₄⋊Dic₇₃

C₂²

yes

—

C₄⋊Dic₇₃

C₂²

yes

characteristic

C₄⋊Dic₇₃

C₂²

yes

characteristic

#11924007

C₂²

yes

—

#11924007

C₂²

yes

—

#11924007

C₂²

yes

—

C₂²⋊Dic₇₃

C₂²

yes

—

#11924005

C₂²

yes

—

#11924005

C₂²

yes

—

#11924005

C₂²

yes

—

C₂×Q₈×C₇₃

C₂²

yes

characteristic

C₂×D₄×C₇₃

C₂²

yes

characteristic

C₂²⋊C₄×C₇₃

C₂²

yes

—

C₂²⋊C₄×C₇₃

C₂²

yes

—

C₄²×C₇₃

C₂²

yes

characteristic

C₂³×C₇₃

C₂³

yes

—

C₂²×D₇₃

C₂³

yes

—

Q₈×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₄×C₇₃

C₂³

yes

—

C₂²⋊D₇₃

—

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂³

yes

—

C₂×Dic₇₃

C₂³

yes

—

C₂×Dic₇₃

C₂³

yes

—

D₂₉₂

—

C₄×D₇₃

—

Dic₁₄₆

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂⁴

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

2₋¹⁺⁴

yes

characteristic

D₇₃

—

C₇₃

#562

yes

characteristic, complemented, Sylow

—

—

—

D₇₃

yes

characteristic, complemented

#123

—

#123

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

—

#121

—

C₂×Q₈

—

C₂×Q₈

C₂×D₇₃

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₇₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₇₃

yes

—

C₂²⋊C₄

C₂×D₇₃

yes

—

C₄²

C₂×D₇₃

yes

characteristic

C₂³

—

100

C₂³

C₂²×D₇₃

yes

—

101

Q₈

—

102

Q₈

—

292

—

103

D₄

—

104

D₄

—

292

—

105

D₄

—

292

—

106

D₄

—

292

—

107

C₂×C₄

—

292

—

108

C₂×C₄

—

292

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂²×D₇₃

yes

characteristic

113

C₂×C₄

C₂²×D₇₃

yes

characteristic

114

C₂×C₄

C₂²×D₇₃

yes

characteristic

115

C₂×C₄

C₂²×D₇₃

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

146

—

120

C₂²

—

146

—

121

C₂²

—

146

—

122

C₂²

C₂³×D₇₃

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

146

—

128

C₄

—

146

—

129

C₄

—

146

—

130

C₂

—

131

C₂

—

292

—

132

C₂

#14556636

yes

characteristic

133

C₂

#14556637

yes

characteristic

134

C₂

#14556638

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇₃
commutator, socle, cosocle	C₂²×C₇₃	C₂⁴
Fitting	#124×C₇₃	C₂
hypercenter	#124	D₇₃
hypocenter	C₇₃	#562
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₇₃

commutator, socle, cosocle

C₂²×C₇₃

C₂⁴

Fitting

#124×C₇₃

C₂

D₇₃

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #14556583	C₂	semidirect
2. #14556579	C₂	semidirect
3. #14556575	C₂	semidirect
4. #14556573	C₂	semidirect
5. #14556568	C₂	semidirect
6. #14556567	C₂	semidirect
7. #14556564	C₂	semidirect
8. #14556563	C₂	semidirect
9. #124×C₇₃	C₂	semidirect
67. C₇₃	#562	semidirect
71. #124	D₇₃	semidirect

H ≅ G/N

type

1. #14556583

C₂

semidirect

2. #14556579

C₂

semidirect

3. #14556575

C₂

semidirect

4. #14556573

C₂

semidirect

5. #14556568

C₂

semidirect

6. #14556567

C₂

semidirect

7. #14556564

C₂

semidirect

8. #14556563

C₂

semidirect

9. #124×C₇₃

C₂

semidirect

67. C₇₃

#562

semidirect

71. #124

D₇₃

semidirect

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

subgroup	index
#14556583	2
#14556579	2
#14556575	2
#14556573	2
#14556568	2
#14556567	2
#14556564	2
#14556563	2
#124×C₇₃	2
#562	73

subgroup

index

#124×C₇₃

Subgroups — #17989331

Conjugacy classes of subgroups

Complements

Maximal subgroups