Group #17989184 — subgroups

Subgroups — #17989184

Order 4672.

218 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 95 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	#14556586	C₂	1	2	yes	maximal, complemented
2	#14556581	C₂	1	2	yes	maximal, complemented
3	#14556576	C₂	1	2	yes	maximal, complemented
4	#14556574	C₂	1	2	yes	maximal, complemented
5	#14556570	C₂	1	1	yes	maximal, characteristic, complemented
6	#14556569	C₂	1	2	yes	maximal, complemented
7	#14556565	C₂	1	1	yes	maximal, characteristic
8	C₄×C₂²⋊D₇₃	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊C₄×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×C₄×D₇₃	C₂²	1	2	yes	—
14	C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
16	#11924007	C₂²	1	2	yes	—
17	#11924007	C₂²	1	2	yes	—
18	#11924007	C₂²	1	2	yes	—
19	#11924007	C₂²	1	2	yes	—
20	C₂²⋊Dic₇₃	C₂²	1	2	yes	—
21	#11924005	C₂²	1	2	yes	—
22	#11924005	C₂²	1	2	yes	—
23	#11924005	C₂²	1	2	yes	—
24	#11924005	C₂²	1	2	yes	—
25	C₄×Dic₇₃	C₂²	1	2	yes	—
26	C₂²×C₄×C₇₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄×C₇₃	C₂²	1	1	yes	characteristic
28	C₂²×C₄×C₇₃	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
30	C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
31	C₂³×C₇₃	C₂³	1	1	yes	characteristic
32	C₂²×D₇₃	C₂³	1	2	yes	—
33	C₂×C₄×C₇₃	—	2	2	no	—
34	C₂×C₄×C₇₃	—	4	1	no	—
35	C₂×C₄×C₇₃	—	4	1	no	—
36	C₂×C₄×C₇₃	C₂³	1	2	yes	—
37	C₂×C₄×C₇₃	C₂³	1	2	yes	—
38	C₂×C₄×C₇₃	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₇₃	C₂³	1	1	yes	characteristic
40	C₂²⋊D₇₃	—	2	4	no	—
41	C₂×Dic₇₃	C₂³	1	2	yes	—
42	C₂×Dic₇₃	C₂³	1	2	yes	—
43	C₂×Dic₇₃	C₂³	1	2	yes	—
44	D₂₉₂	—	4	1	no	—
45	C₄×D₇₃	—	4	2	no	—
46	Dic₁₄₆	—	4	1	no	—
47	C₂²×C₇₃	—	2	1	no	—
48	C₂²×C₇₃	—	2	1	no	—
49	C₂²×C₇₃	#41	1	2	yes	—
50	C₂²×C₇₃	C₂⁴	1	1	yes	characteristic
51	C₂×D₇₃	—	2	2	no	—
52	C₂×D₇₃	—	2	2	no	—
53	C₂×D₇₃	—	2	2	no	—
54	C₄×C₇₃	—	2	2	no	—
55	C₄×C₇₃	—	2	1	no	—
56	C₄×C₇₃	—	2	2	no	—
57	C₄×C₇₃	—	2	1	no	—
58	Dic₇₃	—	2	2	no	—
59	Dic₇₃	—	2	2	no	—
60	Dic₇₃	—	2	2	no	—
61	C₂×C₇₃	—	2	2	no	—
62	C₂×C₇₃	C₂×#41	1	1	yes	characteristic
63	C₂×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
65	D₇₃	—	4	2	no	—
66	C₇₃	#539	1	1	yes	characteristic, complemented, Sylow
67	#539	—	73	1	no	maximal, Sylow
68	#126	—	73	2	no	—
69	#125	—	73	2	no	—
70	#123	—	73	2	no	—
71	#123	—	73	2	no	—
72	C₂²⋊Q₈	—	73	1	no	—
73	C₂²⋊Q₈	—	73	2	no	—
74	#121	—	73	1	no	—
75	C₄×D₄	—	73	2	no	—
76	C₂×C₄⋊C₄	D₇₃	1	1	yes	characteristic, complemented
77	C₂×Q₈	—	73	1	no	—
78	C₂×D₄	—	73	2	no	—
79	C₂×D₄	—	73	1	no	—
80	C₂²×C₄	—	73	2	no	—
81	C₂²×C₄	C₂×D₇₃	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₇₃	1	1	yes	characteristic
83	C₂²×C₄	C₂×D₇₃	1	1	yes	characteristic
84	C₄⋊C₄	—	73	2	no	—
85	C₄⋊C₄	—	73	2	no	—
86	C₄⋊C₄	—	73	2	no	—
87	C₄⋊C₄	—	73	2	no	—
88	C₄⋊C₄	—	73	1	no	—
89	C₄⋊C₄	—	73	1	no	—
90	C₄⋊C₄	C₂×D₇₃	1	2	yes	—
91	C₄⋊C₄	C₂×D₇₃	1	2	yes	—
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	—	73	2	no	—
97	C₄²	—	73	2	no	—
98	C₂³	—	73	2	no	—
99	C₂³	C₂²×D₇₃	1	1	yes	characteristic
100	Q₈	—	292	1	no	—
101	D₄	—	292	1	no	—
102	D₄	—	146	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	292	2	no	—
107	C₂×C₄	—	73	2	no	—
108	C₂×C₄	—	73	2	no	—
109	C₂×C₄	—	73	2	no	—
110	C₂×C₄	C₂²×D₇₃	1	2	yes	—
111	C₂×C₄	C₂²×D₇₃	1	2	yes	—
112	C₂×C₄	C₂²×D₇₃	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₇₃	1	1	yes	characteristic
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	146	2	no	—
117	C₂²	—	146	2	no	—
118	C₂²	—	146	2	no	—
119	C₂²	#11924028	1	2	yes	—
120	C₂²	C₂³×D₇₃	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	2	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	—	146	2	no	—
126	C₄	—	146	2	no	—
127	C₄	—	146	2	no	—
128	C₂	—	2	2	no	—
129	C₂	—	292	2	no	—
130	C₂	C₂×#11924028	1	1	yes	characteristic
131	C₂	#14556636	1	1	yes	characteristic
132	C₂	#14556639	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14556586

C₂

yes

maximal, complemented

#14556581

C₂

yes

maximal, complemented

#14556576

C₂

yes

maximal, complemented

#14556574

C₂

yes

maximal, complemented

#14556570

C₂

yes

maximal, characteristic, complemented

#14556569

C₂

yes

maximal, complemented

#14556565

C₂

yes

maximal, characteristic

C₄×C₂²⋊D₇₃

C₂

yes

maximal, complemented

C₂×C₄⋊C₄×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₄₆

C₂²

yes

characteristic

C₂×D₂₉₂

C₂²

yes

characteristic

C₂×C₂²⋊D₇₃

C₂²

yes

—

C₂×C₄×D₇₃

C₂²

yes

—

C₄⋊Dic₇₃

C₂²

yes

characteristic

C₄⋊Dic₇₃

C₂²

yes

characteristic

#11924007

C₂²

yes

—

#11924007

C₂²

yes

—

#11924007

C₂²

yes

—

#11924007

C₂²

yes

—

C₂²⋊Dic₇₃

C₂²

yes

—

#11924005

C₂²

yes

—

#11924005

C₂²

yes

—

#11924005

C₂²

yes

—

#11924005

C₂²

yes

—

C₄×Dic₇₃

C₂²

yes

—

C₂²×C₄×C₇₃

C₂²

yes

characteristic

C₂²×C₄×C₇₃

C₂²

yes

characteristic

C₂²×C₄×C₇₃

C₂²

yes

characteristic

C₄⋊C₄×C₇₃

C₂²

yes

—

C₄⋊C₄×C₇₃

C₂²

yes

—

C₂³×C₇₃

C₂³

yes

characteristic

C₂²×D₇₃

C₂³

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂³

yes

—

C₂×C₄×C₇₃

C₂³

yes

—

C₂×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₄×C₇₃

C₂³

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

C₂³

yes

—

C₂×Dic₇₃

C₂³

yes

—

C₂×Dic₇₃

C₂³

yes

—

D₂₉₂

—

C₄×D₇₃

—

Dic₁₄₆

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#41

yes

—

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

C₂×#41

yes

characteristic

C₂×C₇₃

H(5,𝔽₂)

yes

characteristic

C₂×C₇₃

2₋¹⁺⁴

yes

characteristic

D₇₃

—

C₇₃

#539

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

D₇₃

yes

characteristic, complemented

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₇₃

yes

characteristic

C₂²×C₄

C₂×D₇₃

yes

characteristic

C₂²×C₄

C₂×D₇₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₇₃

yes

—

C₄⋊C₄

C₂×D₇₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

C₂²×D₇₃

yes

characteristic

100

Q₈

—

292

—

101

D₄

—

292

—

102

D₄

—

146

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

292

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂²×D₇₃

yes

—

111

C₂×C₄

C₂²×D₇₃

yes

—

112

C₂×C₄

C₂²×D₇₃

yes

characteristic

113

C₂×C₄

C₂²×D₇₃

yes

characteristic

114

C₂²

—

115

C₂²

—

116

C₂²

—

146

—

117

C₂²

—

146

—

118

C₂²

—

146

—

119

C₂²

#11924028

yes

—

120

C₂²

C₂³×D₇₃

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

146

—

126

C₄

—

146

—

127

C₄

—

146

—

128

C₂

—

129

C₂

—

292

—

130

C₂

C₂×#11924028

yes

characteristic

131

C₂

#14556636

yes

characteristic

132

C₂

#14556639

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇₃
commutator, socle, cosocle	C₂²×C₇₃	C₂⁴
Fitting	C₂×C₄⋊C₄×C₇₃	C₂
prime radical	#14556565	C₂
hypercenter	C₂×C₄⋊C₄	D₇₃
hypocenter	C₇₃	#539
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₇₃

commutator, socle, cosocle

C₂²×C₇₃

C₂⁴

Fitting

C₂×C₄⋊C₄×C₇₃

C₂

prime radical

#14556565

C₂

hypercenter

C₂×C₄⋊C₄

D₇₃

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
1. #14556586	C₂	semidirect
2. #14556581	C₂	semidirect
3. #14556576	C₂	semidirect
4. #14556574	C₂	semidirect
5. #14556570	C₂	semidirect
6. #14556569	C₂	semidirect
8. C₄×C₂²⋊D₇₃	C₂	semidirect
9. C₂×C₄⋊C₄×C₇₃	C₂	semidirect
66. C₇₃	#539	semidirect
76. C₂×C₄⋊C₄	D₇₃	semidirect

H ≅ G/N

type

1. #14556586

C₂

semidirect

2. #14556581

C₂

semidirect

3. #14556576

C₂

semidirect

4. #14556574

C₂

semidirect

5. #14556570

C₂

semidirect

6. #14556569

C₂

semidirect

8. C₄×C₂²⋊D₇₃

C₂

semidirect

9. C₂×C₄⋊C₄×C₇₃

C₂

semidirect

66. C₇₃

#539

semidirect

76. C₂×C₄⋊C₄

D₇₃

semidirect

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

subgroup	index
#14556586	2
#14556581	2
#14556576	2
#14556574	2
#14556570	2
#14556569	2
#14556565	2
C₄×C₂²⋊D₇₃	2
C₂×C₄⋊C₄×C₇₃	2
#539	73

subgroup

index

C₄×C₂²⋊D₇₃

C₂×C₄⋊C₄×C₇₃

#539

Subgroups — #17989184

Conjugacy classes of subgroups

Complements

Maximal subgroups