Group #34005437 — subgroups

Subgroups — #34005437

Order 18688.

262 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 80 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	#22625856	C₂	1	1	yes	maximal, characteristic
2	#22625853	C₂	1	1	yes	maximal, characteristic
3	#22624785	C₂	1	4	yes	maximal
4	#22608213	C₂	1	1	yes	maximal, characteristic
5	#17989592	C₂²	1	2	yes	—
6	#17989531	C₂²	1	2	yes	—
7	#17989463	—	2	2	no	—
8	#17989461	—	2	2	no	—
9	#17988638	C₂²	1	2	yes	—
10	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
11	#17988309	C₄	1	2	yes	complemented
12	C₈×#10596831	C₄	1	1	yes	characteristic, complemented
13	C₈²×C₇₃	C₄	1	1	yes	characteristic, complemented
14	C₂×#11924038	—	2	2	no	—
15	C₂×C₄⋊#64408	—	2	2	no	—
16	#14556658	—	2	4	no	—
17	(C₂×#64408)⋊C₄	—	2	4	no	—
18	C₂×C₈×D₇₃	D₄	1	2	yes	—
19	C₄²×D₇₃	C₂³	1	1	yes	characteristic
20	#14556520	—	2	2	no	—
21	#14556503	—	2	2	no	—
22	C₄×#10596831	C₂×C₄	1	2	yes	—
23	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
24	C₈×Dic₇₃	Q₈	1	2	yes	—
25	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×#64408	—	4	2	no	—
28	#11924038	—	4	4	no	—
29	C₂×#10596833	—	4	2	no	—
30	C₄⋊#64408	—	4	4	no	—
31	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×D₇₃	#41	1	2	yes	—
33	C₂×#10596831	—	2	2	no	—
34	C₂×#10596831	C₄⋊C₄	1	2	yes	—
35	C₈×D₇₃	—	2	4	no	—
36	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
37	C₄×Dic₇₃	#41	1	2	yes	—
38	C₂×C₈×C₇₃	—	2	2	no	—
39	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
40	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×D₇₃	#124	1	1	yes	characteristic
42	C₂×#64408	—	8	4	no	—
43	C₂×#64408	—	8	2	no	—
44	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₇₃	#117	1	2	yes	—
46	C₂×Dic₇₃	#124	1	1	yes	characteristic
47	C₂×Dic₇₃	#125	1	2	yes	—
48	C₄×D₇₃	—	4	2	no	—
49	C₄×D₇₃	#135	1	4	yes	—
50	#10596833	—	8	2	no	—
51	C₈×C₇₃	—	2	2	no	—
52	C₈×C₇₃	—	4	2	no	—
53	#10596831	—	2	2	no	—
54	#10596831	—	4	2	no	—
55	C₂²×C₇₃	#382	1	1	yes	characteristic
56	C₂×D₇₃	—	2	2	no	—
57	C₂×D₇₃	#488	1	2	yes	—
58	#64408	—	16	2	no	—
59	C₄×C₇₃	—	2	2	no	—
60	C₄×C₇₃	#427	1	2	yes	—
61	Dic₇₃	—	2	2	no	—
62	Dic₇₃	#488	1	2	yes	—
63	#8792	—	73	1	no	maximal, Sylow
64	C₂×C₇₃	#1833	1	2	yes	—
65	C₂×C₇₃	#1835	1	1	yes	characteristic
66	D₇₃	—	2	2	no	—
67	#1837	—	73	1	no	—
68	#1835	—	73	1	no	—
69	#1450	—	73	1	no	—
70	#1286	—	73	4	no	—
71	C₇₃	#8792	1	1	yes	characteristic, complemented, Sylow
72	#431	—	73	2	no	—
73	#423	—	73	2	no	—
74	C₂×C₄×C₈	—	73	1	no	—
75	#382	—	73	2	no	—
76	#341	—	146	2	no	—
77	#340	—	146	2	no	—
78	#322	—	73	2	no	—
79	C₈²	—	73	1	no	—
80	C₈²	#64408	1	1	yes	characteristic, complemented
81	C₂×M₁₆	—	146	2	no	—
82	C₂²×C₈	—	73	2	no	—
83	C₂×C₄⋊C₄	—	146	2	no	—
84	C₂×C₄²	—	73	1	no	—
85	C₄⋊C₈	—	146	2	no	—
86	C₄⋊C₈	—	146	4	no	—
87	C₂²⋊C₈	—	146	2	no	—
88	C₄×C₈	—	73	2	no	—
89	C₄×C₈	—	73	1	no	—
90	C₄×C₈	—	73	2	no	—
91	C₄×C₈	C₂×#64408	1	2	yes	—
92	C₄×C₈	C₂×#64408	1	1	yes	characteristic
93	#95	—	146	4	no	—
94	C₂²×C₄	—	292	2	no	—
95	C₂²×C₄	—	73	2	no	—
96	C₂²×C₄	—	73	1	no	—
97	M₁₆	—	292	4	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	292	2	no	—
100	C₂×C₈	—	73	2	no	—
101	C₂×C₈	—	146	4	no	—
102	C₂×C₈	—	146	2	no	—
103	C₂×C₈	C₄⋊#64408	1	2	yes	—
104	C₄⋊C₄	—	292	4	no	—
105	C₄²	—	73	1	no	—
106	C₄²	—	73	2	no	—
107	C₄²	C₂²×#64408	1	1	yes	characteristic
108	C₂³	—	73	1	no	—
109	C₂×C₄	—	292	2	no	—
110	C₂×C₄	—	584	2	no	—
111	C₂×C₄	—	584	4	no	—
112	C₂×C₄	—	73	1	no	—
113	C₂×C₄	—	73	4	no	—
114	C₂×C₄	—	73	2	no	—
115	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
116	C₂×C₄	#14556670	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	292	2	no	—
120	C₈	—	584	2	no	—
121	C₈	—	146	2	no	—
122	C₂²	—	73	2	no	—
123	C₂²	—	146	2	no	—
124	C₂²	#17989531	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	73	2	no	—
127	C₄	—	1168	2	no	—
128	C₄	—	146	2	no	—
129	C₄	#17989597	1	2	yes	—
130	C₂	—	146	2	no	—
131	C₂	#22625850	1	2	yes	—
132	C₂	#22625853	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625856

C₂

yes

maximal, characteristic

#22625853

C₂

yes

maximal, characteristic

#22624785

C₂

yes

maximal

#22608213

C₂

yes

maximal, characteristic

#17989592

C₂²

yes

—

#17989531

C₂²

yes

—

#17989463

—

#17989461

—

#17988638

C₂²

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988309

C₄

yes

complemented

C₈×#10596831

C₄

yes

characteristic, complemented

C₈²×C₇₃

C₄

yes

characteristic, complemented

C₂×#11924038

—

C₂×C₄⋊#64408

—

#14556658

—

(C₂×#64408)⋊C₄

—

C₂×C₈×D₇₃

D₄

yes

—

C₄²×D₇₃

C₂³

yes

—

—

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₈×Dic₇₃

Q₈

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×#64408

—

#11924038

—

C₂×#10596833

—

C₄⋊#64408

—

C₂×C₄×D₇₃

C₂×D₄

yes

characteristic

C₂×C₄×D₇₃

#41

yes

—

C₂×#10596831

—

C₂×#10596831

C₄⋊C₄

yes

—

C₈×D₇₃

—

C₄×Dic₇₃

C₂×Q₈

yes

characteristic

C₄×Dic₇₃

#41

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

C₂²×C₄

yes

C₂²×D₇₃

yes

—

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

#117

yes

—

C₂×Dic₇₃

#124

yes

characteristic

C₂×Dic₇₃

#125

yes

—

C₄×D₇₃

—

C₄×D₇₃

#135

yes

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

#382

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

#488

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

#427

yes

—

Dic₇₃

—

Dic₇₃

#488

yes

—

#8792

—

maximal, Sylow

C₂×C₇₃

#1833

yes

—

C₂×C₇₃

#1835

yes

characteristic

D₇₃

—

—

—

—

—

C₇₃

#8792

yes

characteristic, complemented, Sylow

#431

—

#423

—

C₂×C₄×C₈

—

#382

—

#341

—

146

—

#340

—

146

—

#322

—

C₈²

—

C₈²

#64408

yes

characteristic, complemented

C₂×M₁₆

—

146

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

—

C₄⋊C₈

—

146

—

C₄⋊C₈

—

146

—

C₂²⋊C₈

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#64408

yes

—

C₄×C₈

C₂×#64408

yes

characteristic

#95

—

146

—

C₂²×C₄

—

292

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

292

—

C₂×C₈

—

C₂×C₈

—

292

—

100

C₂×C₈

—

101

C₂×C₈

—

146

—

102

C₂×C₈

—

146

—

103

C₂×C₈

C₄⋊#64408

yes

—

104

C₄⋊C₄

—

292

—

105

C₄²

—

106

C₄²

—

107

C₄²

C₂²×#64408

yes

characteristic

108

C₂³

—

109

C₂×C₄

—

292

—

110

C₂×C₄

—

584

—

111

C₂×C₄

—

584

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

116

C₂×C₄

#14556670

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

292

—

120

C₈

—

584

—

121

C₈

—

146

—

122

C₂²

—

123

C₂²

—

146

—

124

C₂²

#17989531

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

1168

—

128

C₄

—

146

—

129

C₄

#17989597

yes

—

130

C₂

—

146

—

131

C₂

#22625850

yes

—

132

C₂

#22625853

yes

characteristic

	subgroup	G/H
center	C₂²	#17989531
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×#64408
Fitting	C₈²×C₇₃	C₄
socle	C₂²×C₇₃	#382
prime radical	C₂²×D₇₃	#124
cosocle	C₄²×D₇₃	C₂³
hypercenter	C₈²	#64408
hypocenter	C₇₃	#8792
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#17989531

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×#64408

Fitting

C₈²×C₇₃

C₄

C₂²×C₇₃

C₂²×D₇₃

C₄²×D₇₃

C₂³

C₈²

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
11. #17988309	C₄	semidirect
12. C₈×#10596831	C₄	semidirect
13. C₈²×C₇₃	C₄	semidirect
71. C₇₃	#8792	semidirect
80. C₈²	#64408	semidirect

H ≅ G/N

type

11. #17988309

C₄

semidirect

12. C₈×#10596831

C₄

semidirect

13. C₈²×C₇₃

C₄

semidirect

71. C₇₃

#8792

semidirect

80. C₈²

#64408

semidirect

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

subgroup	index
#22625856	2
#22625853	2
#22624785	2
#22608213	2
#8792	73

subgroup

index

Subgroups — #34005437

Conjugacy classes of subgroups

Complements

Maximal subgroups