Group #34013284 — subgroups

Subgroups — #34013284

Order 18688.

302 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 124 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	#22626263	C₂	1	2	yes	maximal
2	#22624976	C₂	1	4	yes	maximal
3	#22609619	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556665	C₂²	1	4	yes	—
5	C₄⋊#11924031	—	2	4	no	—
6	C₂×#14556501	C₄	1	2	yes	—
7	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
9	#17988433	C₄	1	1	yes	characteristic
10	C₂×#97×C₇₃	C₄	1	1	yes	characteristic
11	#14556665	—	2	8	no	—
12	C₂×#11924031	—	2	4	no	—
13	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
14	C₂²×#10596831	C₂³	1	1	yes	characteristic
15	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
16	#14556501	—	2	4	no	—
17	C₄×#10596831	D₄	1	2	yes	—
18	C₄×#10596831	Q₈	1	2	yes	—
19	C₈×Dic₇₃	—	2	4	no	—
20	#14556498	—	2	2	no	—
21	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
23	#97×C₇₃	C₈	1	4	yes	—
24	#11924031	—	4	4	no	—
25	C₂²×Dic₇₃	—	2	2	no	—
26	C₂×#10596831	—	2	2	no	—
27	C₂×#10596831	C₄⋊C₄	1	2	yes	—
28	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
29	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
30	C₂×#10596831	#41	1	4	yes	—
31	C₄×Dic₇₃	—	2	2	no	—
32	C₄×Dic₇₃	M₁₆	1	4	yes	—
33	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
34	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₇₃	—	2	2	no	—
36	C₂×C₈×C₇₃	—	2	2	no	—
37	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
38	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
39	C₄²×C₇₃	C₂×C₈	1	2	yes	—
40	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
41	C₂×C₄×C₇₃	—	4	2	no	—
42	C₂×C₄×C₇₃	C₄⋊C₈	1	4	yes	—
43	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₃	#117	1	2	yes	—
45	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
46	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
47	C₂×Dic₇₃	—	2	4	no	—
48	C₂×Dic₇₃	—	4	4	no	—
49	C₈×C₇₃	—	2	4	no	—
50	#10596831	—	2	2	no	—
51	#10596831	#124	1	2	yes	—
52	#10596831	#125	1	2	yes	—
53	C₂²×C₇₃	—	2	2	no	—
54	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
55	C₂²×C₇₃	#432	1	2	yes	—
56	C₄×C₇₃	—	2	4	no	—
57	C₄×C₇₃	#382	1	1	yes	characteristic
58	C₄×C₇₃	#423	1	1	yes	characteristic
59	C₄×C₇₃	#431	1	2	yes	—
60	Dic₇₃	—	4	4	no	—
61	#12884	—	73	1	no	maximal, Sylow
62	C₂×C₇₃	—	2	2	no	—
63	C₂×C₇₃	#1831	1	1	yes	characteristic
64	C₂×C₇₃	#2153	1	2	yes	—
65	#2150	—	73	2	no	—
66	#1726	—	73	1	no	—
67	#1376	—	73	4	no	—
68	C₇₃	#12884	1	1	yes	characteristic, complemented, Sylow
69	C₂×M₃₂	—	73	4	no	—
70	C₂×#97	—	73	2	no	—
71	C₂×#97	#64408	1	1	yes	characteristic
72	C₂×C₄×C₈	—	73	1	no	—
73	C₂×C₄×C₈	—	73	2	no	—
74	#376	—	73	1	no	—
75	C₄⋊C₁₆	—	146	4	no	—
76	C₂²×C₈	—	73	1	no	—
77	C₂²×C₈	—	73	1	no	—
78	C₂²×C₈	C₂×#64408	1	2	yes	—
79	C₂×C₄²	—	73	2	no	—
80	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
81	M₃₂	—	146	8	no	—
82	C₂×C₁₆	—	146	4	no	—
83	#97	—	146	4	no	—
84	#97	#10596833	1	4	yes	—
85	C₄×C₈	—	73	2	no	—
86	C₄×C₈	—	73	2	no	—
87	C₄×C₈	—	146	4	no	—
88	#95	—	146	2	no	—
89	C₂²×C₄	—	146	2	no	—
90	C₂²×C₄	#11924038	1	2	yes	—
91	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	73	1	no	—
95	C₂×C₈	—	73	1	no	—
96	C₂×C₈	—	73	4	no	—
97	C₂×C₈	—	73	2	no	—
98	C₂×C₈	—	146	2	no	—
99	C₂×C₈	C₂×#10596833	1	4	yes	—
100	C₄²	—	73	4	no	—
101	C₄²	—	146	2	no	—
102	C₄²	C₄⋊#64408	1	2	yes	—
103	C₄²	C₂×#10596833	1	2	yes	—
104	C₁₆	—	292	4	no	—
105	C₂³	C₂×#11924038	1	1	yes	characteristic
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	292	4	no	—
108	C₂×C₄	—	146	4	no	—
109	C₂×C₄	C₄⋊#10596833	1	4	yes	—
110	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
111	C₂×C₄	#14556670	1	2	yes	—
112	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
113	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
114	C₈	—	2	4	no	—
115	C₈	—	73	2	no	—
116	C₈	—	73	2	no	—
117	C₈	—	146	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	C₂×C₄⋊#10596833	1	1	yes	characteristic
120	C₂²	#17989609	1	2	yes	—
121	C₄	—	2	4	no	—
122	C₄	—	292	4	no	—
123	C₄	#17989531	1	1	yes	characteristic
124	C₄	#17989592	1	1	yes	characteristic
125	C₄	#17989603	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	#22625846	1	1	yes	characteristic
128	C₂	#22626272	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22626263

C₂

yes

maximal

#22624976

C₂

yes

maximal

#22609619

C₂

yes

maximal, characteristic

C₂×#14556665

C₂²

yes

—

C₄⋊#11924031

—

C₂×#14556501

C₄

yes

—

C₂×C₄×#10596831

C₂²

yes

characteristic

C₂×C₈×Dic₇₃

C₂²

yes

—

#17988433

C₄

yes

characteristic

C₂×#97×C₇₃

C₄

yes

—

—

C₂×C₄

yes

characteristic

C₂²×#10596831

C₂³

yes

characteristic

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

#14556501

—

C₄×#10596831

D₄

yes

—

C₄×#10596831

Q₈

yes

—

C₈×Dic₇₃

—

#14556498

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

#97×C₇₃

C₈

yes

—

#11924031

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

C₄⋊C₄

yes

—

C₂×#10596831

C₂×D₄

yes

characteristic

C₂×#10596831

C₂×Q₈

yes

characteristic

C₂×#10596831

#41

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

M₁₆

yes

—

C₂²×C₄×C₇₃

M₁₆

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₄²×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

C₂×C₈

yes

—

C₂³×C₇₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄⋊C₈

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

#117

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

#124

yes

—

#10596831

#125

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₇₃

#432

yes

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

#12884

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1831

yes

characteristic

C₂×C₇₃

#2153

yes

—

#2150

—

#1726

—

#1376

—

C₇₃

#12884

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×#97

—

C₂×#97

#64408

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#376

—

C₄⋊C₁₆

—

146

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×#64408

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×#64408

yes

characteristic

M₃₂

—

146

—

C₂×C₁₆

—

146

—

#97

—

146

—

#97

#10596833

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

146

—

#95

—

146

—

C₂²×C₄

—

146

—

C₂²×C₄

#11924038

yes

—

C₂²×C₄

C₂²×#64408

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

C₂×#10596833

yes

—

100

C₄²

—

101

C₄²

—

146

—

102

C₄²

C₄⋊#64408

yes

—

103

C₄²

C₂×#10596833

yes

—

104

C₁₆

—

292

—

105

C₂³

C₂×#11924038

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

292

—

108

C₂×C₄

—

146

—

109

C₂×C₄

C₄⋊#10596833

yes

—

110

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

111

C₂×C₄

#14556670

yes

—

112

C₂×C₄

C₂²×#10596833

yes

characteristic

113

C₂×C₄

D₇₃⋊M₁₆

yes

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

146

—

118

C₂²

—

119

C₂²

C₂×C₄⋊#10596833

yes

characteristic

120

C₂²

#17989609

yes

—

121

C₄

—

122

C₄

—

292

—

123

C₄

#17989531

yes

characteristic

124

C₄

#17989592

yes

characteristic

125

C₄

#17989603

yes

—

126

C₂

—

127

C₂

#22625846

yes

characteristic

128

C₂

#22626272

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×#97×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄⋊C₈
prime radical	C₂³×C₇₃	C₂×M₁₆
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂×#97	#64408
hypocenter	C₇₃	#12884
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂×C₄

C₂×C₄⋊#64408

commutator

C₂×C₄×C₇₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#64408

Fitting

C₂×#97×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₄⋊C₈

prime radical

C₂³×C₇₃

C₂×M₁₆

C₂³

C₂×#97

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
68. C₇₃	#12884	semidirect

H ≅ G/N

type

68. C₇₃

#12884

semidirect

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

subgroup	index
#22626263	2
#22624976	2
#22609619	2
#12884	73

subgroup

index

Subgroups — #34013284

Conjugacy classes of subgroups

Complements

Maximal subgroups