Group #34013270 — subgroups

Subgroups — #34013270

Order 18688.

322 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 140 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	C₂×C₄⋊#11924031	C₂	1	2	yes	maximal
2	#22624969	C₂	1	4	yes	maximal
3	#22609618	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924031	C₂²	1	4	yes	—
5	C₄⋊#11924031	—	2	4	no	—
6	C₂×#14556502	C₄	1	2	yes	—
7	C₂×#14556502	C₄	1	1	yes	characteristic
8	C₂×C₄×#10596831	C₂²	1	2	yes	—
9	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
10	#376×C₇₃	C₄	1	1	yes	characteristic
11	C₂×#11924031	—	2	4	no	—
12	C₂×#11924031	—	2	8	no	—
13	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
14	C₂²×#10596831	C₂×C₄	1	2	yes	—
15	C₂²×#10596831	C₂³	1	1	yes	characteristic
16	#14556502	—	2	4	no	—
17	#14556502	—	2	2	no	—
18	C₄×#10596831	—	2	4	no	—
19	C₄×#10596831	D₄	1	2	yes	—
20	C₄×#10596831	Q₈	1	2	yes	—
21	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
23	#95×C₇₃	C₈	1	4	yes	—
24	#11924031	—	4	4	no	—
25	C₂×#10596831	—	2	2	no	—
26	C₂×#10596831	—	2	2	no	—
27	C₂×#10596831	—	2	2	no	—
28	C₂×#10596831	C₄⋊C₄	1	2	yes	—
29	C₂×#10596831	M₁₆	1	4	yes	—
30	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
31	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
32	C₂×#10596831	#41	1	4	yes	—
33	C₂²×C₄×C₇₃	C₂×C₈	1	4	yes	—
34	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
35	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₇₃	—	2	2	no	—
37	C₄²×C₇₃	—	2	2	no	—
38	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
39	C₄²×C₇₃	M₁₆	1	2	yes	—
40	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
41	C₂×C₄×C₇₃	—	2	4	no	—
42	C₂×C₄×C₇₃	—	2	4	no	—
43	C₂×C₄×C₇₃	—	2	8	no	—
44	C₂×C₄×C₇₃	C₄⋊C₈	1	4	yes	—
45	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₇₃	#117	1	2	yes	—
47	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
48	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
49	#10596831	—	2	2	no	—
50	#10596831	—	2	4	no	—
51	#10596831	#124	1	2	yes	—
52	#10596831	#125	1	2	yes	—
53	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
54	C₂²×C₇₃	#431	1	2	yes	—
55	C₂²×C₇₃	#504	1	4	yes	—
56	C₄×C₇₃	—	2	4	no	—
57	C₄×C₇₃	—	4	4	no	—
58	C₄×C₇₃	#382	1	1	yes	characteristic
59	C₄×C₇₃	#423	1	1	yes	characteristic
60	C₄×C₇₃	#432	1	2	yes	—
61	#12880	—	73	1	no	maximal, Sylow
62	C₂×C₇₃	#1831	1	1	yes	characteristic
63	C₂×C₇₃	#2151	1	2	yes	—
64	C₂×C₇₃	#2164	1	4	yes	—
65	C₂×C₄⋊C₁₆	—	73	2	no	—
66	#1726	—	73	1	no	—
67	#1374	—	73	4	no	—
68	C₇₃	#12880	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₁₆	—	73	4	no	—
70	C₂×#97	—	73	2	no	—
71	C₂×#97	—	73	1	no	—
72	C₂×C₄×C₈	—	73	1	no	—
73	C₂×C₄×C₈	—	73	2	no	—
74	#376	#64408	1	1	yes	characteristic
75	C₄⋊C₁₆	—	146	4	no	—
76	C₂²×C₈	—	73	1	no	—
77	C₂²×C₈	—	73	2	no	—
78	C₂²×C₈	—	73	1	no	—
79	C₂×C₄²	C₂×#64408	1	2	yes	—
80	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
81	C₂×C₁₆	—	146	4	no	—
82	C₂×C₁₆	—	146	8	no	—
83	#97	—	146	4	no	—
84	#97	—	146	2	no	—
85	C₄×C₈	—	73	2	no	—
86	C₄×C₈	—	73	2	no	—
87	C₄×C₈	—	146	4	no	—
88	#95	#10596833	1	4	yes	—
89	C₂²×C₄	C₂×#10596833	1	4	yes	—
90	C₂²×C₄	C₂×#10596833	1	2	yes	—
91	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
92	C₂×C₈	—	73	1	no	—
93	C₂×C₈	—	73	1	no	—
94	C₂×C₈	—	73	4	no	—
95	C₂×C₈	—	73	4	no	—
96	C₂×C₈	—	73	2	no	—
97	C₂×C₈	—	146	2	no	—
98	C₂×C₈	—	146	2	no	—
99	C₂×C₈	—	146	2	no	—
100	C₄²	—	2	2	no	—
101	C₄²	—	2	2	no	—
102	C₄²	C₄⋊#64408	1	2	yes	—
103	C₄²	#11924038	1	2	yes	—
104	C₁₆	—	292	4	no	—
105	C₂³	C₂²×#10596833	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	8	no	—
108	C₂×C₄	—	2	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	D₇₃⋊M₁₆	1	2	yes	—
113	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
114	C₈	—	73	2	no	—
115	C₈	—	73	2	no	—
116	C₈	—	146	2	no	—
117	C₈	—	146	4	no	—
118	C₂²	C₂×C₄⋊#10596833	1	1	yes	characteristic
119	C₂²	#17989603	1	2	yes	—
120	C₂²	#17989645	1	4	yes	—
121	C₄	—	2	4	no	—
122	C₄	—	4	4	no	—
123	C₄	#17989531	1	1	yes	characteristic
124	C₄	#17989592	1	1	yes	characteristic
125	C₄	#17989609	1	2	yes	—
126	C₂	#22625846	1	1	yes	characteristic
127	C₂	#22626266	1	2	yes	—
128	C₂	#22626296	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11924031

C₂

yes

maximal

#22624969

C₂

yes

maximal

#22609618

C₂

yes

maximal, characteristic

C₂²×#11924031

C₂²

yes

—

C₄⋊#11924031

—

C₂×#14556502

C₄

yes

—

C₂×#14556502

C₄

yes

characteristic

C₂×C₄×#10596831

C₂²

yes

—

C₂×C₄×#10596831

C₂²

yes

characteristic

#376×C₇₃

C₄

yes

—

—

C₂×C₄

yes

characteristic

C₂²×#10596831

C₂×C₄

yes

—

C₂²×#10596831

C₂³

yes

—

—

—

D₄

yes

—

C₄×#10596831

Q₈

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

#95×C₇₃

C₈

yes

—

—

—

—

—

C₄⋊C₄

yes

—

C₂×#10596831

M₁₆

yes

—

C₂×#10596831

C₂×D₄

yes

characteristic

C₂×#10596831

C₂×Q₈

yes

characteristic

C₂×#10596831

#41

yes

—

C₂²×C₄×C₇₃

C₂×C₈

yes

—

C₂²×C₄×C₇₃

C₂×C₈

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

M₁₆

yes

—

C₂³×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

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

C₂×M₁₆

yes

—

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

—

yes

—

#10596831

#125

yes

—

C₂²×C₇₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₇₃

#431

yes

—

C₂²×C₇₃

#504

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#2164

yes

—

C₂×C₄⋊C₁₆

—

#1726

—

#1374

—

C₇₃

#12880

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#376

#64408

yes

characteristic

C₄⋊C₁₆

—

146

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×#64408

yes

—

C₂×C₄²

C₂×#64408

yes

characteristic

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

#97

—

146

—

#97

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

146

—

#95

#10596833

yes

—

C₂²×C₄

C₂×#10596833

yes

—

C₂²×C₄

C₂×#10596833

yes

—

C₂²×C₄

C₂²×#64408

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₄⋊#64408

yes

—

103

C₄²

#11924038

yes

—

104

C₁₆

—

292

—

105

C₂³

C₂²×#10596833

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₄⋊#10596833

yes

—

110

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

111

C₂×C₄

#14556670

yes

—

112

C₂×C₄

D₇₃⋊M₁₆

yes

—

113

C₂×C₄

C₂×#11924038

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

—

146

—

117

C₈

—

146

—

118

C₂²

C₂×C₄⋊#10596833

yes

characteristic

119

C₂²

#17989603

yes

—

120

C₂²

#17989645

yes

—

121

C₄

—

122

C₄

—

123

C₄

#17989531

yes

characteristic

124

C₄

#17989592

yes

characteristic

125

C₄

#17989609

yes

—

126

C₂

#22625846

yes

characteristic

127

C₂

#22626266

yes

—

128

C₂

#22626296

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator, socle, prime radical	C₂³×C₇₃	C₂²×C₈
Fitting	#376×C₇₃	C₄
cosocle	C₂²×#10596831	C₂³
hypercenter	#376	#64408
hypocenter	C₇₃	#12880
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ #376

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×#64408

commutator, socle, prime radical

C₂³×C₇₃

C₂²×C₈

Fitting

#376×C₇₃

C₄

C₂³

C₇₃

G ▷ C₂³×C₇₃ ▷ C₇₃

C₁ ◁ C₂²×C₄ ◁ #376

N	H ≅ G/N	type
68. C₇₃	#12880	semidirect

H ≅ G/N

type

68. C₇₃

#12880

semidirect

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

subgroup	index
C₂×C₄⋊#11924031	2
#22624969	2
#22609618	2
#12880	73

subgroup

index

Subgroups — #34013270

Conjugacy classes of subgroups

Complements

Maximal subgroups