Finite Group Databasebeta

← #13054820

Subgroups — #13054820

Order 1344.

216 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10621432C211yesmaximal, characteristic, complemented
2C7×#948C211yesmaximal, characteristic, complemented
3C3×#6512C211yesmaximal, characteristic, complemented
4C2×#66561C211yesmaximal, characteristic, complemented
5#10620528C211yesmaximal, characteristic, complemented
6#10620495C211yesmaximal, characteristic, complemented
7#10620493C211yesmaximal, characteristic, complemented
8#89914—31nomaximal, Hall
9C3×C7×#41C2211yescharacteristic
10C2×D84C2211yescharacteristic
11C2×C3×D28C2211yescharacteristic
12#66634—21no—
13#66632—21no—
14#66627C2211yescharacteristic
15C7×#208—21no—
16C7×#208C2211yescharacteristic
17C2×C7×C3⋊C8—21no—
18C2×C7×C3⋊C8C2211yescharacteristic
19C3×#1143—21no—
20C3×#1141—21no—
21C3×#1136C2211yescharacteristic
22#66561—21no—
23#66561—21no—
24C7×#131—31no—
25#6512—31no—
26#6512D311yescharacteristic, complemented
27C2×D56—31no—
28#6397—31no—
29#6397—31no—
30#6393—31no—
31#5199—71nomaximal, Hall
32C22×D21—21no—
33C22×C3×D7—21no—
34C3×C7×Q8D411yescharacteristic
35C3×C7×D4—21no—
36C3×C7×D4D411yescharacteristic
37C2×C3×C4×C7—21no—
38C2×C3×C4×C7C2311yescharacteristic
39D84—21no—
40D84—21no—
41C3×D28—21no—
42C3×D28—21no—
43C3⋊(C7⋊C8)—21no—
44C3×C7⋊C8—21no—
45C7×C3⋊C8—21no—
46C7×C3⋊C8D411yescharacteristic
47C7×C3⋊C8D411yescharacteristic
48C7×#41C2×D311yescharacteristic
49C2×D28—31no—
50C2×D28C2×D311yescharacteristic
51C7×M16—31no—
52C7×M16—61no—
53C2×C7×C8—31no—
54C2×C7×C8—61no—
55#1143—21no—
56#1143—61no—
57#1141—21no—
58#1141—61no—
59#1136—31no—
60#1136C2×D311yescharacteristic
61D56—61no—
62D56—61no—
63C3×AGL(1,ℤ/8)—71no—
64#949—71no—
65#948D711yescharacteristic, complemented
66C2×#213—71no—
67#833—71no—
68#823—71no—
69#822—71no—
70C22×C3×C7—21no—
71C22×C3×C7#4111yescharacteristic
72C2×D21—41no—
73C2×D21—81no—
74C2×C3×D7—41no—
75C2×C3×D7—81no—
76C3×C4×C7—21no—
77C3×C4×C7C2×D411yescharacteristic
78C3×C4×C7C2×D411yescharacteristic
79#472—211noSylow
80C22×D7—21no—
81C22×D7—61no—
82C7×Q8#6711yescharacteristic
83C7×D4—21no—
84C7×D4#6711yescharacteristic
85C2×C4×C7—21no—
86C2×C4×C7C22×D311yescharacteristic
87D28—21no—
88D28—21no—
89D28—61no—
90D28—61no—
91C7×C8—31no—
92C7×C8—31no—
93C7×C8—61no—
94C7⋊C8—21no—
95C7⋊C8—61no—
96C3×#41C2×D711yescharacteristic
97C2×C3×D4—71no—
98C2×D12—71no—
99C3×SD16—141no—
100C3×D8—141no—
101C3×M16—71no—
102#215—141no—
103#213—141no—
104#213—141no—
105#213—141no—
106#208—21no—
107#208—71no—
108#208C2×D711yescharacteristic
109C2×C3⋊C8—21no—
110C2×C3⋊C8C2×D711yescharacteristic
111C2×C3×C7—21no—
112C2×C3×C7—41no—
113C2×C3×C7#12111yescharacteristic
114D21—81no—
115C3×D7—81no—
116AGL(1,ℤ/8)—71no—
117AGL(1,ℤ/8)—211no—
118C2×D8—211no—
119#131—31no—
120#108—211no—
121#100—211no—
122#100—211no—
123C22×C7—21no—
124C22×C7#23711yescharacteristic
125C2×D7—41no—
126C2×D7—81no—
127C2×D7—121no—
128C2×D7—241no—
129C4×C7—21no—
130C4×C7D3×D411yescharacteristic
131C4×C7C2×#6711yescharacteristic
132C23×C3—141no—
133C22×D3—141no—
134C3×Q8D2811yescharacteristic
135C3×D4—21no—
136C3×D4—141no—
137C3×D4—141no—
138C3×D4D2811yescharacteristic
139C2×C3×C4—21no—
140C2×C3×C4C22×D711yescharacteristic
141D12—141no—
142D12—141no—
143C3×C8—141no—
144C3⋊C8—21no—
145C3⋊C8—141no—
146C3⋊C8C22⋊D711yescharacteristic
147C3⋊C8C22⋊D711yescharacteristic
148C3×C7#47211yescharacteristic, complemented, Hall
149#41D3×D711yescharacteristic
150C2×D4—71no—
151C2×D4—211no—
152SD16—141no—
153SD16—421no—
154D8—141no—
155D8—421no—
156D8—421no—
157D8—421no—
158M16—31no—
159M16—61no—
160M16—71no—
161M16—211no—
162C2×C8—31no—
163C2×C8—61no—
164C2×C7—21no—
165C2×C7—41no—
166C2×C7#93911yescharacteristic
167D7—81no—
168D7—241no—
169C22×C3—21no—
170C22×C3—281no—
171C22×C3—561no—
172C22×C3#115711yescharacteristic
173C2×D3—281no—
174C2×D3—561no—
175C3×C4—21no—
176C3×C4C2×D2811yescharacteristic
177C3×C4C2×C22⋊D711yescharacteristic
178C23—141no—
179C23—421no—
180Q8#424611yescharacteristic
181D4—21no—
182D4—141no—
183D4—141no—
184D4—421no—
185D4—421no—
186D4#424611yescharacteristic
187C2×C4—21no—
188C2×C4C2×D3×D711yescharacteristic
189C8—31no—
190C8—31no—
191C8—61no—
192C8—141no—
193C8—421no—
194C7#519911yescharacteristic, complemented, Sylow
195C2×C3—21no—
196C2×C3—41no—
197C2×C3—561no—
198C2×C3#649311yescharacteristic
199D3—561no—
200C22—21no—
201C22—281no—
202C22—561no—
203C22—841no—
204C22—1681no—
205C22#6668311yescharacteristic
206C4—21no—
207C4C2×#424611yescharacteristic
208C4D3×C22⋊D711yescharacteristic
209C3#8991411yescharacteristic, complemented, Sylow
210C2—21no—
211C2—41no—
212C2—561no—
213C2—1681no—
214C2#1062113111yescharacteristic
subgroupG/H
centerC2#10621131
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×#41C22
socleC2×C3×C7#121
hypercenter#41D3×D7
hypocenterC3×C7#472
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #10621432C2semidirect
2. C7×#948C2semidirect
3. C3×#6512C2semidirect
4. C2×#66561C2semidirect
5. #10620528C2semidirect
6. #10620495C2semidirect
7. #10620493C2semidirect
26. #6512D3semidirect
65. #948D7semidirect
148. C3×C7#472semidirect
194. C7#5199semidirect
209. C3#89914semidirect

Maximal subgroups

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

subgroupindex
#106214322
C7×#9482
C3×#65122
C2×#665612
#106205282
#106204952
#106204932
#899143
#51997