Finite Group Databasebeta

← #13056407

Subgroups — #13056407

Order 1344.

324 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#66730C211yesmaximal, characteristic
2C2×#66622C211yesmaximal, characteristic, complemented
3C4⋊C4×D21C211yesmaximal, characteristic, complemented
4C3×C7×#102C211yesmaximal, characteristic, complemented
5#10620671C211yesmaximal, characteristic, complemented
6#10620655C211yesmaximal, characteristic, complemented
7#10620649C211yesmaximal, characteristic, complemented
8#89543—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×#4268—21no—
11C22×Dic21—21no—
12#66730—22no—
13#66730C414yescomplemented
14C2×C4×D21—21no—
15C2×C4×D21C2211yescharacteristic
16C2×Dic42C2211yescharacteristic
17C2×C3×C7×C8C2211yescharacteristic
18C3×C7×C4⋊C4C2211yescharacteristic
19C4⋊Dic21C2211yescharacteristic
20#66629—21no—
21C2×C3⋊(C7⋊C8)C2211yescharacteristic
22#66622—22no—
23C2×#1159—31no—
24C2×#1131—31no—
25D7×C4⋊C4—31no—
26C7×#102D311yescharacteristic, complemented
27#6406—31no—
28#6390—31no—
29#6384—31no—
30#4771—71nomaximal, Hall
31C23×C3×C7—21no—
32C22×D21D411yescharacteristic
33C3×C7×D4—21no—
34C3×C7×D4C2×C412yescomplemented
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7C2311yescharacteristic
37#4268—24no—
38C2×Dic21—21no—
39C2×Dic21—24no—
40C2×Dic21—21no—
41C2×Dic21D411yescharacteristic
42C4×D21—42no—
43C4×D21C2×C412yes—
44C4×D21D412yes—
45Dic42—21no—
46Dic42C2×C412yescomplemented
47C3×C7×C8—21no—
48C3⋊(C7⋊C8)—21no—
49C2×C7×D4C2×D311yescharacteristic
50C2×C22⋊D7—61no—
51C22×Dic7—61no—
52#1159—34no—
53#1159—62no—
54C2×C4×D7—31no—
55C2×C4×D7—61no—
56C2×Dic14—31no—
57C2×C7×C8C2×D311yescharacteristic
58C7×C4⋊C4C2×D311yescharacteristic
59C4⋊Dic7—31no—
60Dic7⋊C4—61no—
61C2×C7⋊C8—31no—
62#1131—62no—
63C2×#237—71no—
64C2×ΓL(1,𝔽25)—71no—
65D3×C4⋊C4—71no—
66C3×#102D711yescharacteristic, complemented
67#832—71no—
68#816—71no—
69#810—71no—
70C22×C3×C7—22no—
71C22×C3×C7—41no—
72C22×C3×C7—41no—
73C22×C3×C7C2×D411yescharacteristic
74C2×D21—21no—
75C2×D21—21no—
76C2×D21C22⋊C412yes—
77C3×C4×C7—41no—
78C3×C4×C7C22×C411yescharacteristic
79C3×C4×C7C2×D411yescharacteristic
80Dic21—22no—
81Dic21—41no—
82Dic21C22⋊C412yes—
83#417—211noSylow
84C23×C7—21no—
85C22×D7—31no—
86C7×D4—21no—
87C7×D4C4×D312yescomplemented
88C2×C4×C7—21no—
89C2×C4×C7C22×D311yescharacteristic
90C22⋊D7—64no—
91C2×Dic7—31no—
92C2×Dic7—64no—
93C2×Dic7—61no—
94C2×Dic7—61no—
95C4×D7—32no—
96C4×D7—32no—
97C4×D7—122no—
98Dic14—32no—
99Dic14—61no—
100C7×C8—21no—
101C7⋊C8—61no—
102C2×C3×D4C2×D711yescharacteristic
103C2×#67—141no—
104C22×Dic3—141no—
105#237—74no—
106#237—142no—
107C2×C4×D3—71no—
108C2×C4×D3—141no—
109C2×Dic6—71no—
110C2×C3×C8C2×D711yescharacteristic
111C3×C4⋊C4C2×D711yescharacteristic
112C4⋊Dic3—71no—
113Dic3⋊C4—141no—
114C2×C3⋊C8—71no—
115ΓL(1,𝔽25)—142no—
116C2×C3×C7—42no—
117C2×C3×C7C2×C22⋊C411yescharacteristic
118C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
119C2×C3×C7#13711yescharacteristic
120D21—22no—
121C2×#41—211no—
122C2×M16—211no—
123C2×C4⋊C4—211no—
124#103—211no—
125#103—211no—
126#102—211no—
127#102D2111yescharacteristic, complemented
128C22×C7—22no—
129C22×C7—41no—
130C22×C7—41no—
131C22×C7D3×D411yescharacteristic
132C2×D7—32no—
133C2×D7—61no—
134C2×D7—61no—
135C4×C7—41no—
136C4×C7C2×C4×D311yescharacteristic
137C4×C7D3×D411yescharacteristic
138Dic7—32no—
139Dic7—62no—
140Dic7—121no—
141C23×C3—21no—
142C22×D3—71no—
143C3×D4—21no—
144C3×D4C4×D712yescomplemented
145C2×C3×C4—21no—
146C2×C3×C4C22×D711yescharacteristic
147#67—144no—
148C2×Dic3—71no—
149C2×Dic3—144no—
150C2×Dic3—141no—
151C2×Dic3—141no—
152C4×D3—72no—
153C4×D3—72no—
154C4×D3—282no—
155Dic6—72no—
156Dic6—141no—
157C3×C8—21no—
158C3⋊C8—141no—
159C3×C7#41711yescharacteristic, complemented, Hall
160#41—214no—
161#41—422no—
162C2×Q8—211no—
163C2×D4—421no—
164C2×D4C2×D2111yescharacteristic
165C22×C4—211no—
166C22×C4—421no—
167C22×C4—421no—
168M16—422no—
169C2×C8—211no—
170C2×C8C2×D2111yescharacteristic
171C4⋊C4—211no—
172C4⋊C4—421no—
173C4⋊C4C2×D2111yescharacteristic
174C2×C7—42no—
175C2×C7D3×C22⋊C411yescharacteristic
176C2×C7#91111yescharacteristic
177C2×C7#91511yescharacteristic
178D7—62no—
179C22×C3—22no—
180C22×C3—41no—
181C22×C3—41no—
182C22×C3D4×D711yescharacteristic
183C2×D3—72no—
184C2×D3—141no—
185C2×D3—141no—
186C3×C4—41no—
187C3×C4C2×C4×D711yescharacteristic
188C3×C4D4×D711yescharacteristic
189Dic3—72no—
190Dic3—142no—
191Dic3—281no—
192C23—21no—
193C23—211no—
194Q8—212no—
195Q8—421no—
196D4—21no—
197D4—424no—
198D4C4×D2112yescomplemented
199C2×C4—21no—
200C2×C4—211no—
201C2×C4—212no—
202C2×C4—212no—
203C2×C4—424no—
204C2×C4—421no—
205C2×C4—421no—
206C2×C4—842no—
207C2×C4C22×D2111yescharacteristic
208C8—21no—
209C8—421no—
210C7#477111yescharacteristic, complemented, Sylow
211C2×C3—42no—
212C2×C3D7×C22⋊C411yescharacteristic
213C2×C3#647411yescharacteristic
214C2×C3#647811yescharacteristic
215D3—142no—
216C22—22no—
217C22—41no—
218C22—41no—
219C22—212no—
220C22—421no—
221C22—421no—
222C22D4×D2111yescharacteristic
223C4—41no—
224C4—212no—
225C4—422no—
226C4—841no—
227C4C2×C4×D2111yescharacteristic
228C4D4×D2111yescharacteristic
229C3#8954311yescharacteristic, complemented, Sylow
230C2—42no—
231C2—422no—
232C2C22⋊C4×D2111yescharacteristic
233C2#1062139411yescharacteristic
234C2#1062139811yescharacteristic
subgroupG/H
centerC22D4×D21
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C22×D21
FittingC3×C7×#102C2
socleC22×C3×C7C2×D4
prime radicalC2×#66730C2
cosocleC2×C3×C4×C7C23
hypercenter#102D21
hypocenterC3×C7#417
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. C2×#66622C2semidirect
3. C4⋊C4×D21C2semidirect
4. C3×C7×#102C2semidirect
5. #10620671C2semidirect
6. #10620655C2semidirect
7. #10620649C2semidirect
13. #66730C4semidirect
26. C7×#102D3semidirect
34. C3×C7×D4C2×C4semidirect
46. Dic42C2×C4semidirect
66. C3×#102D7semidirect
87. C7×D4C4×D3semidirect
127. #102D21semidirect
144. C3×D4C4×D7semidirect
159. C3×C7#417semidirect
198. D4C4×D21semidirect
210. C7#4771semidirect
229. C3#89543semidirect

Maximal subgroups

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

subgroupindex
C2×#667302
C2×#666222
C4⋊C4×D212
C3×C7×#1022
#106206712
#106206552
#106206492
#895433
#47717