Finite Group Databasebeta

← #13059509

Subgroups — #13059509

Order 1344.

588 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 144 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×#66686C212yesmaximal, complemented
2#10621413C211yesmaximal, characteristic, complemented
3C7×#930C211yesmaximal, characteristic, complemented
4C2×C3×Dic7⋊C4C211yesmaximal, characteristic, complemented
5#10621128C212yesmaximal, complemented
6#10620982C212yesmaximal, complemented
7#10620981C212yesmaximal, complemented
8#10620979C212yesmaximal, complemented
9#10620978C212yesmaximal, complemented
10#90288—31nomaximal, Hall
11C22×C3×C4×C7C2211yescharacteristic
12C2×#4268C2212yes—
13C2×D84C2211yescharacteristic
14C2×C7×#67C2212yes—
15C2×C7×D12C2211yescharacteristic
16C22×C3×Dic7C2211yescharacteristic
17C22×C3×Dic7C2211yescharacteristic
18C2×#4248C2212yes—
19C2×#4247C2212yes—
20C2×#4247C2212yes—
21C2×D21⋊C4C2212yes—
22#66686—28no—
23C2×D3×Dic7C2212yes—
24#66631C2212yes—
25C4⋊Dic21C2211yescharacteristic
26C7×(C2×D3)⋊C4C2212yes—
27C7×C4⋊Dic3C2211yescharacteristic
28C3×Dic7⋊C4C2212yes—
29C3×Dic7⋊C4C2212yes—
30Dic7⋊Dic3C2212yes—
31#66575C2212yes—
32(C2×D3)⋊Dic7C2212yes—
33C2×#1159—32no—
34C7×#121—31no—
35#6502—32no—
36#6502—32no—
37#6493—31no—
38C2×Dic7⋊C4D311yescharacteristic, complemented
39#6459—32no—
40#6447—32no—
41#6441—32no—
42#5510—71nomaximal, Hall
43C23×C3×C7C2311yescharacteristic
44C22×D21C2312yes—
45C22×D3×C7C2312yes—
46C2×C3×C4×C7—41no—
47C2×C3×C4×C7C2311yescharacteristic
48C2×C3×C4×C7C2311yescharacteristic
49#4268—24no—
50C2×Dic21C2312yes—
51D84—41no—
52C7×#67—24no—
53C2×C7×Dic3C2312yes—
54C7×D12—41no—
55C2×C3×Dic7—41no—
56C2×C3×Dic7D414yescomplemented
57C2×C3×Dic7C2312yes—
58C2×C3×Dic7C2312yes—
59#4248—24no—
60#4247—24no—
61#4247—42no—
62D21⋊C4—24no—
63D3×Dic7—24no—
64C2×C7×D4—32no—
65C2×C7×D4—31no—
66C22×C4×C7C2×D311yescharacteristic
67C2×C22⋊D7—32no—
68C2×C22⋊D7—32no—
69C2×C22⋊D7—32no—
70C22×Dic7—32no—
71C22×Dic7C2×D311yescharacteristic
72C22×Dic7C2×D311yescharacteristic
73#1159—68no—
74C2×D28—31no—
75C2×C4×D7—32no—
76C2×Dic14—32no—
77C7×C4⋊C4—31no—
78C7×C22⋊C4—32no—
79C22⋊Dic7—32no—
80(C2×D7)⋊C4—32no—
81(C2×D7)⋊C4—32no—
82C4⋊Dic7—31no—
83Dic7⋊C4—32no—
84Dic7⋊C4C2×D312yes—
85Dic7⋊C4C2×D312yes—
86C2×#235—72no—
87C2×C3×C4⋊C4—71no—
88#930—71no—
89#930—72no—
90#930D711yescharacteristic, complemented
91#897—72no—
92#897—72no—
93#895—72no—
94#895—72no—
95C22×C3×C7—21no—
96C22×C3×C7—21no—
97C22×C3×C7C2×D412yes—
98C22×C3×C7C2411yescharacteristic
99C2×D21—22no—
100C2×D21—22no—
101C2×D21—22no—
102C2×D3×C7—22no—
103C2×D3×C7—22no—
104C2×D3×C7—22no—
105C3×C4×C7—21no—
106C3×C4×C7—21no—
107Dic21—22no—
108C3×Dic7—22no—
109C3×Dic7C2×D414yes—
110C7×Dic3—22no—
111#537—211noSylow
112C23×C7—32no—
113C23×C7C22×D311yescharacteristic
114C22×D7—32no—
115C7×D4—64no—
116C7×D4—121no—
117C2×C4×C7—32no—
118C2×C4×C7—41no—
119C2×C4×C7C22×D311yescharacteristic
120C2×C4×C7C22×D311yescharacteristic
121C22⋊D7—64no—
122C22⋊D7—64no—
123C22⋊D7—122no—
124C2×Dic7—32no—
125C2×Dic7—41no—
126C2×Dic7—64no—
127C2×Dic7D1214yescomplemented
128C2×Dic7C22×D312yes—
129C2×Dic7C22×D312yes—
130D28—121no—
131C4×D7—64no—
132Dic14—64no—
133C22×C3×C4—71no—
134C22×C3×C4—71no—
135C22×C3×C4C2×D711yescharacteristic
136C2×#67—72no—
137C2×#67C2×D712yes—
138#235—148no—
139C2×D12—72no—
140C2×D12—72no—
141C2×D12—71no—
142C2×D12C2×D711yescharacteristic
143C2×C4×D3—72no—
144C2×C4×D3—72no—
145C2×Dic6—72no—
146C3×C4⋊C4—72no—
147C3×C4⋊C4—72no—
148(C2×D3)⋊C4—72no—
149(C2×D3)⋊C4—72no—
150(C2×D3)⋊C4—72no—
151(C2×D3)⋊C4C2×D712yes—
152C4⋊Dic3—71no—
153C4⋊Dic3—72no—
154C4⋊Dic3C2×D711yescharacteristic
155C2×C3×C7—22no—
156C2×C3×C7C22×D411yescharacteristic
157C2×C3×C7H(5,𝔽2)11yescharacteristic
158C2×C3×C72−1+411yescharacteristic
159D21—42no—
160D3×C7—42no—
161C2×#41—212no—
162C22⋊Q8—212no—
163C22⋊Q8—212no—
164#121—31no—
165#121—212no—
166#121—212no—
167#121—212no—
168#121—211no—
169C2×C4⋊C4—71no—
170C22×C7—21no—
171C22×C7—21no—
172C22×C7—62no—
173C22×C7—62no—
174C22×C7—62no—
175C22×C7C2×D1212yes—
176C22×C7C23×D311yescharacteristic
177C2×D7—62no—
178C2×D7—62no—
179C2×D7—62no—
180C4×C7—21no—
181C4×C7—21no—
182C4×C7—62no—
183Dic7—22no—
184Dic7—62no—
185Dic7C2×D1214yes—
186C23×C3C22×D711yescharacteristic
187C22×D3—72no—
188C22×D3C22×D712yes—
189C2×C3×C4—41no—
190C2×C3×C4—74no—
191C2×C3×C4—72no—
192C2×C3×C4—72no—
193C2×C3×C4—281no—
194C2×C3×C4C22×D711yescharacteristic
195C2×C3×C4C22×D711yescharacteristic
196#67—24no—
197#67—144no—
198C2×Dic3—72no—
199C2×Dic3C22×D712yes—
200D12—41no—
201D12—144no—
202D12—281no—
203D12—282no—
204C4×D3—144no—
205C4×D3—144no—
206Dic6—144no—
207C3×C7#53711yescharacteristic, complemented, Hall
208#41—428no—
209C2×Q8—212no—
210C2×D4—32no—
211C2×D4—31no—
212C2×D4—212no—
213C2×D4—212no—
214C2×D4—212no—
215C2×D4—211no—
216C22×C4—71no—
217C22×C4—71no—
218C22×C4—212no—
219C22×C4—212no—
220C22×C4D3×D711yescharacteristic
221C4⋊C4—31no—
222C4⋊C4—72no—
223C4⋊C4—72no—
224C4⋊C4—212no—
225C4⋊C4—211no—
226C22⋊C4—32no—
227C22⋊C4—212no—
228C22⋊C4—212no—
229C22⋊C4—212no—
230C2×C7—22no—
231C2×C7—122no—
232C2×C7C22×D1211yescharacteristic
233C2×C7#100411yescharacteristic
234C2×C7#100711yescharacteristic
235D7—122no—
236C22×C3—21no—
237C22×C3—21no—
238C22×C3D4×D712yes—
239C22×C3C23×D711yescharacteristic
240C2×D3—22no—
241C2×D3—22no—
242C2×D3—22no—
243C2×D3—142no—
244C2×D3—142no—
245C2×D3—142no—
246C3×C4—21no—
247C3×C4—21no—
248C3×C4—74no—
249C3×C4—142no—
250Dic3—22no—
251Dic3—142no—
252C23—32no—
253C23—212no—
254C23C2×D3×D711yescharacteristic
255Q8—424no—
256D4—64no—
257D4—121no—
258D4—424no—
259D4—424no—
260D4—841no—
261D4—842no—
262C2×C4—32no—
263C2×C4—41no—
264C2×C4—74no—
265C2×C4—72no—
266C2×C4—72no—
267C2×C4—212no—
268C2×C4—281no—
269C2×C4—424no—
270C2×C4—424no—
271C2×C4C2×D3×D711yescharacteristic
272C2×C4C2×D3×D711yescharacteristic
273C7#551011yescharacteristic, complemented, Sylow
274C2×C3—22no—
275C2×C3C2×D4×D711yescharacteristic
276C2×C3#654811yescharacteristic
277C2×C3#655411yescharacteristic
278D3—42no—
279D3—282no—
280C22—21no—
281C22—21no—
282C22—62no—
283C22—62no—
284C22—62no—
285C22—422no—
286C22—422no—
287C22—422no—
288C22D7×D1212yes—
289C22C22×D3×D711yescharacteristic
290C4—21no—
291C4—21no—
292C4—62no—
293C4—74no—
294C4—142no—
295C4—422no—
296C3#9028811yescharacteristic, complemented, Sylow
297C2—22no—
298C2—122no—
299C2—842no—
300C2#1062155111yescharacteristic
301C2C2×D7×D1211yescharacteristic
302C2#1062159611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC22×C3×C4×C7C22
hypercenterC22×C4D3×D7
hypocenterC3×C7#537
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#66686C2semidirect
2. #10621413C2semidirect
3. C7×#930C2semidirect
4. C2×C3×Dic7⋊C4C2semidirect
5. #10621128C2semidirect
6. #10620982C2semidirect
7. #10620981C2semidirect
8. #10620979C2semidirect
9. #10620978C2semidirect
38. C2×Dic7⋊C4D3semidirect
56. C2×C3×Dic7D4semidirect
90. #930D7semidirect
127. C2×Dic7D12semidirect
207. C3×C7#537semidirect
273. C7#5510semidirect
296. C3#90288semidirect

Maximal subgroups

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

subgroupindex
C2×#666862
#106214132
C7×#9302
C2×C3×Dic7⋊C42
#106211282
#106209822
#106209812
#106209792
#106209782
#902883
#55107