Finite Group Databasebeta

← #13059552

Subgroups — #13059552

Order 1344.

588 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 136 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×#66728C211yesmaximal, characteristic, complemented
2C2×#66670C211yesmaximal, characteristic, complemented
3C7×#924C211yesmaximal, characteristic, complemented
4C3×#6493C211yesmaximal, characteristic, complemented
5#10621112C212yesmaximal, complemented
6#10621095C212yesmaximal, complemented
7#10621027C211yesmaximal, characteristic, complemented
8#10621021C212yesmaximal, complemented
9#10620999C211yesmaximal, characteristic, complemented
10#10620993C212yesmaximal, complemented
11#10620945C211yesmaximal, characteristic, complemented
12#90320—31nomaximal, Hall
13C22×C3×C4×C7C2211yescharacteristic
14C2×#4268C2211yescharacteristic
15C2×#4268C2211yescharacteristic
16#66728—24no—
17C2×D84C2211yescharacteristic
18C2×C4×D21C2211yescharacteristic
19C2×C4×D21C2211yescharacteristic
20C2×Dic42C2211yescharacteristic
21C2×C7×Dic6C2211yescharacteristic
22C2×C3×C22⋊D7C2212yes—
23C2×C3×D28C2211yescharacteristic
24C2×#4246C2212yes—
25C2×#4246C2212yes—
26C2×D21⋊C4C2212yes—
27C2×Dic3×D7C2212yes—
28#66670—24no—
29C7×#217C2212yes—
30C7×C4⋊Dic3C2211yescharacteristic
31C7×Dic3⋊C4C2212yes—
32C3×(C2×D7)⋊C4C2212yes—
33C3×C4⋊Dic7C2211yescharacteristic
34#66577C2212yes—
35Dic7⋊Dic3C2212yes—
36#66575C2212yes—
37(C2×D7)⋊Dic3C2212yes—
38C2×#1161—31no—
39C2×#1157—31no—
40C7×C22⋊Q8—31no—
41#6509—31no—
42#6493D311yescharacteristic, complemented
43#6459—32no—
44#6458—31no—
45#6458—32no—
46D7×C4⋊C4—31no—
47#6447—32no—
48#6447—32no—
49#5608—71nomaximal, Hall
50C23×C3×C7C2311yescharacteristic
51C22×D21C2311yescharacteristic
52C22×D21C2311yescharacteristic
53C22×C3×D7C2312yes—
54C2×C3×C4×C7—22no—
55C2×C3×C4×C7C2311yescharacteristic
56C2×C3×C4×C7C2311yescharacteristic
57#4268—22no—
58#4268—22no—
59C2×Dic21C2311yescharacteristic
60C2×Dic21C2311yescharacteristic
61D84—22no—
62C4×D21—22no—
63C4×D21D414yescomplemented
64Dic42—22no—
65C2×C7×Dic3C2312yes—
66C2×C7×Dic3C2312yes—
67C7×Dic6—22no—
68C3×C22⋊D7—42no—
69C2×C3×Dic7C2312yes—
70C3×D28—22no—
71#4246—24no—
72#4246—42no—
73D21⋊C4—42no—
74Dic3×D7—24no—
75C2×C7×Q8—31no—
76C22×C4×C7C2×D311yescharacteristic
77C2×C22⋊D7—31no—
78C2×C22⋊D7—31no—
79C2×C22⋊D7C2×D312yes—
80#1161—64no—
81#1157—64no—
82C2×D28—32no—
83C2×D28—31no—
84C2×D28—32no—
85C2×D28C2×D311yescharacteristic
86C2×C4×D7—31no—
87C2×C4×D7—32no—
88C2×C4×D7—31no—
89C2×C4×D7—32no—
90C2×Dic14—31no—
91C7×C4⋊C4—32no—
92C7×C4⋊C4—31no—
93C7×C22⋊C4—32no—
94(C2×D7)⋊C4—32no—
95(C2×D7)⋊C4—32no—
96(C2×D7)⋊C4C2×D312yes—
97C4⋊Dic7C2×D311yescharacteristic
98Dic7⋊C4—32no—
99Dic7⋊C4—32no—
100C2×#237—71no—
101C2×#235—71no—
102C3×#121—71no—
103#939—72no—
104#938—72no—
105#938—71no—
106#924D711yescharacteristic, complemented
107#897—71no—
108D3×C4⋊C4—71no—
109#884—72no—
110#878—72no—
111C22×C3×C7—21no—
112C22×C3×C7—21no—
113C22×C3×C7—21no—
114C22×C3×C7C2411yescharacteristic
115C2×D21—21no—
116C2×D21—21no—
117C2×D21—21no—
118C2×D21—21no—
119C2×D21—21no—
120C2×D21C2×D412yes—
121C2×C3×D7—22no—
122C2×C3×D7—22no—
123C2×C3×D7—22no—
124C3×C4×C7—21no—
125C3×C4×C7C2×D412yes—
126Dic21—21no—
127Dic21C2×D412yes—
128C3×Dic7—22no—
129C7×Dic3—22no—
130C7×Dic3—22no—
131#537—211noSylow
132C23×C7C22×D311yescharacteristic
133C22×D7—31no—
134C22×D7—31no—
135C22×D7C22×D312yes—
136C7×Q8—62no—
137C2×C4×C7—22no—
138C2×C4×C7—32no—
139C2×C4×C7—32no—
140C2×C4×C7C22×D311yescharacteristic
141C2×C4×C7C22×D311yescharacteristic
142C22⋊D7—42no—
143C22⋊D7—62no—
144C22⋊D7—62no—
145C2×Dic7—31no—
146C2×Dic7—31no—
147C2×Dic7C22×D312yes—
148D28—22no—
149D28—62no—
150D28—64no—
151D28—122no—
152C4×D7—34no—
153C4×D7—64no—
154C4×D7—62no—
155C4×D7—122no—
156Dic14—62no—
157C2×C3×D4—72no—
158C2×C3×D4—71no—
159C22×C3×C4C2×D711yescharacteristic
160C2×#67—71no—
161C2×#67—71no—
162C2×#67—72no—
163C2×#67—72no—
164C22×Dic3—72no—
165#237—144no—
166#235—144no—
167C2×D12—71no—
168C2×C4×D3—71no—
169C2×C4×D3—71no—
170C2×C4×D3—72no—
171C2×Dic6—71no—
172C2×Dic6C2×D711yescharacteristic
173C3×C4⋊C4—71no—
174C3×C22⋊C4—72no—
175#217—72no—
176#217C2×D712yes—
177(C2×D3)⋊C4—72no—
178C4⋊Dic3—72no—
179C4⋊Dic3C2×D711yescharacteristic
180Dic3⋊C4—72no—
181Dic3⋊C4C2×D712yes—
182C2×C3×C7—41no—
183C2×C3×C7C22×D411yescharacteristic
184C2×C3×C7H(5,𝔽2)11yescharacteristic
185C2×C3×C72−1+411yescharacteristic
186D21—22no—
187D21—41no—
188C3×D7—42no—
189C2×#41—211no—
190C2×#41—211no—
191C22⋊Q8—31no—
192C22⋊Q8—212no—
193C22⋊Q8—211no—
194#121—71no—
195#121—212no—
196#121—212no—
197#121—212no—
198#121—211no—
199C2×C4⋊C4—211no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7C23×D311yescharacteristic
204C2×D7—22no—
205C2×D7—22no—
206C2×D7—22no—
207C2×D7—32no—
208C2×D7—61no—
209C2×D7—61no—
210C2×D7—61no—
211C2×D7—61no—
212C2×D7—61no—
213C4×C7—21no—
214C4×C7—62no—
215C4×C7—62no—
216C4×C7D3×D412yes—
217Dic7—22no—
218Dic7—32no—
219Dic7—61no—
220C23×C3—72no—
221C23×C3C22×D711yescharacteristic
222C22×D3—71no—
223C22×D3—71no—
224C3×D4—142no—
225C3×D4—282no—
226C2×C3×C4—22no—
227C2×C3×C4—72no—
228C2×C3×C4C22×D711yescharacteristic
229C2×C3×C4C22×D711yescharacteristic
230#67—144no—
231#67—142no—
232#67—142no—
233#67—282no—
234C2×Dic3—71no—
235C2×Dic3—71no—
236C2×Dic3—144no—
237C2×Dic3C22×D712yes—
238C2×Dic3C22×D712yes—
239D12—142no—
240C4×D3—74no—
241C4×D3—142no—
242C4×D3—282no—
243Dic6—22no—
244Dic6—142no—
245C3×C7#53711yescharacteristic, complemented, Hall
246#41—424no—
247#41—424no—
248C2×Q8—31no—
249C2×Q8—211no—
250C2×D4—72no—
251C2×D4—71no—
252C2×D4—212no—
253C2×D4—212no—
254C2×D4—211no—
255C2×D4—211no—
256C2×D4—211no—
257C22×C4—212no—
258C22×C4—212no—
259C22×C4—211no—
260C22×C4—211no—
261C22×C4D3×D711yescharacteristic
262C4⋊C4—31no—
263C4⋊C4—32no—
264C4⋊C4—71no—
265C4⋊C4—212no—
266C4⋊C4—212no—
267C22⋊C4—32no—
268C22⋊C4—72no—
269C22⋊C4—212no—
270C22⋊C4—212no—
271C2×C7—41no—
272C2×C7C2×D3×D411yescharacteristic
273C2×C7#100411yescharacteristic
274C2×C7#101011yescharacteristic
275D7—42no—
276D7—62no—
277D7—121no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—142no—
282C22×C3—142no—
283C22×C3—142no—
284C22×C3C23×D711yescharacteristic
285C2×D3—72no—
286C2×D3—141no—
287C2×D3—141no—
288C2×D3—141no—
289C2×D3—141no—
290C2×D3—141no—
291C3×C4—21no—
292C3×C4—142no—
293C3×C4D4×D712yes—
294Dic3—22no—
295Dic3—22no—
296Dic3—72no—
297Dic3—141no—
298C23—72no—
299C23—211no—
300C23—211no—
301C23C2×D3×D711yescharacteristic
302Q8—62no—
303Q8—422no—
304D4—142no—
305D4—282no—
306D4—422no—
307D4—424no—
308D4—422no—
309D4—422no—
310D4—842no—
311C2×C4—22no—
312C2×C4—32no—
313C2×C4—32no—
314C2×C4—72no—
315C2×C4—211no—
316C2×C4—214no—
317C2×C4—211no—
318C2×C4—424no—
319C2×C4—422no—
320C2×C4—842no—
321C2×C4C2×D3×D711yescharacteristic
322C2×C4C2×D3×D711yescharacteristic
323C7#560811yescharacteristic, complemented, Sylow
324C2×C3—41no—
325C2×C3—282no—
326C2×C3C2×D4×D711yescharacteristic
327C2×C3#655111yescharacteristic
328C2×C3#655311yescharacteristic
329D3—142no—
330D3—281no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—142no—
335C22—142no—
336C22—142no—
337C22—212no—
338C22—421no—
339C22—421no—
340C22—421no—
341C22—421no—
342C22—421no—
343C22C22×D3×D711yescharacteristic
344C4—21no—
345C4—62no—
346C4—62no—
347C4—142no—
348C4—212no—
349C4—421no—
350C4#6669412yes—
351C3#9032011yescharacteristic, complemented, Sylow
352C2—41no—
353C2—282no—
354C2—422no—
355C2—841no—
356C2#1062154711yescharacteristic
357C2#1062156411yescharacteristic
358C2C2×#6669411yescharacteristic
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×#66728C2semidirect
2. C2×#66670C2semidirect
3. C7×#924C2semidirect
4. C3×#6493C2semidirect
5. #10621112C2semidirect
6. #10621095C2semidirect
7. #10621027C2semidirect
8. #10621021C2semidirect
9. #10620999C2semidirect
10. #10620993C2semidirect
11. #10620945C2semidirect
42. #6493D3semidirect
63. C4×D21D4semidirect
106. #924D7semidirect
245. C3×C7#537semidirect
323. C7#5608semidirect
351. C3#90320semidirect

Maximal subgroups

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

subgroupindex
C2×#667282
C2×#666702
C7×#9242
C3×#64932
#106211122
#106210952
#106210272
#106210212
#106209992
#106209932
#106209452
#903203
#56087