Finite Group Databasebeta

← #13251117

Subgroups — #13251117

Order 1792.

640 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2543C211yesmaximal, characteristic
2#11739746C211yesmaximal, characteristic, complemented
3#11729429C212yesmaximal, complemented
4#11729373C211yesmaximal, characteristic, complemented
5#11725418C212yesmaximal, complemented
6C2×C7×#121—21no—
7C2×C7×#121C2211yescharacteristic
8C2×C4×C7×D4C2211yescharacteristic
9C7×#398C411yescharacteristic, complemented
10C7×#397C411yescharacteristic, complemented
11C7×#394C411yescharacteristic, complemented
12C7×#392C411yescharacteristic, complemented
13C7×#388—21no—
14C7×#386—21no—
15C7×#386—21no—
16C2×C7×#95C2211yescharacteristic
17C2×#6408—21no—
18#89984C2212yes—
19C2×C22⋊(C7⋊C8)—21no—
20C2×C22⋊(C7⋊C8)C2212yes—
21D4×C7⋊C8—24no—
22#89697—21no—
23#89332—24no—
24#89321—21no—
25#9384—71nomaximal, Sylow
26C22×C7×D4—21no—
27C22×C7×D4—21no—
28C22×C7×D4C2×C411yescharacteristic
29C22×C7×D4D411yescharacteristic
30C23×C4×C7—21no—
31C23×C4×C7D411yescharacteristic
32C23×C4×C7C2311yescharacteristic
33C7×#121—22no—
34C7×#121—22no—
35C7×#121—42no—
36C7×#121—42no—
37C4×C7×D4—22no—
38C4×C7×D4—22no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4C2×C411yescharacteristic
41C2×C7×C4⋊C4C2×C411yescharacteristic
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4—21no—
47C2×C7×C22⋊C4C2×C411yescharacteristic
48C2×C7×C22⋊C4D411yescharacteristic
49C2×C42×C7D411yescharacteristic
50C2×C22⋊Dic7—22no—
51C2×C4⋊Dic7—22no—
52C22×C7⋊C8—22no—
53C22×C7⋊C8—22no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95—21no—
57C7×#95C2×C411yescharacteristic
58C7×#95C2×C411yescharacteristic
59#6408—42no—
60C7⋊#95—22no—
61C22⋊(C7⋊C8)—22no—
62C22⋊(C7⋊C8)—22no—
63C22⋊(C7⋊C8)D414yescomplemented
64C4⋊(C7⋊C8)—22no—
65C4×C7⋊C8—22no—
66#2972—71no—
67#2543D711yescharacteristic
68#1799—71no—
69#1519—72no—
70#1294—72no—
71C24×C7—21no—
72C24×C7C22⋊C411yescharacteristic
73C24×C7C2×D411yescharacteristic
74C2×C7×D4—22no—
75C2×C7×D4—22no—
76C2×C7×D4—22no—
77C2×C7×D4—42no—
78C2×C7×D4—42no—
79C2×C7×D4—42no—
80C2×C7×D4—41no—
81C2×C7×D4—41no—
82C2×C7×D4—42no—
83C2×C7×D4—81no—
84C2×C7×D4—81no—
85C2×C7×D4D812yes—
86C2×C7×D4SD1612yes—
87C22×C4×C7—21no—
88C22×C4×C7—21no—
89C22×C4×C7—21no—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—42no—
93C22×C4×C7—41no—
94C22×C4×C7—41no—
95C22×C4×C7—81no—
96C22×C4×C7—81no—
97C22×C4×C7—81no—
98C22×C4×C7C22⋊C411yescharacteristic
99C22×C4×C7C22⋊C411yescharacteristic
100C22×C4×C7C22⋊C411yescharacteristic
101C22×C4×C7C22×C411yescharacteristic
102C22×C4×C7C2×D412yes—
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7#4112yes—
105C22×Dic7—42no—
106C7×C4⋊C4—22no—
107C7×C4⋊C4—22no—
108C7×C4⋊C4—42no—
109C7×C22⋊C4—22no—
110C7×C22⋊C4—22no—
111C7×C22⋊C4—42no—
112C7×C22⋊C4—42no—
113C7×C22⋊C4—42no—
114C7×C22⋊C4—41no—
115C7×C22⋊C4—41no—
116C7×C22⋊C4—81no—
117C42×C7—21no—
118C42×C7—21no—
119C22⋊Dic7—42no—
120C22⋊Dic7—42no—
121C4⋊Dic7—82no—
122C2×C7⋊C8—24no—
123C2×C7⋊C8—82no—
124C2×C7⋊C8—82no—
125C2×#121—21no—
126C2×#121C2×D711yescharacteristic
127C2×C4×D4C2×D711yescharacteristic
128C8×D4—144no—
129#431—141no—
130C2×#99—141no—
131C2×C22⋊C8—72no—
132C2×C22⋊C8—141no—
133#398Dic711yescharacteristic, complemented
134#397Dic711yescharacteristic, complemented
135#394Dic711yescharacteristic, complemented
136#392Dic711yescharacteristic, complemented
137#388—21no—
138#386—21no—
139#386—21no—
140#380—72no—
141C2×#95C2×D711yescharacteristic
142#342—141no—
143#327—144no—
144C23×C7—21no—
145C23×C7—21no—
146C23×C7—41no—
147C23×C7—41no—
148C23×C7—41no—
149C23×C7—41no—
150C23×C7—41no—
151C23×C7—41no—
152C23×C7—41no—
153C23×C7—41no—
154C23×C7—41no—
155C23×C7—41no—
156C23×C7—41no—
157C23×C7#10212yes—
158C23×C7C2×C22⋊C411yescharacteristic
159C23×C7#12112yes—
160C23×C7C2×D811yescharacteristic
161C23×C7C2×SD1611yescharacteristic
162C23×C7#13512yes—
163C7×D4—24no—
164C7×D4—24no—
165C7×D4—82no—
166C7×D4—84no—
167C7×D4—82no—
168C7×D4—82no—
169C2×C4×C7—21no—
170C2×C4×C7—21no—
171C2×C4×C7—22no—
172C2×C4×C7—21no—
173C2×C4×C7—21no—
174C2×C4×C7—42no—
175C2×C4×C7—42no—
176C2×C4×C7—41no—
177C2×C4×C7—41no—
178C2×C4×C7—42no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—41no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—81no—
190C2×C4×C7—82no—
191C2×C4×C7—82no—
192C2×C4×C7#10212yes—
193C2×C4×C7C4×D412yes—
194C2×C4×C7#12011yescharacteristic
195C2×C4×C7#12311yescharacteristic
196C2×Dic7—82no—
197C2×Dic7—82no—
198C2×Dic7—82no—
199C7⋊C8—44no—
200C22×D4—21no—
201C22×D4—21no—
202C22×D4C2×Dic711yescharacteristic
203C22×D4C22⋊D711yescharacteristic
204C23×C4—21no—
205C23×C4C22⋊D711yescharacteristic
206C23×C4C22×D711yescharacteristic
207C22×C8—142no—
208C22×C8—142no—
209#121—22no—
210#121—22no—
211#121—42no—
212#121—42no—
213C4×D4—22no—
214C4×D4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—142no—
217C2×C4⋊C4C2×Dic711yescharacteristic
218C2×C4⋊C4C2×Dic711yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—142no—
225C2×C22⋊C4C2×Dic711yescharacteristic
226C2×C22⋊C4C22⋊D711yescharacteristic
227C2×C42C22⋊D711yescharacteristic
228C4⋊C8—142no—
229#99—282no—
230C22⋊C8—74no—
231C22⋊C8—142no—
232C22⋊C8—142no—
233C4×C8—142no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—142no—
238#95C2×Dic711yescharacteristic
239#95C2×Dic711yescharacteristic
240C22×C7—22no—
241C22×C7—21no—
242C22×C7—22no—
243C22×C7—21no—
244C22×C7—22no—
245C22×C7—41no—
246C22×C7—41no—
247C22×C7—42no—
248C22×C7—42no—
249C22×C7—41no—
250C22×C7—41no—
251C22×C7—81no—
252C22×C7—81no—
253C22×C7—81no—
254C22×C7—81no—
255C22×C7—81no—
256C22×C7—81no—
257C22×C7—81no—
258C22×C7—81no—
259C22×C7#38611yescharacteristic
260C22×C7C2×#10211yescharacteristic
261C22×C7#41611yescharacteristic
262C22×C7#46512yes—
263C22×C7#46612yes—
264C4×C7—22no—
265C4×C7—21no—
266C4×C7—21no—
267C4×C7—22no—
268C4×C7—42no—
269C4×C7—41no—
270C4×C7—41no—
271C4×C7—81no—
272C4×C7—81no—
273Dic7—162no—
274C24—21no—
275C24C22⋊Dic711yescharacteristic
276C24C2×C22⋊D711yescharacteristic
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—81no—
287C2×D4—81no—
288C2×D4#114112yes—
289C2×D4#114212yes—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—81no—
299C22×C4—81no—
300C22×C4—81no—
301C22×C4—282no—
302C22×C4C22⋊Dic711yescharacteristic
303C22×C4C22⋊Dic711yescharacteristic
304C22×C4C22⋊Dic711yescharacteristic
305C22×C4D4×D712yes—
306C22×C4#115912yes—
307C22×C4C22×Dic711yescharacteristic
308C22×C4C2×C22⋊D711yescharacteristic
309C2×C8—144no—
310C2×C8—562no—
311C2×C8—562no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—42no—
315C4⋊C4—562no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—81no—
324C22⋊C4—282no—
325C22⋊C4—282no—
326C42—21no—
327C42—21no—
328C2×C7—22no—
329C2×C7—21no—
330C2×C7—21no—
331C2×C7—22no—
332C2×C7—42no—
333C2×C7—161no—
334C2×C7#188411yescharacteristic
335C2×C7#189311yescharacteristic
336C2×C7#191311yescharacteristic
337C23—21no—
338C23—21no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23#640612yes—
351C23C2×#114111yescharacteristic
352C23C2×#114211yescharacteristic
353C23#650212yes—
354C23#651312yes—
355C23C2×C22⋊Dic711yescharacteristic
356D4—24no—
357D4—24no—
358D4—82no—
359D4—84no—
360D4—82no—
361D4—82no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—81no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—562no—
386C2×C4—562no—
387C2×C4—562no—
388C2×C4#640612yes—
389C2×C4D4×Dic712yes—
390C2×C4#649811yescharacteristic
391C2×C4#650011yescharacteristic
392C8—284no—
393C7#938411yescharacteristic, complemented, Sylow
394C22—21no—
395C22—22no—
396C22—21no—
397C22—22no—
398C22—22no—
399C22—41no—
400C22—41no—
401C22—42no—
402C22—42no—
403C22—41no—
404C22—41no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22#8981012yes—
414C22#8981312yes—
415C22C2×#640611yescharacteristic
416C22#8999211yescharacteristic
417C22#9000511yescharacteristic
418C4—22no—
419C4—21no—
420C4—21no—
421C4—22no—
422C4—42no—
423C4—41no—
424C4—41no—
425C4—81no—
426C4—81no—
427C4—1122no—
428C2—21no—
429C2—22no—
430C2—21no—
431C2—22no—
432C2—42no—
433C2—161no—
434C2#1172943511yescharacteristic
435C2#1172957111yescharacteristic
436C2#1172960211yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
Fitting, prime radicalC7×#2543C2
socleC22×C7#386
cosocleC23×C4×C7C23
hypercenter#2543D7
hypocenterC7#9384
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2543

Complements

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

NH ≅ G/Ntype
2. #11739746C2semidirect
3. #11729429C2semidirect
4. #11729373C2semidirect
5. #11725418C2semidirect
9. C7×#398C4semidirect
10. C7×#397C4semidirect
11. C7×#394C4semidirect
12. C7×#392C4semidirect
63. C22⋊(C7⋊C8)D4semidirect
133. #398Dic7semidirect
134. #397Dic7semidirect
135. #394Dic7semidirect
136. #392Dic7semidirect
393. C7#9384semidirect

Maximal subgroups

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

subgroupindex
C7×#25432
#117397462
#117294292
#117293732
#117254182
#93847