Finite Group Databasebeta

← #13249747

Subgroups — #13249747

Order 1792.

548 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3036C211yesmaximal, characteristic
2#11739783C211yesmaximal, characteristic, complemented
3#11729568C211yesmaximal, characteristic, complemented
4#11729565C211yesmaximal, characteristic, complemented
5#11729351C211yesmaximal, characteristic, complemented
6#11725521C211yesmaximal, characteristic, complemented
7#11725519C211yesmaximal, characteristic, complemented
8C2×C7×AGL(1,ℤ/8)—21no—
9C7×#532C2211yescharacteristic
10C2×C4×C7×D4C2211yescharacteristic
11C7×#463—22no—
12C7×#461C412yescomplemented
13C7×#459C412yescomplemented
14C7×#423C2211yescharacteristic
15C7×#417—21no—
16#89993—21no—
17#89983—21no—
18C4×C22⋊Dic7—21no—
19C4⋊C4×Dic7—21no—
20#89723—22no—
21#89708C2211yescharacteristic
22C4×C4⋊Dic7C2211yescharacteristic
23#89697C2211yescharacteristic
24#89697C2211yescharacteristic
25#89332—21no—
26#89332—21no—
27#89332—22no—
28D4⋊(C7⋊C8)—22no—
29#9181—71nomaximal, Sylow
30C2×C7×#41—21no—
31C22×C7×D4D411yescharacteristic
32C23×C4×C7—21no—
33C7×AGL(1,ℤ/8)—42no—
34C7×AGL(1,ℤ/8)—42no—
35C2×C7×SD16—22no—
36C2×C7×D8—22no—
37C2×C7×M16—21no—
38C7×C4⋊Q8C2×C411yescharacteristic
39C7×#127C2×C411yescharacteristic
40C7×#124—21no—
41C7×#121—21no—
42C7×#121—21no—
43C4×C7×D4—21no—
44C4×C7×D4—21no—
45C4×C7×D4—21no—
46C4×C7×D4—21no—
47C4×C7×D4C2×C412yes—
48C4×C7×D4D412yes—
49C2×C7×C4⋊C4D411yescharacteristic
50C2×C7×C22⋊C4—21no—
51C2×C42×C7C2311yescharacteristic
52C2×C22⋊Dic7—22no—
53C2×C22⋊Dic7—22no—
54C2×C4⋊Dic7—21no—
55C2×C4⋊Dic7—21no—
56C2×C4×Dic7—21no—
57C2×C4×Dic7—22no—
58C7×C4⋊C8—21no—
59C7×C4⋊C8C2×C412yes—
60C7×#103—22no—
61C7×#102—22no—
62C7⋊#95—22no—
63C7⋊#95—21no—
64C7⋊#95D411yescharacteristic
65C7⋊#95D411yescharacteristic
66C7⋊#95D412yes—
67C22⋊(C7⋊C8)—21no—
68C22⋊(C7⋊C8)—21no—
69C4⋊(C7⋊C8)—21no—
70C4⋊(C7⋊C8)—21no—
71C4×C7⋊C8—21no—
72C4×C7⋊C8—21no—
73#3036D711yescharacteristic
74#2357—71no—
75#1517—71no—
76#1502—71no—
77#1478—71no—
78#1303—71no—
79#1286—71no—
80C24×C7—21no—
81C7×#41—41no—
82C7×#41—41no—
83C7×#41—41no—
84C7×#41—41no—
85C2×C7×Q8—21no—
86C2×C7×D4—21no—
87C2×C7×D4—21no—
88C2×C7×D4—21no—
89C2×C7×D4—21no—
90C2×C7×D4—21no—
91C2×C7×D4—21no—
92C2×C7×D4—21no—
93C2×C7×D4—21no—
94C2×C7×D4—41no—
95C2×C7×D4C22⋊C412yes—
96C22×C4×C7—21no—
97C22×C4×C7—22no—
98C22×C4×C7—22no—
99C22×C4×C7—41no—
100C22×C4×C7—82no—
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7C2×D411yescharacteristic
103C22×C4×C7C2×D411yescharacteristic
104C22×Dic7—21no—
105C22×Dic7—22no—
106C22×Dic7—21no—
107C7×SD16—44no—
108C7×D8—44no—
109C7×M16—41no—
110C7×M16—41no—
111C2×C7×C8—22no—
112C7×C4⋊C4—21no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—21no—
115C7×C4⋊C4C22⋊C412yes—
116C7×C22⋊C4—22no—
117C7×C22⋊C4—22no—
118C7×C22⋊C4—41no—
119C7×C22⋊C4—41no—
120C42×C7C22×C411yescharacteristic
121C42×C7C2×D411yescharacteristic
122C42×C7#4111yescharacteristic
123C42×C7#4111yescharacteristic
124C22⋊Dic7—44no—
125C22⋊Dic7—82no—
126C4⋊Dic7—42no—
127C4⋊Dic7—41no—
128C4⋊Dic7—41no—
129C4×Dic7—41no—
130C4×Dic7—41no—
131C4×Dic7—82no—
132C2×C7⋊C8—41no—
133C2×C7⋊C8—41no—
134C2×AGL(1,ℤ/8)—21no—
135#532C2×D711yescharacteristic
136C2×C4×D4C2×D711yescharacteristic
137#463—22no—
138#461Dic712yescomplemented
139#459Dic712yescomplemented
140#431—71no—
141#431—71no—
142#423C2×D711yescharacteristic
143#417—21no—
144#390—141no—
145#388—142no—
146#382—71no—
147#381—141no—
148C4×C4⋊C4—71no—
149C4×C4⋊C4—141no—
150C4×C22⋊C4—141no—
151#327—141no—
152#327—141no—
153#327—142no—
154D4⋊C8—142no—
155C23×C7—22no—
156C23×C7—42no—
157C23×C7—42no—
158C23×C7—41no—
159C23×C7—41no—
160C23×C7—41no—
161C23×C7#12011yescharacteristic
162C7×Q8—41no—
163C7×Q8—41no—
164C7×D4—22no—
165C7×D4—22no—
166C7×D4—41no—
167C7×D4—41no—
168C7×D4—41no—
169C7×D4—41no—
170C7×D4—81no—
171C7×D4—81no—
172C7×D4—81no—
173C7×D4—81no—
174C2×C4×C7—22no—
175C2×C4×C7—41no—
176C2×C4×C7—41no—
177C2×C4×C7—42no—
178C2×C4×C7—41no—
179C2×C4×C7—82no—
180C2×C4×C7—82no—
181C2×C4×C7—81no—
182C2×C4×C7—81no—
183C2×C4×C7C2×C22⋊C411yescharacteristic
184C2×C4×C7C4×D411yescharacteristic
185C2×C4×C7C4×D411yescharacteristic
186C2×C4×C7#12111yescharacteristic
187C2×C4×C7#12111yescharacteristic
188C2×C4×C7#12311yescharacteristic
189C2×C4×C7#13511yescharacteristic
190C2×C4×C7#13511yescharacteristic
191C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
192C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
193C2×Dic7—22no—
194C2×Dic7—41no—
195C2×Dic7—41no—
196C2×Dic7—42no—
197C2×Dic7—41no—
198C2×Dic7—42no—
199C2×Dic7—41no—
200C2×Dic7—41no—
201C2×Dic7—42no—
202C7×C8—42no—
203C7⋊C8—81no—
204C7⋊C8—81no—
205C2×#41—21no—
206C22×D4C22⋊D711yescharacteristic
207C23×C4—21no—
208AGL(1,ℤ/8)—42no—
209AGL(1,ℤ/8)—42no—
210C2×SD16—22no—
211C2×D8—22no—
212C2×M16—21no—
213C4⋊Q8C2×Dic711yescharacteristic
214#127C2×Dic711yescharacteristic
215#124—21no—
216#121—21no—
217#121—21no—
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4C2×Dic712yes—
223C4×D4C22⋊D712yes—
224C2×C4⋊C4—141no—
225C2×C4⋊C4—141no—
226C2×C4⋊C4C22⋊D711yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—142no—
229C2×C22⋊C4—142no—
230C2×C42—141no—
231C2×C42—142no—
232C2×C42C22×D711yescharacteristic
233C4⋊C8—21no—
234C4⋊C8—141no—
235C4⋊C8—141no—
236C4⋊C8C2×Dic712yes—
237#103—22no—
238#102—22no—
239C22⋊C8—141no—
240C22⋊C8—141no—
241C4×C8—141no—
242C4×C8—141no—
243#95—71no—
244#95—71no—
245#95—72no—
246#95—142no—
247#95—141no—
248C22×C7—21no—
249C22×C7—21no—
250C22×C7—42no—
251C22×C7—42no—
252C22×C7—42no—
253C22×C7—82no—
254C22×C7—82no—
255C22×C7—81no—
256C22×C7—81no—
257C22×C7—81no—
258C22×C7#38611yescharacteristic
259C22×C7#44911yescharacteristic
260C22×C7#44911yescharacteristic
261C4×C7—21no—
262C4×C7—21no—
263C4×C7—21no—
264C4×C7—21no—
265C4×C7—42no—
266C4×C7—81no—
267C4×C7#41611yescharacteristic
268C4×C7#41811yescharacteristic
269C4×C7#48211yescharacteristic
270C4×C7#48211yescharacteristic
271Dic7—42no—
272Dic7—81no—
273Dic7—82no—
274C24—21no—
275#41—41no—
276#41—41no—
277#41—41no—
278#41—41no—
279C2×Q8—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4C22⋊Dic712yes—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—41no—
294C22×C4—82no—
295C22×C4—141no—
296C22×C4—142no—
297C22×C4—141no—
298C22×C4D4×D711yescharacteristic
299C22×C4D4×D711yescharacteristic
300C22×C4C2×C22⋊D711yescharacteristic
301SD16—44no—
302D8—44no—
303M16—41no—
304M16—41no—
305C2×C8—22no—
306C2×C8—281no—
307C2×C8—281no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—282no—
312C4⋊C4—281no—
313C4⋊C4—281no—
314C4⋊C4C22⋊Dic712yes—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—284no—
320C22⋊C4—562no—
321C42—281no—
322C42—281no—
323C42—562no—
324C42#115911yescharacteristic
325C42#115911yescharacteristic
326C42C22×Dic711yescharacteristic
327C42C2×C22⋊D711yescharacteristic
328C2×C7—21no—
329C2×C7—21no—
330C2×C7—82no—
331C2×C7—161no—
332C2×C7#187511yescharacteristic
333C2×C7#190411yescharacteristic
334C2×C7#190611yescharacteristic
335C23—22no—
336C23—42no—
337C23—42no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23#650011yescharacteristic
342Q8—41no—
343Q8—41no—
344D4—22no—
345D4—22no—
346D4—41no—
347D4—41no—
348D4—41no—
349D4—41no—
350D4—81no—
351D4—81no—
352D4—81no—
353D4—81no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—142no—
364C2×C4—281no—
365C2×C4—281no—
366C2×C4—282no—
367C2×C4—282no—
368C2×C4—281no—
369C2×C4—281no—
370C2×C4—281no—
371C2×C4—282no—
372C2×C4#649511yescharacteristic
373C2×C4#649511yescharacteristic
374C2×C4D4×Dic711yescharacteristic
375C2×C4D4×Dic711yescharacteristic
376C2×C4#649811yescharacteristic
377C2×C4#650211yescharacteristic
378C2×C4#650211yescharacteristic
379C2×C4#651311yescharacteristic
380C2×C4#651311yescharacteristic
381C2×C4C2×C22⋊Dic711yescharacteristic
382C8—42no—
383C8—561no—
384C8—561no—
385C7#918111yescharacteristic, complemented, Sylow
386C22—21no—
387C22—21no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—82no—
392C22—82no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22#8980911yescharacteristic
397C22#8980911yescharacteristic
398C22#8999211yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—81no—
405C4—282no—
406C4—561no—
407C4—562no—
408C4#8980711yescharacteristic
409C4#8980711yescharacteristic
410C4#8998711yescharacteristic
411C4#9000511yescharacteristic
412C2—21no—
413C2—21no—
414C2—82no—
415C2—161no—
416C2#1172943211yescharacteristic
417C2#1172984311yescharacteristic
418C2#1172986811yescharacteristic
subgroupG/H
centerC22#89992
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
Fitting, prime radicalC7×#3036C2
socleC22×C7#386
cosocleC2×C42×C7C23
hypercenter#3036D7
hypocenterC7#9181
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
2. #11739783C2semidirect
3. #11729568C2semidirect
4. #11729565C2semidirect
5. #11729351C2semidirect
6. #11725521C2semidirect
7. #11725519C2semidirect
12. C7×#461C4semidirect
13. C7×#459C4semidirect
138. #461Dic7semidirect
139. #459Dic7semidirect
385. C7#9181semidirect

Maximal subgroups

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

subgroupindex
C7×#30362
#117397832
#117295682
#117295652
#117293512
#117255212
#117255192
#91817