Finite Group Databasebeta

← #13250868

Subgroups — #13250868

Order 1792.

470 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3094C211yesmaximal, characteristic
2#11739655C211yesmaximal, characteristic, complemented
3#11729430C211yesmaximal, characteristic
4#11729429C211yesmaximal, characteristic
5#11729329C211yesmaximal, characteristic, complemented
6#11725494C211yesmaximal, characteristic, complemented
7#11725490C211yesmaximal, characteristic, complemented
8C2×C7×#121C2211yescharacteristic
9C2×C7×#117C2211yescharacteristic
10C7×#482—21no—
11C7×#482—21no—
12C7×#481—21no—
13C7×#480—21no—
14C7×#428C411yescharacteristic, complemented
15C7×#428C411yescharacteristic, complemented
16C7×#418C411yescharacteristic, complemented
17C7×#418C411yescharacteristic, complemented
18C7×#407C2211yescharacteristic
19#89984C2211yescharacteristic
20#89979C2211yescharacteristic
21C2×C22⋊(C7⋊C8)C2211yescharacteristic
22C2×C22⋊(C7⋊C8)C2211yescharacteristic
23#89772—22no—
24#89771—21no—
25#89769—21no—
26#89698—21no—
27#89697—21no—
28#89332—21no—
29#89332—21no—
30#89324—21no—
31#89324—21no—
32#89320—22no—
33#9348—71nomaximal, Sylow
34C22×C7×D4—21no—
35C22×C7×D4C2×C411yescharacteristic
36C23×C4×C7C2311yescharacteristic
37C2×C7×M16C2×C411yescharacteristic
38C2×C7×M16C2×C411yescharacteristic
39C7×#121—21no—
40C7×#121—21no—
41C7×#121—21no—
42C7×#121—21no—
43C7×#117—21no—
44C7×#117—21no—
45C7×#117C2×C411yescharacteristic
46C7×#117C2×C411yescharacteristic
47C7×#117D411yescharacteristic
48C7×#117D411yescharacteristic
49C2×C7×C4⋊C4C2×C411yescharacteristic
50C2×C7×C4⋊C4D411yescharacteristic
51C2×C7×C4⋊C4D411yescharacteristic
52C2×C7×C22⋊C4—21no—
53C2×C7×C22⋊C4—21no—
54C2×C42×C7—21no—
55C2×C22⋊Dic7—21no—
56C2×C4⋊Dic7—21no—
57C2×#1136—21no—
58C22×C7⋊C8—21no—
59C22×C7⋊C8—21no—
60C7×#107—21no—
61C7×#107—21no—
62C7×#106—21no—
63C7×#106—21no—
64C7×#102—21no—
65C7×#102—21no—
66C7×#102—21no—
67C7×#102—21no—
68C7×C22⋊C8—21no—
69C7×C22⋊C8—21no—
70C7⋊#95—21no—
71C22⋊(C7⋊C8)—21no—
72C22⋊(C7⋊C8)—21no—
73C22⋊(C7⋊C8)—21no—
74C22⋊(C7⋊C8)—21no—
75C22⋊(C7⋊C8)D411yescharacteristic
76C22⋊(C7⋊C8)D411yescharacteristic
77C22⋊(C7⋊C8)D412yes—
78C4⋊(C7⋊C8)—22no—
79C4⋊(C7⋊C8)—22no—
80#6377—22no—
81C4×C7⋊C8—22no—
82#3094D711yescharacteristic
83#2924—71no—
84#1825—71no—
85#1519—71no—
86#1471—71no—
87#1305—71no—
88#1298—71no—
89C24×C7—21no—
90C24×C7#4111yescharacteristic
91C2×C7×D4—21no—
92C2×C7×D4—21no—
93C2×C7×D4—41no—
94C2×C7×D4—41no—
95C2×C7×D4—41no—
96C2×C7×D4—41no—
97C2×C7×D4—81no—
98C2×C7×D4—81no—
99C22×C4×C7—21no—
100C22×C4×C7—21no—
101C22×C4×C7—21no—
102C22×C4×C7—41no—
103C22×C4×C7—41no—
104C22×C4×C7C22×C411yescharacteristic
105C22×C4×C7C2×D411yescharacteristic
106C22×C4×C7C2×D411yescharacteristic
107C22×C4×C7C2×D411yescharacteristic
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7#4111yescharacteristic
110C22×Dic7—41no—
111C7×M16—41no—
112C7×M16—41no—
113C2×C7×C8—21no—
114C2×C7×C8—21no—
115C7×C4⋊C4—21no—
116C7×C4⋊C4—21no—
117C7×C4⋊C4—21no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4C22⋊C411yescharacteristic
120C7×C4⋊C4C22⋊C411yescharacteristic
121C7×C4⋊C4C22⋊C411yescharacteristic
122C7×C4⋊C4C22⋊C411yescharacteristic
123C7×C22⋊C4—21no—
124C7×C22⋊C4—21no—
125C7×C22⋊C4—21no—
126C7×C22⋊C4—21no—
127C7×C22⋊C4—41no—
128C7×C22⋊C4—41no—
129C42×C7—21no—
130C42×C7—21no—
131C42×C7—21no—
132C42×C7—21no—
133C22⋊Dic7—81no—
134C4⋊Dic7—41no—
135C4⋊Dic7—41no—
136#1136—41no—
137#1136—41no—
138C2×C7⋊C8—21no—
139C2×C7⋊C8—21no—
140C2×C7⋊C8—22no—
141C2×C7⋊C8—41no—
142C2×C7⋊C8—41no—
143C2×C7⋊C8—41no—
144C2×C7⋊C8—41no—
145C2×C7⋊C8—41no—
146C2×#121C2×D711yescharacteristic
147C2×#117C2×D711yescharacteristic
148#482—21no—
149#482—21no—
150#481—21no—
151#480—21no—
152#433—142no—
153#432—141no—
154#431—141no—
155#428Dic711yescharacteristic, complemented
156#428Dic711yescharacteristic, complemented
157#424—141no—
158#418Dic711yescharacteristic, complemented
159#418Dic711yescharacteristic, complemented
160#407—71no—
161#407C2×D711yescharacteristic
162C2×C22⋊C8—71no—
163C2×C22⋊C8—71no—
164#405—141no—
165#380—71no—
166#340—141no—
167#340—141no—
168#327—141no—
169#327—141no—
170#323—142no—
171C23×C7—21no—
172C23×C7—21no—
173C23×C7—41no—
174C23×C7—41no—
175C23×C7—42no—
176C23×C7—41no—
177C23×C7—81no—
178C23×C7—81no—
179C23×C7C4×D411yescharacteristic
180C23×C7#12111yescharacteristic
181C23×C7#12311yescharacteristic
182C7×D4—41no—
183C7×D4—41no—
184C7×D4—81no—
185C7×D4—82no—
186C7×D4—81no—
187C7×D4—81no—
188C2×C4×C7—21no—
189C2×C4×C7—21no—
190C2×C4×C7—21no—
191C2×C4×C7—21no—
192C2×C4×C7—21no—
193C2×C4×C7—21no—
194C2×C4×C7—21no—
195C2×C4×C7—21no—
196C2×C4×C7—21no—
197C2×C4×C7—21no—
198C2×C4×C7—41no—
199C2×C4×C7—41no—
200C2×C4×C7—41no—
201C2×C4×C7—41no—
202C2×C4×C7—41no—
203C2×C4×C7—41no—
204C2×C4×C7—81no—
205C2×C4×C7—81no—
206C2×C4×C7C2×C22⋊C411yescharacteristic
207C2×C4×C7C4×D411yescharacteristic
208C2×C4×C7#12011yescharacteristic
209C2×C4×C7#12111yescharacteristic
210C2×Dic7—81no—
211C2×Dic7—81no—
212C2×Dic7—81no—
213C7×C8—41no—
214C7×C8—41no—
215C7⋊C8—41no—
216C7⋊C8—41no—
217C7⋊C8—42no—
218C7⋊C8—81no—
219C22×D4—21no—
220C22×D4C2×Dic711yescharacteristic
221C23×C4C22×D711yescharacteristic
222C2×M16—141no—
223C2×M16C2×Dic711yescharacteristic
224C2×M16C2×Dic711yescharacteristic
225C22×C8—141no—
226C22×C8—141no—
227#121—21no—
228#121—21no—
229#121—21no—
230#121—21no—
231#117—21no—
232#117—21no—
233#117C2×Dic711yescharacteristic
234#117C2×Dic711yescharacteristic
235#117C22⋊D711yescharacteristic
236#117C22⋊D711yescharacteristic
237C2×C4⋊C4—141no—
238C2×C4⋊C4C2×Dic711yescharacteristic
239C2×C4⋊C4C22⋊D711yescharacteristic
240C2×C4⋊C4C22⋊D711yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—141no—
244C2×C42—21no—
245#107—21no—
246#107—21no—
247#106—21no—
248#106—21no—
249C4⋊C8—142no—
250C4⋊C8—142no—
251#102—21no—
252#102—21no—
253#102—21no—
254#102—21no—
255C22⋊C8—21no—
256C22⋊C8—21no—
257C22⋊C8—72no—
258C22⋊C8—71no—
259C22⋊C8—71no—
260C22⋊C8—141no—
261C22⋊C8—141no—
262C22⋊C8—141no—
263C22⋊C8—141no—
264#97—142no—
265C4×C8—142no—
266#95—141no—
267C22×C7—21no—
268C22×C7—21no—
269C22×C7—21no—
270C22×C7—21no—
271C22×C7—21no—
272C22×C7—21no—
273C22×C7—21no—
274C22×C7—21no—
275C22×C7—81no—
276C22×C7—81no—
277C22×C7—81no—
278C22×C7—81no—
279C22×C7—81no—
280C22×C7—162no—
281C22×C7#38611yescharacteristic
282C22×C7#47111yescharacteristic
283C22×C7#47211yescharacteristic
284C4×C7—21no—
285C4×C7—21no—
286C4×C7—21no—
287C4×C7—41no—
288C4×C7—41no—
289C4×C7—41no—
290C4×C7—41no—
291C4×C7—81no—
292C4×C7#41011yescharacteristic
293C4×C7#41011yescharacteristic
294Dic7—161no—
295C24—21no—
296C24#115911yescharacteristic
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—81no—
304C2×D4—81no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—281no—
311C22×C4D4×D711yescharacteristic
312C22×C4D4×D711yescharacteristic
313C22×C4#115911yescharacteristic
314C22×C4C22×Dic711yescharacteristic
315C22×C4C2×C22⋊D711yescharacteristic
316C22×C4C2×C22⋊D711yescharacteristic
317M16—41no—
318M16—41no—
319M16—281no—
320M16—281no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—141no—
324C2×C8—142no—
325C2×C8—141no—
326C2×C8—281no—
327C2×C8—281no—
328C2×C8—281no—
329C2×C8—281no—
330C2×C8—281no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—281no—
336C4⋊C4—281no—
337C4⋊C4C22⋊Dic711yescharacteristic
338C4⋊C4C22⋊Dic711yescharacteristic
339C4⋊C4C22⋊Dic711yescharacteristic
340C4⋊C4C22⋊Dic711yescharacteristic
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—21no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—561no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—21no—
352C2×C7—21no—
353C2×C7—21no—
354C2×C7—41no—
355C2×C7—41no—
356C2×C7—161no—
357C2×C7#188111yescharacteristic
358C2×C7#189911yescharacteristic
359C2×C7#190711yescharacteristic
360C23—21no—
361C23—21no—
362C23—41no—
363C23—41no—
364C23—42no—
365C23—41no—
366C23—81no—
367C23—81no—
368C23D4×Dic711yescharacteristic
369C23#649811yescharacteristic
370C23#650211yescharacteristic
371D4—41no—
372D4—41no—
373D4—81no—
374D4—82no—
375D4—81no—
376D4—81no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—561no—
396C2×C4—561no—
397C2×C4—561no—
398C2×C4D4×Dic711yescharacteristic
399C2×C4#650011yescharacteristic
400C2×C4#650211yescharacteristic
401C2×C4C2×C22⋊Dic711yescharacteristic
402C8—41no—
403C8—41no—
404C8—281no—
405C8—281no—
406C8—282no—
407C8—561no—
408C7#934811yescharacteristic, complemented, Sylow
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—162no—
423C22#8997111yescharacteristic
424C22#8997211yescharacteristic
425C22#8999211yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—81no—
434C4—1121no—
435C4#9001211yescharacteristic
436C4#9001211yescharacteristic
437C2—21no—
438C2—21no—
439C2—41no—
440C2—41no—
441C2—161no—
442C2#1172945911yescharacteristic
443C2#1172984511yescharacteristic
444C2#1172987011yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#3094C2
socleC22×C7#386
prime radicalC2×C7×#121C22
cosocleC23×C4×C7C23
hypercenter#3094D7
hypocenterC7#9348
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3094

Complements

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

NH ≅ G/Ntype
2. #11739655C2semidirect
5. #11729329C2semidirect
6. #11725494C2semidirect
7. #11725490C2semidirect
14. C7×#428C4semidirect
15. C7×#428C4semidirect
16. C7×#418C4semidirect
17. C7×#418C4semidirect
155. #428Dic7semidirect
156. #428Dic7semidirect
158. #418Dic7semidirect
159. #418Dic7semidirect
408. C7#9348semidirect

Maximal subgroups

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

subgroupindex
C7×#30942
#117396552
#117294302
#117294292
#117293292
#117254942
#117254902
#93487