Finite Group Databasebeta

← #13250854

Subgroups — #13250854

Order 1792.

444 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3095C211yesmaximal, characteristic
2#11739655C211yesmaximal, characteristic
3#11729445C211yesmaximal, characteristic
4#11729444C211yesmaximal, characteristic
5#11729329C211yesmaximal, characteristic
6#11725494C211yesmaximal, characteristic
7#11725490C211yesmaximal, characteristic
8C2×C7×C22⋊Q8C2211yescharacteristic
9C2×C7×#117C2211yescharacteristic
10C7×#485—21no—
11C7×#485—21no—
12C7×#484—21no—
13C7×#483—21no—
14C7×#428C411yescharacteristic, complemented
15C7×#428C411yescharacteristic, complemented
16C7×#419C411yescharacteristic, complemented
17C7×#419C411yescharacteristic, 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#89333—21no—
29#89333—21no—
30#89324—21no—
31#89324—21no—
32#89323—22no—
33#9346—71nomaximal, Sylow
34C22×C7×Q8C2×C411yescharacteristic
35C23×C4×C7C2311yescharacteristic
36C2×C7×M16C2×C411yescharacteristic
37C2×C7×M16C2×C411yescharacteristic
38C7×C22⋊Q8—21no—
39C7×C22⋊Q8—21no—
40C7×C22⋊Q8—21no—
41C7×C22⋊Q8—21no—
42C7×#117—21no—
43C7×#117—21no—
44C7×#117C2×C411yescharacteristic
45C7×#117C2×C411yescharacteristic
46C7×#117D411yescharacteristic
47C7×#117D411yescharacteristic
48C2×C7×C4⋊C4—21no—
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×#103—21no—
65C7×#103—21no—
66C7×#103—21no—
67C7×#103—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#3095D711yescharacteristic
83#2924—71no—
84#1825—71no—
85#1520—71no—
86#1472—71no—
87#1305—71no—
88#1298—71no—
89C24×C7#4111yescharacteristic
90C2×C7×Q8—21no—
91C2×C7×Q8—21no—
92C2×C7×Q8—41no—
93C2×C7×Q8—41no—
94C22×C4×C7—21no—
95C22×C4×C7—21no—
96C22×C4×C7—21no—
97C22×C4×C7—21no—
98C22×C4×C7—41no—
99C22×C4×C7—41no—
100C22×C4×C7C22×C411yescharacteristic
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7C2×D411yescharacteristic
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7C2×D411yescharacteristic
105C22×C4×C7#4111yescharacteristic
106C22×Dic7—41no—
107C7×M16—41no—
108C7×M16—41no—
109C2×C7×C8—21no—
110C2×C7×C8—21no—
111C7×C4⋊C4—21no—
112C7×C4⋊C4—21no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—21no—
115C7×C4⋊C4—41no—
116C7×C4⋊C4—41no—
117C7×C4⋊C4C22⋊C411yescharacteristic
118C7×C4⋊C4C22⋊C411yescharacteristic
119C7×C4⋊C4C22⋊C411yescharacteristic
120C7×C4⋊C4C22⋊C411yescharacteristic
121C7×C22⋊C4—21no—
122C7×C22⋊C4—21no—
123C7×C22⋊C4—21no—
124C7×C22⋊C4—21no—
125C7×C22⋊C4—41no—
126C7×C22⋊C4—41no—
127C42×C7—21no—
128C42×C7—21no—
129C42×C7—21no—
130C42×C7—21no—
131C22⋊Dic7—81no—
132C4⋊Dic7—41no—
133C4⋊Dic7—41no—
134#1136—41no—
135#1136—41no—
136C2×C7⋊C8—21no—
137C2×C7⋊C8—21no—
138C2×C7⋊C8—22no—
139C2×C7⋊C8—41no—
140C2×C7⋊C8—41no—
141C2×C7⋊C8—41no—
142C2×C7⋊C8—41no—
143C2×C7⋊C8—41no—
144C2×C22⋊Q8C2×D711yescharacteristic
145C2×#117C2×D711yescharacteristic
146#485—21no—
147#485—21no—
148#484—21no—
149#483—21no—
150#433—142no—
151#432—141no—
152#431—141no—
153#428Dic711yescharacteristic, complemented
154#428Dic711yescharacteristic, complemented
155#424—141no—
156#419Dic711yescharacteristic, complemented
157#419Dic711yescharacteristic, complemented
158#407—71no—
159#407C2×D711yescharacteristic
160C2×C22⋊C8—71no—
161C2×C22⋊C8—71no—
162#405—141no—
163#380—71no—
164#340—141no—
165#340—141no—
166#328—141no—
167#328—141no—
168#324—142no—
169C23×C7—21no—
170C23×C7—21no—
171C23×C7—41no—
172C23×C7—41no—
173C23×C7C4×D411yescharacteristic
174C23×C7#12111yescharacteristic
175C23×C7#12311yescharacteristic
176C7×Q8—41no—
177C7×Q8—41no—
178C7×Q8—81no—
179C2×C4×C7—21no—
180C2×C4×C7—21no—
181C2×C4×C7—21no—
182C2×C4×C7—21no—
183C2×C4×C7—21no—
184C2×C4×C7—21no—
185C2×C4×C7—21no—
186C2×C4×C7—21no—
187C2×C4×C7—21no—
188C2×C4×C7—21no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
193C2×C4×C7—41no—
194C2×C4×C7—42no—
195C2×C4×C7—41no—
196C2×C4×C7—41no—
197C2×C4×C7—81no—
198C2×C4×C7—81no—
199C2×C4×C7C2×C22⋊C411yescharacteristic
200C2×C4×C7C4×D411yescharacteristic
201C2×C4×C7#12011yescharacteristic
202C2×C4×C7#12111yescharacteristic
203C2×Dic7—81no—
204C2×Dic7—81no—
205C2×Dic7—81no—
206C7×C8—41no—
207C7×C8—41no—
208C7⋊C8—41no—
209C7⋊C8—41no—
210C7⋊C8—42no—
211C7⋊C8—81no—
212C22×Q8C2×Dic711yescharacteristic
213C23×C4C22×D711yescharacteristic
214C2×M16—141no—
215C2×M16C2×Dic711yescharacteristic
216C2×M16C2×Dic711yescharacteristic
217C22×C8—141no—
218C22×C8—141no—
219C22⋊Q8—21no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222C22⋊Q8—21no—
223#117—21no—
224#117—21no—
225#117C2×Dic711yescharacteristic
226#117C2×Dic711yescharacteristic
227#117C22⋊D711yescharacteristic
228#117C22⋊D711yescharacteristic
229C2×C4⋊C4—21no—
230C2×C4⋊C4—141no—
231C2×C4⋊C4C2×Dic711yescharacteristic
232C2×C4⋊C4C22⋊D711yescharacteristic
233C2×C4⋊C4C22⋊D711yescharacteristic
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—141no—
237C2×C42—21no—
238#107—21no—
239#107—21no—
240#106—21no—
241#106—21no—
242C4⋊C8—142no—
243C4⋊C8—142no—
244#103—21no—
245#103—21no—
246#103—21no—
247#103—21no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250C22⋊C8—72no—
251C22⋊C8—71no—
252C22⋊C8—71no—
253C22⋊C8—141no—
254C22⋊C8—141no—
255C22⋊C8—141no—
256C22⋊C8—141no—
257#97—142no—
258C4×C8—142no—
259#95—141no—
260C22×C7—21no—
261C22×C7—21no—
262C22×C7—21no—
263C22×C7—21no—
264C22×C7—21no—
265C22×C7—21no—
266C22×C7—21no—
267C22×C7—21no—
268C22×C7—81no—
269C22×C7—81no—
270C22×C7#38611yescharacteristic
271C22×C7#47111yescharacteristic
272C22×C7#47311yescharacteristic
273C4×C7—21no—
274C4×C7—21no—
275C4×C7—21no—
276C4×C7—41no—
277C4×C7—41no—
278C4×C7—41no—
279C4×C7—41no—
280C4×C7—81no—
281C4×C7—81no—
282C4×C7#41011yescharacteristic
283C4×C7#41011yescharacteristic
284Dic7—161no—
285C24#115911yescharacteristic
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—41no—
289C2×Q8—41no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—281no—
297C22×C4D4×D711yescharacteristic
298C22×C4D4×D711yescharacteristic
299C22×C4#115911yescharacteristic
300C22×C4C22×Dic711yescharacteristic
301C22×C4C2×C22⋊D711yescharacteristic
302C22×C4C2×C22⋊D711yescharacteristic
303M16—41no—
304M16—41no—
305M16—281no—
306M16—281no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—141no—
310C2×C8—142no—
311C2×C8—141no—
312C2×C8—281no—
313C2×C8—281no—
314C2×C8—281no—
315C2×C8—281no—
316C2×C8—281no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—281no—
324C4⋊C4—281no—
325C4⋊C4C22⋊Dic711yescharacteristic
326C4⋊C4C22⋊Dic711yescharacteristic
327C4⋊C4C22⋊Dic711yescharacteristic
328C4⋊C4C22⋊Dic711yescharacteristic
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—561no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C2×C7—21no—
341C2×C7—21no—
342C2×C7—41no—
343C2×C7—41no—
344C2×C7#188111yescharacteristic
345C2×C7#189911yescharacteristic
346C2×C7#190811yescharacteristic
347C23—21no—
348C23—21no—
349C23—41no—
350C23—41no—
351C23D4×Dic711yescharacteristic
352C23#649811yescharacteristic
353C23#650211yescharacteristic
354Q8—41no—
355Q8—41no—
356Q8—81no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—561no—
378C2×C4—561no—
379C2×C4—561no—
380C2×C4D4×Dic711yescharacteristic
381C2×C4#650011yescharacteristic
382C2×C4#650211yescharacteristic
383C2×C4C2×C22⋊Dic711yescharacteristic
384C8—41no—
385C8—41no—
386C8—281no—
387C8—281no—
388C8—282no—
389C8—561no—
390C7#934611yescharacteristic, complemented, Sylow
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—81no—
400C22—81no—
401C22#8997511yescharacteristic
402C22#8997611yescharacteristic
403C22#8999211yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—81no—
412C4—81no—
413C4—1121no—
414C4#9001211yescharacteristic
415C4#9001211yescharacteristic
416C2—21no—
417C2—21no—
418C2—41no—
419C2—41no—
420C2#1172945911yescharacteristic
421C2#1172985711yescharacteristic
422C2#1172987011yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#3095C2
socleC22×C7#386
prime radicalC24×C7#41
cosocleC23×C4×C7C23
hypercenter#3095D7
hypocenterC7#9346
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3095

Complements

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

NH ≅ G/Ntype
14. C7×#428C4semidirect
15. C7×#428C4semidirect
16. C7×#419C4semidirect
17. C7×#419C4semidirect
153. #428Dic7semidirect
154. #428Dic7semidirect
156. #419Dic7semidirect
157. #419Dic7semidirect
390. C7#9346semidirect

Maximal subgroups

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

subgroupindex
C7×#30952
#117396552
#117294452
#117294442
#117293292
#117254942
#117254902
#93467