Finite Group Databasebeta

← #13248914

Subgroups — #13248914

Order 1792.

468 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3091C211yesmaximal, characteristic
2#11740364C211yesmaximal, characteristic
3#11729461C211yesmaximal, characteristic
4#11729344C211yesmaximal, characteristic, complemented
5#11729338C211yesmaximal, characteristic, complemented
6#11725486C211yesmaximal, characteristic, complemented
7#11725484C211yesmaximal, characteristic, complemented
8C7×#525C2211yescharacteristic
9C2×C7×#117C2211yescharacteristic
10C7×#485—21no—
11C7×#484C411yescharacteristic, complemented
12C7×#483C411yescharacteristic, complemented
13C7×#482—21no—
14C7×#481C411yescharacteristic, complemented
15C7×#480C411yescharacteristic, complemented
16C7×#427—21no—
17C7×#416—21no—
18C2×C7×C22⋊C8C2211yescharacteristic
19#90006—21no—
20C4×C22⋊Dic7C2211yescharacteristic
21#89979C2211yescharacteristic
22C2×C22⋊(C7⋊C8)C2211yescharacteristic
23C2×C22⋊(C7⋊C8)C2211yescharacteristic
24#89784—21no—
25#89783—21no—
26#89783—21no—
27D4×C7⋊C8—21no—
28#89772—21no—
29#89333—21no—
30#89332—21no—
31#89327—21no—
32#89327—21no—
33#89319—21no—
34#89319—21no—
35#9038—71nomaximal, Sylow
36C2×C7×#41D411yescharacteristic
37C23×C4×C7C2311yescharacteristic
38C22×C7×C8—21no—
39C7×#123—21no—
40C7×C22⋊Q8C2×C411yescharacteristic
41C7×C22⋊Q8D411yescharacteristic
42C7×#121C2×C411yescharacteristic
43C7×#121D411yescharacteristic
44C7×#120—21no—
45C4×C7×D4—21no—
46C4×C7×D4—21no—
47C7×#117—21no—
48C7×#117—21no—
49C7×#117—21no—
50C7×#117—21no—
51C7×#117D411yescharacteristic
52C2×C7×C4⋊C4C2×C411yescharacteristic
53C2×C7×C4⋊C4C2×C411yescharacteristic
54C2×C7×C22⋊C4—21no—
55C2×C42×C7—21no—
56C2×C22⋊Dic7—21no—
57C2×C4×Dic7—21no—
58C2×#1136—21no—
59C2×#1136D411yescharacteristic
60C2×#1136D411yescharacteristic
61C22×C7⋊C8—21no—
62C22×C7⋊C8—21no—
63C22×C7⋊C8D411yescharacteristic
64C22×C7⋊C8D411yescharacteristic
65C7×#107—21no—
66C7×#107—21no—
67C7×#106—21no—
68C7×#106—21no—
69C7×#103—21no—
70C7×#103—21no—
71C7×#102—21no—
72C7×#102—21no—
73C7×C22⋊C8—21no—
74C7×C22⋊C8C2×C411yescharacteristic
75C7×C22⋊C8C2×C411yescharacteristic
76C7⋊#95—21no—
77C22⋊(C7⋊C8)—21no—
78C22⋊(C7⋊C8)—21no—
79C22⋊(C7⋊C8)—21no—
80C22⋊(C7⋊C8)—21no—
81C22⋊(C7⋊C8)—21no—
82C22⋊(C7⋊C8)—21no—
83C22⋊(C7⋊C8)—21no—
84C22⋊(C7⋊C8)—21no—
85C4⋊(C7⋊C8)—21no—
86C4⋊(C7⋊C8)—21no—
87#6377—21no—
88C4×C7⋊C8—21no—
89#3091D711yescharacteristic
90#2976—71no—
91#1826—71no—
92#1818—71no—
93#1501—71no—
94#1298—71no—
95#1282—71no—
96C24×C7#4111yescharacteristic
97C7×#41—41no—
98C7×#41—41no—
99C2×C7×Q8C22⋊C411yescharacteristic
100C2×C7×D4—21no—
101C2×C7×D4—21no—
102C2×C7×D4—21no—
103C2×C7×D4C22⋊C411yescharacteristic
104C22×C4×C7—21no—
105C22×C4×C7—21no—
106C22×C4×C7—21no—
107C22×C4×C7—41no—
108C22×C4×C7—41no—
109C22×C4×C7C22×C411yescharacteristic
110C22×C4×C7C2×D411yescharacteristic
111C22×C4×C7C2×D411yescharacteristic
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7C2×D411yescharacteristic
114C22×C4×C7#4111yescharacteristic
115C22×Dic7—41no—
116C2×C7×C8—21no—
117C2×C7×C8—21no—
118C2×C7×C8—41no—
119C2×C7×C8—41no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—21no—
124C7×C4⋊C4—21no—
125C7×C4⋊C4—21no—
126C7×C4⋊C4C22⋊C411yescharacteristic
127C7×C4⋊C4C22⋊C411yescharacteristic
128C7×C22⋊C4—21no—
129C7×C22⋊C4—21no—
130C7×C22⋊C4—21no—
131C7×C22⋊C4—21no—
132C7×C22⋊C4—21no—
133C7×C22⋊C4—41no—
134C7×C22⋊C4—41no—
135C42×C7—21no—
136C42×C7—41no—
137C42×C7—41no—
138C22⋊Dic7—22no—
139C22⋊Dic7—41no—
140C4×Dic7—41no—
141C4×Dic7—41no—
142#1136—41no—
143#1136—41no—
144#1136—81no—
145C2×C7⋊C8—21no—
146C2×C7⋊C8—21no—
147C2×C7⋊C8—21no—
148C2×C7⋊C8—21no—
149C2×C7⋊C8—41no—
150C2×C7⋊C8—41no—
151C2×C7⋊C8—41no—
152C2×C7⋊C8—41no—
153C2×C7⋊C8—41no—
154C2×C7⋊C8—81no—
155#525C2×D711yescharacteristic
156C2×#117C2×D711yescharacteristic
157#485—21no—
158#484Dic711yescharacteristic, complemented
159#483Dic711yescharacteristic, complemented
160#482—21no—
161#481Dic711yescharacteristic, complemented
162#480Dic711yescharacteristic, complemented
163#436—141no—
164#435—141no—
165#435—141no—
166C8×D4—141no—
167#433—141no—
168#427—21no—
169#416—21no—
170#408—141no—
171#407—71no—
172C2×C22⋊C8—71no—
173C2×C22⋊C8—71no—
174C2×C22⋊C8C2×D711yescharacteristic
175C4×C22⋊C4—71no—
176#339—141no—
177#339—141no—
178#336—141no—
179#336—141no—
180#328—141no—
181#327—141no—
182C23×C7—21no—
183C23×C7—21no—
184C23×C7—21no—
185C23×C7—41no—
186C23×C7—41no—
187C23×C7C4×D411yescharacteristic
188C23×C7#12111yescharacteristic
189C23×C7#12311yescharacteristic
190C7×Q8—41no—
191C7×D4—41no—
192C7×D4—41no—
193C7×D4—41no—
194C7×D4—81no—
195C7×D4—81no—
196C2×C4×C7—21no—
197C2×C4×C7—21no—
198C2×C4×C7—21no—
199C2×C4×C7—21no—
200C2×C4×C7—21no—
201C2×C4×C7—21no—
202C2×C4×C7—21no—
203C2×C4×C7—21no—
204C2×C4×C7—21no—
205C2×C4×C7—41no—
206C2×C4×C7—41no—
207C2×C4×C7—41no—
208C2×C4×C7—41no—
209C2×C4×C7—41no—
210C2×C4×C7—42no—
211C2×C4×C7—41no—
212C2×C4×C7—41no—
213C2×C4×C7—41no—
214C2×C4×C7—41no—
215C2×C4×C7C2×C22⋊C411yescharacteristic
216C2×C4×C7C4×D411yescharacteristic
217C2×C4×C7#12011yescharacteristic
218C2×C4×C7#12111yescharacteristic
219C2×Dic7—42no—
220C2×Dic7—81no—
221C2×Dic7—81no—
222C7×C8—41no—
223C7×C8—41no—
224C7⋊C8—41no—
225C7⋊C8—41no—
226C7⋊C8—41no—
227C7⋊C8—41no—
228C7⋊C8—81no—
229C2×#41C22⋊D711yescharacteristic
230C23×C4C22×D711yescharacteristic
231C2×M16—71no—
232C2×M16—71no—
233C2×M16—141no—
234C22×C8—21no—
235C22×C8—71no—
236C22×C8—71no—
237C22×C8—141no—
238C22×C8—141no—
239#123—21no—
240C22⋊Q8C2×Dic711yescharacteristic
241C22⋊Q8C22⋊D711yescharacteristic
242#121C2×Dic711yescharacteristic
243#121C22⋊D711yescharacteristic
244#120—21no—
245C4×D4—21no—
246C4×D4—21no—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117C22⋊D711yescharacteristic
252C2×C4⋊C4C2×Dic711yescharacteristic
253C2×C4⋊C4C2×Dic711yescharacteristic
254C2×C22⋊C4—21no—
255C2×C22⋊C4—141no—
256C2×C42—21no—
257C2×C42—141no—
258#107—21no—
259#107—21no—
260#106—21no—
261#106—21no—
262C4⋊C8—141no—
263C4⋊C8—141no—
264#103—21no—
265#103—21no—
266#102—21no—
267#102—21no—
268C22⋊C8—21no—
269C22⋊C8—141no—
270C22⋊C8—141no—
271C22⋊C8—141no—
272C22⋊C8—141no—
273C22⋊C8—141no—
274C22⋊C8—141no—
275C22⋊C8—141no—
276C22⋊C8—141no—
277C22⋊C8C2×Dic711yescharacteristic
278C22⋊C8C2×Dic711yescharacteristic
279#97—141no—
280C4×C8—141no—
281#95—141no—
282C22×C7—21no—
283C22×C7—21no—
284C22×C7—21no—
285C22×C7—21no—
286C22×C7—21no—
287C22×C7—21no—
288C22×C7—21no—
289C22×C7—21no—
290C22×C7—41no—
291C22×C7—41no—
292C22×C7—41no—
293C22×C7—81no—
294C22×C7—81no—
295C22×C7#38611yescharacteristic
296C22×C7#44311yescharacteristic
297C22×C7#44411yescharacteristic
298C4×C7—21no—
299C4×C7—21no—
300C4×C7—21no—
301C4×C7—41no—
302C4×C7—41no—
303C4×C7—41no—
304C4×C7—81no—
305C4×C7—81no—
306C4×C7#45811yescharacteristic
307C4×C7#45811yescharacteristic
308Dic7—82no—
309C24#115911yescharacteristic
310#41—41no—
311#41—41no—
312C2×Q8C22⋊Dic711yescharacteristic
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4C22⋊Dic711yescharacteristic
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—281no—
323C22×C4D4×D711yescharacteristic
324C22×C4D4×D711yescharacteristic
325C22×C4#115911yescharacteristic
326C22×C4C22×Dic711yescharacteristic
327C22×C4C2×C22⋊D711yescharacteristic
328C22×C4C2×C22⋊D711yescharacteristic
329M16—281no—
330M16—281no—
331M16—561no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—141no—
337C2×C8—141no—
338C2×C8—141no—
339C2×C8—141no—
340C2×C8—281no—
341C2×C8—281no—
342C2×C8—281no—
343C2×C8—281no—
344C2×C8—281no—
345C2×C8—561no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4C22⋊Dic711yescharacteristic
353C4⋊C4C22⋊Dic711yescharacteristic
354C22⋊C4—21no—
355C22⋊C4—21no—
356C22⋊C4—21no—
357C22⋊C4—21no—
358C22⋊C4—21no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—142no—
362C22⋊C4—281no—
363C42—21no—
364C42—41no—
365C42—41no—
366C42—281no—
367C42—281no—
368C2×C7—21no—
369C2×C7—21no—
370C2×C7—41no—
371C2×C7—41no—
372C2×C7—81no—
373C2×C7#187611yescharacteristic
374C2×C7#188111yescharacteristic
375C2×C7#189011yescharacteristic
376C23—21no—
377C23—21no—
378C23—21no—
379C23—41no—
380C23—41no—
381C23D4×Dic711yescharacteristic
382C23#649811yescharacteristic
383C23#650211yescharacteristic
384Q8—41no—
385D4—41no—
386D4—41no—
387D4—41no—
388D4—81no—
389D4—81no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—282no—
410C2×C4—561no—
411C2×C4—561no—
412C2×C4D4×Dic711yescharacteristic
413C2×C4#650011yescharacteristic
414C2×C4#650211yescharacteristic
415C2×C4C2×C22⋊Dic711yescharacteristic
416C8—41no—
417C8—41no—
418C8—281no—
419C8—281no—
420C8—281no—
421C8—281no—
422C8—561no—
423C7#903811yescharacteristic, complemented, Sylow
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—81no—
436C22—81no—
437C22#8996811yescharacteristic
438C22#8996911yescharacteristic
439C22#8999211yescharacteristic
440C4—21no—
441C4—21no—
442C4—21no—
443C4—41no—
444C4—41no—
445C4—41no—
446C4—81no—
447C4—81no—
448C4—562no—
449C4#8982511yescharacteristic
450C4#8982511yescharacteristic
451C2—21no—
452C2—21no—
453C2—41no—
454C2—41no—
455C2—81no—
456C2#1172945811yescharacteristic
457C2#1172984111yescharacteristic
458C2#1172987011yescharacteristic
subgroupG/H
centerC2×C4#6500
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#3091C2
socleC22×C7#386
prime radicalC7×#525C22
cosocleC23×C4×C7C23
hypercenter#3091D7
hypocenterC7#9038
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #3091

Complements

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

NH ≅ G/Ntype
4. #11729344C2semidirect
5. #11729338C2semidirect
6. #11725486C2semidirect
7. #11725484C2semidirect
11. C7×#484C4semidirect
12. C7×#483C4semidirect
14. C7×#481C4semidirect
15. C7×#480C4semidirect
158. #484Dic7semidirect
159. #483Dic7semidirect
161. #481Dic7semidirect
162. #480Dic7semidirect
423. C7#9038semidirect

Maximal subgroups

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

subgroupindex
C7×#30912
#117403642
#117294612
#117293442
#117293382
#117254862
#117254842
#90387