Finite Group Databasebeta

← #13249607

Subgroups — #13249607

Order 1792.

474 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3115C211yesmaximal, characteristic
2#11739963C211yesmaximal, characteristic, complemented
3#11729566C211yesmaximal, characteristic, complemented
4#11729551C211yesmaximal, characteristic, complemented
5#11729397C211yesmaximal, characteristic, complemented
6#11725559C211yesmaximal, characteristic, complemented
7#11725554C211yesmaximal, characteristic, complemented
8C7×#532C2211yescharacteristic
9C4×C7×#41C2211yescharacteristic
10C7×#482—21no—
11C7×#475C411yescharacteristic, complemented
12C7×#474C411yescharacteristic, complemented
13C7×#460C411yescharacteristic, complemented
14C7×#459C411yescharacteristic, complemented
15C7×#449—21no—
16C7×#442—21no—
17C7×#440—21no—
18C7×#432C2211yescharacteristic
19C2×#6411—21no—
20#89796—21no—
21#89783—22no—
22D4×C7⋊C8—21no—
23#89704C2211yescharacteristic
24#89698—21no—
25#89698C2211yescharacteristic
26C4×#1136C2211yescharacteristic
27C4×#1136C2211yescharacteristic
28C2×#6377—21no—
29#89335—21no—
30#89335—21no—
31#89332—22no—
32#89327—21no—
33#9161—71nomaximal, Sylow
34C2×C7×#41—21no—
35C2×C7×#41D411yescharacteristic
36C2×C7×SD16—21no—
37C2×C7×D8—21no—
38C7×C4⋊Q8C2×C411yescharacteristic
39C7×#127C2×C411yescharacteristic
40C7×#124—21no—
41C7×#121—21no—
42C7×#121—21no—
43C4×C7×Q8C2×C411yescharacteristic
44C4×C7×Q8D411yescharacteristic
45C4×C7×D4—21no—
46C4×C7×D4—21no—
47C4×C7×D4—21no—
48C4×C7×D4—21no—
49C4×C7×D4C2×C411yescharacteristic
50C4×C7×D4D411yescharacteristic
51C7×#117—21no—
52C7×#117D411yescharacteristic
53C2×C42×C7—21no—
54C2×C42×C7C2311yescharacteristic
55C2×C4×Dic7—21no—
56C2×#1136—21no—
57C2×#1136—21no—
58C22×C7⋊C8—22no—
59C7×#107—21no—
60C7×#106—21no—
61C7×C4⋊C8C2×C411yescharacteristic
62C7×C4⋊C8C2×C411yescharacteristic
63C7×#103—21no—
64C7×#102—21no—
65C7×#102—21no—
66C7×#102—21no—
67C7×C22⋊C8—21no—
68C7×#97—21no—
69#6411—24no—
70C7⋊#95—21no—
71C7⋊#95—21no—
72C22⋊(C7⋊C8)—22no—
73C22⋊(C7⋊C8)D412yes—
74C4⋊(C7⋊C8)—22no—
75C4⋊(C7⋊C8)—21no—
76#6377—22no—
77#6377—21no—
78#6377—21no—
79#6377D411yescharacteristic
80#6377D411yescharacteristic
81C4×C7⋊C8—21no—
82C4×C7⋊C8—21no—
83#3115D711yescharacteristic
84#2970—71no—
85#1759—71no—
86#1525—71no—
87#1505—71no—
88#1300—71no—
89#1291—71no—
90C7×#41—22no—
91C7×#41—41no—
92C7×#41—82no—
93C2×C7×Q8—21no—
94C2×C7×Q8C22⋊C411yescharacteristic
95C2×C7×D4—21no—
96C2×C7×D4—21no—
97C2×C7×D4—21no—
98C2×C7×D4—21no—
99C2×C7×D4—41no—
100C2×C7×D4C22⋊C411yescharacteristic
101C22×C4×C7—21no—
102C22×C4×C7—21no—
103C22×C4×C7—21no—
104C22×C4×C7—41no—
105C22×C4×C7C2×D411yescharacteristic
106C22×C4×C7C2×D411yescharacteristic
107C22×C4×C7#4111yescharacteristic
108C22×Dic7—41no—
109C7×SD16—42no—
110C7×D8—42no—
111C2×C7×C8—21no—
112C2×C7×C8—21no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—21no—
115C7×C4⋊C4—21no—
116C7×C4⋊C4—21no—
117C7×C4⋊C4C22⋊C411yescharacteristic
118C7×C4⋊C4C22⋊C411yescharacteristic
119C7×C22⋊C4—21no—
120C7×C22⋊C4—21no—
121C7×C22⋊C4—21no—
122C7×C22⋊C4—41no—
123C7×C22⋊C4—41no—
124C42×C7—21no—
125C42×C7—21no—
126C42×C7—21no—
127C42×C7C22×C411yescharacteristic
128C42×C7C2×D411yescharacteristic
129C42×C7C2×D411yescharacteristic
130C42×C7#4111yescharacteristic
131C4×Dic7—22no—
132C4×Dic7—41no—
133#1136—22no—
134#1136—42no—
135#1136—41no—
136C2×C7⋊C8—21no—
137C2×C7⋊C8—21no—
138C2×C7⋊C8—22no—
139C2×C7⋊C8—44no—
140C2×C7⋊C8—41no—
141#532C2×D711yescharacteristic
142C4×#41C2×D711yescharacteristic
143#482—21no—
144#475Dic711yescharacteristic, complemented
145#474Dic711yescharacteristic, complemented
146#460Dic711yescharacteristic, complemented
147#459Dic711yescharacteristic, complemented
148#449—21no—
149#446—141no—
150#442—21no—
151#440—21no—
152#435—142no—
153C8×D4—141no—
154#432—71no—
155#432—141no—
156#432C2×D711yescharacteristic
157C2×#104—141no—
158C4×M16—71no—
159C4×M16—71no—
160C2×#97—141no—
161#376—71no—
162#339—141no—
163#329—141no—
164#329—141no—
165#327—142no—
166C23×C7—21no—
167C23×C7—41no—
168C23×C7#12111yescharacteristic
169C7×Q8—21no—
170C7×Q8—21no—
171C7×Q8—81no—
172C7×D4—21no—
173C7×D4—42no—
174C7×D4—41no—
175C7×D4—41no—
176C7×D4—81no—
177C7×D4—81no—
178C7×D4—82no—
179C7×D4#10412yes—
180C2×C4×C7—21no—
181C2×C4×C7—21no—
182C2×C4×C7—21no—
183C2×C4×C7—21no—
184C2×C4×C7—21no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—42no—
188C2×C4×C7—42no—
189C2×C4×C7—41no—
190C2×C4×C7—81no—
191C2×C4×C7—82no—
192C2×C4×C7C2×C22⋊C411yescharacteristic
193C2×C4×C7C4×D411yescharacteristic
194C2×C4×C7C4×D411yescharacteristic
195C2×C4×C7#12011yescharacteristic
196C2×C4×C7#12111yescharacteristic
197C2×C4×C7#12311yescharacteristic
198C2×Dic7—42no—
199C2×Dic7—81no—
200C2×Dic7—81no—
201C7×C8—41no—
202C7×C8—41no—
203C7⋊C8—22no—
204C7⋊C8—42no—
205C7⋊C8—41no—
206C7⋊C8—42no—
207C2×#41—21no—
208C2×#41C22⋊D711yescharacteristic
209C2×SD16—21no—
210C2×D8—21no—
211C2×M16—141no—
212C2×M16—141no—
213C22×C8—142no—
214C4⋊Q8C2×Dic711yescharacteristic
215#127C2×Dic711yescharacteristic
216#124—21no—
217#121—21no—
218#121—21no—
219C4×Q8C2×Dic711yescharacteristic
220C4×Q8C22⋊D711yescharacteristic
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—21no—
224C4×D4—21no—
225C4×D4C2×Dic711yescharacteristic
226C4×D4C22⋊D711yescharacteristic
227#117—21no—
228#117C22⋊D711yescharacteristic
229C2×C42—21no—
230C2×C42—141no—
231C2×C42C22×D711yescharacteristic
232#107—21no—
233#106—21no—
234C4⋊C8—142no—
235C4⋊C8—141no—
236C4⋊C8C2×Dic711yescharacteristic
237C4⋊C8C2×Dic711yescharacteristic
238#104—144no—
239#103—21no—
240#102—21no—
241#102—21no—
242#102—21no—
243C22⋊C8—21no—
244C22⋊C8—72no—
245C22⋊C8—142no—
246#97—21no—
247#97—71no—
248#97—71no—
249#97—142no—
250#97—141no—
251#97—141no—
252C4×C8—141no—
253C4×C8—141no—
254#95—141no—
255#95—141no—
256C22×C7—21no—
257C22×C7—22no—
258C22×C7—21no—
259C22×C7—21no—
260C22×C7—41no—
261C22×C7—41no—
262C22×C7—81no—
263C22×C7—81no—
264C22×C7—81no—
265C22×C7#38611yescharacteristic
266C4×C7—21no—
267C4×C7—21no—
268C4×C7—21no—
269C4×C7—21no—
270C4×C7—21no—
271C4×C7—41no—
272C4×C7—41no—
273C4×C7—41no—
274C4×C7—81no—
275C4×C7C2×#10411yescharacteristic
276C4×C7#42111yescharacteristic
277Dic7—82no—
278#41—22no—
279#41—41no—
280#41—82no—
281C2×Q8—21no—
282C2×Q8C22⋊Dic711yescharacteristic
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—41no—
288C2×D4C22⋊Dic711yescharacteristic
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—41no—
293C22×C4—281no—
294C22×C4D4×D711yescharacteristic
295C22×C4#115911yescharacteristic
296C22×C4C2×C22⋊D711yescharacteristic
297SD16—42no—
298D8—42no—
299M16—142no—
300M16—282no—
301M16—281no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—141no—
305C2×C8—141no—
306C2×C8—142no—
307C2×C8—284no—
308C2×C8—281no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4C22⋊Dic711yescharacteristic
314C4⋊C4C22⋊Dic711yescharacteristic
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—142no—
324C42—281no—
325C42D4×D711yescharacteristic
326C42#115911yescharacteristic
327C42C22×Dic711yescharacteristic
328C42C2×C22⋊D711yescharacteristic
329C2×C7—41no—
330C2×C7—42no—
331C2×C7—161no—
332C2×C7#187111yescharacteristic
333C2×C7#190311yescharacteristic
334C2×C7#190611yescharacteristic
335C23—21no—
336C23—41no—
337C23#650211yescharacteristic
338Q8—21no—
339Q8—21no—
340Q8—81no—
341D4—21no—
342D4—42no—
343D4—41no—
344D4—41no—
345D4—81no—
346D4—81no—
347D4—82no—
348D4#641112yes—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—81no—
360C2×C4—82no—
361C2×C4—282no—
362C2×C4—561no—
363C2×C4—561no—
364C2×C4D4×Dic711yescharacteristic
365C2×C4D4×Dic711yescharacteristic
366C2×C4#649811yescharacteristic
367C2×C4#650011yescharacteristic
368C2×C4#650211yescharacteristic
369C2×C4C2×C22⋊Dic711yescharacteristic
370C8—41no—
371C8—41no—
372C8—142no—
373C8—282no—
374C8—281no—
375C8—282no—
376C7#916111yescharacteristic, complemented, Sylow
377C22—21no—
378C22—22no—
379C22—21no—
380C22—21no—
381C22—41no—
382C22—41no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#8999211yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—81no—
396C4—562no—
397C4C2×#641111yescharacteristic
398C4#9000811yescharacteristic
399C2—41no—
400C2—42no—
401C2—161no—
402C2#1172946711yescharacteristic
403C2#1172984411yescharacteristic
404C2#1172986811yescharacteristic
subgroupG/H
centerC22#89992
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
Fitting, prime radicalC7×#3115C2
socleC22×C7#386
cosocleC2×C42×C7C23
hypercenter#3115D7
hypocenterC7#9161
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3115

Complements

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

NH ≅ G/Ntype
2. #11739963C2semidirect
3. #11729566C2semidirect
4. #11729551C2semidirect
5. #11729397C2semidirect
6. #11725559C2semidirect
7. #11725554C2semidirect
11. C7×#475C4semidirect
12. C7×#474C4semidirect
13. C7×#460C4semidirect
14. C7×#459C4semidirect
144. #475Dic7semidirect
145. #474Dic7semidirect
146. #460Dic7semidirect
147. #459Dic7semidirect
376. C7#9161semidirect

Maximal subgroups

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

subgroupindex
C7×#31152
#117399632
#117295662
#117295512
#117293972
#117255592
#117255542
#91617