Finite Group Databasebeta

← #13249593

Subgroups — #13249593

Order 1792.

452 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3116C211yesmaximal, characteristic
2#11739963C211yesmaximal, characteristic
3#11729590C211yesmaximal, characteristic, complemented
4#11729575C211yesmaximal, characteristic, complemented
5#11729397C211yesmaximal, characteristic
6#11725559C211yesmaximal, characteristic, complemented
7#11725554C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9C4×C7×#41C2211yescharacteristic
10C7×#485—21no—
11C7×#477C411yescharacteristic, complemented
12C7×#476C411yescharacteristic, complemented
13C7×#462C411yescharacteristic, complemented
14C7×#461C411yescharacteristic, complemented
15C7×#452—21no—
16C7×#441—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#89338—21no—
30#89338—21no—
31#89333—22no—
32#89327—21no—
33#9159—71nomaximal, Sylow
34C2×C7×#41D411yescharacteristic
35C2×C7×Q16—21no—
36C2×C7×SD16—21no—
37C7×C4⋊Q8C2×C411yescharacteristic
38C7×C4⋊Q8C2×C411yescharacteristic
39C7×#125—21no—
40C7×C22⋊Q8—21no—
41C7×C22⋊Q8—21no—
42C4×C7×Q8—21no—
43C4×C7×Q8—21no—
44C4×C7×Q8C2×C411yescharacteristic
45C4×C7×Q8D411yescharacteristic
46C4×C7×D4—21no—
47C4×C7×D4—21no—
48C4×C7×D4C2×C411yescharacteristic
49C4×C7×D4D411yescharacteristic
50C7×#117—21no—
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×#103—21no—
65C7×#103—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#3116D711yescharacteristic
84#2970—71no—
85#1759—71no—
86#1526—71no—
87#1506—71no—
88#1300—71no—
89#1291—71no—
90C7×#41—22no—
91C7×#41—41no—
92C2×C7×Q8—21no—
93C2×C7×Q8—21no—
94C2×C7×Q8C22⋊C411yescharacteristic
95C2×C7×D4—21no—
96C2×C7×D4C22⋊C411yescharacteristic
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7—21no—
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7#4111yescharacteristic
103C22×Dic7—41no—
104C7×Q16—42no—
105C7×SD16—42no—
106C2×C7×C8—21no—
107C2×C7×C8—21no—
108C7×C4⋊C4—21no—
109C7×C4⋊C4—21no—
110C7×C4⋊C4—21no—
111C7×C4⋊C4—21no—
112C7×C4⋊C4—21no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—41no—
115C7×C4⋊C4—41no—
116C7×C4⋊C4C22⋊C411yescharacteristic
117C7×C4⋊C4C22⋊C411yescharacteristic
118C7×C22⋊C4—21no—
119C7×C22⋊C4—21no—
120C7×C22⋊C4—21no—
121C7×C22⋊C4—41no—
122C42×C7—21no—
123C42×C7—21no—
124C42×C7—21no—
125C42×C7—41no—
126C42×C7C22×C411yescharacteristic
127C42×C7C2×D411yescharacteristic
128C42×C7C2×D411yescharacteristic
129C42×C7#4111yescharacteristic
130C4×Dic7—22no—
131C4×Dic7—41no—
132#1136—22no—
133#1136—42no—
134#1136—41no—
135C2×C7⋊C8—21no—
136C2×C7⋊C8—21no—
137C2×C7⋊C8—22no—
138C2×C7⋊C8—44no—
139C2×C7⋊C8—41no—
140#533C2×D711yescharacteristic
141C4×#41C2×D711yescharacteristic
142#485—21no—
143#477Dic711yescharacteristic, complemented
144#476Dic711yescharacteristic, complemented
145#462Dic711yescharacteristic, complemented
146#461Dic711yescharacteristic, complemented
147#452—21no—
148#446—141no—
149#441—21no—
150#440—21no—
151#435—142no—
152C8×D4—141no—
153#432—71no—
154#432—141no—
155#432C2×D711yescharacteristic
156C2×#104—141no—
157C4×M16—71no—
158C4×M16—71no—
159C2×#97—141no—
160#376—71no—
161#339—141no—
162#330—141no—
163#330—141no—
164#328—142no—
165C23×C7—21no—
166C23×C7#12111yescharacteristic
167C7×Q8—21no—
168C7×Q8—21no—
169C7×Q8—41no—
170C7×Q8—41no—
171C7×Q8—81no—
172C7×D4—21no—
173C7×D4—42no—
174C7×D4#10412yes—
175C2×C4×C7—21no—
176C2×C4×C7—21no—
177C2×C4×C7—21no—
178C2×C4×C7—21no—
179C2×C4×C7—21no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—42no—
183C2×C4×C7—42no—
184C2×C4×C7—41no—
185C2×C4×C7—41no—
186C2×C4×C7—81no—
187C2×C4×C7C2×C22⋊C411yescharacteristic
188C2×C4×C7C4×D411yescharacteristic
189C2×C4×C7C4×D411yescharacteristic
190C2×C4×C7#12011yescharacteristic
191C2×C4×C7#12111yescharacteristic
192C2×C4×C7#12311yescharacteristic
193C2×Dic7—42no—
194C2×Dic7—81no—
195C2×Dic7—81no—
196C7×C8—41no—
197C7×C8—41no—
198C7⋊C8—22no—
199C7⋊C8—42no—
200C7⋊C8—41no—
201C7⋊C8—42no—
202C2×#41C22⋊D711yescharacteristic
203C2×Q16—21no—
204C2×SD16—21no—
205C2×M16—141no—
206C2×M16—141no—
207C22×C8—142no—
208C4⋊Q8C2×Dic711yescharacteristic
209C4⋊Q8C2×Dic711yescharacteristic
210#125—21no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213C4×Q8—21no—
214C4×Q8—21no—
215C4×Q8C2×Dic711yescharacteristic
216C4×Q8C22⋊D711yescharacteristic
217C4×D4—21no—
218C4×D4—21no—
219C4×D4C2×Dic711yescharacteristic
220C4×D4C22⋊D711yescharacteristic
221#117—21no—
222#117—21no—
223#117C22⋊D711yescharacteristic
224C2×C42—21no—
225C2×C42—141no—
226C2×C42C22×D711yescharacteristic
227#107—21no—
228#106—21no—
229C4⋊C8—142no—
230C4⋊C8—141no—
231C4⋊C8C2×Dic711yescharacteristic
232C4⋊C8C2×Dic711yescharacteristic
233#104—144no—
234#103—21no—
235#103—21no—
236#103—21no—
237#102—21no—
238C22⋊C8—21no—
239C22⋊C8—72no—
240C22⋊C8—142no—
241#97—21no—
242#97—71no—
243#97—71no—
244#97—142no—
245#97—141no—
246#97—141no—
247C4×C8—141no—
248C4×C8—141no—
249#95—141no—
250#95—141no—
251C22×C7—21no—
252C22×C7—22no—
253C22×C7—21no—
254C22×C7—21no—
255C22×C7—41no—
256C22×C7—41no—
257C22×C7#38611yescharacteristic
258C4×C7—21no—
259C4×C7—21no—
260C4×C7—21no—
261C4×C7—21no—
262C4×C7—21no—
263C4×C7—41no—
264C4×C7—41no—
265C4×C7—41no—
266C4×C7—81no—
267C4×C7—81no—
268C4×C7C2×#10411yescharacteristic
269C4×C7#42111yescharacteristic
270Dic7—82no—
271#41—22no—
272#41—41no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8C22⋊Dic711yescharacteristic
276C2×D4—21no—
277C2×D4C22⋊Dic711yescharacteristic
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—281no—
282C22×C4D4×D711yescharacteristic
283C22×C4#115911yescharacteristic
284C22×C4C2×C22⋊D711yescharacteristic
285Q16—42no—
286SD16—42no—
287M16—142no—
288M16—282no—
289M16—281no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—141no—
293C2×C8—141no—
294C2×C8—142no—
295C2×C8—284no—
296C2×C8—281no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4C22⋊Dic711yescharacteristic
306C4⋊C4C22⋊Dic711yescharacteristic
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—41no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—41no—
315C42—142no—
316C42—281no—
317C42D4×D711yescharacteristic
318C42#115911yescharacteristic
319C42C22×Dic711yescharacteristic
320C42C2×C22⋊D711yescharacteristic
321C2×C7—41no—
322C2×C7—42no—
323C2×C7#187111yescharacteristic
324C2×C7#190211yescharacteristic
325C2×C7#190611yescharacteristic
326C23—21no—
327C23#650211yescharacteristic
328Q8—21no—
329Q8—21no—
330Q8—41no—
331Q8—41no—
332Q8—81no—
333D4—21no—
334D4—42no—
335D4#641112yes—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—81no—
348C2×C4—282no—
349C2×C4—561no—
350C2×C4—561no—
351C2×C4D4×Dic711yescharacteristic
352C2×C4D4×Dic711yescharacteristic
353C2×C4#649811yescharacteristic
354C2×C4#650011yescharacteristic
355C2×C4#650211yescharacteristic
356C2×C4C2×C22⋊Dic711yescharacteristic
357C8—41no—
358C8—41no—
359C8—142no—
360C8—282no—
361C8—281no—
362C8—282no—
363C7#915911yescharacteristic, complemented, Sylow
364C22—21no—
365C22—22no—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22#8999211yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—81no—
380C4—81no—
381C4—562no—
382C4C2×#641111yescharacteristic
383C4#9000811yescharacteristic
384C2—41no—
385C2—42no—
386C2#1172946711yescharacteristic
387C2#1172985611yescharacteristic
388C2#1172986811yescharacteristic
subgroupG/H
centerC22#89992
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#3116C2
socleC22×C7#386
prime radicalC2×C7×#41D4
cosocleC2×C42×C7C23
hypercenter#3116D7
hypocenterC7#9159
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3116

Complements

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

NH ≅ G/Ntype
3. #11729590C2semidirect
4. #11729575C2semidirect
6. #11725559C2semidirect
7. #11725554C2semidirect
11. C7×#477C4semidirect
12. C7×#476C4semidirect
13. C7×#462C4semidirect
14. C7×#461C4semidirect
143. #477Dic7semidirect
144. #476Dic7semidirect
145. #462Dic7semidirect
146. #461Dic7semidirect
363. C7#9159semidirect

Maximal subgroups

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

subgroupindex
C7×#31162
#117399632
#117295902
#117295752
#117293972
#117255592
#117255542
#91597