Finite Group Databasebeta

← #11743337

Subgroups — #11743337

Order 896.

910 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#533C211yesmaximal, characteristic, complemented
2#90423C211yesmaximal, characteristic, complemented
3#90421C211yesmaximal, characteristic, complemented
4#90413C211yesmaximal, characteristic, complemented
5#90412C211yesmaximal, characteristic, complemented
6#90388C212yesmaximal, complemented
7#90380C212yesmaximal, complemented
8#90337C214yesmaximal, complemented
9#90334C212yesmaximal
10#90316C212yesmaximal, complemented
11#90274C214yesmaximal, complemented
12#90260C212yesmaximal, complemented
13#90259C212yesmaximal, complemented
14#90224C212yesmaximal, complemented
15#90220C212yesmaximal, complemented
16#90170C211yesmaximal, characteristic
17#90161C211yesmaximal, characteristic, complemented
18C7×C4⋊Q8C2211yescharacteristic
19C7×C4⋊Q8C2211yescharacteristic
20C7×#125C2212yes—
21C7×C22⋊Q8C2212yes—
22C7×C22⋊Q8C2212yes—
23C4×C7×Q8C2212yes—
24C4×C7×Q8C2212yes—
25C7×#117C2212yes—
26C2×C42×C7C2211yescharacteristic
27#6510C2212yes—
28#6509C2214yes—
29Q8×Dic7C2212yes—
30Q8×Dic7C2212yes—
31#6507C2212yes—
32#6507C2214yes—
33#6493C2211yescharacteristic
34#6492C2212yes—
35C4×C22⋊D7C2212yes—
36#6489C2212yes—
37#6487C2211yescharacteristic
38#6487C2211yescharacteristic
39#6487C2212yes—
40#6487C2211yescharacteristic
41#6461C2214yes—
42#6461C2214yes—
43#6461C2214yes—
44#6460C2214yes—
45#6460C2212yes—
46#6460C2212yes—
47#6459C2214yes—
48#6456C2212yes—
49#6455C2212yes—
50D7×C4⋊C4C2212yes—
51#6453C2212yes—
52#6453C2214yes—
53#6453C2214yes—
54#6453C2212yes—
55#6452C2214yes—
56#6452C2214yes—
57#6451C2212yes—
58#6451C2214yes—
59#6451C2212yes—
60#6450C2212yes—
61#6450C2212yes—
62#6450C2212yes—
63#6448C2214yes—
64#6446C2214yes—
65#6442C2214yes—
66#6442C2214yes—
67#6439C2212yes—
68#6439C2214yes—
69#6438C2211yescharacteristic
70C4×D28C2211yescharacteristic
71#6435C2212yes—
72#6435C2214yes—
73#6433C2212yes—
74#6433C2211yescharacteristic
75#6433C2214yes—
76#6432C2211yescharacteristic
77#6432C2211yescharacteristic
78C4×Dic14C2211yescharacteristic
79C4×Dic14C2211yescharacteristic
80C4×Dic14C2211yescharacteristic
81C4×Dic14C2212yes—
82C4×Dic14C2214yes—
83#3555—71nomaximal, Sylow
84C2×C7×Q8C2312yes—
85C2×C7×Q8C2312yes—
86C22×C4×C7C2311yescharacteristic
87C22×C4×C7C2311yescharacteristic
88C22×C4×C7C2311yescharacteristic
89C2×C22⋊D7C2312yes—
90C2×D28C2311yescharacteristic
91C2×C4×D7C2312yes—
92C2×C4×D7C2314yes—
93C2×Dic14C2311yescharacteristic
94C2×Dic14C2311yescharacteristic
95C2×Dic14C2311yescharacteristic
96C2×Dic14C2312yes—
97C2×Dic14C2314yes—
98C7×C4⋊C4C2312yes—
99C7×C4⋊C4C2312yes—
100C7×C4⋊C4C2312yes—
101C7×C4⋊C4C2314yes—
102C7×C4⋊C4C2312yes—
103C7×C4⋊C4C2314yes—
104C7×C22⋊C4C2314yes—
105C42×C7C2311yescharacteristic
106C42×C7C2311yescharacteristic
107C42×C7C2314yes—
108C42×C7C2311yescharacteristic
109C42×C7C2311yescharacteristic
110C22⋊Dic7C2312yes—
111C22⋊Dic7C2312yes—
112C22⋊Dic7C2312yes—
113(C2×D7)⋊C4C2312yes—
114(C2×D7)⋊C4C2312yes—
115(C2×D7)⋊C4C2312yes—
116(C2×D7)⋊C4C2314yes—
117(C2×D7)⋊C4C2312yes—
118(C2×D7)⋊C4C2314yes—
119(C2×D7)⋊C4C2314yes—
120C4⋊Dic7C2311yescharacteristic
121C4⋊Dic7C2311yescharacteristic
122C4⋊Dic7C2312yes—
123C4⋊Dic7C2311yescharacteristic
124C4⋊Dic7C2314yes—
125C4⋊Dic7C2312yes—
126C4⋊Dic7C2312yes—
127C4⋊Dic7C2311yescharacteristic
128C4⋊Dic7C2314yes—
129C4⋊Dic7C2314yes—
130Dic7⋊C4C2312yes—
131Dic7⋊C4C2312yes—
132Dic7⋊C4C2312yes—
133Dic7⋊C4C2312yes—
134Dic7⋊C4C2312yes—
135Dic7⋊C4C2312yes—
136Dic7⋊C4C2312yes—
137Dic7⋊C4C2314yes—
138Dic7⋊C4C2314yes—
139Dic7⋊C4C2312yes—
140Dic7⋊C4C2314yes—
141Dic7⋊C4C2314yes—
142Dic7⋊C4C2314yes—
143C4×Dic7C2312yes—
144C4×Dic7C2312yes—
145C4×Dic7C2312yes—
146C4×Dic7C2314yes—
147C4×Dic7C2314yes—
148C4×Dic7C2314yes—
149#563—74no—
150Q82—71no—
151#557—71no—
152#557—72no—
153#556—71no—
154#556—72no—
155#556—72no—
156#554—71no—
157#552—72no—
158#544—72no—
159#542—72no—
160#541—72no—
161#541—72no—
162#541—74no—
163#533—71no—
164#533D711yescharacteristic, complemented
165#529—71no—
166C23×C7C2411yescharacteristic
167C22×D7C2412yes—
168C7×Q8—24no—
169C7×Q8—42no—
170C2×C4×C7—22no—
171C2×C4×C7—22no—
172C2×C4×C7—41no—
173C2×C4×C7C2414yes—
174C2×C4×C7C2414yes—
175C2×C4×C7C2411yescharacteristic
176C2×C4×C7C2411yescharacteristic
177C2×C4×C7C2411yescharacteristic
178C2×C4×C7C2411yescharacteristic
179C2×C4×C7C2411yescharacteristic
180C2×C4×C7C2411yescharacteristic
181C22⋊D7—42no—
182C2×Dic7C2412yes—
183C2×Dic7C2412yes—
184C2×Dic7C2412yes—
185C2×Dic7C2412yes—
186C2×Dic7C2412yes—
187C2×Dic7C2412yes—
188C2×Dic7C2412yes—
189D28—22no—
190C4×D7—24no—
191C4×D7—44no—
192Dic14—22no—
193Dic14—22no—
194Dic14—22no—
195Dic14—44no—
196Dic14—42no—
197C4⋊Q8—72no—
198C4⋊Q8—74no—
199C4⋊Q8—72no—
200C4⋊Q8—72no—
201C4⋊Q8—74no—
202C4⋊Q8—71no—
203C4⋊Q8—71no—
204C4⋊Q8C2×D711yescharacteristic
205C4⋊Q8C2×D711yescharacteristic
206#126—74no—
207#126—74no—
208#126—74no—
209#126—74no—
210#126—74no—
211#126—72no—
212#126—74no—
213#125—74no—
214#125—74no—
215#125—72no—
216#125—74no—
217#125—74no—
218#125—72no—
219#125—71no—
220#125—72no—
221#125—74no—
222#125C2×D712yes—
223#124—72no—
224#124—74no—
225#124—71no—
226#123—72no—
227#123—74no—
228C22⋊Q8—74no—
229C22⋊Q8—71no—
230C22⋊Q8—71no—
231C22⋊Q8—71no—
232C22⋊Q8—72no—
233C22⋊Q8—74no—
234C22⋊Q8—72no—
235C22⋊Q8—72no—
236C22⋊Q8—74no—
237C22⋊Q8C2×D712yes—
238C22⋊Q8C2×D712yes—
239#121—71no—
240C4×Q8—72no—
241C4×Q8—72no—
242C4×Q8—72no—
243C4×Q8—72no—
244C4×Q8—72no—
245C4×Q8—71no—
246C4×Q8—71no—
247C4×Q8—72no—
248C4×Q8—71no—
249C4×Q8—74no—
250C4×Q8C2×D712yes—
251C4×Q8C2×D712yes—
252C4×D4—72no—
253C4×D4—72no—
254C4×D4—71no—
255#117—72no—
256#117—72no—
257#117—72no—
258#117—74no—
259#117C2×D712yes—
260C2×C4⋊C4—72no—
261C2×C42C2×D711yescharacteristic
262C22×C7—21no—
263C22×C7—21no—
264C22×C7—21no—
265C22×C7C2511yescharacteristic
266C2×D7—22no—
267C2×D7—22no—
268C2×D7—22no—
269C4×C7—21no—
270C4×C7—21no—
271C4×C7—24no—
272C4×C7—21no—
273C4×C7—24no—
274C4×C7—21no—
275C4×C72−1+412yes—
276C4×C72−1+412yes—
277Dic7—22no—
278Dic7—22no—
279Dic7—22no—
280Dic7—22no—
281Dic7—22no—
282Dic7—22no—
283Dic7—22no—
284C2×Q8—71no—
285C2×Q8—71no—
286C2×Q8—71no—
287C2×Q8—72no—
288C2×Q8—74no—
289C2×Q8C22×D712yes—
290C2×Q8C22×D712yes—
291C2×D4—72no—
292C2×D4—71no—
293C22×C4—72no—
294C22×C4—74no—
295C22×C4C22×D711yescharacteristic
296C22×C4C22×D711yescharacteristic
297C22×C4C22×D711yescharacteristic
298C4⋊C4—74no—
299C4⋊C4—72no—
300C4⋊C4—72no—
301C4⋊C4—74no—
302C4⋊C4—72no—
303C4⋊C4—72no—
304C4⋊C4—72no—
305C4⋊C4—72no—
306C4⋊C4—74no—
307C4⋊C4—72no—
308C4⋊C4—74no—
309C4⋊C4—72no—
310C4⋊C4—74no—
311C4⋊C4—71no—
312C4⋊C4—71no—
313C4⋊C4—72no—
314C4⋊C4—72no—
315C4⋊C4—71no—
316C4⋊C4—71no—
317C4⋊C4—74no—
318C4⋊C4—74no—
319C4⋊C4—74no—
320C4⋊C4—72no—
321C4⋊C4C22×D712yes—
322C4⋊C4C22×D712yes—
323C4⋊C4C22×D712yes—
324C4⋊C4C22×D714yes—
325C4⋊C4C22×D714yes—
326C4⋊C4C22×D712yes—
327C22⋊C4—72no—
328C22⋊C4—72no—
329C22⋊C4—72no—
330C22⋊C4—72no—
331C22⋊C4—72no—
332C22⋊C4—72no—
333C22⋊C4—74no—
334C22⋊C4—74no—
335C22⋊C4—72no—
336C22⋊C4—74no—
337C22⋊C4C22×D714yes—
338C42—72no—
339C42—72no—
340C42—72no—
341C42—74no—
342C42—74no—
343C42—74no—
344C42C22×D711yescharacteristic
345C42C22×D711yescharacteristic
346C42C22×D711yescharacteristic
347C42C22×D711yescharacteristic
348C42C22×D714yes—
349C2×C7—41no—
350C2×C7C2×2−1+411yescharacteristic
351C2×C7C2×2−1+411yescharacteristic
352C2×C7#58511yescharacteristic
353D7—42no—
354C23—72no—
355C23C23×D711yescharacteristic
356Q8—24no—
357Q8—42no—
358Q8—142no—
359Q8—142no—
360Q8—142no—
361Q8—282no—
362Q8—284no—
363D4—142no—
364D4—282no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—72no—
369C2×C4—72no—
370C2×C4—72no—
371C2×C4—72no—
372C2×C4—72no—
373C2×C4—72no—
374C2×C4—72no—
375C2×C4—144no—
376C2×C4—284no—
377C2×C4C23×D714yes—
378C2×C4C23×D714yes—
379C2×C4C23×D711yescharacteristic
380C2×C4C23×D711yescharacteristic
381C2×C4C23×D711yescharacteristic
382C2×C4C23×D711yescharacteristic
383C2×C4C23×D711yescharacteristic
384C2×C4C23×D711yescharacteristic
385C7#355511yescharacteristic, complemented, Sylow
386C22—21no—
387C22—21no—
388C22—21no—
389C22—142no—
390C22—142no—
391C22—142no—
392C22C24×D711yescharacteristic
393C4—21no—
394C4—21no—
395C4—24no—
396C4—21no—
397C4—21no—
398C4—24no—
399C4—142no—
400C4—142no—
401C4—142no—
402C4—142no—
403C4—142no—
404C4—142no—
405C4—142no—
406C4#655112yes—
407C4#655412yes—
408C2—41no—
409C2—282no—
410C2C2×#655111yescharacteristic
411C2C2×#655411yescharacteristic
412C2#9061611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D7
commutator, socle, cosocleC22×C7C25
FittingC7×#533C2
prime radical#6493C22
hypercenter#533D7
hypocenterC7#3555
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #533

Complements

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

NH ≅ G/Ntype
1. C7×#533C2semidirect
2. #90423C2semidirect
3. #90421C2semidirect
4. #90413C2semidirect
5. #90412C2semidirect
6. #90388C2semidirect
7. #90380C2semidirect
8. #90337C2semidirect
10. #90316C2semidirect
11. #90274C2semidirect
12. #90260C2semidirect
13. #90259C2semidirect
14. #90224C2semidirect
15. #90220C2semidirect
17. #90161C2semidirect
164. #533D7semidirect
385. C7#3555semidirect

Maximal subgroups

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

subgroupindex
C7×#5332
#904232
#904212
#904132
#904122
#903882
#903802
#903372
#903342
#903162
#902742
#902602
#902592
#902242
#902202
#901702
#901612
#35557