Finite Group Databasebeta

← #11744104

Subgroups — #11744104

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×#557C211yesmaximal, characteristic, complemented
2#90423C211yesmaximal, characteristic, complemented
3#90423C212yesmaximal, complemented
4#90422C212yesmaximal, complemented
5#90419C211yesmaximal, characteristic, complemented
6#90390C214yesmaximal, complemented
7#90387C214yesmaximal, complemented
8#90385C212yesmaximal, complemented
9#90380C212yesmaximal, complemented
10#90275C212yesmaximal, complemented
11#90274C212yesmaximal, complemented
12#90273C212yesmaximal
13#90271C211yesmaximal, characteristic
14#90270C211yesmaximal, characteristic, complemented
15#90269C211yesmaximal, characteristic, complemented
16#90260C212yesmaximal, complemented
17Q8×Dic14C211yesmaximal, characteristic, complemented
18C7×C4⋊Q8C2211yescharacteristic
19C7×C4⋊Q8C2212yes—
20C7×#125C2212yes—
21C7×#125C2214yes—
22C4×C7×Q8C2211yescharacteristic
23C4×C7×Q8C2212yes—
24C4×C7×Q8C2211yescharacteristic
25C4×C7×Q8C2212yes—
26#6510C2211yescharacteristic
27#6510C2212yes—
28#6509C2212yes—
29#6509C2211yescharacteristic
30#6509C2211yescharacteristic
31#6509C2212yes—
32Q8×Dic7C2212yes—
33Q8×Dic7C2212yes—
34Q8×Dic7C2212yes—
35#6507C2211yescharacteristic
36#6461C2214yes—
37#6461C2212yes—
38#6461C2214yes—
39#6461C2214yes—
40#6461C2214yes—
41#6461C2214yes—
42#6460C2214yes—
43#6460C2211yescharacteristic
44#6460C2212yes—
45#6460C2212yes—
46#6459C2214yes—
47#6459C2214yes—
48#6459C2212yes—
49#6458C2211yescharacteristic
50#6457C2214yes—
51#6457C2212yes—
52#6455C2212yes—
53#6455C2214yes—
54#6455C2212yes—
55D7×C4⋊C4C2212yes—
56#6453C2212yes—
57#6453C2214yes—
58#6453C2211yescharacteristic
59#6452C2214yes—
60#6452C2212yes—
61#6452C2214yes—
62#6451C2211yescharacteristic
63#6451C2214yes—
64#6451C2214yes—
65#6451C2212yes—
66#6450C2212yes—
67#6439C2214yes—
68#6438C2214yes—
69C4×D28C2212yes—
70C4×D28C2211yescharacteristic
71C4×D28C2212yes—
72#6435C2212yes—
73#6435C2212yes—
74C42×D7C2211yescharacteristic
75#6433C2214yes—
76#6433C2212yes—
77#6432C2212yes—
78#6432C2211yescharacteristic
79C4×Dic14C2212yes—
80C4×Dic14C2214yes—
81C4×Dic14C2211yescharacteristic
82C4×Dic14C2212yes—
83#3555—71nomaximal, Sylow
84C2×C7×Q8C2311yescharacteristic
85C2×C7×Q8C2312yes—
86C2×C7×Q8C2311yescharacteristic
87C2×D28C2311yescharacteristic
88C2×D28C2312yes—
89C2×C4×D7C2311yescharacteristic
90C2×C4×D7C2311yescharacteristic
91C2×C4×D7C2312yes—
92C2×C4×D7C2311yescharacteristic
93C2×C4×D7C2314yes—
94C2×Dic14C2311yescharacteristic
95C2×Dic14C2314yes—
96C2×Dic14C2312yes—
97C2×Dic14C2312yes—
98C7×C4⋊C4C2312yes—
99C7×C4⋊C4C2311yescharacteristic
100C7×C4⋊C4C2312yes—
101C7×C4⋊C4C2314yes—
102C7×C4⋊C4C2311yescharacteristic
103C7×C4⋊C4C2314yes—
104C7×C4⋊C4C2312yes—
105C7×C4⋊C4C2314yes—
106C7×C4⋊C4C2312yes—
107C42×C7C2311yescharacteristic
108C42×C7C2312yes—
109C42×C7C2312yes—
110C42×C7C2314yes—
111(C2×D7)⋊C4C2312yes—
112(C2×D7)⋊C4C2312yes—
113(C2×D7)⋊C4C2312yes—
114(C2×D7)⋊C4C2312yes—
115(C2×D7)⋊C4C2314yes—
116(C2×D7)⋊C4C2312yes—
117(C2×D7)⋊C4C2312yes—
118(C2×D7)⋊C4C2312yes—
119(C2×D7)⋊C4C2314yes—
120(C2×D7)⋊C4C2314yes—
121(C2×D7)⋊C4C2314yes—
122C4⋊Dic7C2312yes—
123C4⋊Dic7C2311yescharacteristic
124C4⋊Dic7C2312yes—
125C4⋊Dic7C2312yes—
126C4⋊Dic7C2312yes—
127C4⋊Dic7C2314yes—
128C4⋊Dic7C2311yescharacteristic
129C4⋊Dic7C2314yes—
130C4⋊Dic7C2314yes—
131Dic7⋊C4C2312yes—
132Dic7⋊C4C2314yes—
133Dic7⋊C4C2312yes—
134Dic7⋊C4C2314yes—
135Dic7⋊C4C2312yes—
136Dic7⋊C4C2312yes—
137Dic7⋊C4C2314yes—
138Dic7⋊C4C2312yes—
139Dic7⋊C4C2312yes—
140Dic7⋊C4C2312yes—
141Dic7⋊C4C2314yes—
142C4×Dic7C2311yescharacteristic
143C4×Dic7C2312yes—
144C4×Dic7C2314yes—
145C4×Dic7C2314yes—
146C4×Dic7C2314yes—
147C4×Dic7C2311yescharacteristic
148C4×Dic7C2311yescharacteristic
149#563—74no—
150Q82—71no—
151#557—72no—
152#557D711yescharacteristic, complemented
153#556—72no—
154#556—71no—
155#556—72no—
156#554—71no—
157#552—72no—
158#544—72no—
159#542—72no—
160#541—72no—
161#541—74no—
162#541—72no—
163#533—71no—
164#533—71no—
165#529—71no—
166C22×D7C2411yescharacteristic
167C22×D7C2412yes—
168C7×Q8—22no—
169C7×Q8—24no—
170C7×Q8—22no—
171C2×C4×C7C2414yes—
172C2×C4×C7C2412yes—
173C2×C4×C7C2411yescharacteristic
174C2×C4×C7C2412yes—
175C2×C4×C7C2412yes—
176C2×C4×C7C2412yes—
177C2×C4×C7C2411yescharacteristic
178C2×C4×C7C2411yescharacteristic
179C2×Dic7C2411yescharacteristic
180C2×Dic7C2412yes—
181C2×Dic7C2412yes—
182C2×Dic7C2411yescharacteristic
183C2×Dic7C2412yes—
184C2×Dic7C2411yescharacteristic
185C2×Dic7C2414yes—
186D28—22no—
187D28—42no—
188C4×D7—22no—
189C4×D7—24no—
190C4×D7—22no—
191C4×D7—41no—
192C4×D7—44no—
193Dic14—22no—
194Dic14—44no—
195Dic14—42no—
196Dic14—42no—
197C4⋊Q8—71no—
198C4⋊Q8—71no—
199C4⋊Q8—74no—
200C4⋊Q8—72no—
201C4⋊Q8—74no—
202C4⋊Q8—71no—
203C4⋊Q8—72no—
204C4⋊Q8C2×D711yescharacteristic
205C4⋊Q8C2×D712yes—
206#126—72no—
207#126—74no—
208#126—74no—
209#126—74no—
210#126—74no—
211#126—74no—
212#126—74no—
213#125—71no—
214#125—72no—
215#125—72no—
216#125—74no—
217#125—74no—
218#125—74no—
219#125—72no—
220#125—74no—
221#125C2×D714yes—
222#125C2×D712yes—
223#124—71no—
224#124—72no—
225#124—74no—
226#123—72no—
227#123—74no—
228C22⋊Q8—71no—
229C22⋊Q8—72no—
230C22⋊Q8—71no—
231C22⋊Q8—72no—
232C22⋊Q8—71no—
233C22⋊Q8—72no—
234C22⋊Q8—72no—
235C22⋊Q8—74no—
236C22⋊Q8—72no—
237C22⋊Q8—74no—
238C22⋊Q8—74no—
239#121—71no—
240C4×Q8—72no—
241C4×Q8—72no—
242C4×Q8—72no—
243C4×Q8—72no—
244C4×Q8—72no—
245C4×Q8—71no—
246C4×Q8—72no—
247C4×Q8—74no—
248C4×Q8C2×D711yescharacteristic
249C4×Q8C2×D711yescharacteristic
250C4×Q8C2×D712yes—
251C4×Q8C2×D712yes—
252C4×D4—72no—
253C4×D4—71no—
254C4×D4—72no—
255#117—72no—
256#117—74no—
257#117—72no—
258#117—72no—
259#117—72no—
260C2×C4⋊C4—72no—
261C2×C42—71no—
262C22×C7C2511yescharacteristic
263C2×D7—21no—
264C2×D7—22no—
265C2×D7—21no—
266C2×D7—22no—
267C2×D7—22no—
268C2×D7—21no—
269C4×C7—22no—
270C4×C7—24no—
271C4×C7—22no—
272C4×C7—21no—
273C4×C7—22no—
274C4×C7—22no—
275C4×C72−1+412yes—
276C4×C72−1+412yes—
277Dic7—22no—
278Dic7—21no—
279Dic7—22no—
280Dic7—21no—
281Dic7—22no—
282Dic7—24no—
283Dic7—21no—
284C2×Q8—71no—
285C2×Q8—72no—
286C2×Q8—72no—
287C2×Q8—74no—
288C2×Q8C22×D712yes—
289C2×Q8C22×D711yescharacteristic
290C2×Q8C22×D711yescharacteristic
291C2×D4—71no—
292C2×D4—72no—
293C22×C4—71no—
294C22×C4—71no—
295C22×C4—72no—
296C22×C4—71no—
297C22×C4—74no—
298C4⋊C4—74no—
299C4⋊C4—72no—
300C4⋊C4—72no—
301C4⋊C4—72no—
302C4⋊C4—72no—
303C4⋊C4—72no—
304C4⋊C4—74no—
305C4⋊C4—74no—
306C4⋊C4—72no—
307C4⋊C4—72no—
308C4⋊C4—74no—
309C4⋊C4—72no—
310C4⋊C4—72no—
311C4⋊C4—71no—
312C4⋊C4—71no—
313C4⋊C4—72no—
314C4⋊C4—72no—
315C4⋊C4—74no—
316C4⋊C4—74no—
317C4⋊C4—74no—
318C4⋊C4C22×D711yescharacteristic
319C4⋊C4C22×D712yes—
320C4⋊C4C22×D714yes—
321C4⋊C4C22×D712yes—
322C4⋊C4C22×D711yescharacteristic
323C4⋊C4C22×D712yes—
324C4⋊C4C22×D714yes—
325C4⋊C4C22×D712yes—
326C4⋊C4C22×D714yes—
327C22⋊C4—72no—
328C22⋊C4—72no—
329C22⋊C4—72no—
330C22⋊C4—72no—
331C22⋊C4—74no—
332C22⋊C4—74no—
333C22⋊C4—72no—
334C22⋊C4—74no—
335C22⋊C4—72no—
336C22⋊C4—74no—
337C22⋊C4—72no—
338C42—71no—
339C42—71no—
340C42—71no—
341C42—74no—
342C42—72no—
343C42—74no—
344C42—74no—
345C42C22×D711yescharacteristic
346C42C22×D712yes—
347C42C22×D712yes—
348C42C22×D714yes—
349C2×C7C2×2−1+411yescharacteristic
350C2×C7C2×2−1+411yescharacteristic
351C2×C7#58511yescharacteristic
352D7—41no—
353D7—42no—
354C23—71no—
355C23—72no—
356Q8—22no—
357Q8—24no—
358Q8—22no—
359Q8—142no—
360Q8—282no—
361Q8—282no—
362Q8—284no—
363D4—142no—
364D4—282no—
365C2×C4—71no—
366C2×C4—72no—
367C2×C4—72no—
368C2×C4—72no—
369C2×C4—71no—
370C2×C4—71no—
371C2×C4—74no—
372C2×C4—142no—
373C2×C4—144no—
374C2×C4—142no—
375C2×C4—284no—
376C2×C4—281no—
377C2×C4C23×D714yes—
378C2×C4C23×D712yes—
379C2×C4C23×D711yescharacteristic
380C2×C4C23×D712yes—
381C2×C4C23×D712yes—
382C2×C4C23×D712yes—
383C2×C4C23×D711yescharacteristic
384C2×C4C23×D711yescharacteristic
385C7#355511yescharacteristic, complemented, Sylow
386C22—141no—
387C22—142no—
388C22—141no—
389C22—142no—
390C22—142no—
391C22—141no—
392C22C24×D711yescharacteristic
393C4—22no—
394C4—24no—
395C4—22no—
396C4—22no—
397C4—21no—
398C4—22no—
399C4—141no—
400C4—142no—
401C4—142no—
402C4—141no—
403C4—141no—
404C4—144no—
405C4—142no—
406C4#655112yes—
407C4#655412yes—
408C2—281no—
409C2—282no—
410C2C2×#655111yescharacteristic
411C2C2×#655411yescharacteristic
412C2#9062011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D7
commutator, socle, cosocleC22×C7C25
FittingC7×#557C2
prime radical#6458C22
hypercenter#557D7
hypocenterC7#3555
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. C7×#557C2semidirect
2. #90423C2semidirect
3. #90423C2semidirect
4. #90422C2semidirect
5. #90419C2semidirect
6. #90390C2semidirect
7. #90387C2semidirect
8. #90385C2semidirect
9. #90380C2semidirect
10. #90275C2semidirect
11. #90274C2semidirect
14. #90270C2semidirect
15. #90269C2semidirect
16. #90260C2semidirect
17. Q8×Dic14C2semidirect
152. #557D7semidirect
385. C7#3555semidirect

Maximal subgroups

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

subgroupindex
C7×#5572
#904232
#904222
#904192
#903902
#903872
#903852
#903802
#902752
#902742
#902732
#902712
#902702
#902692
#902602
Q8×Dic142
#35557