Finite Group Databasebeta

← #11743788

Subgroups — #11743788

Order 896.

1010 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#548C211yesmaximal, characteristic, complemented
2#90522C211yesmaximal, characteristic, complemented
3#41×Dic7C211yesmaximal, characteristic, complemented
4Q8×C22⋊D7C211yesmaximal, characteristic, complemented
5#90503C211yesmaximal, characteristic, complemented
6#90373C213yesmaximal, complemented
7#90362C213yesmaximal, complemented
8#90332C213yesmaximal, complemented
9#90313C213yesmaximal, complemented
10#90307C213yesmaximal, complemented
11#90294C213yesmaximal, complemented
12#90263C211yesmaximal, characteristic, complemented
13Q8×Dic14C211yesmaximal, characteristic, complemented
14#90233C213yesmaximal, complemented
15#90229C213yesmaximal, complemented
16C2×C7×#41C2211yescharacteristic
17C22×C7×Q8C2211yescharacteristic
18C2×Q8×D7C2211yescharacteristic
19C7×#124C2213yes—
20C7×C22⋊Q8C2213yes—
21C7×#121C2213yes—
22C4×C7×Q8C2211yescharacteristic
23C4×C7×D4C2213yes—
24#6509C2213yes—
25Q8×Dic7C2213yes—
26Q8×Dic7C2211yescharacteristic
27Q8×Dic7C2211yescharacteristic
28Q8×Dic7C2213yes—
29Q8×Dic7C2211yescharacteristic
30#6507C2213yes—
31#6502C2213yes—
32#6501C2213yes—
33#6499C2216yes—
34#6499C2213yes—
35#6498C2213yes—
36D4×Dic7C2213yes—
37D4×Dic7C2213yes—
38C4×C22⋊D7C2213yes—
39C4×C22⋊D7C2213yes—
40#6489C2213yes—
41#6489C2213yes—
42#6489C2213yes—
43#6487C2213yes—
44#6487C2213yes—
45C2×C4×Dic7C2213yes—
46#6461C2213yes—
47#6455C2213yes—
48#6452C2213yes—
49#6451C2213yes—
50#6451C2213yes—
51#6450C2216yes—
52#6450C2213yes—
53#6449C2213yes—
54#6448C2216yes—
55#6448C2211yescharacteristic
56#6442C2216yes—
57#6442C2213yes—
58#6442C2216yes—
59#6441C2216yes—
60#6441C2213yes—
61#6440C2216yes—
62#6435C2213yes—
63#6432C2213yes—
64C4×Dic14C2213yes—
65C4×Dic14C2213yes—
66#3518—71nomaximal, Sylow
67C7×#41—24no—
68C2×C7×Q8—24no—
69C2×C7×Q8C2313yes—
70C2×C7×Q8C2311yescharacteristic
71C2×C7×D4C2313yes—
72C2×C7×D4C2313yes—
73C22×C4×C7C2313yes—
74C22×C4×C7C2313yes—
75C2×C22⋊D7C2311yescharacteristic
76C2×C22⋊D7C2313yes—
77C22×Dic7C2313yes—
78C22×Dic7C2313yes—
79Q8×D7—24no—
80C2×C4×D7C2313yes—
81C2×Dic14C2311yescharacteristic
82C2×Dic14C2313yes—
83C2×Dic14C2316yes—
84C7×C4⋊C4C2313yes—
85C7×C4⋊C4C2313yes—
86C7×C22⋊C4C2316yes—
87C7×C22⋊C4C2313yes—
88C7×C22⋊C4C2311yescharacteristic
89C42×C7C2313yes—
90C22⋊Dic7C2316yes—
91C22⋊Dic7C2313yes—
92C22⋊Dic7C2316yes—
93C22⋊Dic7C2313yes—
94C22⋊Dic7C2313yes—
95C22⋊Dic7C2311yescharacteristic
96(C2×D7)⋊C4C2311yescharacteristic
97(C2×D7)⋊C4C2313yes—
98(C2×D7)⋊C4C2313yes—
99(C2×D7)⋊C4C2311yescharacteristic
100C4⋊Dic7C2313yes—
101C4⋊Dic7C2313yes—
102C4⋊Dic7C2313yes—
103C4⋊Dic7C2313yes—
104C4⋊Dic7C2313yes—
105C4⋊Dic7C2313yes—
106Dic7⋊C4C2313yes—
107Dic7⋊C4C2313yes—
108Dic7⋊C4C2313yes—
109Dic7⋊C4C2313yes—
110Dic7⋊C4C2313yes—
111Dic7⋊C4C2316yes—
112Dic7⋊C4C2313yes—
113C4×Dic7C2316yes—
114C4×Dic7C2313yes—
115C4×Dic7C2313yes—
116C4×Dic7C2313yes—
117C4×Dic7C2316yes—
118C4×Dic7C2313yes—
119C4×Dic7C2311yescharacteristic
120Q82—71no—
121#556—73no—
122#556—73no—
123#556—73no—
124#555—73no—
125D4×Q8—71no—
126#548—71no—
127#548D711yescharacteristic, complemented
128#542—73no—
129#542—73no—
130#533—73no—
131#529—73no—
132#519—71no—
133#519—71no—
134C4×#41—71no—
135C23×C7C2411yescharacteristic
136C23×C7C2413yes—
137C22×D7C2411yescharacteristic
138C7×Q8—26no—
139C7×Q8#4114yes—
140C7×D4—26no—
141C7×D4—43no—
142C2×C4×C7—26no—
143C2×C4×C7—26no—
144C2×C4×C7C2411yescharacteristic
145C2×C4×C7C2413yes—
146C2×C4×C7C2413yes—
147C2×C4×C7C2411yescharacteristic
148C2×C4×C7C2413yes—
149C22⋊D7—22no—
150C22⋊D7—43no—
151C2×Dic7—26no—
152C2×Dic7—43no—
153C2×Dic7C2413yes—
154C2×Dic7C2411yescharacteristic
155C2×Dic7C2411yescharacteristic
156C2×Dic7C2413yes—
157C2×Dic7C2413yes—
158C2×Dic7C2413yes—
159C2×Dic7C2411yescharacteristic
160C4×D7—26no—
161Dic14—26no—
162Dic14—22no—
163Dic14—46no—
164C2×#41C2×D711yescharacteristic
165C22×Q8—71no—
166C22×Q8C2×D711yescharacteristic
167C4⋊Q8—73no—
168C4⋊Q8—73no—
169C4⋊Q8—73no—
170C4⋊Q8—73no—
171#126—73no—
172#126—73no—
173#126—76no—
174#126—76no—
175#125—73no—
176#124—73no—
177#124—76no—
178#124—71no—
179#124—76no—
180#124C2×D713yes—
181#123—73no—
182#123—73no—
183C22⋊Q8—73no—
184C22⋊Q8—73no—
185C22⋊Q8—73no—
186C22⋊Q8—73no—
187C22⋊Q8—76no—
188C22⋊Q8C2×D713yes—
189#121—73no—
190#121—73no—
191#121C2×D713yes—
192C4×Q8—71no—
193C4×Q8—71no—
194C4×Q8—73no—
195C4×Q8—71no—
196C4×Q8—73no—
197C4×Q8—73no—
198C4×Q8—76no—
199C4×Q8—73no—
200C4×Q8—73no—
201C4×Q8C2×D711yescharacteristic
202C4×D4—73no—
203C4×D4—73no—
204C4×D4—73no—
205C4×D4—73no—
206C4×D4C2×D713yes—
207#117—73no—
208#117—73no—
209#117—73no—
210#117—73no—
211#117—76no—
212#117—73no—
213C2×C42—73no—
214C22×C7—21no—
215C22×C7—23no—
216C22×C7—23no—
217C22×C7—21no—
218C22×C7—23no—
219C22×C7—21no—
220C22×C7C2511yescharacteristic
221C2×D7—21no—
222C2×D7—21no—
223C2×D7—21no—
224C4×C7—21no—
225C4×C7—23no—
226C4×C7—21no—
227C4×C7—23no—
228C4×C7C2×#4116yes—
229Dic7—23no—
230Dic7—21no—
231Dic7—23no—
232Dic7—23no—
233Dic7—23no—
234Dic7—21no—
235Dic72−1+412yes—
236#41—24no—
237C2×Q8—24no—
238C2×Q8—71no—
239C2×Q8—73no—
240C2×Q8—76no—
241C2×Q8—144no—
242C2×Q8C22×D713yes—
243C2×Q8C22×D711yescharacteristic
244C2×D4—71no—
245C2×D4—73no—
246C2×D4C22×D713yes—
247C2×D4C22×D713yes—
248C22×C4—73no—
249C22×C4—73no—
250C22×C4—73no—
251C22×C4C22×D713yes—
252C22×C4C22×D713yes—
253C4⋊C4—73no—
254C4⋊C4—73no—
255C4⋊C4—73no—
256C4⋊C4—73no—
257C4⋊C4—73no—
258C4⋊C4—76no—
259C4⋊C4—73no—
260C4⋊C4—73no—
261C4⋊C4—73no—
262C4⋊C4—73no—
263C4⋊C4—73no—
264C4⋊C4—73no—
265C4⋊C4—73no—
266C4⋊C4C22×D713yes—
267C4⋊C4C22×D713yes—
268C22⋊C4—73no—
269C22⋊C4—71no—
270C22⋊C4—76no—
271C22⋊C4—76no—
272C22⋊C4—73no—
273C22⋊C4—73no—
274C22⋊C4—73no—
275C22⋊C4—71no—
276C22⋊C4—71no—
277C22⋊C4—73no—
278C22⋊C4C22×D716yes—
279C22⋊C4C22×D711yescharacteristic
280C22⋊C4C22×D713yes—
281C42—71no—
282C42—73no—
283C42—73no—
284C42—73no—
285C42—76no—
286C42—76no—
287C42—73no—
288C42C22×D713yes—
289C2×C7—41no—
290C2×C7—43no—
291C2×C7C22×#4111yescharacteristic
292C2×C7C2×2−1+411yescharacteristic
293C2×C7#58511yescharacteristic
294D7—41no—
295C23—71no—
296C23C23×D711yescharacteristic
297C23C23×D713yes—
298Q8—26no—
299Q8—142no—
300Q8—146no—
301Q8—286no—
302Q8#115914yes—
303D4—26no—
304D4—43no—
305D4—142no—
306D4—283no—
307C2×C4—26no—
308C2×C4—26no—
309C2×C4—71no—
310C2×C4—73no—
311C2×C4—71no—
312C2×C4—71no—
313C2×C4—73no—
314C2×C4—73no—
315C2×C4—73no—
316C2×C4—146no—
317C2×C4—146no—
318C2×C4—283no—
319C2×C4C23×D711yescharacteristic
320C2×C4C23×D713yes—
321C2×C4C23×D713yes—
322C2×C4C23×D711yescharacteristic
323C2×C4C23×D713yes—
324C7#351811yescharacteristic, complemented, Sylow
325C22—21no—
326C22—23no—
327C22—23no—
328C22—21no—
329C22—23no—
330C22—21no—
331C22—141no—
332C22—141no—
333C22—141no—
334C22C24×D711yescharacteristic
335C4—21no—
336C4—23no—
337C4—23no—
338C4—21no—
339C4—72no—
340C4—143no—
341C4—141no—
342C4—143no—
343C4—143no—
344C4—143no—
345C4—141no—
346C4C2×#115916yes—
347C2—41no—
348C2—43no—
349C2—281no—
350C2C22×#115911yescharacteristic
351C2#9061611yescharacteristic
352C2D7×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D7
commutator, socle, cosocleC22×C7C25
FittingC7×#548C2
hypercenter#548D7
hypocenterC7#3518
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C7×#548C2semidirect
2. #90522C2semidirect
3. #41×Dic7C2semidirect
4. Q8×C22⋊D7C2semidirect
5. #90503C2semidirect
6. #90373C2semidirect
7. #90362C2semidirect
8. #90332C2semidirect
9. #90313C2semidirect
10. #90307C2semidirect
11. #90294C2semidirect
12. #90263C2semidirect
13. Q8×Dic14C2semidirect
14. #90233C2semidirect
15. #90229C2semidirect
127. #548D7semidirect
324. C7#3518semidirect

Maximal subgroups

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

subgroupindex
C7×#5482
#905222
#41×Dic72
Q8×C22⋊D72
#905032
#903732
#903622
#903322
#903132
#903072
#902942
#902632
Q8×Dic142
#902332
#902292
#35187