Finite Group Databasebeta

← #11742952

Subgroups — #11742952

Order 896.

992 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#519C211yesmaximal, characteristic, complemented
2Q8×C22⋊D7C211yesmaximal, characteristic, complemented
3Q8×C22⋊D7C211yesmaximal, characteristic, complemented
4#90274C213yesmaximal, complemented
5#90272C213yesmaximal, complemented
6#90269C213yesmaximal, complemented
7Q8×D28C211yesmaximal, characteristic, complemented
8#90263C211yesmaximal, characteristic, complemented
9C4×Q8×D7C211yesmaximal, characteristic, complemented
10#90260C211yesmaximal, characteristic, complemented
11#90258C213yesmaximal, complemented
12#90224C213yesmaximal, complemented
13#90220C213yesmaximal, complemented
14#90219C213yesmaximal
15#90209C213yesmaximal, complemented
16C22×C7×Q8C2211yescharacteristic
17C2×Q8×D7C2211yescharacteristic
18C2×Q8×D7C2211yescharacteristic
19C4×C7×Q8C2213yes—
20C4×C7×Q8C2211yescharacteristic
21C4×C7×Q8C2211yescharacteristic
22C4×C7×Q8C2213yes—
23C7×#117C2213yes—
24C7×#117C2213yes—
25#6509C2213yes—
26#6509C2213yes—
27Q8×Dic7C2211yescharacteristic
28Q8×Dic7C2211yescharacteristic
29#6507C2213yes—
30#6507C2213yes—
31C4×C22⋊D7C2213yes—
32C4×C22⋊D7C2213yes—
33#6487C2213yes—
34#6487C2213yes—
35#6461C2213yes—
36#6460C2213yes—
37#6460C2213yes—
38#6459C2213yes—
39#6457C2213yes—
40#6455C2213yes—
41D7×C4⋊C4C2213yes—
42#6453C2213yes—
43#6452C2216yes—
44#6452C2216yes—
45#6452C2213yes—
46#6451C2213yes—
47#6450C2213yes—
48#6450C2213yes—
49#6448C2213yes—
50#6447C2211yescharacteristic
51#6446C2213yes—
52#6442C2213yes—
53#6442C2216yes—
54#6439C2216yes—
55C4×D28C2213yes—
56#6435C2213yes—
57#6435C2216yes—
58C42×D7C2213yes—
59#6433C2213yes—
60#6433C2216yes—
61#6432C2213yes—
62C4×Dic14C2213yes—
63C4×Dic14C2213yes—
64C4×Dic14C2213yes—
65C4×Dic14C2216yes—
66#3523—71nomaximal, Sylow
67C2×C7×Q8—24no—
68C2×C7×Q8C2313yes—
69C2×C7×Q8C2311yescharacteristic
70C22×C4×C7C2313yes—
71C2×C22⋊D7C2311yescharacteristic
72C2×C22⋊D7C2311yescharacteristic
73Q8×D7—24no—
74Q8×D7—24no—
75C2×D28C2311yescharacteristic
76C2×C4×D7C2311yescharacteristic
77C2×C4×D7C2313yes—
78C2×C4×D7C2313yes—
79C2×C4×D7C2313yes—
80C2×Dic14C2313yes—
81C2×Dic14C2313yes—
82C2×Dic14C2313yes—
83C7×C4⋊C4C2313yes—
84C7×C4⋊C4C2313yes—
85C7×C4⋊C4C2313yes—
86C7×C4⋊C4C2313yes—
87C7×C22⋊C4C2311yescharacteristic
88C7×C22⋊C4C2313yes—
89C42×C7C2313yes—
90C42×C7C2313yes—
91C42×C7C2316yes—
92C22⋊Dic7C2313yes—
93C22⋊Dic7C2313yes—
94(C2×D7)⋊C4C2311yescharacteristic
95(C2×D7)⋊C4C2313yes—
96(C2×D7)⋊C4C2313yes—
97(C2×D7)⋊C4C2313yes—
98(C2×D7)⋊C4C2313yes—
99(C2×D7)⋊C4C2313yes—
100C4⋊Dic7C2316yes—
101C4⋊Dic7C2313yes—
102C4⋊Dic7C2313yes—
103C4⋊Dic7C2313yes—
104C4⋊Dic7C2313yes—
105Dic7⋊C4C2311yescharacteristic
106Dic7⋊C4C2313yes—
107Dic7⋊C4C2313yes—
108Dic7⋊C4C2313yes—
109Dic7⋊C4C2316yes—
110Dic7⋊C4C2316yes—
111Dic7⋊C4C2313yes—
112Dic7⋊C4C2313yes—
113Dic7⋊C4C2316yes—
114Dic7⋊C4C2316yes—
115C4×Dic7C2313yes—
116C4×Dic7C2313yes—
117C4×Dic7C2316yes—
118C4×Dic7C2313yes—
119C4×Dic7C2313yes—
120#557—73no—
121#557—71no—
122#556—73no—
123#552—73no—
124#552—73no—
125D4×Q8—71no—
126D4×Q8—71no—
127D4×Q8—71no—
128#541—73no—
129#541—73no—
130#533—73no—
131#529—73no—
132#519—71no—
133#519D711yescharacteristic, complemented
134C2×C4×Q8—71no—
135C23×C7C2411yescharacteristic
136C22×D7C2411yescharacteristic
137C22×D7C2411yescharacteristic
138C7×Q8—26no—
139C7×Q8#4114yes—
140C2×C4×C7—26no—
141C2×C4×C7C2411yescharacteristic
142C2×C4×C7C2413yes—
143C2×C4×C7C2413yes—
144C2×C4×C7C2411yescharacteristic
145C2×C4×C7C2413yes—
146C2×C4×C7C2413yes—
147C22⋊D7—22no—
148C22⋊D7—22no—
149C2×Dic7C2411yescharacteristic
150C2×Dic7C2413yes—
151C2×Dic7C2413yes—
152C2×Dic7C2413yes—
153C2×Dic7C2413yes—
154C2×Dic7C2411yescharacteristic
155D28—22no—
156C4×D7—26no—
157C4×D7—26no—
158C4×D7—26no—
159C4×D7—22no—
160Dic14—26no—
161Dic14—26no—
162Dic14—26no—
163C22×Q8—71no—
164C22×Q8—71no—
165C22×Q8C2×D711yescharacteristic
166C4⋊Q8—73no—
167C4⋊Q8—73no—
168C4⋊Q8—73no—
169C4⋊Q8—73no—
170#126—73no—
171#126—73no—
172#126—76no—
173#126—76no—
174#125—73no—
175#125—76no—
176#125—76no—
177#125—73no—
178#125—76no—
179#125—73no—
180#124—73no—
181#123—73no—
182#123—73no—
183C22⋊Q8—73no—
184C22⋊Q8—73no—
185C22⋊Q8—73no—
186C22⋊Q8—73no—
187C22⋊Q8—73no—
188C22⋊Q8—73no—
189C22⋊Q8—73no—
190#121—71no—
191C4×Q8—71no—
192C4×Q8—71no—
193C4×Q8—73no—
194C4×Q8—73no—
195C4×Q8—73no—
196C4×Q8—73no—
197C4×Q8—73no—
198C4×Q8—76no—
199C4×Q8C2×D711yescharacteristic
200C4×Q8C2×D713yes—
201C4×Q8C2×D711yescharacteristic
202C4×Q8C2×D713yes—
203C4×D4—73no—
204C4×D4—73no—
205C4×D4—73no—
206#117—73no—
207#117—73no—
208#117—76no—
209#117C2×D713yes—
210#117C2×D713yes—
211C2×C4⋊C4—73no—
212C2×C42—73no—
213C22×C7—21no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7C2511yescharacteristic
217C2×D7—21no—
218C2×D7—21no—
219C2×D7—21no—
220C2×D7—21no—
221C2×D7—21no—
222C2×D72−1+412yes—
223C4×C7—23no—
224C4×C7—23no—
225C4×C7—23no—
226C4×C7—21no—
227C4×C7—21no—
228C4×C7C2×#4116yes—
229Dic7—23no—
230Dic7—23no—
231Dic7—23no—
232Dic7—23no—
233Dic7—21no—
234Dic72−1+412yes—
235C2×Q8—24no—
236C2×Q8—73no—
237C2×Q8—73no—
238C2×Q8—73no—
239C2×Q8—144no—
240C2×Q8—144no—
241C2×Q8C22×D713yes—
242C2×Q8C22×D711yescharacteristic
243C2×D4—71no—
244C2×D4—71no—
245C2×D4—71no—
246C22×C4—71no—
247C22×C4—73no—
248C22×C4—73no—
249C22×C4—73no—
250C22×C4C22×D713yes—
251C4⋊C4—76no—
252C4⋊C4—73no—
253C4⋊C4—73no—
254C4⋊C4—76no—
255C4⋊C4—76no—
256C4⋊C4—73no—
257C4⋊C4—73no—
258C4⋊C4—71no—
259C4⋊C4—76no—
260C4⋊C4—73no—
261C4⋊C4—73no—
262C4⋊C4—73no—
263C4⋊C4—73no—
264C4⋊C4—76no—
265C4⋊C4—73no—
266C4⋊C4C22×D713yes—
267C4⋊C4C22×D713yes—
268C4⋊C4C22×D713yes—
269C4⋊C4C22×D713yes—
270C22⋊C4—73no—
271C22⋊C4—73no—
272C22⋊C4—71no—
273C22⋊C4—73no—
274C22⋊C4—73no—
275C22⋊C4—73no—
276C22⋊C4—73no—
277C22⋊C4—73no—
278C22⋊C4C22×D711yescharacteristic
279C22⋊C4C22×D713yes—
280C42—73no—
281C42—73no—
282C42—73no—
283C42—76no—
284C42—73no—
285C42C22×D713yes—
286C42C22×D716yes—
287C42C22×D713yes—
288C2×C7—41no—
289C2×C7C22×#4111yescharacteristic
290C2×C7C2×2−1+411yescharacteristic
291C2×C7C2×2−1+411yescharacteristic
292D7—22no—
293D7—41no—
294C23—71no—
295C23—71no—
296C23C23×D711yescharacteristic
297Q8—26no—
298Q8—146no—
299Q8—146no—
300Q8—146no—
301Q8#115714yes—
302D4—142no—
303D4—142no—
304D4—142no—
305C2×C4—26no—
306C2×C4—71no—
307C2×C4—73no—
308C2×C4—71no—
309C2×C4—73no—
310C2×C4—73no—
311C2×C4—73no—
312C2×C4—142no—
313C2×C4—146no—
314C2×C4—146no—
315C2×C4—146no—
316C2×C4C23×D713yes—
317C2×C4C23×D711yescharacteristic
318C2×C4C23×D713yes—
319C2×C4C23×D711yescharacteristic
320C2×C4C23×D713yes—
321C2×C4C23×D713yes—
322C7#352311yescharacteristic, complemented, Sylow
323C22—21no—
324C22—21no—
325C22—21no—
326C22—72no—
327C22—141no—
328C22—141no—
329C22—141no—
330C22—141no—
331C22—141no—
332C22C24×D711yescharacteristic
333C4—23no—
334C4—21no—
335C4—23no—
336C4—21no—
337C4—23no—
338C4—72no—
339C4—143no—
340C4—143no—
341C4—143no—
342C4—141no—
343C4—143no—
344C4C2×#115716yes—
345C2—41no—
346C2—142no—
347C2—281no—
348C2C22×#115711yescharacteristic
349C2D7×2−1+411yescharacteristic
350C2D7×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D7
commutator, socle, cosocleC22×C7C25
FittingC7×#519C2
prime radical#6447C22
hypercenter#519D7
hypocenterC7#3523
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
1. C7×#519C2semidirect
2. Q8×C22⋊D7C2semidirect
3. Q8×C22⋊D7C2semidirect
4. #90274C2semidirect
5. #90272C2semidirect
6. #90269C2semidirect
7. Q8×D28C2semidirect
8. #90263C2semidirect
9. C4×Q8×D7C2semidirect
10. #90260C2semidirect
11. #90258C2semidirect
12. #90224C2semidirect
13. #90220C2semidirect
15. #90209C2semidirect
133. #519D7semidirect
322. C7#3523semidirect

Maximal subgroups

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

subgroupindex
C7×#5192
Q8×C22⋊D72
#902742
#902722
#902692
Q8×D282
#902632
C4×Q8×D72
#902602
#902582
#902242
#902202
#902192
#902092
#35237