Finite Group Databasebeta

← #11743840

Subgroups — #11743840

Order 896.

1122 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#550C211yesmaximal, characteristic, complemented
2#90525C211yesmaximal, characteristic, complemented
3#90525C211yesmaximal, characteristic, complemented
4#90518C211yesmaximal, characteristic, complemented
5#41×Dic7C211yesmaximal, characteristic, complemented
6#90410C213yesmaximal, complemented
7#90409C213yesmaximal, complemented
8#90310C213yesmaximal, complemented
9#90305C213yesmaximal, complemented
10#90302C213yesmaximal, complemented
11#90287C213yesmaximal, complemented
12#90265C211yesmaximal, characteristic, complemented
13#90260C211yesmaximal, characteristic, complemented
14#90256C213yesmaximal, complemented
15#90229C213yesmaximal, complemented
16C2×C7×#41C2211yescharacteristic
17C2×C7×#41C2211yescharacteristic
18C2×#1161C2211yescharacteristic
19C2×#1159C2216yes—
20C7×#127C2213yes—
21C7×#121C2213yes—
22C7×#121C2213yes—
23C4×C7×Q8C2211yescharacteristic
24C4×C7×D4C2213yes—
25Q8×Dic7C2211yescharacteristic
26Q8×Dic7C2211yescharacteristic
27Q8×Dic7C2211yescharacteristic
28#6503C2213yes—
29#6503C2213yes—
30#6502C2213yes—
31#6502C2216yes—
32#6502C2216yes—
33#6501C2216yes—
34#6501C2213yes—
35#6501C2213yes—
36#6498C2213yes—
37#6498C2216yes—
38D4×Dic7C2213yes—
39D4×Dic7C2213yes—
40D4×Dic7C2213yes—
41D4×Dic7C2213yes—
42#6493C2213yes—
43#6493C2213yes—
44C4×C22⋊D7C2213yes—
45C4×C22⋊D7C2213yes—
46#6489C2213yes—
47#6489C2213yes—
48C2×C4⋊Dic7C2213yes—
49C2×C4×Dic7C2213yes—
50#6461C2213yes—
51#6456C2213yes—
52D7×C4⋊C4C2213yes—
53#6453C2213yes—
54#6452C2211yescharacteristic
55#6451C2213yes—
56#6449C2213yes—
57#6448C2213yes—
58#6446C2216yes—
59(C22⋊D7)⋊C4C2216yes—
60#6442C2213yes—
61#6441C2213yes—
62C4×D28C2213yes—
63#6435C2213yes—
64#6433C2213yes—
65C4×Dic14C2213yes—
66#3517—71nomaximal, Sylow
67C7×#41—24no—
68C7×#41—24no—
69C2×C7×Q8C2311yescharacteristic
70C2×C7×D4C2313yes—
71C2×C7×D4C2313yes—
72C2×C7×D4C2316yes—
73C2×C7×D4C2313yes—
74C22×C4×C7C2313yes—
75C22×C4×C7C2313yes—
76C2×C22⋊D7C2313yes—
77C2×C22⋊D7C2316yes—
78C2×C22⋊D7C2316yes—
79C2×C22⋊D7C2313yes—
80C22×Dic7C2313yes—
81C22×Dic7C2316yes—
82C22×Dic7C2313yes—
83#1161—24no—
84#1159—412no—
85C2×D28C2313yes—
86C2×C4×D7C2313yes—
87C2×C4×D7C2316yes—
88C2×Dic14C2313yes—
89C7×C4⋊C4C2313yes—
90C7×C4⋊C4C2311yescharacteristic
91C7×C22⋊C4C2313yes—
92C7×C22⋊C4C2313yes—
93C42×C7C2313yes—
94C22⋊Dic7C2313yes—
95C22⋊Dic7C2313yes—
96C22⋊Dic7C2313yes—
97C22⋊Dic7C2316yes—
98C22⋊Dic7C2313yes—
99(C2×D7)⋊C4C2313yes—
100(C2×D7)⋊C4C2313yes—
101(C2×D7)⋊C4C2313yes—
102(C2×D7)⋊C4C2313yes—
103C4⋊Dic7C2313yes—
104C4⋊Dic7C2313yes—
105C4⋊Dic7C2313yes—
106C4⋊Dic7C2313yes—
107C4⋊Dic7C2313yes—
108C4⋊Dic7C2311yescharacteristic
109Dic7⋊C4C2311yescharacteristic
110Dic7⋊C4C2313yes—
111Dic7⋊C4C2316yes—
112Dic7⋊C4C2311yescharacteristic
113Dic7⋊C4C2313yes—
114Dic7⋊C4C2311yescharacteristic
115Dic7⋊C4C2311yescharacteristic
116C4×Dic7C2313yes—
117C4×Dic7C2313yes—
118C4×Dic7C2313yes—
119C4×Dic7C2313yes—
120C4×Dic7C2311yescharacteristic
121#557—71no—
122#553—73no—
123#553—73no—
124#550—71no—
125#550—71no—
126#550—73no—
127#550D711yescharacteristic, complemented
128#547—73no—
129#540—73no—
130#540—73no—
131#532—73no—
132#529—73no—
133#520—71no—
134#520—71no—
135C4×#41—71no—
136C23×C7C2413yes—
137C23×C7C2413yes—
138C22×D7C2413yes—
139C7×Q8#4114yescomplemented
140C7×D4—26no—
141C7×D4—26no—
142C7×D4—46no—
143C7×D4—43no—
144C2×C4×C7—26no—
145C2×C4×C7—26no—
146C2×C4×C7C2411yescharacteristic
147C2×C4×C7C2411yescharacteristic
148C2×C4×C7C2413yes—
149C2×C4×C7C2411yescharacteristic
150C2×C4×C7C2413yes—
151C22⋊D7—26no—
152C22⋊D7—46no—
153C22⋊D7—46no—
154C22⋊D7—43no—
155C2×Dic7—26no—
156C2×Dic7—46no—
157C2×Dic7—43no—
158C2×Dic7C2411yescharacteristic
159C2×Dic7C2411yescharacteristic
160C2×Dic7C2413yes—
161C2×Dic7C2413yes—
162C2×Dic7C2413yes—
163C2×Dic7C2411yescharacteristic
164C2×Dic7C2411yescharacteristic
165D28—26no—
166C4×D7—26no—
167C4×D7—46no—
168Dic14—26no—
169C2×#41—71no—
170C2×#41—76no—
171C2×#41C2×D711yescharacteristic
172C2×#41C2×D711yescharacteristic
173C4⋊Q8—73no—
174#127—73no—
175#127—73no—
176#127C2×D713yes—
177#126—73no—
178#126—73no—
179#125—71no—
180#125—73no—
181#125—73no—
182#124—73no—
183#123—73no—
184#123—76no—
185#123—73no—
186#123—76no—
187C22⋊Q8—73no—
188#121—73no—
189#121—76no—
190#121—73no—
191#121—76no—
192#121—73no—
193#121—76no—
194#121—73no—
195#121—73no—
196#121C2×D713yes—
197#121C2×D713yes—
198C4×Q8—71no—
199C4×Q8—71no—
200C4×Q8—71no—
201C4×Q8—73no—
202C4×Q8C2×D711yescharacteristic
203C4×D4—73no—
204C4×D4—73no—
205C4×D4—73no—
206C4×D4—73no—
207C4×D4—73no—
208C4×D4—73no—
209C4×D4—73no—
210C4×D4—76no—
211C4×D4—73no—
212C4×D4C2×D713yes—
213#117—73no—
214#117—73no—
215#117—73no—
216C2×C4⋊C4—73no—
217C2×C4⋊C4—73no—
218C2×C42—73no—
219C22×C7—23no—
220C22×C7—23no—
221C22×C7—23no—
222C22×C7—23no—
223C22×C7—23no—
224C22×C7—23no—
225C22×C7C2511yescharacteristic
226C2×D7—23no—
227C2×D7—23no—
228C2×D7—23no—
229C4×C7—21no—
230C4×C7—21no—
231C4×C7—23no—
232C4×C7—21no—
233C4×C7C2×#4116yes—
234Dic7—21no—
235Dic7—23no—
236Dic7—23no—
237Dic7—21no—
238Dic7—23no—
239Dic7—21no—
240Dic7H(5,𝔽2)12yes—
241#41—24no—
242#41—24no—
243#41—144no—
244#41—2812no—
245C2×Q8—73no—
246C2×Q8C22×D711yescharacteristic
247C2×D4—73no—
248C2×D4—76no—
249C2×D4—73no—
250C2×D4—76no—
251C2×D4—73no—
252C2×D4C22×D713yes—
253C2×D4C22×D713yes—
254C2×D4C22×D713yes—
255C2×D4C22×D716yes—
256C22×C4—73no—
257C22×C4—73no—
258C22×C4—76no—
259C22×C4—73no—
260C22×C4—76no—
261C22×C4C22×D713yes—
262C22×C4C22×D713yes—
263C4⋊C4—73no—
264C4⋊C4—71no—
265C4⋊C4—71no—
266C4⋊C4—71no—
267C4⋊C4—76no—
268C4⋊C4—71no—
269C4⋊C4—73no—
270C4⋊C4—73no—
271C4⋊C4—73no—
272C4⋊C4—73no—
273C4⋊C4—71no—
274C4⋊C4—73no—
275C4⋊C4—73no—
276C4⋊C4C22×D713yes—
277C4⋊C4C22×D711yescharacteristic
278C22⋊C4—73no—
279C22⋊C4—76no—
280C22⋊C4—73no—
281C22⋊C4—73no—
282C22⋊C4—73no—
283C22⋊C4—73no—
284C22⋊C4—73no—
285C22⋊C4—73no—
286C22⋊C4—73no—
287C22⋊C4C22×D713yes—
288C22⋊C4C22×D713yes—
289C42—71no—
290C42—73no—
291C42—73no—
292C42—73no—
293C42—73no—
294C42C22×D713yes—
295C2×C7—43no—
296C2×C7—43no—
297C2×C7C22×#4111yescharacteristic
298C2×C7C2×H(5,𝔽2)11yescharacteristic
299C2×C7#58511yescharacteristic
300D7—43no—
301C23—73no—
302C23C23×D713yes—
303C23C23×D713yes—
304Q8—146no—
305Q8#115914yescomplemented
306D4—26no—
307D4—26no—
308D4—43no—
309D4—46no—
310D4—146no—
311D4—146no—
312D4—286no—
313D4—283no—
314D4—286no—
315C2×C4—26no—
316C2×C4—26no—
317C2×C4—71no—
318C2×C4—71no—
319C2×C4—71no—
320C2×C4—73no—
321C2×C4—73no—
322C2×C4—73no—
323C2×C4—71no—
324C2×C4—146no—
325C2×C4—146no—
326C2×C4—286no—
327C2×C4—283no—
328C2×C4—286no—
329C2×C4C23×D711yescharacteristic
330C2×C4C23×D711yescharacteristic
331C2×C4C23×D713yes—
332C2×C4C23×D711yescharacteristic
333C2×C4C23×D713yes—
334C7#351711yescharacteristic, complemented, Sylow
335C22—23no—
336C22—23no—
337C22—23no—
338C22—23no—
339C22—23no—
340C22—23no—
341C22—143no—
342C22—143no—
343C22—143no—
344C22C24×D711yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—23no—
349C4—72no—
350C4—141no—
351C4—141no—
352C4—143no—
353C4—141no—
354C4—143no—
355C4—143no—
356C4C2×#115916yes—
357C2—43no—
358C2—43no—
359C2—283no—
360C2C22×#115911yescharacteristic
361C2#9061611yescharacteristic
362C2D7×H(5,𝔽2)11yescharacteristic
subgroupG/H
center, FrattiniC22C24×D7
commutator, socle, cosocleC22×C7C25
FittingC7×#550C2
hypercenter#550D7
hypocenterC7#3517
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C7×#550C2semidirect
2. #90525C2semidirect
3. #90525C2semidirect
4. #90518C2semidirect
5. #41×Dic7C2semidirect
6. #90410C2semidirect
7. #90409C2semidirect
8. #90310C2semidirect
9. #90305C2semidirect
10. #90302C2semidirect
11. #90287C2semidirect
12. #90265C2semidirect
13. #90260C2semidirect
14. #90256C2semidirect
15. #90229C2semidirect
127. #550D7semidirect
139. C7×Q8#41semidirect
305. Q8#1159semidirect
334. C7#3517semidirect

Maximal subgroups

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

subgroupindex
C7×#5502
#905252
#905182
#41×Dic72
#904102
#904092
#903102
#903052
#903022
#902872
#902652
#902602
#902562
#902292
#35177