Finite Group Databasebeta

← #11731932

Subgroups — #11731932

Order 896.

564 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊D7C211yesmaximal, characteristic
2C2×#6450C211yesmaximal, characteristic, complemented
3C7×#385C211yesmaximal, characteristic, complemented
4#89982C211yesmaximal, characteristic, complemented
5#89764C211yesmaximal, characteristic, complemented
6#89761C211yesmaximal, characteristic, complemented
7#89744C211yesmaximal, characteristic, complemented
8#89729C211yesmaximal, characteristic
9#89729C211yesmaximal, characteristic
10#89721C211yesmaximal, characteristic, complemented
11#89719C211yesmaximal, characteristic, complemented
12#89445C211yesmaximal, characteristic, complemented
13#89440C211yesmaximal, characteristic, complemented
14#89422C211yesmaximal, characteristic, complemented
15#89419C211yesmaximal, characteristic, complemented
16C23×C4×C7C2211yescharacteristic
17C22×C22⋊D7C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
19C22×Dic14C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×C7×C4⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C4×C22⋊D7—24no—
28C2×(C2×D7)⋊C4C2211yescharacteristic
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×(C2×D7)⋊C4C2211yescharacteristic
33C2×C4⋊Dic7C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C2×Dic7⋊C4C2211yescharacteristic
40C2×Dic7⋊C4C2211yescharacteristic
41C2×C4×Dic7C2211yescharacteristic
42C2×C4×Dic7C2211yescharacteristic
43C2×C4×Dic7C2211yescharacteristic
44#6450—24no—
45C7×#95C2211yescharacteristic
46C7×#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53#2580—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C23×D7C2311yescharacteristic
56C22×C4×C7—22no—
57C22×C4×C7—41no—
58C22×C4×C7—41no—
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C2×C22⋊D7—22no—
66C2×C22⋊D7—22no—
67C2×C22⋊D7Q814yescomplemented
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C22×Dic7C2311yescharacteristic
75C2×C4×D7—22no—
76C2×C4×D7—41no—
77C2×C4×D7—41no—
78C2×Dic14—22no—
79C2×Dic14—22no—
80C2×Dic14—22no—
81C7×C4⋊C4—22no—
82C7×C4⋊C4—41no—
83C7×C22⋊C4—22no—
84C7×C22⋊C4—41no—
85C22⋊Dic7—22no—
86C22⋊Dic7—22no—
87C22⋊Dic7—41no—
88(C2×D7)⋊C4—22no—
89(C2×D7)⋊C4—22no—
90(C2×D7)⋊C4—22no—
91(C2×D7)⋊C4—41no—
92(C2×D7)⋊C4—41no—
93C4⋊Dic7—41no—
94Dic7⋊C4—22no—
95Dic7⋊C4—22no—
96Dic7⋊C4—22no—
97Dic7⋊C4—22no—
98Dic7⋊C4—22no—
99Dic7⋊C4—41no—
100Dic7⋊C4—41no—
101C4×Dic7—22no—
102C4×Dic7—22no—
103C4×Dic7D414yes—
104C2×C4×Q8—71no—
105C2×C4×D4—71no—
106#399—71no—
107#399—71no—
108#393—71no—
109#393—71no—
110#390—71no—
111#390—71no—
112#389—71no—
113#389—71no—
114#385—71no—
115#385—71no—
116#385—71no—
117#385D711yescharacteristic, complemented
118#382—71no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7C2×Q812yes—
126C23×C7C2411yescharacteristic
127C22×D7—21no—
128C22×D7—21no—
129C22×D7—21no—
130C22×D7—21no—
131C22×D7—21no—
132C22×D7—21no—
133C22×D7C2×Q812yes—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—44no—
151C2×C4×C7C2×D412yes—
152C2×C4×C7#4112yes—
153C22⋊D7—28no—
154C2×Dic7—21no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—21no—
158C2×Dic7—21no—
159C2×Dic7—21no—
160C2×Dic7—21no—
161C2×Dic7—21no—
162C2×Dic7—21no—
163C2×Dic7—21no—
164C2×Dic7—21no—
165C2×Dic7—21no—
166C2×Dic7—21no—
167C2×Dic7—21no—
168C2×Dic7—21no—
169C2×Dic7—21no—
170C2×Dic7C2×D412yes—
171C2×Dic7C2×D412yes—
172C2×Dic7C2×Q812yes—
173C2×Dic7#4112yes—
174C2×Dic7#4112yes—
175C4×D7—44no—
176Dic14—28no—
177C22×Q8—71no—
178C22×D4—71no—
179C23×C4—71no—
180C23×C4C2×D711yescharacteristic
181C4×Q8—144no—
182C4×D4—144no—
183C2×C4⋊C4—71no—
184C2×C4⋊C4—71no—
185C2×C4⋊C4—71no—
186C2×C4⋊C4—71no—
187C2×C4⋊C4—71no—
188C2×C4⋊C4—71no—
189C2×C4⋊C4—71no—
190C2×C4⋊C4—71no—
191C2×C4⋊C4C2×D711yescharacteristic
192C2×C4⋊C4C2×D711yescharacteristic
193C2×C22⋊C4—71no—
194C2×C22⋊C4—71no—
195C2×C22⋊C4—71no—
196C2×C22⋊C4—71no—
197C2×C22⋊C4—71no—
198C2×C22⋊C4—71no—
199C2×C22⋊C4—71no—
200C2×C22⋊C4—71no—
201C2×C22⋊C4C2×D711yescharacteristic
202C2×C22⋊C4C2×D711yescharacteristic
203C2×C42—71no—
204C2×C42—71no—
205C2×C42—71no—
206#95—71no—
207#95—71no—
208#95—71no—
209#95—71no—
210#95—71no—
211#95—71no—
212#95C2×D711yescharacteristic
213#95C2×D711yescharacteristic
214C22×C7—22no—
215C22×C7—22no—
216C22×C7—22no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—41no—
220C22×C7—41no—
221C22×C7C22×D411yescharacteristic
222C22×C7C22×Q811yescharacteristic
223C22×C7C2×#4111yescharacteristic
224C22×C7C2×#4111yescharacteristic
225C22×C7C2×#4111yescharacteristic
226C22×C7H(5,𝔽2)11yescharacteristic
227C22×C7H(5,𝔽2)11yescharacteristic
228C2×D7—22no—
229C2×D7—22no—
230C2×D7—22no—
231C2×D7—41no—
232C2×D7—41no—
233C2×D7—41no—
234C2×D7—41no—
235C4×C7—22no—
236C4×C7—22no—
237C4×C7—41no—
238C4×C7—41no—
239C4×C7—41no—
240C4×C7—41no—
241Dic7—22no—
242Dic7—22no—
243Dic7—41no—
244Dic7—41no—
245Dic7—41no—
246Dic7—41no—
247Dic7C22⋊Q814yes—
248C24—71no—
249C24C22×D711yescharacteristic
250C2×Q8—142no—
251C2×Q8—142no—
252C2×Q8—142no—
253C2×D4—74no—
254C2×D4—142no—
255C2×D4—142no—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—71no—
260C22×C4—71no—
261C22×C4—71no—
262C22×C4—71no—
263C22×C4—71no—
264C22×C4—71no—
265C22×C4—71no—
266C22×C4—142no—
267C22×C4—281no—
268C22×C4—281no—
269C22×C4C22×D711yescharacteristic
270C22×C4C22×D711yescharacteristic
271C22×C4C22×D711yescharacteristic
272C22×C4C22×D711yescharacteristic
273C22×C4C22×D711yescharacteristic
274C22×C4C22×D711yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—41no—
277C4⋊C4—142no—
278C4⋊C4—142no—
279C4⋊C4—142no—
280C4⋊C4—142no—
281C4⋊C4—142no—
282C4⋊C4—281no—
283C4⋊C4—281no—
284C4⋊C4—281no—
285C22⋊C4—22no—
286C22⋊C4—41no—
287C22⋊C4—142no—
288C22⋊C4—142no—
289C22⋊C4—142no—
290C22⋊C4—142no—
291C22⋊C4—142no—
292C22⋊C4—281no—
293C22⋊C4—281no—
294C22⋊C4—281no—
295C42—74no—
296C42—142no—
297C42—142no—
298C2×C7—42no—
299C2×C7C2×C22⋊Q811yescharacteristic
300C2×C7#52511yescharacteristic
301C2×C7#53511yescharacteristic
302C2×C7#55411yescharacteristic
303C2×C7#55411yescharacteristic
304C2×C7#55911yescharacteristic
305C2×C7#55911yescharacteristic
306D7—42no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—72no—
314C23—141no—
315C23—141no—
316C23—141no—
317C23—141no—
318C23—141no—
319C23—141no—
320C23Q8×D712yes—
321C23C23×D711yescharacteristic
322Q8—148no—
323D4—148no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—44no—
341C2×C4—72no—
342C2×C4—72no—
343C2×C4—72no—
344C2×C4—72no—
345C2×C4—72no—
346C2×C4—141no—
347C2×C4—141no—
348C2×C4—141no—
349C2×C4—141no—
350C2×C4—141no—
351C2×C4—141no—
352C2×C4—141no—
353C2×C4—141no—
354C2×C4—141no—
355C2×C4—141no—
356C2×C4—141no—
357C2×C4—141no—
358C2×C4—141no—
359C2×C4—141no—
360C2×C4—141no—
361C2×C4—141no—
362C2×C4—284no—
363C2×C4#115712yes—
364C2×C4D4×D712yes—
365C7#258011yescharacteristic, complemented, Sylow
366C22—22no—
367C22—22no—
368C22—22no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—142no—
374C22—142no—
375C22—142no—
376C22—281no—
377C22—281no—
378C22—281no—
379C22—281no—
380C22C2×#115711yescharacteristic
381C22C2×D4×D711yescharacteristic
382C22#654811yescharacteristic
383C22#654811yescharacteristic
384C22C2×Q8×D711yescharacteristic
385C22D7×#4111yescharacteristic
386C22D7×#4111yescharacteristic
387C4—22no—
388C4—22no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—74no—
394C4—142no—
395C4—142no—
396C4—281no—
397C4—281no—
398C4—281no—
399C4—281no—
400C2—42no—
401C2—282no—
402C2#9020211yescharacteristic
403C2#9023811yescharacteristic
404C2#9025011yescharacteristic
405C2#9028311yescharacteristic
406C2D7×C22⋊Q811yescharacteristic
407C2#9032711yescharacteristic
408C2#9035511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#385C2
prime radicalC22×C22⋊D7C22
hypercenter#385D7
hypocenterC7#2580
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#6450C2semidirect
3. C7×#385C2semidirect
4. #89982C2semidirect
5. #89764C2semidirect
6. #89761C2semidirect
7. #89744C2semidirect
10. #89721C2semidirect
11. #89719C2semidirect
12. #89445C2semidirect
13. #89440C2semidirect
14. #89422C2semidirect
15. #89419C2semidirect
67. C2×C22⋊D7Q8semidirect
117. #385D7semidirect
365. C7#2580semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D72
C2×#64502
C7×#3852
#899822
#897642
#897612
#897442
#897292
#897212
#897192
#894452
#894402
#894222
#894192
#25807