Finite Group Databasebeta

← #11734705

Subgroups — #11734705

Order 896.

562 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
1C2×#6510C211yesmaximal, characteristic
2C2×#6507C211yesmaximal, characteristic, complemented
3C2×#6459C211yesmaximal, characteristic, complemented
4C2×#6457C211yesmaximal, characteristic
5C7×#395C211yesmaximal, characteristic, complemented
6#89764C211yesmaximal, characteristic, complemented
7#89762C211yesmaximal, characteristic, complemented
8#89761C211yesmaximal, characteristic, complemented
9#89744C211yesmaximal, characteristic, complemented
10#89448C211yesmaximal, characteristic, complemented
11#89447C211yesmaximal, characteristic, complemented
12#89445C211yesmaximal, characteristic, complemented
13#89444C211yesmaximal, characteristic
14#89441C211yesmaximal, characteristic, complemented
15#89424C211yesmaximal, characteristic, complemented
16C22×C7×Q8C2211yescharacteristic
17C22×D28C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
19C22×Dic14C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×C7×C4⋊C4C2211yescharacteristic
22C2×C7×C4⋊C4C2211yescharacteristic
23#6510—24no—
24#6507—24no—
25C2×(C2×D7)⋊C4C2211yescharacteristic
26C2×(C2×D7)⋊C4C2211yescharacteristic
27C2×(C2×D7)⋊C4C2211yescharacteristic
28C2×(C2×D7)⋊C4C2211yescharacteristic
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×(C2×D7)⋊C4C2211yescharacteristic
33C2×(C2×D7)⋊C4C2211yescharacteristic
34C2×(C2×D7)⋊C4C2211yescharacteristic
35C2×(C2×D7)⋊C4C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C2×Dic7⋊C4C2211yescharacteristic
40C2×Dic7⋊C4C2211yescharacteristic
41C2×Dic7⋊C4C2211yescharacteristic
42C2×Dic7⋊C4C2211yescharacteristic
43C2×C4×Dic7C2211yescharacteristic
44C2×C4×Dic7C2211yescharacteristic
45#6459—24no—
46#6457—24no—
47C7×#95C2211yescharacteristic
48C7×#95C2211yescharacteristic
49C7×#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53C7⋊#95C2211yescharacteristic
54C7⋊#95C2211yescharacteristic
55#2674—71nomaximal, Sylow
56C23×D7C2311yescharacteristic
57C23×D7C2311yescharacteristic
58C2×C7×Q8—22no—
59C2×C7×Q8—22no—
60C2×C7×Q8—41no—
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C22×C4×C7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C2×D28—22no—
75C2×D28—22no—
76C2×D28—41no—
77C2×C4×D7—22no—
78C2×C4×D7—22no—
79C2×C4×D7—41no—
80C2×Dic14—22no—
81C2×Dic14—22no—
82C2×Dic14—41no—
83C7×C4⋊C4—22no—
84C7×C4⋊C4—22no—
85C7×C4⋊C4—41no—
86(C2×D7)⋊C4—22no—
87(C2×D7)⋊C4—22no—
88(C2×D7)⋊C4—22no—
89(C2×D7)⋊C4—22no—
90(C2×D7)⋊C4—22no—
91(C2×D7)⋊C4—22no—
92(C2×D7)⋊C4—22no—
93(C2×D7)⋊C4—22no—
94(C2×D7)⋊C4—22no—
95(C2×D7)⋊C4—41no—
96(C2×D7)⋊C4D414yes—
97Dic7⋊C4—22no—
98Dic7⋊C4—22no—
99Dic7⋊C4—22no—
100Dic7⋊C4—22no—
101Dic7⋊C4—22no—
102Dic7⋊C4—22no—
103Dic7⋊C4D414yes—
104C4×Dic7—22no—
105C4×Dic7—22no—
106C2×C4⋊Q8—71no—
107C2×#124—71no—
108C2×#123—71no—
109C2×C22⋊Q8—71no—
110#399—71no—
111#397—71no—
112#396—71no—
113#395D711yescharacteristic, complemented
114#394—71no—
115#393—71no—
116#389—71no—
117#388—71no—
118#388—71no—
119#387—71no—
120#385—71no—
121C23×C7C2411yescharacteristic
122C22×D7—21no—
123C22×D7—21no—
124C22×D7—21no—
125C22×D7—21no—
126C22×D7—21no—
127C22×D7—21no—
128C22×D7—21no—
129C22×D7—21no—
130C22×D7—21no—
131C22×D7—21no—
132C22×D7—21no—
133C22×D7—21no—
134C22×D7—21no—
135C22×D7C2×D412yes—
136C7×Q8—44no—
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—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7C2×D412yes—
157C2×C4×C7C2×D412yes—
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×Dic7—21no—
171C2×Dic7—21no—
172C2×Dic7C2×D412yes—
173C2×Dic7C2×D412yes—
174C2×Dic7C2×D412yes—
175C2×Dic7#4112yes—
176D28—44no—
177C4×D7—44no—
178Dic14—44no—
179C22×Q8—71no—
180C22×Q8C2×D711yescharacteristic
181C22×D4—71no—
182C23×C4—71no—
183C4⋊Q8—144no—
184#124—144no—
185#123—144no—
186C22⋊Q8—144no—
187C2×C4⋊C4—71no—
188C2×C4⋊C4—71no—
189C2×C4⋊C4—71no—
190C2×C4⋊C4—71no—
191C2×C4⋊C4—71no—
192C2×C4⋊C4—71no—
193C2×C4⋊C4—71no—
194C2×C4⋊C4C2×D711yescharacteristic
195C2×C4⋊C4C2×D711yescharacteristic
196C2×C4⋊C4C2×D711yescharacteristic
197C2×C22⋊C4—71no—
198C2×C22⋊C4—71no—
199C2×C22⋊C4—71no—
200C2×C22⋊C4—71no—
201C2×C22⋊C4—71no—
202C2×C22⋊C4—71no—
203C2×C22⋊C4—71no—
204C2×C22⋊C4—71no—
205C2×C22⋊C4—71no—
206C2×C22⋊C4—71no—
207C2×C22⋊C4—71no—
208C2×C42—71no—
209C2×C42—71no—
210#95—71no—
211#95—71no—
212#95—71no—
213#95—71no—
214#95—71no—
215#95C2×D711yescharacteristic
216#95C2×D711yescharacteristic
217#95C2×D711yescharacteristic
218C22×C7C22×D411yescharacteristic
219C22×C7C22×D411yescharacteristic
220C22×C7C2×#4111yescharacteristic
221C22×C7H(5,𝔽2)11yescharacteristic
222C22×C72−1+411yescharacteristic
223C22×C72−1+411yescharacteristic
224C22×C72−1+411yescharacteristic
225C2×D7—22no—
226C2×D7—22no—
227C2×D7—22no—
228C2×D7—41no—
229C2×D7—41no—
230C2×D7—41no—
231C2×D7—41no—
232C2×D7—41no—
233C2×D7—41no—
234C2×D7—41no—
235C2×D7—41no—
236C2×D7—41no—
237C2×D7—41no—
238C2×D7—41no—
239C4×C7—22no—
240C4×C7—22no—
241C4×C7—41no—
242C4×C7—41no—
243C4×C7—41no—
244C4×C7—41no—
245C4×C7—41no—
246Dic7—22no—
247Dic7—22no—
248Dic7—22no—
249Dic7—22no—
250Dic7—41no—
251Dic7—41no—
252C24—71no—
253C24—71no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—41no—
257C2×Q8—142no—
258C2×Q8—142no—
259C2×Q8—281no—
260C2×D4—142no—
261C2×D4—142no—
262C2×D4—281no—
263C22×C4—71no—
264C22×C4—71no—
265C22×C4—71no—
266C22×C4—71no—
267C22×C4—71no—
268C22×C4—71no—
269C22×C4—142no—
270C22×C4—142no—
271C22×C4—281no—
272C22×C4C22×D711yescharacteristic
273C22×C4C22×D711yescharacteristic
274C22×C4C22×D711yescharacteristic
275C22×C4C22×D711yescharacteristic
276C22×C4C22×D711yescharacteristic
277C22×C4C22×D711yescharacteristic
278C22×C4C22×D711yescharacteristic
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—41no—
282C4⋊C4—74no—
283C4⋊C4—142no—
284C4⋊C4—142no—
285C4⋊C4—142no—
286C4⋊C4—142no—
287C4⋊C4—142no—
288C4⋊C4—142no—
289C22⋊C4—74no—
290C22⋊C4—142no—
291C22⋊C4—142no—
292C22⋊C4—142no—
293C22⋊C4—142no—
294C22⋊C4—142no—
295C22⋊C4—142no—
296C22⋊C4—142no—
297C22⋊C4—142no—
298C22⋊C4—142no—
299C22⋊C4—281no—
300C42—142no—
301C42—142no—
302C2×C7#53611yescharacteristic
303C2×C7#53611yescharacteristic
304C2×C7#54211yescharacteristic
305C2×C7D4211yescharacteristic
306C2×C7#54711yescharacteristic
307C2×C7#54811yescharacteristic
308C2×C7#56311yescharacteristic
309D7—42no—
310D7—81no—
311C23—72no—
312C23—141no—
313C23—141no—
314C23—141no—
315C23—141no—
316C23—141no—
317C23—141no—
318C23—141no—
319C23—141no—
320C23—141no—
321C23—141no—
322C23—141no—
323C23—141no—
324C23—141no—
325C23C23×D711yescharacteristic
326Q8—44no—
327Q8—284no—
328D4—284no—
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—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—72no—
349C2×C4—72no—
350C2×C4—72no—
351C2×C4—72no—
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—141no—
363C2×C4—141no—
364C2×C4—141no—
365C2×C4—141no—
366C2×C4—284no—
367C2×C4D4×D712yes—
368C2×C4D4×D712yes—
369C7#267411yescharacteristic, complemented, Sylow
370C22—142no—
371C22—142no—
372C22—142no—
373C22—281no—
374C22—281no—
375C22—281no—
376C22—281no—
377C22—281no—
378C22—281no—
379C22—281no—
380C22—281no—
381C22—281no—
382C22—281no—
383C22—281no—
384C22C2×D4×D711yescharacteristic
385C22C2×D4×D711yescharacteristic
386C22#654811yescharacteristic
387C22#655111yescharacteristic
388C22#655111yescharacteristic
389C22#655111yescharacteristic
390C22D7×#4111yescharacteristic
391C4—22no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—142no—
399C4—142no—
400C4—142no—
401C4—142no—
402C4—281no—
403C4—281no—
404C2—282no—
405C2—561no—
406C2#9028111yescharacteristic
407C2#9032211yescharacteristic
408C2#9032411yescharacteristic
409C2#9033111yescharacteristic
410C2#9041511yescharacteristic
411C2#9041511yescharacteristic
412C2#9042611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#395C2
prime radicalC22×D28C22
hypercenter#395D7
hypocenterC7#2674
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#6507C2semidirect
3. C2×#6459C2semidirect
5. C7×#395C2semidirect
6. #89764C2semidirect
7. #89762C2semidirect
8. #89761C2semidirect
9. #89744C2semidirect
10. #89448C2semidirect
11. #89447C2semidirect
12. #89445C2semidirect
14. #89441C2semidirect
15. #89424C2semidirect
113. #395D7semidirect
369. C7#2674semidirect

Maximal subgroups

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

subgroupindex
C2×#65102
C2×#65072
C2×#64592
C2×#64572
C7×#3952
#897642
#897622
#897612
#897442
#894482
#894472
#894452
#894442
#894412
#894242
#26747