Finite Group Databasebeta

← #11733367

Subgroups — #11733367

Order 896.

512 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6459C211yesmaximal, characteristic, complemented
2C2×#6449C211yesmaximal, characteristic
3C7×#388C211yesmaximal, characteristic, complemented
4#89755C211yesmaximal, characteristic, complemented
5#89731C211yesmaximal, characteristic
6#89728C211yesmaximal, characteristic
7#89719C211yesmaximal, characteristic, complemented
8#89718C211yesmaximal, characteristic, complemented
9#89714C211yesmaximal, characteristic, complemented
10#89712C211yesmaximal, characteristic, complemented
11#89701C211yesmaximal, characteristic, complemented
12#89447C211yesmaximal, characteristic, complemented
13#89436C211yesmaximal, characteristic, complemented
14#89432C211yesmaximal, characteristic, complemented
15#89429C211yesmaximal, characteristic, complemented
16C22×C22⋊D7C2211yescharacteristic
17C23×Dic7C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
19C22×Dic14C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C42×C7C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×(C2×D7)⋊C4C2211yescharacteristic
28C2×(C2×D7)⋊C4C2211yescharacteristic
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×C4⋊Dic7C2211yescharacteristic
33C2×C4⋊Dic7C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39#6459—24no—
40#6449—24no—
41C7×#95C2211yescharacteristic
42C7×#95C2211yescharacteristic
43C7⋊#95C2211yescharacteristic
44C7⋊#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53#2687—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C23×D7C2311yescharacteristic
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C2×C22⋊D7—22no—
63C2×C22⋊D7—22no—
64C2×C22⋊D7—41no—
65C22×Dic7—22no—
66C22×Dic7—22no—
67C22×Dic7—41no—
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C22×Dic7C2311yescharacteristic
75C2×C4×D7—22no—
76C2×C4×D7—22no—
77C2×C4×D7—41no—
78C2×Dic14—22no—
79C2×Dic14—22no—
80C2×Dic14—41no—
81C7×C4⋊C4—22no—
82C7×C22⋊C4—22no—
83C7×C22⋊C4—22no—
84C7×C22⋊C4—41no—
85C42×C7—41no—
86C22⋊Dic7—22no—
87C22⋊Dic7—41no—
88(C2×D7)⋊C4—22no—
89(C2×D7)⋊C4—22no—
90(C2×D7)⋊C4—41no—
91(C2×D7)⋊C4—41no—
92(C2×D7)⋊C4D414yes—
93C4⋊Dic7—22no—
94C4⋊Dic7—22no—
95Dic7⋊C4—22no—
96Dic7⋊C4—22no—
97Dic7⋊C4—41no—
98Dic7⋊C4—41no—
99Dic7⋊C4—41no—
100C2×#123—71no—
101C2×C22⋊Q8—71no—
102#400—71no—
103#398—71no—
104#398—71no—
105#397—71no—
106#394—71no—
107#393—71no—
108#391—71no—
109#388—71no—
110#388D711yescharacteristic, complemented
111#386—71no—
112#385—71no—
113#381—71no—
114#380—71no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7#4112yes—
122C23×C7C2411yescharacteristic
123C22×D7—21no—
124C22×D7—21no—
125C22×D7—21no—
126C22×D7—21no—
127C22×D7—21no—
128C22×D7—21no—
129C22×D7C2×D412yes—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
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×C7C2×D412yes—
148C22⋊D7—44no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7—21no—
153C2×Dic7—21no—
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—44no—
169C2×Dic7C2×D412yes—
170C2×Dic7#4112yes—
171C4×D7—44no—
172Dic14—44no—
173C22×Q8—71no—
174C22×D4—71no—
175C23×C4—71no—
176C23×C4—71no—
177#123—144no—
178C22⋊Q8—144no—
179C2×C4⋊C4—71no—
180C2×C4⋊C4—71no—
181C2×C4⋊C4—71no—
182C2×C4⋊C4—71no—
183C2×C4⋊C4—71no—
184C2×C4⋊C4—71no—
185C2×C4⋊C4—71no—
186C2×C4⋊C4C2×D711yescharacteristic
187C2×C22⋊C4—71no—
188C2×C22⋊C4—71no—
189C2×C22⋊C4—71no—
190C2×C22⋊C4—71no—
191C2×C22⋊C4—71no—
192C2×C22⋊C4—71no—
193C2×C22⋊C4—71no—
194C2×C22⋊C4C2×D711yescharacteristic
195C2×C22⋊C4C2×D711yescharacteristic
196C2×C22⋊C4C2×D711yescharacteristic
197C2×C42C2×D711yescharacteristic
198#95—71no—
199#95—71no—
200#95—71no—
201#95—71no—
202#95—71no—
203#95—71no—
204#95—71no—
205#95—71no—
206#95—71no—
207#95—71no—
208#95C2×D711yescharacteristic
209#95C2×D711yescharacteristic
210C22×C7—22no—
211C22×C7—22no—
212C22×C7—22no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7C22×D411yescharacteristic
218C22×C7C2×#4111yescharacteristic
219C22×C7C2×#4111yescharacteristic
220C22×C7H(5,𝔽2)11yescharacteristic
221C22×C7H(5,𝔽2)11yescharacteristic
222C22×C72−1+411yescharacteristic
223C22×C72−1+411yescharacteristic
224C2×D7—22no—
225C2×D7—22no—
226C2×D7—22no—
227C2×D7—41no—
228C2×D7—41no—
229C2×D7—41no—
230C2×D7—41no—
231C4×C7—22no—
232C4×C7—41no—
233C4×C7—41no—
234C4×C7—41no—
235C4×C7—41no—
236C4×C7—41no—
237Dic7—22no—
238Dic7—22no—
239Dic7—41no—
240Dic7—41no—
241Dic7—41no—
242Dic7—41no—
243Dic7—41no—
244C24—71no—
245C24C22×D711yescharacteristic
246C2×Q8—142no—
247C2×Q8—142no—
248C2×Q8—281no—
249C2×D4—142no—
250C2×D4—142no—
251C2×D4—281no—
252C22×C4—71no—
253C22×C4—71no—
254C22×C4—71no—
255C22×C4—71no—
256C22×C4—71no—
257C22×C4—71no—
258C22×C4—71no—
259C22×C4—142no—
260C22×C4—142no—
261C22×C4—142no—
262C22×C4—142no—
263C22×C4—281no—
264C22×C4—281no—
265C22×C4C22×D711yescharacteristic
266C22×C4C22×D711yescharacteristic
267C22×C4C22×D711yescharacteristic
268C22×C4C22×D711yescharacteristic
269C22×C4C22×D711yescharacteristic
270C22×C4C22×D711yescharacteristic
271C4⋊C4—22no—
272C4⋊C4—142no—
273C4⋊C4—142no—
274C4⋊C4—142no—
275C4⋊C4—142no—
276C4⋊C4—281no—
277C4⋊C4—281no—
278C4⋊C4—281no—
279C22⋊C4—22no—
280C22⋊C4—22no—
281C22⋊C4—41no—
282C22⋊C4—74no—
283C22⋊C4—142no—
284C22⋊C4—142no—
285C22⋊C4—142no—
286C22⋊C4—281no—
287C22⋊C4—281no—
288C22⋊C4—281no—
289C42—41no—
290C2×C7—42no—
291C2×C7#53611yescharacteristic
292C2×C7#53911yescharacteristic
293C2×C7#54211yescharacteristic
294C2×C7#54611yescharacteristic
295C2×C7#54611yescharacteristic
296C2×C7#55211yescharacteristic
297C2×C7#56211yescharacteristic
298D7—42no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—72no—
306C23—141no—
307C23—141no—
308C23—141no—
309C23—141no—
310C23—141no—
311C23—141no—
312C23#115712yes—
313C23C23×D711yescharacteristic
314Q8—284no—
315D4—284no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
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—72no—
334C2×C4—72no—
335C2×C4—141no—
336C2×C4—141no—
337C2×C4—141no—
338C2×C4—141no—
339C2×C4—141no—
340C2×C4—141no—
341C2×C4—141no—
342C2×C4—141no—
343C2×C4—141no—
344C2×C4—141no—
345C2×C4—141no—
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—284no—
355C2×C4—284no—
356C2×C4D4×D712yes—
357C7#268711yescharacteristic, complemented, Sylow
358C22—22no—
359C22—22no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—142no—
366C22—142no—
367C22—142no—
368C22—281no—
369C22—281no—
370C22—281no—
371C22—281no—
372C22C2×#115711yescharacteristic
373C22C2×D4×D711yescharacteristic
374C22#654811yescharacteristic
375C22D7×#4111yescharacteristic
376C22#655311yescharacteristic
377C22#655411yescharacteristic
378C22#655411yescharacteristic
379C4—22no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—142no—
386C4—142no—
387C4—281no—
388C4—281no—
389C4—281no—
390C4—281no—
391C4—281no—
392C2—42no—
393C2—282no—
394C2#9022211yescharacteristic
395C2#9024111yescharacteristic
396C2#9025511yescharacteristic
397C2#9030111yescharacteristic
398C2#9035811yescharacteristic
399C2#9036611yescharacteristic
400C2#9040111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#388C2
prime radicalC22×C22⋊D7C22
hypercenter#388D7
hypocenterC7#2687
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#6459C2semidirect
3. C7×#388C2semidirect
4. #89755C2semidirect
7. #89719C2semidirect
8. #89718C2semidirect
9. #89714C2semidirect
10. #89712C2semidirect
11. #89701C2semidirect
12. #89447C2semidirect
13. #89436C2semidirect
14. #89432C2semidirect
15. #89429C2semidirect
110. #388D7semidirect
357. C7#2687semidirect

Maximal subgroups

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

subgroupindex
C2×#64592
C2×#64492
C7×#3882
#897552
#897312
#897282
#897192
#897182
#897142
#897122
#897012
#894472
#894362
#894322
#894292
#26877