Finite Group Databasebeta

← #11740433

Subgroups — #11740433

Order 896.

498 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#124C211yesmaximal, characteristic
2C2×#6442C211yesmaximal, characteristic, complemented
3#90002C211yesmaximal, characteristic, complemented
4#90001C211yesmaximal, characteristic, complemented
5#89994C211yesmaximal, characteristic
6#89992C211yesmaximal, characteristic
7#89725C211yesmaximal, characteristic, complemented
8#89724C211yesmaximal, characteristic, complemented
9#89722C211yesmaximal, characteristic, complemented
10#89720C211yesmaximal, characteristic, complemented
11#89720C211yesmaximal, characteristic, complemented
12#89716C211yesmaximal, characteristic, complemented
13#89714C211yesmaximal, characteristic, complemented
14#89705C211yesmaximal, characteristic, complemented
15#89700C211yesmaximal, characteristic, complemented
16C22×C7×Q8C2211yescharacteristic
17C22×C7×D4C2211yescharacteristic
18C23×Dic7C2211yescharacteristic
19C7×#124—24no—
20C2×C7×C22⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C42×C7C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31C2×C22⋊Dic7C2211yescharacteristic
32C2×C4⋊Dic7C2211yescharacteristic
33C2×C4⋊Dic7C2211yescharacteristic
34C2×C4⋊Dic7C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C2×Dic7⋊C4C2211yescharacteristic
40C2×Dic7⋊C4C2211yescharacteristic
41C2×C4×Dic7C2211yescharacteristic
42#6442—24no—
43C7⋊#95C2211yescharacteristic
44C7⋊#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53#2715—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C24×C7C2311yescharacteristic
56C2×C7×Q8—22no—
57C2×C7×Q8—22no—
58C2×C7×Q8—41no—
59C2×C7×D4—22no—
60C2×C7×D4—22no—
61C2×C7×D4—41no—
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C22×Dic7—22no—
68C22×Dic7—41no—
69C22×Dic7—41no—
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C22×Dic7C2311yescharacteristic
75C22×Dic7C2311yescharacteristic
76C22×Dic7C2311yescharacteristic
77C22×Dic7C2311yescharacteristic
78C7×C22⋊C4—22no—
79C7×C22⋊C4—22no—
80C7×C22⋊C4—22no—
81C7×C22⋊C4D414yes—
82C42×C7—22no—
83C22⋊Dic7—22no—
84C22⋊Dic7—22no—
85C22⋊Dic7—22no—
86C22⋊Dic7—41no—
87C22⋊Dic7—41no—
88C22⋊Dic7—41no—
89C22⋊Dic7—41no—
90C4⋊Dic7—22no—
91C4⋊Dic7—22no—
92C4⋊Dic7—41no—
93Dic7⋊C4—22no—
94Dic7⋊C4—22no—
95Dic7⋊C4—22no—
96Dic7⋊C4—41no—
97Dic7⋊C4—41no—
98Dic7⋊C4—41no—
99C4×Dic7—22no—
100C2×#126—71no—
101C2×#124D711yescharacteristic
102#397—71no—
103#397—71no—
104#396—71no—
105#396—71no—
106#395—71no—
107#394—71no—
108#391—71no—
109#389—71no—
110#388—71no—
111#387—71no—
112#386—71no—
113#385—71no—
114#380—71no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7C2×D412yes—
129C23×C7C2411yescharacteristic
130C7×Q8—44no—
131C7×D4—44no—
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×C7C2×D412yes—
145C2×C4×C7C2×D412yes—
146C2×C4×C7#4112yes—
147C2×Dic7—21no—
148C2×Dic7—21no—
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—21no—
169C2×Dic7—21no—
170C2×Dic7—44no—
171C2×Dic7#4112yes—
172C22×Q8C2×D711yescharacteristic
173C22×D4C2×D711yescharacteristic
174C23×C4—71no—
175#126—144no—
176#124—24no—
177C2×C4⋊C4—71no—
178C2×C4⋊C4—71no—
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×C22⋊C4—71no—
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⋊C4C2×D711yescharacteristic
194C2×C22⋊C4C2×D711yescharacteristic
195C2×C22⋊C4C2×D711yescharacteristic
196C2×C22⋊C4C2×D711yescharacteristic
197C2×C42—71no—
198C2×C42C2×D711yescharacteristic
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#95—71no—
209C22×C7—22no—
210C22×C7—22no—
211C22×C7—22no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—41no—
220C22×C7—41no—
221C22×C7—41no—
222C22×C7—41no—
223C22×C7C22×D411yescharacteristic
224C22×C7C2×#4111yescharacteristic
225C22×C7C2×#4111yescharacteristic
226C22×C7H(5,𝔽2)11yescharacteristic
227C22×C7H(5,𝔽2)11yescharacteristic
228C22×C72−1+411yescharacteristic
229C22×C72−1+411yescharacteristic
230C4×C7—22no—
231C4×C7—22no—
232C4×C7—22no—
233C4×C7—41no—
234C4×C7—41no—
235Dic7—22no—
236Dic7—41no—
237Dic7—41no—
238Dic7—41no—
239Dic7—41no—
240Dic7—41no—
241Dic7—41no—
242Dic7—41no—
243C24C22×D711yescharacteristic
244C24C22×D711yescharacteristic
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—41no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—41no—
251C22×C4—71no—
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—281no—
261C22×C4—281no—
262C22×C4C22×D711yescharacteristic
263C22×C4C22×D711yescharacteristic
264C22×C4C22×D711yescharacteristic
265C22×C4C22×D711yescharacteristic
266C22×C4C22×D711yescharacteristic
267C4⋊C4—142no—
268C4⋊C4—142no—
269C4⋊C4—142no—
270C4⋊C4—142no—
271C4⋊C4—142no—
272C4⋊C4—281no—
273C4⋊C4—281no—
274C4⋊C4—281no—
275C4⋊C4—281no—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—22no—
279C22⋊C4—142no—
280C22⋊C4—142no—
281C22⋊C4—142no—
282C22⋊C4—281no—
283C22⋊C4—281no—
284C22⋊C4—281no—
285C22⋊C4—281no—
286C22⋊C4C22⋊D714yes—
287C42—22no—
288C42—142no—
289C2×C7—42no—
290C2×C7—81no—
291C2×C7#53711yescharacteristic
292C2×C7#53811yescharacteristic
293C2×C7#54211yescharacteristic
294C2×C7#54611yescharacteristic
295C2×C7#54711yescharacteristic
296C2×C7#55611yescharacteristic
297C2×C7#56311yescharacteristic
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23C2×C22⋊D712yes—
312C23C23×D711yescharacteristic
313Q8—44no—
314D4—44no—
315C2×C4—21no—
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—72no—
328C2×C4—141no—
329C2×C4—141no—
330C2×C4—141no—
331C2×C4—141no—
332C2×C4—141no—
333C2×C4—141no—
334C2×C4—141no—
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—284no—
352C2×C4#115912yes—
353C2×C4C2×C22⋊D712yes—
354C2×C4C2×C22⋊D712yes—
355C7#271511yescharacteristic, complemented, Sylow
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22C2×#115911yescharacteristic
371C22#654811yescharacteristic
372C22D7×#4111yescharacteristic
373C22#655311yescharacteristic
374C22#655411yescharacteristic
375C22#655411yescharacteristic
376C22C22×C22⋊D711yescharacteristic
377C4—22no—
378C4—22no—
379C4—22no—
380C4—41no—
381C4—41no—
382C4—142no—
383C4—281no—
384C4—281no—
385C4—281no—
386C4—281no—
387C4—281no—
388C4—281no—
389C4—281no—
390C2—42no—
391C2—81no—
392C2#9036211yescharacteristic
393C2#9036311yescharacteristic
394C2#9036911yescharacteristic
395C2#9037311yescharacteristic
396C2#9049711yescharacteristic
397C2#9051511yescharacteristic
398C2#9052411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#124C2
prime radicalC22×C7×D4C22
hypercenterC2×#124D7
hypocenterC7#2715
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×#6442C2semidirect
3. #90002C2semidirect
4. #90001C2semidirect
7. #89725C2semidirect
8. #89724C2semidirect
9. #89722C2semidirect
10. #89720C2semidirect
11. #89720C2semidirect
12. #89716C2semidirect
13. #89714C2semidirect
14. #89705C2semidirect
15. #89700C2semidirect
355. C7#2715semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1242
C2×#64422
#900022
#900012
#899942
#899922
#897252
#897242
#897222
#897202
#897162
#897142
#897052
#897002
#27157