Finite Group Databasebeta

← #11732315

Subgroups — #11732315

Order 896.

582 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
1C22×C4×Dic7C211yesmaximal, characteristic, complemented
2C2×#6442C211yesmaximal, characteristic, complemented
3C7×#386C211yesmaximal, characteristic
4#89992C211yesmaximal, characteristic
5#89992C211yesmaximal, characteristic
6#89982C211yesmaximal, characteristic, complemented
7#89723C211yesmaximal, characteristic, complemented
8#89722C211yesmaximal, characteristic, complemented
9#89718C211yesmaximal, characteristic, complemented
10#89717C211yesmaximal, characteristic, complemented
11#89716C211yesmaximal, characteristic, complemented
12#89427C211yesmaximal, characteristic, complemented
13#89425C211yesmaximal, characteristic, complemented
14#89424C211yesmaximal, characteristic, complemented
15#89423C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C23×C4×C7C2211yescharacteristic
18C23×Dic7C2211yescharacteristic
19C23×Dic7C2211yescharacteristic
20C2×C7×C22⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C22⋊Dic7C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C4⋊Dic7C2211yescharacteristic
30C2×C4⋊Dic7C2211yescharacteristic
31C2×Dic7⋊C4C2211yescharacteristic
32C2×Dic7⋊C4C2211yescharacteristic
33C2×Dic7⋊C4C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×C4×Dic7—24no—
36C2×C4×Dic7C2211yescharacteristic
37C2×C4×Dic7C2211yescharacteristic
38C2×C4×Dic7C2211yescharacteristic
39C2×C4×Dic7C2211yescharacteristic
40#6442—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#2852—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C24×C7C2311yescharacteristic
56C2×C7×D4—22no—
57C2×C7×D4—22no—
58C2×C7×D4—22no—
59C22×C4×C7—22no—
60C22×C4×C7—22no—
61C22×C4×C7—22no—
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C22×Dic7—22no—
68C22×Dic7—22no—
69C22×Dic7—22no—
70C22×Dic7—22no—
71C22×Dic7—22no—
72C22×Dic7—22no—
73C22×Dic7C2311yescharacteristic
74C22×Dic7C2311yescharacteristic
75C22×Dic7C2311yescharacteristic
76C22×Dic7C2311yescharacteristic
77C22×Dic7C2311yescharacteristic
78C22×Dic7C2311yescharacteristic
79C22×Dic7C2311yescharacteristic
80C22×Dic7C2311yescharacteristic
81C7×C22⋊C4—22no—
82C7×C22⋊C4—22no—
83C7×C22⋊C4—22no—
84C22⋊Dic7—22no—
85C22⋊Dic7—22no—
86C22⋊Dic7—22no—
87C22⋊Dic7—22no—
88C22⋊Dic7—22no—
89C22⋊Dic7—22no—
90C4⋊Dic7—22no—
91C4⋊Dic7—22no—
92Dic7⋊C4—22no—
93Dic7⋊C4—22no—
94Dic7⋊C4—22no—
95Dic7⋊C4—22no—
96C4×Dic7—22no—
97C4×Dic7—22no—
98C4×Dic7—22no—
99C4×Dic7D414yescomplemented
100C2×#126—71no—
101C22×C42—71no—
102#388—71no—
103#388—71no—
104#388—71no—
105#387—71no—
106#387—71no—
107#387—71no—
108#386—71no—
109#386—71no—
110#386D711yescharacteristic
111#385—71no—
112#385—71no—
113#385—71no—
114#383—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#4112yes—
127C23×C7#4112yes—
128C23×C7#4112yes—
129C23×C7C2411yescharacteristic
130C7×D4—44no—
131C2×C4×C7—28no—
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×C7C2×D412yes—
146C2×C4×C7#4112yes—
147C2×Dic7—28no—
148C2×Dic7—28no—
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×Dic7C2×D412yes—
170C2×Dic7C2×D412yes—
171C2×Dic7#4112yes—
172C2×Dic7#4112yes—
173C22×D4C2×D711yescharacteristic
174C23×C4—71no—
175C23×C4—71no—
176C23×C4C2×D711yescharacteristic
177#126—144no—
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×C22⋊C4—71no—
185C2×C22⋊C4—71no—
186C2×C22⋊C4—71no—
187C2×C22⋊C4—71no—
188C2×C22⋊C4—71no—
189C2×C22⋊C4—71no—
190C2×C22⋊C4C2×D711yescharacteristic
191C2×C22⋊C4C2×D711yescharacteristic
192C2×C22⋊C4C2×D711yescharacteristic
193C2×C42—71no—
194C2×C42—71no—
195C2×C42—71no—
196C2×C42—71no—
197C2×C42—144no—
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—22no—
214C22×C7—22no—
215C22×C7—22no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—41no—
220C22×C7—41no—
221C22×C7—41no—
222C22×C7—41no—
223C22×C7#12614yescomplemented
224C22×C7C22×D411yescharacteristic
225C22×C7C2×#4111yescharacteristic
226C22×C7C2×#4111yescharacteristic
227C22×C7C2×#4111yescharacteristic
228C22×C7C2×#4111yescharacteristic
229C22×C7C2×#4111yescharacteristic
230C22×C7C2×#4111yescharacteristic
231C4×C7—22no—
232C4×C7—22no—
233C4×C7—41no—
234C4×C7—41no—
235C4×C7—41no—
236Dic7—22no—
237Dic7—22no—
238Dic7—22no—
239Dic7—22no—
240Dic7—41no—
241Dic7—41no—
242Dic7—41no—
243Dic7—41no—
244C24C22×D711yescharacteristic
245C24C22×D711yescharacteristic
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—71no—
253C22×C4—71no—
254C22×C4—71no—
255C22×C4—71no—
256C22×C4—71no—
257C22×C4—71no—
258C22×C4—71no—
259C22×C4—71no—
260C22×C4—142no—
261C22×C4—142no—
262C22×C4—142no—
263C22×C4—142no—
264C22×C4—142no—
265C22×C4—142no—
266C22×C4C22×D711yescharacteristic
267C22×C4C22×D711yescharacteristic
268C22×C4C22×D711yescharacteristic
269C22×C4C22×D711yescharacteristic
270C22×C4C22×D711yescharacteristic
271C4⋊C4—142no—
272C4⋊C4—142no—
273C4⋊C4—142no—
274C4⋊C4—142no—
275C4⋊C4—142no—
276C4⋊C4—142no—
277C22⋊C4—22no—
278C22⋊C4—22no—
279C22⋊C4—22no—
280C22⋊C4—142no—
281C22⋊C4—142no—
282C22⋊C4—142no—
283C22⋊C4—142no—
284C22⋊C4—142no—
285C22⋊C4—142no—
286C42—74no—
287C42—142no—
288C42—142no—
289C42—142no—
290C2×C7—24no—
291C2×C7—81no—
292C2×C7#52511yescharacteristic
293C2×C7#52511yescharacteristic
294C2×C7#52511yescharacteristic
295C2×C7C2×#12611yescharacteristic
296C2×C7#52911yescharacteristic
297C2×C7#52911yescharacteristic
298C2×C7#52911yescharacteristic
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23#115712yes—
311C23#115912yes—
312C23#115912yes—
313C23C23×D711yescharacteristic
314D4—44no—
315C2×C4—28no—
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—72no—
330C2×C4—72no—
331C2×C4—72no—
332C2×C4—72no—
333C2×C4—148no—
334C2×C4—148no—
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—141no—
355C2×C4D4×D712yes—
356C2×C4#115912yes—
357C7#285211yescharacteristic, complemented, Sylow
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22#644214yescomplemented
372C22C2×#115711yescharacteristic
373C22C2×D4×D711yescharacteristic
374C22C2×#115911yescharacteristic
375C22C2×#115911yescharacteristic
376C22C2×#115911yescharacteristic
377C22D7×#4111yescharacteristic
378C22D7×#4111yescharacteristic
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—142no—
385C4—142no—
386C4—142no—
387C4—142no—
388C4—281no—
389C4—281no—
390C4—281no—
391C4—281no—
392C2—24no—
393C2—81no—
394C2C2×#644211yescharacteristic
395C2#9022911yescharacteristic
396C2#9023811yescharacteristic
397C2#9027711yescharacteristic
398C2#9029111yescharacteristic
399C2#9029711yescharacteristic
400C2#9034011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#386C2
prime radicalC22×C7×D4C22
hypercenter#386D7
hypocenterC7#2852
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C22×C4×Dic7C2semidirect
2. C2×#6442C2semidirect
6. #89982C2semidirect
7. #89723C2semidirect
8. #89722C2semidirect
9. #89718C2semidirect
10. #89717C2semidirect
11. #89716C2semidirect
12. #89427C2semidirect
13. #89425C2semidirect
14. #89424C2semidirect
15. #89423C2semidirect
99. C4×Dic7D4semidirect
223. C22×C7#126semidirect
357. C7#2852semidirect
371. C22#6442semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic72
C2×#64422
C7×#3862
#899922
#899822
#897232
#897222
#897182
#897172
#897162
#894272
#894252
#894242
#894232
#28527