Finite Group Databasebeta

← #11731963

Subgroups — #11731963

Order 896.

490 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6492C211yesmaximal, characteristic
2C2×#6452C211yesmaximal, characteristic, complemented
3C7×#385C211yesmaximal, characteristic, complemented
4#89984C211yesmaximal, characteristic, complemented
5#89766C211yesmaximal, characteristic, complemented
6#89762C211yesmaximal, characteristic, complemented
7#89745C211yesmaximal, characteristic, complemented
8#89731C211yesmaximal, characteristic
9#89729C211yesmaximal, characteristic
10#89723C211yesmaximal, characteristic, complemented
11#89721C211yesmaximal, characteristic, complemented
12#89449C211yesmaximal, characteristic, complemented
13#89447C211yesmaximal, characteristic, complemented
14#89429C211yesmaximal, characteristic, complemented
15#89427C211yesmaximal, characteristic, complemented
16C23×C4×C7C2211yescharacteristic
17C22×C22⋊D7C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20C2×C7×C22⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C22⋊Dic7C2211yescharacteristic
23C2×C22⋊Dic7C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25#6492—24no—
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×C4⋊Dic7C2211yescharacteristic
33C2×C4⋊Dic7C2211yescharacteristic
34C2×C4⋊Dic7C2211yescharacteristic
35C2×C4⋊Dic7C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C2×Dic7⋊C4C2211yescharacteristic
40C2×Dic7⋊C4C2211yescharacteristic
41C2×Dic7⋊C4C2211yescharacteristic
42C2×C4×Dic7C2211yescharacteristic
43C2×C4×Dic7C2211yescharacteristic
44#6452—24no—
45C7×#95C2211yescharacteristic
46C7×#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53#2694—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C23×D7C2311yescharacteristic
56C22×C4×C7—22no—
57C22×C4×C7—22no—
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⋊D7—41no—
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C22×Dic7C2311yescharacteristic
75C7×C4⋊C4—22no—
76C7×C4⋊C4—41no—
77C7×C22⋊C4—22no—
78C7×C22⋊C4—22no—
79C22⋊Dic7—22no—
80C22⋊Dic7—41no—
81C22⋊Dic7—41no—
82(C2×D7)⋊C4—22no—
83(C2×D7)⋊C4—22no—
84(C2×D7)⋊C4—22no—
85(C2×D7)⋊C4—41no—
86(C2×D7)⋊C4—41no—
87(C2×D7)⋊C4—41no—
88C4⋊Dic7—22no—
89C4⋊Dic7—22no—
90C4⋊Dic7—41no—
91C4⋊Dic7—41no—
92Dic7⋊C4—22no—
93Dic7⋊C4—22no—
94Dic7⋊C4—22no—
95Dic7⋊C4—22no—
96Dic7⋊C4—41no—
97Dic7⋊C4D414yes—
98C4×Dic7—22no—
99C4×Dic7—41no—
100C2×#125—71no—
101C2×#123—71no—
102#399—71no—
103#399—71no—
104#398—71no—
105#397—71no—
106#396—71no—
107#394—71no—
108#390—71no—
109#389—71no—
110#388—71no—
111#388—71no—
112#387—71no—
113#385D711yescharacteristic, complemented
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×D7—21no—
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—44no—
147C2×C4×C7C2×D412yes—
148C2×C4×C7#4112yes—
149C22⋊D7—44no—
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—
171C22×D4—71no—
172C23×C4C2×D711yescharacteristic
173#125—144no—
174#123—144no—
175C2×C4⋊C4—71no—
176C2×C4⋊C4—71no—
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⋊C4C2×D711yescharacteristic
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⋊C4—71no—
195C2×C22⋊C4—71no—
196C2×C22⋊C4C2×D711yescharacteristic
197C2×C22⋊C4C2×D711yescharacteristic
198C2×C42—71no—
199C2×C42—71no—
200#95—71no—
201#95—71no—
202#95—71no—
203#95—71no—
204#95—71no—
205#95—71no—
206#95C2×D711yescharacteristic
207#95C2×D711yescharacteristic
208C22×C7—22no—
209C22×C7—22no—
210C22×C7—22no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7C22×D411yescharacteristic
216C22×C7C2×#4111yescharacteristic
217C22×C7C2×#4111yescharacteristic
218C22×C7H(5,𝔽2)11yescharacteristic
219C22×C7H(5,𝔽2)11yescharacteristic
220C22×C72−1+411yescharacteristic
221C22×C72−1+411yescharacteristic
222C2×D7—41no—
223C2×D7—41no—
224C2×D7—41no—
225C2×D7—41no—
226C2×D7—41no—
227C2×D7—41no—
228C2×D7—41no—
229C4×C7—22no—
230C4×C7—22no—
231C4×C7—41no—
232C4×C7—41no—
233C4×C7—41no—
234C4×C7—41no—
235Dic7—22no—
236Dic7—22no—
237Dic7—41no—
238Dic7—41no—
239Dic7—41no—
240Dic7—41no—
241Dic7—41no—
242C24—71no—
243C24C22×D711yescharacteristic
244C2×D4—142no—
245C2×D4—142no—
246C2×D4—281no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—41no—
250C22×C4—71no—
251C22×C4—71no—
252C22×C4—71no—
253C22×C4—71no—
254C22×C4—71no—
255C22×C4—71no—
256C22×C4—71no—
257C22×C4C22×D711yescharacteristic
258C22×C4C22×D711yescharacteristic
259C22×C4C22×D711yescharacteristic
260C22×C4C22×D711yescharacteristic
261C22×C4C22×D711yescharacteristic
262C22×C4C22×D711yescharacteristic
263C4⋊C4—22no—
264C4⋊C4—41no—
265C4⋊C4—74no—
266C4⋊C4—142no—
267C4⋊C4—142no—
268C4⋊C4—142no—
269C4⋊C4—142no—
270C4⋊C4—142no—
271C4⋊C4—142no—
272C4⋊C4—281no—
273C4⋊C4—281no—
274C4⋊C4—281no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—142no—
278C22⋊C4—142no—
279C22⋊C4—142no—
280C22⋊C4—142no—
281C22⋊C4—281no—
282C22⋊C4—281no—
283C22⋊C4—281no—
284C22⋊C4—281no—
285C22⋊C4—281no—
286C42—142no—
287C42—281no—
288C2×C7—42no—
289C2×C7#53611yescharacteristic
290C2×C7#53911yescharacteristic
291C2×C7#53911yescharacteristic
292C2×C7#54611yescharacteristic
293C2×C7#55011yescharacteristic
294C2×C7#55211yescharacteristic
295C2×C7#56211yescharacteristic
296D7—81no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—141no—
304C23—141no—
305C23—141no—
306C23—141no—
307C23—141no—
308C23—141no—
309C23—141no—
310C23#115912yes—
311C23C23×D711yescharacteristic
312D4—284no—
313C2×C4—21no—
314C2×C4—21no—
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—21no—
328C2×C4—21no—
329C2×C4—44no—
330C2×C4—72no—
331C2×C4—72no—
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#115712yes—
352C2×C4D4×D712yes—
353C7#269411yescharacteristic, complemented, Sylow
354C22—22no—
355C22—22no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—281no—
362C22—281no—
363C22—281no—
364C22—281no—
365C22—281no—
366C22—281no—
367C22—281no—
368C22C2×#115711yescharacteristic
369C22C2×D4×D711yescharacteristic
370C22C2×#115911yescharacteristic
371C22#654811yescharacteristic
372C22#655111yescharacteristic
373C22#655311yescharacteristic
374C22#655411yescharacteristic
375C4—22no—
376C4—22no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—142no—
382C4—142no—
383C4—281no—
384C4—281no—
385C4—281no—
386C4—281no—
387C4—281no—
388C2—42no—
389C2—561no—
390C2#9020411yescharacteristic
391C2#9024411yescharacteristic
392C2#9025511yescharacteristic
393C2#9028211yescharacteristic
394C2#9031911yescharacteristic
395C2#9033811yescharacteristic
396C2#9035611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#385C2
prime radicalC22×C22⋊D7C22
hypercenter#385D7
hypocenterC7#2694
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#6452C2semidirect
3. C7×#385C2semidirect
4. #89984C2semidirect
5. #89766C2semidirect
6. #89762C2semidirect
7. #89745C2semidirect
10. #89723C2semidirect
11. #89721C2semidirect
12. #89449C2semidirect
13. #89447C2semidirect
14. #89429C2semidirect
15. #89427C2semidirect
113. #385D7semidirect
353. C7#2694semidirect

Maximal subgroups

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

subgroupindex
C2×#64922
C2×#64522
C7×#3852
#899842
#897662
#897622
#897452
#897312
#897292
#897232
#897212
#894492
#894472
#894292
#894272
#26947