Finite Group Databasebeta

← #11732296

Subgroups — #11732296

Order 896.

520 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#386C211yesmaximal, characteristic, complemented
2#89995C211yesmaximal, characteristic
3#89992C211yesmaximal, characteristic, complemented
4#89985C211yesmaximal, characteristic, complemented
5#89983C211yesmaximal, characteristic, complemented
6#89730C211yesmaximal, characteristic, complemented
7#89728C211yesmaximal, characteristic, complemented
8#89728C211yesmaximal, characteristic, complemented
9#89719C211yesmaximal, characteristic, complemented
10#89719C211yesmaximal, characteristic, complemented
11#89715C211yesmaximal, characteristic, complemented
12#89445C211yesmaximal, characteristic, complemented
13#89436C211yesmaximal, characteristic, complemented
14C7⋊#401C211yesmaximal, characteristic, complemented
15C7⋊#401C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C23×C4×C7C2211yescharacteristic
18C22×C22⋊D7C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C23×Dic7C2211yescharacteristic
21C22×C4×D7C2211yescharacteristic
22C22×Dic14C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×(C2×D7)⋊C4C2211yescharacteristic
33C2×(C2×D7)⋊C4C2211yescharacteristic
34C2×(C2×D7)⋊C4C2211yescharacteristic
35C7×#95C2211yescharacteristic
36C7×#95C2211yescharacteristic
37C7⋊#95C2211yescharacteristic
38C7⋊#95C2211yescharacteristic
39C7⋊#95C2211yescharacteristic
40C7⋊#95C2211yescharacteristic
41C7⋊#95C2211yescharacteristic
42C7⋊#95C2211yescharacteristic
43C7⋊#95C2211yescharacteristic
44C7⋊#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51#2760—71nomaximal, Sylow
52C24×C7C2311yescharacteristic
53C24×C7C2311yescharacteristic
54C23×D7C2311yescharacteristic
55C2×C7×D4—22no—
56C2×C7×D4—22no—
57C2×C7×D4—41no—
58C22×C4×C7—22no—
59C22×C4×C7—41no—
60C22×C4×C7—41no—
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C2×C22⋊D7—22no—
67C2×C22⋊D7—22no—
68C2×C22⋊D7—22no—
69C2×C22⋊D7—22no—
70C2×C22⋊D7—41no—
71C2×C22⋊D7—41no—
72C22×Dic7—22no—
73C22×Dic7—41no—
74C22×Dic7—41no—
75C22×Dic7C2311yescharacteristic
76C22×Dic7C2311yescharacteristic
77C22×Dic7C2311yescharacteristic
78C22×Dic7C2311yescharacteristic
79C22×Dic7C2311yescharacteristic
80C22×Dic7C2311yescharacteristic
81C22×Dic7C2311yescharacteristic
82C2×C4×D7—22no—
83C2×C4×D7—41no—
84C2×C4×D7—41no—
85C2×Dic14—41no—
86C2×Dic14—41no—
87C2×Dic14—41no—
88C7×C22⋊C4—41no—
89C7×C22⋊C4—41no—
90C7×C22⋊C4—41no—
91C22⋊Dic7—41no—
92C22⋊Dic7—41no—
93C22⋊Dic7—41no—
94C22⋊Dic7—41no—
95C22⋊Dic7—41no—
96(C2×D7)⋊C4—41no—
97(C2×D7)⋊C4—41no—
98(C2×D7)⋊C4—41no—
99(C2×D7)⋊C4—41no—
100#401—71no—
101#401—71no—
102#393—71no—
103#393—71no—
104#393—71no—
105#392—71no—
106#386—71no—
107#386—71no—
108#386—71no—
109#386—71no—
110#386—71no—
111#386D711yescharacteristic, complemented
112#381—71no—
113#381—71no—
114#381—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#4112yes—
128C23×C7#4112yes—
129C23×C7C2411yescharacteristic
130C22×D7—21no—
131C22×D7—21no—
132C22×D7—21no—
133C22×D7—21no—
134C22×D7—21no—
135C22×D7—21no—
136C22×D7#4112yes—
137C7×D4—44no—
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—44no—
154C22⋊D7—44no—
155C22⋊D7—44no—
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—21no—
171C2×Dic7—21no—
172C2×Dic7—21no—
173C2×Dic7—21no—
174C2×Dic7—21no—
175C2×Dic7—21no—
176C2×Dic7—21no—
177C2×Dic7—44no—
178C4×D7—44no—
179Dic14—44no—
180C22×Q8—71no—
181C22×D4—71no—
182C22×D4—71no—
183C22×D4C2×D711yescharacteristic
184C23×C4—71no—
185C23×C4—71no—
186C23×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×C22⋊C4C2×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—
209#95—71no—
210#95—71no—
211#95—71no—
212#95—71no—
213#95C2×D711yescharacteristic
214#95C2×D711yescharacteristic
215C22×C7—22no—
216C22×C7—22no—
217C22×C7—22no—
218C22×C7—22no—
219C22×C7—22no—
220C22×C7—22no—
221C22×C7—41no—
222C22×C7—41no—
223C22×C7—41no—
224C22×C7—41no—
225C22×C7—41no—
226C22×C7—41no—
227C22×C7—41no—
228C22×C7—41no—
229C22×C7C2×#4111yescharacteristic
230C22×C7C2×#4111yescharacteristic
231C22×C7C2×#4111yescharacteristic
232C22×C7H(5,𝔽2)11yescharacteristic
233C22×C7H(5,𝔽2)11yescharacteristic
234C22×C7H(5,𝔽2)11yescharacteristic
235C22×C7H(5,𝔽2)11yescharacteristic
236C2×D7—22no—
237C2×D7—22no—
238C2×D7—22no—
239C2×D7—41no—
240C2×D7—41no—
241C2×D7—41no—
242C2×D7—41no—
243C4×C7—41no—
244C4×C7—41no—
245C4×C7—41no—
246C4×C7—41no—
247C4×C7—41no—
248Dic7—41no—
249Dic7—41no—
250Dic7—41no—
251Dic7—41no—
252Dic7—41no—
253Dic7—41no—
254Dic7—41no—
255C24—71no—
256C24C22×D711yescharacteristic
257C24C22×D711yescharacteristic
258C2×Q8—281no—
259C2×Q8—281no—
260C2×Q8—281no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—41no—
264C2×D4—142no—
265C2×D4—142no—
266C2×D4—142no—
267C2×D4—142no—
268C2×D4—281no—
269C2×D4—281no—
270C22×C4—22no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—71no—
274C22×C4—71no—
275C22×C4—71no—
276C22×C4—71no—
277C22×C4—71no—
278C22×C4—71no—
279C22×C4—71no—
280C22×C4—142no—
281C22×C4—142no—
282C22×C4—281no—
283C22×C4—281no—
284C22×C4—281no—
285C22×C4—281no—
286C22×C4C22×D711yescharacteristic
287C22×C4C22×D711yescharacteristic
288C22×C4C22×D711yescharacteristic
289C22×C4C22×D711yescharacteristic
290C22×C4C22×D711yescharacteristic
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—281no—
295C22⋊C4—281no—
296C22⋊C4—281no—
297C22⋊C4—281no—
298C22⋊C4—281no—
299C22⋊C4—281no—
300C22⋊C4—281no—
301C22⋊C4—281no—
302C22⋊C4—281no—
303C2×C7—42no—
304C2×C7—42no—
305C2×C7#53811yescharacteristic
306C2×C7#53811yescharacteristic
307C2×C7#53811yescharacteristic
308C2×C7#55111yescharacteristic
309C2×C7#55111yescharacteristic
310C2×C7#55111yescharacteristic
311C2×C7#56011yescharacteristic
312D7—42no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—72no—
326C23—141no—
327C23—141no—
328C23—141no—
329C23—141no—
330C23—141no—
331C23—141no—
332C23#115712yes—
333C23#115912yes—
334C23C23×D711yescharacteristic
335Q8—284no—
336D4—44no—
337D4—284no—
338D4—284no—
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—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—44no—
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—141no—
367C2×C4—141no—
368C2×C4—141no—
369C2×C4—141no—
370C2×C4—141no—
371C2×C4—141no—
372C2×C4—141no—
373C2×C4—141no—
374C2×C4—141no—
375C2×C4—141no—
376C2×C4—284no—
377C2×C4—284no—
378C7#276011yescharacteristic, complemented, Sylow
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—142no—
394C22—142no—
395C22—142no—
396C22—281no—
397C22—281no—
398C22—281no—
399C22—281no—
400C22C2×#115711yescharacteristic
401C22C2×#115911yescharacteristic
402C22#654811yescharacteristic
403C22#654811yescharacteristic
404C22#654811yescharacteristic
405C22D7×#4111yescharacteristic
406C22#655311yescharacteristic
407C4—41no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—281no—
413C4—281no—
414C4—281no—
415C4—281no—
416C4—281no—
417C4—281no—
418C4—281no—
419C2—42no—
420C2—42no—
421C2—282no—
422C2#9018611yescharacteristic
423C2#9024711yescharacteristic
424C2#9025111yescharacteristic
425C2#9028411yescharacteristic
426C2#9030411yescharacteristic
427C2#9031111yescharacteristic
428C2#9034911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#386C2
prime radical#89995C2
hypercenter#386D7
hypocenterC7#2760
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C7×#386C2semidirect
3. #89992C2semidirect
4. #89985C2semidirect
5. #89983C2semidirect
6. #89730C2semidirect
7. #89728C2semidirect
8. #89728C2semidirect
9. #89719C2semidirect
10. #89719C2semidirect
11. #89715C2semidirect
12. #89445C2semidirect
13. #89436C2semidirect
14. C7⋊#401C2semidirect
15. C7⋊#401C2semidirect
111. #386D7semidirect
378. C7#2760semidirect

Maximal subgroups

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

subgroupindex
C7×#3862
#899952
#899922
#899852
#899832
#897302
#897282
#897192
#897152
#894452
#894362
C7⋊#4012
#27607