Finite Group Databasebeta

← #11733253

Subgroups — #11733253

Order 896.

470 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#388C211yesmaximal, characteristic, complemented
2#89760C211yesmaximal, characteristic, complemented
3#89750C211yesmaximal, characteristic, complemented
4#89733C211yesmaximal, characteristic, complemented
5#89732C211yesmaximal, characteristic
6#89731C211yesmaximal, characteristic, complemented
7#89731C211yesmaximal, characteristic, complemented
8#89729C211yesmaximal, characteristic, complemented
9#89717C211yesmaximal, characteristic, complemented
10#89712C211yesmaximal, characteristic, complemented
11#89712C211yesmaximal, characteristic, complemented
12#89446C211yesmaximal, characteristic, complemented
13#89444C211yesmaximal, characteristic, complemented
14#89441C211yesmaximal, characteristic, complemented
15#89419C211yesmaximal, characteristic, complemented
16C22×C22⋊D7C2211yescharacteristic
17C22×C22⋊D7C2211yescharacteristic
18C22×D28C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20C2×C7×C22⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C42×C7C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
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×(C2×D7)⋊C4C2211yescharacteristic
33C2×(C2×D7)⋊C4C2211yescharacteristic
34C2×(C2×D7)⋊C4C2211yescharacteristic
35C2×(C2×D7)⋊C4C2211yescharacteristic
36C2×C4⋊Dic7C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C2×Dic7⋊C4C2211yescharacteristic
40C2×Dic7⋊C4C2211yescharacteristic
41C2×C4×Dic7C2211yescharacteristic
42C2×C4×Dic7C2211yescharacteristic
43C7×#95C2211yescharacteristic
44C7×#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51#2803—71nomaximal, Sylow
52C24×C7C2311yescharacteristic
53C23×D7C2311yescharacteristic
54C23×D7C2311yescharacteristic
55C22×C4×C7C2311yescharacteristic
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C2×C22⋊D7—22no—
62C2×C22⋊D7—41no—
63C2×C22⋊D7—41no—
64C2×C22⋊D7—41no—
65C2×C22⋊D7—41no—
66C2×C22⋊D7—41no—
67C22×Dic7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C2×D28—22no—
74C2×D28—41no—
75C2×D28—41no—
76C7×C4⋊C4—41no—
77C7×C22⋊C4—22no—
78C7×C22⋊C4—41no—
79C7×C22⋊C4—41no—
80C42×C7—41no—
81C22⋊Dic7—22no—
82C22⋊Dic7—41no—
83(C2×D7)⋊C4—22no—
84(C2×D7)⋊C4—22no—
85(C2×D7)⋊C4—22no—
86(C2×D7)⋊C4—41no—
87(C2×D7)⋊C4—41no—
88(C2×D7)⋊C4—41no—
89(C2×D7)⋊C4—41no—
90(C2×D7)⋊C4—41no—
91(C2×D7)⋊C4—41no—
92(C2×D7)⋊C4—41no—
93C4⋊Dic7—41no—
94Dic7⋊C4—22no—
95Dic7⋊C4—22no—
96Dic7⋊C4—41no—
97Dic7⋊C4—41no—
98C4×Dic7—41no—
99C4×Dic7—41no—
100#397—71no—
101#394—71no—
102#394—71no—
103#394—71no—
104#394—71no—
105#392—71no—
106#390—71no—
107#390—71no—
108#389—71no—
109#388—71no—
110#388—71no—
111#388—71no—
112#388—71no—
113#388D711yescharacteristic, complemented
114#387—71no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7C2411yescharacteristic
123C22×D7—21no—
124C22×D7—21no—
125C22×D7—21no—
126C22×D7—21no—
127C22×D7—21no—
128C22×D7—21no—
129C22×D7—21no—
130C22×D7—21no—
131C22×D7—21no—
132C22×D7—21no—
133C22×D7—21no—
134C22×D7—21no—
135C22×D7—21no—
136C22×D7—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×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—21no—
154C2×C4×C7#4112yes—
155C22⋊D7—44no—
156C22⋊D7—44no—
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#4112yes—
174C2×Dic7#4112yes—
175D28—44no—
176C22×D4—71no—
177C22×D4—71no—
178C22×D4—71no—
179C2×C4⋊C4—71no—
180C2×C4⋊C4—71no—
181C2×C4⋊C4—71no—
182C2×C4⋊C4—71no—
183C2×C4⋊C4—71no—
184C2×C4⋊C4C2×D711yescharacteristic
185C2×C22⋊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⋊C4—71no—
194C2×C22⋊C4—71no—
195C2×C22⋊C4—71no—
196C2×C22⋊C4—71no—
197C2×C22⋊C4C2×D711yescharacteristic
198C2×C22⋊C4C2×D711yescharacteristic
199C2×C22⋊C4C2×D711yescharacteristic
200C2×C42—71no—
201C2×C42—71no—
202C2×C42C2×D711yescharacteristic
203#95—71no—
204#95—71no—
205#95—71no—
206#95—71no—
207#95—71no—
208#95—71no—
209#95C2×D711yescharacteristic
210#95C2×D711yescharacteristic
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7C2×#4111yescharacteristic
219C22×C7C2×#4111yescharacteristic
220C22×C7C2×#4111yescharacteristic
221C22×C7H(5,𝔽2)11yescharacteristic
222C22×C7H(5,𝔽2)11yescharacteristic
223C22×C7H(5,𝔽2)11yescharacteristic
224C22×C7H(5,𝔽2)11yescharacteristic
225C2×D7—41no—
226C2×D7—41no—
227C2×D7—41no—
228C2×D7—41no—
229C2×D7—41no—
230C2×D7—41no—
231C2×D7—41no—
232C2×D7—41no—
233C2×D7—41no—
234C2×D7—41no—
235C2×D7—41no—
236C2×D7—41no—
237C2×D7—41no—
238C2×D7—41no—
239C4×C7—22no—
240C4×C7—41no—
241C4×C7—41no—
242C4×C7—41no—
243C4×C7—41no—
244C4×C7—41no—
245Dic7—22no—
246Dic7—22no—
247Dic7—41no—
248Dic7—41no—
249Dic7—41no—
250Dic7—41no—
251C24—71no—
252C24—71no—
253C24C22×D711yescharacteristic
254C2×D4—142no—
255C2×D4—142no—
256C2×D4—281no—
257C2×D4—281no—
258C2×D4—281no—
259C2×D4—281no—
260C2×D4—281no—
261C2×D4—281no—
262C2×D4—281no—
263C22×C4—71no—
264C22×C4—71no—
265C22×C4—71no—
266C22×C4—71no—
267C22×C4—71no—
268C22×C4—71no—
269C22×C4C22×D711yescharacteristic
270C22×C4C22×D711yescharacteristic
271C22×C4C22×D711yescharacteristic
272C22×C4C22×D711yescharacteristic
273C22×C4C22×D711yescharacteristic
274C22×C4C22×D711yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—142no—
277C4⋊C4—142no—
278C4⋊C4—281no—
279C4⋊C4—281no—
280C4⋊C4—281no—
281C22⋊C4—22no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—142no—
285C22⋊C4—142no—
286C22⋊C4—142no—
287C22⋊C4—142no—
288C22⋊C4—281no—
289C22⋊C4—281no—
290C22⋊C4—281no—
291C22⋊C4—281no—
292C22⋊C4—281no—
293C22⋊C4—281no—
294C22⋊C4—281no—
295C22⋊C4—281no—
296C42—41no—
297C42—281no—
298C42—281no—
299C2×C7—81no—
300C2×C7#53811yescharacteristic
301C2×C7#53811yescharacteristic
302C2×C7#54011yescharacteristic
303C2×C7#55311yescharacteristic
304C2×C7#55511yescharacteristic
305C2×C7#55911yescharacteristic
306C2×C7#56011yescharacteristic
307D7—81no—
308D7—81no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—141no—
317C23—141no—
318C23—141no—
319C23—141no—
320C23—141no—
321C23—141no—
322C23—141no—
323C23—141no—
324C23—141no—
325C23—141no—
326C23—141no—
327C23—141no—
328C23—141no—
329C23—141no—
330C23C23×D711yescharacteristic
331D4—284no—
332D4—284no—
333D4—284no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
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—72no—
352C2×C4—72no—
353C2×C4—141no—
354C2×C4—141no—
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#115712yes—
370C7#280311yescharacteristic, complemented, Sylow
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—281no—
379C22—281no—
380C22—281no—
381C22—281no—
382C22—281no—
383C22—281no—
384C22—281no—
385C22—281no—
386C22—281no—
387C22—281no—
388C22—281no—
389C22—281no—
390C22—281no—
391C22—281no—
392C22C2×#115711yescharacteristic
393C22#654811yescharacteristic
394C22#654811yescharacteristic
395C22D7×#4111yescharacteristic
396C22D7×#4111yescharacteristic
397C22#655311yescharacteristic
398C22#655311yescharacteristic
399C4—22no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—142no—
406C4—142no—
407C4—281no—
408C4—281no—
409C4—281no—
410C4—281no—
411C2—81no—
412C2—561no—
413C2—561no—
414C2#9022311yescharacteristic
415C2#9025011yescharacteristic
416C2#9025111yescharacteristic
417C2#9030611yescharacteristic
418C2#9035711yescharacteristic
419C2#9036911yescharacteristic
420C2#9040011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#388C2
prime radical#89732C2
hypercenter#388D7
hypocenterC7#2803
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C7×#388C2semidirect
2. #89760C2semidirect
3. #89750C2semidirect
4. #89733C2semidirect
6. #89731C2semidirect
7. #89731C2semidirect
8. #89729C2semidirect
9. #89717C2semidirect
10. #89712C2semidirect
11. #89712C2semidirect
12. #89446C2semidirect
13. #89444C2semidirect
14. #89441C2semidirect
15. #89419C2semidirect
113. #388D7semidirect
370. C7#2803semidirect

Maximal subgroups

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

subgroupindex
C7×#3882
#897602
#897502
#897332
#897322
#897312
#897292
#897172
#897122
#894462
#894442
#894412
#894192
#28037