Finite Group Databasebeta

← #11732723

Subgroups — #11732723

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×#387C211yesmaximal, characteristic, complemented
2#89763C211yesmaximal, characteristic, complemented
3#89762C211yesmaximal, characteristic, complemented
4#89762C211yesmaximal, characteristic, complemented
5#89760C211yesmaximal, characteristic, complemented
6#89712C211yesmaximal, characteristic, complemented
7#89711C211yesmaximal, characteristic, complemented
8#89447C211yesmaximal, characteristic, complemented
9#89444C211yesmaximal, characteristic, complemented
10#89444C211yesmaximal, characteristic, complemented
11#89444C211yesmaximal, characteristic, complemented
12#89443C211yesmaximal, characteristic
13#89442C211yesmaximal, characteristic, complemented
14#89441C211yesmaximal, characteristic, complemented
15#89419C211yesmaximal, characteristic, complemented
16C22×D28C2211yescharacteristic
17C22×D28C2211yescharacteristic
18C22×D28C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×C42×C7C2211yescharacteristic
22C2×(C2×D7)⋊C4C2211yescharacteristic
23C2×(C2×D7)⋊C4C2211yescharacteristic
24C2×(C2×D7)⋊C4C2211yescharacteristic
25C2×(C2×D7)⋊C4C2211yescharacteristic
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×(C2×D7)⋊C4C2211yescharacteristic
37C2×C4⋊Dic7C2211yescharacteristic
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
52C23×D7C2311yescharacteristic
53C23×D7C2311yescharacteristic
54C23×D7C2311yescharacteristic
55C22×C4×C7C2311yescharacteristic
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×Dic7C2311yescharacteristic
63C22×Dic7C2311yescharacteristic
64C22×Dic7C2311yescharacteristic
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C2×D28—22no—
68C2×D28—22no—
69C2×D28—41no—
70C2×D28—41no—
71C2×D28—41no—
72C2×D28—41no—
73C2×D28—41no—
74C2×D28—41no—
75C2×D28—41no—
76C7×C4⋊C4—22no—
77C7×C4⋊C4—41no—
78C42×C7—41no—
79(C2×D7)⋊C4—22no—
80(C2×D7)⋊C4—22no—
81(C2×D7)⋊C4—22no—
82(C2×D7)⋊C4—22no—
83(C2×D7)⋊C4—22no—
84(C2×D7)⋊C4—41no—
85(C2×D7)⋊C4—41no—
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—
93(C2×D7)⋊C4—41no—
94C4⋊Dic7—41no—
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#388—71no—
114#387D711yescharacteristic, complemented
115C23×C7C2411yescharacteristic
116C22×D7—21no—
117C22×D7—21no—
118C22×D7—21no—
119C22×D7—21no—
120C22×D7—21no—
121C22×D7—21no—
122C22×D7—21no—
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—21no—
155C2×C4×C7—21no—
156C2×C4×C7#4112yes—
157C2×C4×C7#4112yes—
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#4112yes—
173D28—44no—
174D28—44no—
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⋊C4C2×D711yescharacteristic
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⋊C4—71no—
198C2×C22⋊C4—71no—
199C2×C22⋊C4—71no—
200C2×C42—71no—
201C2×C42—71no—
202C2×C42C2×D711yescharacteristic
203#95—71no—
204#95—71no—
205#95—71no—
206#95—71no—
207#95C2×D711yescharacteristic
208#95C2×D711yescharacteristic
209#95C2×D711yescharacteristic
210#95C2×D711yescharacteristic
211C22×C7C2×#4111yescharacteristic
212C22×C7C2×#4111yescharacteristic
213C22×C7C2×#4111yescharacteristic
214C22×C7H(5,𝔽2)11yescharacteristic
215C22×C7H(5,𝔽2)11yescharacteristic
216C22×C7H(5,𝔽2)11yescharacteristic
217C22×C7H(5,𝔽2)11yescharacteristic
218C2×D7—41no—
219C2×D7—41no—
220C2×D7—41no—
221C2×D7—41no—
222C2×D7—41no—
223C2×D7—41no—
224C2×D7—41no—
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—22no—
241C4×C7—41no—
242C4×C7—41no—
243C4×C7—41no—
244C4×C7—41no—
245C4×C7—41no—
246Dic7—22no—
247Dic7—41no—
248Dic7—41no—
249Dic7—41no—
250Dic7—41no—
251C24—71no—
252C24—71no—
253C24—71no—
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×C4C22×D711yescharacteristic
269C22×C4C22×D711yescharacteristic
270C22×C4C22×D711yescharacteristic
271C22×C4C22×D711yescharacteristic
272C22×C4C22×D711yescharacteristic
273C22×C4C22×D711yescharacteristic
274C22×C4C22×D711yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—41no—
277C4⋊C4—142no—
278C4⋊C4—281no—
279C4⋊C4—281no—
280C4⋊C4—281no—
281C22⋊C4—142no—
282C22⋊C4—142no—
283C22⋊C4—142no—
284C22⋊C4—142no—
285C22⋊C4—142no—
286C22⋊C4—281no—
287C22⋊C4—281no—
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#53811yescharacteristic
300C2×C7#53811yescharacteristic
301C2×C7#54011yescharacteristic
302C2×C7#55311yescharacteristic
303C2×C7#55511yescharacteristic
304C2×C7#55911yescharacteristic
305C2×C7#56011yescharacteristic
306D7—81no—
307D7—81no—
308D7—81no—
309C23—141no—
310C23—141no—
311C23—141no—
312C23—141no—
313C23—141no—
314C23—141no—
315C23—141no—
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—21no—
352C2×C4—21no—
353C2×C4—72no—
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#115712yes—
369C2×C4#116112yes—
370C7#280311yescharacteristic, complemented, Sylow
371C22—281no—
372C22—281no—
373C22—281no—
374C22—281no—
375C22—281no—
376C22—281no—
377C22—281no—
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
394C22C2×#116111yescharacteristic
395C22D7×#4111yescharacteristic
396C22#655311yescharacteristic
397C22#655311yescharacteristic
398C22#655311yescharacteristic
399C4—22no—
400C4—22no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—142no—
407C4—281no—
408C4—281no—
409C4—281no—
410C4—281no—
411C2—561no—
412C2—561no—
413C2—561no—
414C2#9022311yescharacteristic
415C2#9025111yescharacteristic
416C2#9027611yescharacteristic
417C2#9033511yescharacteristic
418C2#9035411yescharacteristic
419C2#9038611yescharacteristic
420C2#9040211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#387C2
prime radical#89443C2
hypercenter#387D7
hypocenterC7#2803
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
1. C7×#387C2semidirect
2. #89763C2semidirect
3. #89762C2semidirect
4. #89762C2semidirect
5. #89760C2semidirect
6. #89712C2semidirect
7. #89711C2semidirect
8. #89447C2semidirect
9. #89444C2semidirect
10. #89444C2semidirect
11. #89444C2semidirect
13. #89442C2semidirect
14. #89441C2semidirect
15. #89419C2semidirect
114. #387D7semidirect
370. C7#2803semidirect

Maximal subgroups

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

subgroupindex
C7×#3872
#897632
#897622
#897602
#897122
#897112
#894472
#894442
#894432
#894422
#894412
#894192
#28037