Finite Group Databasebeta

← #11732014

Subgroups — #11732014

Order 896.

472 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
1C7×#385C211yesmaximal, characteristic, complemented
2#89985C211yesmaximal, characteristic
3#89984C211yesmaximal, characteristic, complemented
4#89766C211yesmaximal, characteristic, complemented
5#89759C211yesmaximal, characteristic, complemented
6#89753C211yesmaximal, characteristic, complemented
7#89750C211yesmaximal, characteristic, complemented
8#89729C211yesmaximal, characteristic
9#89728C211yesmaximal, characteristic
10#89720C211yesmaximal, characteristic, complemented
11#89719C211yesmaximal, characteristic, complemented
12#89445C211yesmaximal, characteristic, complemented
13#89440C211yesmaximal, characteristic, complemented
14#89430C211yesmaximal, characteristic, complemented
15#89428C211yesmaximal, characteristic, complemented
16C23×C4×C7C2211yescharacteristic
17C22×C22⋊D7C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
19C22×Dic14C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×C7×C4⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
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×Dic7⋊C4C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×C4×Dic7C2211yescharacteristic
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#2784—71nomaximal, Sylow
52C24×C7C2311yescharacteristic
53C23×D7C2311yescharacteristic
54C22×C4×C7—22no—
55C22×C4×C7—41no—
56C22×C4×C7—41no—
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C2×C22⋊D7—41no—
64C2×C22⋊D7—41no—
65C2×C22⋊D7Q814yescomplemented
66C22×Dic7C2311yescharacteristic
67C22×Dic7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C2×C4×D7—22no—
74C2×C4×D7—41no—
75C2×C4×D7—41no—
76C2×Dic14—41no—
77C2×Dic14—41no—
78C2×Dic14—41no—
79C7×C4⋊C4—22no—
80C7×C4⋊C4—41no—
81C7×C22⋊C4—22no—
82C7×C22⋊C4—41no—
83C22⋊Dic7—41no—
84C22⋊Dic7—41no—
85C22⋊Dic7—41no—
86(C2×D7)⋊C4—22no—
87(C2×D7)⋊C4—41no—
88(C2×D7)⋊C4—41no—
89(C2×D7)⋊C4—41no—
90(C2×D7)⋊C4—41no—
91C4⋊Dic7—41no—
92C4⋊Dic7—41no—
93C4⋊Dic7—41no—
94Dic7⋊C4—22no—
95Dic7⋊C4—41no—
96Dic7⋊C4—41no—
97C4×Dic7—41no—
98#400—71no—
99#396—71no—
100#396—71no—
101#395—71no—
102#393—71no—
103#393—71no—
104#390—71no—
105#387—71no—
106#387—71no—
107#386—71no—
108#386—71no—
109#385—71no—
110#385D711yescharacteristic, complemented
111#380—71no—
112#380—71no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7C2×Q812yes—
120C23×C7C2411yescharacteristic
121C22×D7—21no—
122C22×D7—21no—
123C22×D7—21no—
124C22×D7—21no—
125C22×D7—21no—
126C22×D7—21no—
127C22×D7C2×Q812yes—
128C2×C4×C7—21no—
129C2×C4×C7—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—44no—
145C2×C4×C7#4112yes—
146C2×C4×C7#4112yes—
147C22⋊D7—44no—
148C2×Dic7—21no—
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×Dic7C2×Q812yes—
169C4×D7—44no—
170Dic14—44no—
171C22×Q8—71no—
172C22×D4—71no—
173C23×C4—71no—
174C23×C4C2×D711yescharacteristic
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⋊C4C2×D711yescharacteristic
182C2×C4⋊C4C2×D711yescharacteristic
183C2×C22⋊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⋊C4—71no—
191C2×C22⋊C4C2×D711yescharacteristic
192C2×C22⋊C4C2×D711yescharacteristic
193C2×C42—71no—
194#95—71no—
195#95—71no—
196#95—71no—
197#95—71no—
198#95—71no—
199#95—71no—
200#95—71no—
201#95—71no—
202#95—71no—
203#95—71no—
204#95C2×D711yescharacteristic
205#95C2×D711yescharacteristic
206C22×C7—22no—
207C22×C7—22no—
208C22×C7—22no—
209C22×C7—41no—
210C22×C7—41no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7C22×Q811yescharacteristic
214C22×C7C2×#4111yescharacteristic
215C22×C7C2×#4111yescharacteristic
216C22×C7H(5,𝔽2)11yescharacteristic
217C22×C7H(5,𝔽2)11yescharacteristic
218C22×C7H(5,𝔽2)11yescharacteristic
219C22×C7H(5,𝔽2)11yescharacteristic
220C2×D7—22no—
221C2×D7—22no—
222C2×D7—22no—
223C2×D7—41no—
224C2×D7—41no—
225C2×D7—41no—
226C2×D7—41no—
227C4×C7—22no—
228C4×C7—22no—
229C4×C7—41no—
230C4×C7—41no—
231C4×C7—41no—
232C4×C7—41no—
233Dic7—22no—
234Dic7—41no—
235Dic7—41no—
236Dic7—41no—
237Dic7—41no—
238Dic7—41no—
239Dic7—41no—
240C24—71no—
241C24C22×D711yescharacteristic
242C2×Q8—281no—
243C2×Q8—281no—
244C2×Q8—281no—
245C2×D4—74no—
246C2×D4—281no—
247C2×D4—281no—
248C22×C4—22no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—71no—
252C22×C4—71no—
253C22×C4—71no—
254C22×C4—71no—
255C22×C4—71no—
256C22×C4—71no—
257C22×C4—71no—
258C22×C4—142no—
259C22×C4—281no—
260C22×C4—281no—
261C22×C4C22×D711yescharacteristic
262C22×C4C22×D711yescharacteristic
263C22×C4C22×D711yescharacteristic
264C22×C4C22×D711yescharacteristic
265C22×C4C22×D711yescharacteristic
266C22×C4C22×D711yescharacteristic
267C4⋊C4—22no—
268C4⋊C4—41no—
269C4⋊C4—142no—
270C4⋊C4—281no—
271C4⋊C4—281no—
272C4⋊C4—281no—
273C4⋊C4—281no—
274C4⋊C4—281no—
275C22⋊C4—22no—
276C22⋊C4—41no—
277C22⋊C4—142no—
278C22⋊C4—281no—
279C22⋊C4—281no—
280C22⋊C4—281no—
281C22⋊C4—281no—
282C22⋊C4—281no—
283C22⋊C4—281no—
284C22⋊C4—281no—
285C42—281no—
286C2×C7—42no—
287C2×C7#53811yescharacteristic
288C2×C7#53811yescharacteristic
289C2×C7#54311yescharacteristic
290C2×C7#55411yescharacteristic
291C2×C7#55411yescharacteristic
292C2×C7#55511yescharacteristic
293C2×C7#56011yescharacteristic
294D7—42no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—72no—
302C23—141no—
303C23—141no—
304C23—141no—
305C23—141no—
306C23—141no—
307C23—141no—
308C23Q8×D712yes—
309C23C23×D711yescharacteristic
310Q8—284no—
311D4—284no—
312C2×C4—21no—
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—44no—
329C2×C4—72no—
330C2×C4—141no—
331C2×C4—141no—
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—284no—
351C2×C4#115712yes—
352C2×C4#115912yes—
353C7#278411yescharacteristic, complemented, Sylow
354C22—22no—
355C22—22no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—142no—
362C22—142no—
363C22—142no—
364C22—281no—
365C22—281no—
366C22—281no—
367C22—281no—
368C22C2×#115711yescharacteristic
369C22C2×#115911yescharacteristic
370C22#654811yescharacteristic
371C22#654811yescharacteristic
372C22C2×Q8×D711yescharacteristic
373C22#655311yescharacteristic
374C22#655311yescharacteristic
375C4—22no—
376C4—22no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—142no—
382C4—281no—
383C4—281no—
384C4—281no—
385C4—281no—
386C4—281no—
387C4—281no—
388C2—42no—
389C2—282no—
390C2#9020211yescharacteristic
391C2#9025111yescharacteristic
392C2#9025411yescharacteristic
393C2#9028411yescharacteristic
394C2#9032511yescharacteristic
395C2#9032611yescharacteristic
396C2#9035711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#385C2
prime radicalC22×C22⋊D7C22
hypercenter#385D7
hypocenterC7#2784
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C7×#385C2semidirect
3. #89984C2semidirect
4. #89766C2semidirect
5. #89759C2semidirect
6. #89753C2semidirect
7. #89750C2semidirect
10. #89720C2semidirect
11. #89719C2semidirect
12. #89445C2semidirect
13. #89440C2semidirect
14. #89430C2semidirect
15. #89428C2semidirect
65. C2×C22⋊D7Q8semidirect
110. #385D7semidirect
353. C7#2784semidirect

Maximal subgroups

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

subgroupindex
C7×#3852
#899852
#899842
#897662
#897592
#897532
#897502
#897292
#897282
#897202
#897192
#894452
#894402
#894302
#894282
#27847