Finite Group Databasebeta

← #11735111

Subgroups — #11735111

Order 896.

366 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#397C211yesmaximal, characteristic
2#89751C211yesmaximal, characteristic, complemented
3#89744C211yesmaximal, characteristic, complemented
4#89725C211yesmaximal, characteristic
5#89724C211yesmaximal, characteristic
6#89721C211yesmaximal, characteristic
7#89720C211yesmaximal, characteristic
8#89719C211yesmaximal, characteristic
9#89717C211yesmaximal, characteristic
10C7⋊#401C211yesmaximal, characteristic, complemented
11#89432C211yesmaximal, characteristic, complemented
12#89432C211yesmaximal, characteristic, complemented
13#89429C211yesmaximal, characteristic, complemented
14#89428C211yesmaximal, characteristic, complemented
15#89425C211yesmaximal, characteristic, complemented
16C22×Dic14C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×C7×C22⋊C4C2211yescharacteristic
19C2×C7×C22⋊C4C2211yescharacteristic
20C2×C7×C22⋊C4C2211yescharacteristic
21C2×C22⋊Dic7C2211yescharacteristic
22C2×C22⋊Dic7C2211yescharacteristic
23C2×C22⋊Dic7C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25C2×C4⋊Dic7C2211yescharacteristic
26C2×C4⋊Dic7C2211yescharacteristic
27C2×C4⋊Dic7C2211yescharacteristic
28C2×Dic7⋊C4C2211yescharacteristic
29C2×Dic7⋊C4C2211yescharacteristic
30C2×Dic7⋊C4C2211yescharacteristic
31C2×Dic7⋊C4C2211yescharacteristic
32C2×Dic7⋊C4C2211yescharacteristic
33C2×Dic7⋊C4C2211yescharacteristic
34C2×C4×Dic7C2211yescharacteristic
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#2838—71nomaximal, Sylow
52C24×C7C2311yescharacteristic
53C22×C4×C7C2311yescharacteristic
54C22×C4×C7C2311yescharacteristic
55C22×C4×C7C2311yescharacteristic
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×Dic7C2311yescharacteristic
60C22×Dic7C2311yescharacteristic
61C22×Dic7C2311yescharacteristic
62C22×Dic7C2311yescharacteristic
63C22×Dic7C2311yescharacteristic
64C22×Dic7C2311yescharacteristic
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C2×Dic14—41no—
68C2×Dic14—41no—
69C2×Dic14—41no—
70C7×C4⋊C4—41no—
71C7×C22⋊C4—41no—
72C7×C22⋊C4—41no—
73C7×C22⋊C4—41no—
74C22⋊Dic7—22no—
75C22⋊Dic7—41no—
76C22⋊Dic7—41no—
77C22⋊Dic7—41no—
78C4⋊Dic7—22no—
79C4⋊Dic7—41no—
80C4⋊Dic7—41no—
81Dic7⋊C4—22no—
82Dic7⋊C4—41no—
83Dic7⋊C4—41no—
84Dic7⋊C4—41no—
85Dic7⋊C4—41no—
86Dic7⋊C4—41no—
87C4×Dic7—41no—
88#401—71no—
89#400—71no—
90#400—71no—
91#399—71no—
92#398—71no—
93#397—71no—
94#397—71no—
95#397D711yescharacteristic
96#396—71no—
97#395—71no—
98#395—71no—
99#393—71no—
100#389—71no—
101#388—71no—
102#387—71no—
103C23×C7—21no—
104C23×C7—21no—
105C23×C7—21no—
106C23×C7—21no—
107C23×C7—21no—
108C23×C7—21no—
109C23×C7—21no—
110C23×C7C2411yescharacteristic
111C2×C4×C7—21no—
112C2×C4×C7—21no—
113C2×C4×C7—21no—
114C2×C4×C7—21no—
115C2×C4×C7—21no—
116C2×C4×C7—21no—
117C2×C4×C7—21no—
118C2×C4×C7—21no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×Dic7—21no—
130C2×Dic7—21no—
131C2×Dic7—21no—
132C2×Dic7—21no—
133C2×Dic7—21no—
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7#4112yes—
153Dic14—44no—
154C22×Q8—71no—
155C2×C4⋊C4—71no—
156C2×C4⋊C4—71no—
157C2×C4⋊C4—71no—
158C2×C4⋊C4—71no—
159C2×C4⋊C4—71no—
160C2×C4⋊C4—71no—
161C2×C4⋊C4—71no—
162C2×C4⋊C4—71no—
163C2×C4⋊C4—71no—
164C2×C4⋊C4C2×D711yescharacteristic
165C2×C22⋊C4—71no—
166C2×C22⋊C4—71no—
167C2×C22⋊C4—71no—
168C2×C22⋊C4—71no—
169C2×C22⋊C4C2×D711yescharacteristic
170C2×C22⋊C4C2×D711yescharacteristic
171C2×C22⋊C4C2×D711yescharacteristic
172C2×C42—71no—
173#95—71no—
174#95—71no—
175#95—71no—
176#95—71no—
177#95—71no—
178#95—71no—
179#95—71no—
180#95—71no—
181#95—71no—
182#95—71no—
183#95—71no—
184#95—71no—
185#95—71no—
186#95C2×D711yescharacteristic
187#95C2×D711yescharacteristic
188#95C2×D711yescharacteristic
189C22×C7—41no—
190C22×C7—41no—
191C22×C7—41no—
192C22×C7—41no—
193C22×C7—41no—
194C22×C7—41no—
195C22×C7—41no—
196C22×C7C2×#4111yescharacteristic
197C22×C7H(5,𝔽2)11yescharacteristic
198C22×C7H(5,𝔽2)11yescharacteristic
199C22×C7H(5,𝔽2)11yescharacteristic
200C22×C72−1+411yescharacteristic
201C22×C72−1+411yescharacteristic
202C22×C72−1+411yescharacteristic
203C4×C7—41no—
204C4×C7—41no—
205C4×C7—41no—
206C4×C7—41no—
207C4×C7—41no—
208C4×C7—41no—
209Dic7—22no—
210Dic7—41no—
211Dic7—41no—
212Dic7—41no—
213Dic7—41no—
214Dic7—41no—
215Dic7—41no—
216Dic7—41no—
217C24C22×D711yescharacteristic
218C2×Q8—281no—
219C2×Q8—281no—
220C2×Q8—281no—
221C22×C4—71no—
222C22×C4—71no—
223C22×C4—71no—
224C22×C4—71no—
225C22×C4—71no—
226C22×C4—71no—
227C22×C4—71no—
228C22×C4—71no—
229C22×C4C22×D711yescharacteristic
230C22×C4C22×D711yescharacteristic
231C22×C4C22×D711yescharacteristic
232C22×C4C22×D711yescharacteristic
233C22×C4C22×D711yescharacteristic
234C22×C4C22×D711yescharacteristic
235C4⋊C4—41no—
236C4⋊C4—142no—
237C4⋊C4—142no—
238C4⋊C4—281no—
239C4⋊C4—281no—
240C4⋊C4—281no—
241C4⋊C4—281no—
242C4⋊C4—281no—
243C4⋊C4—281no—
244C4⋊C4—281no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—142no—
249C22⋊C4—281no—
250C22⋊C4—281no—
251C22⋊C4—281no—
252C42—281no—
253C2×C7—81no—
254C2×C7#53911yescharacteristic
255C2×C7#54211yescharacteristic
256C2×C7#54211yescharacteristic
257C2×C7#56011yescharacteristic
258C2×C7#56311yescharacteristic
259C2×C7#56311yescharacteristic
260C2×C7#56311yescharacteristic
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23C23×D711yescharacteristic
269Q8—284no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—72no—
289C2×C4—141no—
290C2×C4—141no—
291C2×C4—141no—
292C2×C4—141no—
293C2×C4—141no—
294C2×C4—141no—
295C2×C4—141no—
296C2×C4—141no—
297C2×C4—141no—
298C2×C4—141no—
299C2×C4—141no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—141no—
305C2×C4—141no—
306C2×C4—141no—
307C2×C4—141no—
308C2×C4—141no—
309C2×C4—141no—
310C2×C4—141no—
311C2×C4—141no—
312C7#283811yescharacteristic, complemented, Sylow
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22#654811yescharacteristic
321C22#654811yescharacteristic
322C22D7×#4111yescharacteristic
323C22#655311yescharacteristic
324C22#655411yescharacteristic
325C22#655411yescharacteristic
326C22#655411yescharacteristic
327C4—41no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—142no—
334C4—281no—
335C4—281no—
336C4—281no—
337C4—281no—
338C4—281no—
339C4—281no—
340C4—281no—
341C2—81no—
342C2#9031111yescharacteristic
343C2#9034211yescharacteristic
344C2#9035311yescharacteristic
345C2#9035311yescharacteristic
346C2#9036311yescharacteristic
347C2#9039811yescharacteristic
348C2#9040311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#397C2
prime radicalC24×C7C23
hypercenter#397D7
hypocenterC7#2838
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #89751C2semidirect
3. #89744C2semidirect
10. C7⋊#401C2semidirect
11. #89432C2semidirect
12. #89432C2semidirect
13. #89429C2semidirect
14. #89428C2semidirect
15. #89425C2semidirect
312. C7#2838semidirect

Maximal subgroups

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

subgroupindex
C7×#3972
#897512
#897442
#897252
#897242
#897212
#897202
#897192
#897172
C7⋊#4012
#894322
#894292
#894282
#894252
#28387