Finite Group Databasebeta

← #11733263

Subgroups — #11733263

Order 896.

400 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#388C211yesmaximal, characteristic, complemented
2#89764C211yesmaximal, characteristic, complemented
3#89756C211yesmaximal, characteristic, complemented
4#89733C211yesmaximal, characteristic
5#89731C211yesmaximal, characteristic
6#89731C211yesmaximal, characteristic
7#89725C211yesmaximal, characteristic, complemented
8#89720C211yesmaximal, characteristic, complemented
9#89720C211yesmaximal, characteristic, complemented
10#89712C211yesmaximal, characteristic, complemented
11#89700C211yesmaximal, characteristic, complemented
12#89449C211yesmaximal, characteristic, complemented
13#89447C211yesmaximal, characteristic, complemented
14#89432C211yesmaximal, characteristic, complemented
15#89429C211yesmaximal, characteristic, complemented
16C22×C22⋊D7C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×C7×C22⋊C4C2211yescharacteristic
19C2×C7×C22⋊C4C2211yescharacteristic
20C2×C7×C22⋊C4C2211yescharacteristic
21C2×C42×C7C2211yescharacteristic
22C2×C22⋊Dic7C2211yescharacteristic
23C2×C22⋊Dic7C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
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×C4⋊Dic7C2211yescharacteristic
32C2×C4⋊Dic7C2211yescharacteristic
33C2×C4⋊Dic7C2211yescharacteristic
34C2×C4⋊Dic7C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C2×Dic7⋊C4C2211yescharacteristic
40C2×Dic7⋊C4C2211yescharacteristic
41C7×#95C2211yescharacteristic
42C7×#95C2211yescharacteristic
43C7⋊#95C2211yescharacteristic
44C7⋊#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51#2837—71nomaximal, Sylow
52C24×C7C2311yescharacteristic
53C23×D7C2311yescharacteristic
54C22×C4×C7C2311yescharacteristic
55C22×C4×C7C2311yescharacteristic
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C2×C22⋊D7—41no—
61C2×C22⋊D7—41no—
62C2×C22⋊D7—41no—
63C22×Dic7C2311yescharacteristic
64C22×Dic7C2311yescharacteristic
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C22×Dic7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C7×C4⋊C4—41no—
71C7×C22⋊C4—22no—
72C7×C22⋊C4—41no—
73C7×C22⋊C4—41no—
74C42×C7—41no—
75C22⋊Dic7—41no—
76C22⋊Dic7—41no—
77C22⋊Dic7—41no—
78(C2×D7)⋊C4—22no—
79(C2×D7)⋊C4—41no—
80(C2×D7)⋊C4—41no—
81(C2×D7)⋊C4—41no—
82(C2×D7)⋊C4—41no—
83(C2×D7)⋊C4—41no—
84C4⋊Dic7—22no—
85C4⋊Dic7—41no—
86C4⋊Dic7—41no—
87C4⋊Dic7—41no—
88Dic7⋊C4—41no—
89Dic7⋊C4—41no—
90Dic7⋊C4—41no—
91Dic7⋊C4—41no—
92Dic7⋊C4—41no—
93Dic7⋊C4—41no—
94#400—71no—
95#399—71no—
96#399—71no—
97#398—71no—
98#398—71no—
99#397—71no—
100#397—71no—
101#396—71no—
102#396—71no—
103#394—71no—
104#394—71no—
105#394—71no—
106#389—71no—
107#388—71no—
108#388D711yescharacteristic, complemented
109C23×C7—21no—
110C23×C7—21no—
111C23×C7—21no—
112C23×C7—21no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7—21no—
116C23×C7C2411yescharacteristic
117C22×D7—21no—
118C22×D7—21no—
119C22×D7—21no—
120C22×D7—21no—
121C22×D7—21no—
122C22×D7—21no—
123C22×D7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
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#4112yes—
142C22⋊D7—44no—
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—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—
164C22×D4—71no—
165C2×C4⋊C4—71no—
166C2×C4⋊C4—71no—
167C2×C4⋊C4—71no—
168C2×C4⋊C4—71no—
169C2×C4⋊C4—71no—
170C2×C4⋊C4—71no—
171C2×C4⋊C4—71no—
172C2×C4⋊C4—71no—
173C2×C4⋊C4—71no—
174C2×C4⋊C4—71no—
175C2×C4⋊C4C2×D711yescharacteristic
176C2×C22⋊C4—71no—
177C2×C22⋊C4—71no—
178C2×C22⋊C4—71no—
179C2×C22⋊C4—71no—
180C2×C22⋊C4—71no—
181C2×C22⋊C4—71no—
182C2×C22⋊C4—71no—
183C2×C22⋊C4—71no—
184C2×C22⋊C4—71no—
185C2×C22⋊C4C2×D711yescharacteristic
186C2×C22⋊C4C2×D711yescharacteristic
187C2×C22⋊C4C2×D711yescharacteristic
188C2×C42C2×D711yescharacteristic
189#95—71no—
190#95—71no—
191#95—71no—
192#95—71no—
193#95—71no—
194#95—71no—
195#95—71no—
196#95—71no—
197#95C2×D711yescharacteristic
198#95C2×D711yescharacteristic
199C22×C7—41no—
200C22×C7—41no—
201C22×C7—41no—
202C22×C7—41no—
203C22×C7—41no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7C2×#4111yescharacteristic
207C22×C7H(5,𝔽2)11yescharacteristic
208C22×C7H(5,𝔽2)11yescharacteristic
209C22×C7H(5,𝔽2)11yescharacteristic
210C22×C7H(5,𝔽2)11yescharacteristic
211C22×C72−1+411yescharacteristic
212C22×C72−1+411yescharacteristic
213C2×D7—41no—
214C2×D7—41no—
215C2×D7—41no—
216C2×D7—41no—
217C2×D7—41no—
218C2×D7—41no—
219C2×D7—41no—
220C4×C7—22no—
221C4×C7—41no—
222C4×C7—41no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7—41no—
226Dic7—41no—
227Dic7—41no—
228Dic7—41no—
229Dic7—41no—
230Dic7—41no—
231Dic7—41no—
232Dic7—41no—
233C24—71no—
234C24C22×D711yescharacteristic
235C2×D4—281no—
236C2×D4—281no—
237C2×D4—281no—
238C22×C4—71no—
239C22×C4—71no—
240C22×C4—71no—
241C22×C4—71no—
242C22×C4—71no—
243C22×C4—71no—
244C22×C4—71no—
245C22×C4C22×D711yescharacteristic
246C22×C4C22×D711yescharacteristic
247C22×C4C22×D711yescharacteristic
248C22×C4C22×D711yescharacteristic
249C22×C4C22×D711yescharacteristic
250C22×C4C22×D711yescharacteristic
251C4⋊C4—41no—
252C4⋊C4—142no—
253C4⋊C4—281no—
254C4⋊C4—281no—
255C4⋊C4—281no—
256C4⋊C4—281no—
257C4⋊C4—281no—
258C4⋊C4—281no—
259C4⋊C4—281no—
260C4⋊C4—281no—
261C4⋊C4—281no—
262C22⋊C4—22no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—142no—
266C22⋊C4—281no—
267C22⋊C4—281no—
268C22⋊C4—281no—
269C22⋊C4—281no—
270C22⋊C4—281no—
271C22⋊C4—281no—
272C22⋊C4—281no—
273C22⋊C4—281no—
274C42—41no—
275C2×C7—81no—
276C2×C7#53911yescharacteristic
277C2×C7#54011yescharacteristic
278C2×C7#54211yescharacteristic
279C2×C7#56011yescharacteristic
280C2×C7#56211yescharacteristic
281C2×C7#56211yescharacteristic
282C2×C7#56311yescharacteristic
283D7—81no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—141no—
292C23—141no—
293C23—141no—
294C23—141no—
295C23—141no—
296C23—141no—
297C23—141no—
298C23C23×D711yescharacteristic
299D4—284no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—141no—
320C2×C4—141no—
321C2×C4—141no—
322C2×C4—141no—
323C2×C4—141no—
324C2×C4—141no—
325C2×C4—141no—
326C2×C4—141no—
327C2×C4—141no—
328C2×C4—141no—
329C2×C4—141no—
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#115712yes—
339C7#283711yescharacteristic, complemented, Sylow
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—281no—
348C22—281no—
349C22—281no—
350C22—281no—
351C22—281no—
352C22—281no—
353C22—281no—
354C22C2×#115711yescharacteristic
355C22#654811yescharacteristic
356C22#654811yescharacteristic
357C22#655311yescharacteristic
358C22#655311yescharacteristic
359C22#655411yescharacteristic
360C22#655411yescharacteristic
361C4—22no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—281no—
368C4—281no—
369C4—281no—
370C4—281no—
371C4—281no—
372C4—281no—
373C4—281no—
374C2—81no—
375C2—561no—
376C2#9022611yescharacteristic
377C2#9025211yescharacteristic
378C2#9025511yescharacteristic
379C2#9031111yescharacteristic
380C2#9035811yescharacteristic
381C2#9037511yescharacteristic
382C2#9040311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#388C2
prime radicalC22×C22⋊D7C22
hypercenter#388D7
hypocenterC7#2837
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. #89764C2semidirect
3. #89756C2semidirect
7. #89725C2semidirect
8. #89720C2semidirect
9. #89720C2semidirect
10. #89712C2semidirect
11. #89700C2semidirect
12. #89449C2semidirect
13. #89447C2semidirect
14. #89432C2semidirect
15. #89429C2semidirect
108. #388D7semidirect
339. C7#2837semidirect

Maximal subgroups

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

subgroupindex
C7×#3882
#897642
#897562
#897332
#897312
#897252
#897202
#897122
#897002
#894492
#894472
#894322
#894292
#28377