Finite Group Databasebeta

← #11735487

Subgroups — #11735487

Order 896.

436 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#399C211yesmaximal, characteristic
2#89755C211yesmaximal, characteristic, complemented
3#89754C211yesmaximal, characteristic, complemented
4#89753C211yesmaximal, characteristic, complemented
5#89750C211yesmaximal, characteristic, complemented
6C4⋊C4×Dic7C211yesmaximal, characteristic, complemented
7#89746C211yesmaximal, characteristic, complemented
8#89722C211yesmaximal, characteristic
9#89719C211yesmaximal, characteristic
10#89718C211yesmaximal, characteristic
11#89717C211yesmaximal, characteristic
12#89716C211yesmaximal, characteristic
13C22⋊C4×Dic7C211yesmaximal, characteristic
14#89419C211yesmaximal, characteristic, complemented
15#89418C211yesmaximal, characteristic, complemented
16C23×Dic7C2211yescharacteristic
17C22×Dic14C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C4⋊Dic7C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic
29C2×Dic7⋊C4C2211yescharacteristic
30C2×Dic7⋊C4C2211yescharacteristic
31C2×Dic7⋊C4C2211yescharacteristic
32C2×Dic7⋊C4C2211yescharacteristic
33C2×C4×Dic7C2211yescharacteristic
34C2×C4×Dic7C2211yescharacteristic
35C2×C4×Dic7C2211yescharacteristic
36C2×C4×Dic7C2211yescharacteristic
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#2589—71nomaximal, Sylow
52C24×C7C2311yescharacteristic
53C22×C4×C7C2311yescharacteristic
54C22×C4×C7C2311yescharacteristic
55C22×C4×C7C2311yescharacteristic
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×Dic7—22no—
60C22×Dic7—41no—
61C22×Dic7—41no—
62C22×Dic7C2311yescharacteristic
63C22×Dic7C2311yescharacteristic
64C22×Dic7C2311yescharacteristic
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C22×Dic7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C2×Dic14—22no—
71C2×Dic14—22no—
72C2×Dic14—41no—
73C7×C4⋊C4—22no—
74C7×C4⋊C4—22no—
75C7×C4⋊C4—41no—
76C7×C22⋊C4—22no—
77C7×C22⋊C4—22no—
78C7×C22⋊C4—41no—
79C22⋊Dic7—22no—
80C22⋊Dic7—41no—
81C22⋊Dic7Q814yescomplemented
82C4⋊Dic7—22no—
83C4⋊Dic7—41no—
84Dic7⋊C4—22no—
85Dic7⋊C4—22no—
86Dic7⋊C4—41no—
87Dic7⋊C4—41no—
88C4×Dic7—22no—
89C4×Dic7—41no—
90C4×Dic7—41no—
91C4×Dic7—41no—
92#400—71no—
93#399D711yescharacteristic
94#398—71no—
95#393—71no—
96#391—71no—
97#391—71no—
98#389—71no—
99#388—71no—
100#388—71no—
101#387—71no—
102#387—71no—
103#385—71no—
104#385—71no—
105C4×C4⋊C4—71no—
106C4×C22⋊C4—71no—
107C23×C7—21no—
108C23×C7—21no—
109C23×C7—21no—
110C23×C7—21no—
111C23×C7—21no—
112C23×C7—21no—
113C23×C7C2×Q812yes—
114C23×C7C2411yescharacteristic
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×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7#4112yes—
132C2×C4×C7#4112yes—
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—21no—
153C2×Dic7—44no—
154C2×Dic7C2×Q812yes—
155C2×Dic7C2×Q812yes—
156C2×Dic7#4112yes—
157C2×Dic7#4112yes—
158Dic14—44no—
159C22×Q8—71no—
160C23×C4—71no—
161C2×C4⋊C4—71no—
162C2×C4⋊C4—71no—
163C2×C4⋊C4—71no—
164C2×C4⋊C4—71no—
165C2×C4⋊C4—71no—
166C2×C4⋊C4—71no—
167C2×C4⋊C4C2×D711yescharacteristic
168C2×C4⋊C4C2×D711yescharacteristic
169C2×C4⋊C4C2×D711yescharacteristic
170C2×C22⋊C4—71no—
171C2×C22⋊C4—71no—
172C2×C22⋊C4—71no—
173C2×C22⋊C4C2×D711yescharacteristic
174C2×C22⋊C4C2×D711yescharacteristic
175C2×C22⋊C4C2×D711yescharacteristic
176C2×C42—71no—
177C2×C42—71no—
178C2×C42—71no—
179C2×C42—71no—
180#95—71no—
181#95—71no—
182#95—71no—
183#95—71no—
184#95—71no—
185#95—71no—
186#95—71no—
187#95—71no—
188#95—71no—
189#95—71no—
190#95—71no—
191#95—71no—
192#95—71no—
193#95C2×D711yescharacteristic
194C22×C7—22no—
195C22×C7—22no—
196C22×C7—22no—
197C22×C7—41no—
198C22×C7—41no—
199C22×C7—41no—
200C22×C7—41no—
201C22×C7C22×Q811yescharacteristic
202C22×C7C2×#4111yescharacteristic
203C22×C7C2×#4111yescharacteristic
204C22×C7C2×#4111yescharacteristic
205C22×C7C2×#4111yescharacteristic
206C22×C7H(5,𝔽2)11yescharacteristic
207C22×C7H(5,𝔽2)11yescharacteristic
208C4×C7—22no—
209C4×C7—22no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C7—41no—
214Dic7—22no—
215Dic7—22no—
216Dic7—22no—
217Dic7—22no—
218Dic7—41no—
219Dic7—41no—
220Dic7—41no—
221Dic7—41no—
222C24C22×D711yescharacteristic
223C2×Q8—142no—
224C2×Q8—142no—
225C2×Q8—281no—
226C22×C4—71no—
227C22×C4—71no—
228C22×C4—71no—
229C22×C4—71no—
230C22×C4—71no—
231C22×C4—71no—
232C22×C4—71no—
233C22×C4—71no—
234C22×C4—142no—
235C22×C4—281no—
236C22×C4—281no—
237C22×C4C22×D711yescharacteristic
238C22×C4C22×D711yescharacteristic
239C22×C4C22×D711yescharacteristic
240C22×C4C22×D711yescharacteristic
241C22×C4C22×D711yescharacteristic
242C22×C4C22×D711yescharacteristic
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—41no—
246C4⋊C4—142no—
247C4⋊C4—142no—
248C4⋊C4—142no—
249C4⋊C4—281no—
250C4⋊C4—281no—
251C4⋊C4—281no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—41no—
255C22⋊C4—74no—
256C22⋊C4—142no—
257C22⋊C4—281no—
258C42—142no—
259C42—281no—
260C42—281no—
261C42—281no—
262C2×C7—42no—
263C2×C7#52911yescharacteristic
264C2×C7#53311yescharacteristic
265C2×C7#53811yescharacteristic
266C2×C7#55111yescharacteristic
267C2×C7#55411yescharacteristic
268C2×C7#55411yescharacteristic
269C2×C7#55911yescharacteristic
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23Q8×D712yes—
277C23C23×D711yescharacteristic
278Q8—284no—
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—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—72no—
296C2×C4—72no—
297C2×C4—72no—
298C2×C4—72no—
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—
312C2×C4—141no—
313C2×C4—141no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—141no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—284no—
320C2×C4#115912yes—
321C2×C4#115912yes—
322C7#258911yescharacteristic, complemented, Sylow
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22C2×#115911yescharacteristic
331C22C2×#115911yescharacteristic
332C22#654811yescharacteristic
333C22C2×Q8×D711yescharacteristic
334C22D7×#4111yescharacteristic
335C22D7×#4111yescharacteristic
336C22#655311yescharacteristic
337C4—22no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—142no—
344C4—142no—
345C4—142no—
346C4—142no—
347C4—281no—
348C4—281no—
349C4—281no—
350C4—281no—
351C2—42no—
352C2#9031311yescharacteristic
353C2#9032611yescharacteristic
354C2#9032711yescharacteristic
355C2#9034011yescharacteristic
356C2#9034811yescharacteristic
357C2#9034911yescharacteristic
358C2#9040411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#399C2
prime radicalC24×C7C23
hypercenter#399D7
hypocenterC7#2589
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
2. #89755C2semidirect
3. #89754C2semidirect
4. #89753C2semidirect
5. #89750C2semidirect
6. C4⋊C4×Dic7C2semidirect
7. #89746C2semidirect
14. #89419C2semidirect
15. #89418C2semidirect
81. C22⋊Dic7Q8semidirect
322. C7#2589semidirect

Maximal subgroups

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

subgroupindex
C7×#3992
#897552
#897542
#897532
#897502
C4⋊C4×Dic72
#897462
#897222
#897192
#897182
#897172
#897162
C22⋊C4×Dic72
#894192
#894182
#25897