Finite Group Databasebeta

← #11735306

Subgroups — #11735306

Order 896.

472 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6459C211yesmaximal, characteristic
2C2×#6452C211yesmaximal, characteristic, complemented
3C7×#398C211yesmaximal, characteristic, complemented
4#89766C211yesmaximal, characteristic
5#89764C211yesmaximal, characteristic
6#89761C211yesmaximal, characteristic
7#89751C211yesmaximal, characteristic, complemented
8#89748C211yesmaximal, characteristic, complemented
9#89745C211yesmaximal, characteristic, complemented
10#89442C211yesmaximal, characteristic
11#89441C211yesmaximal, characteristic
12#89440C211yesmaximal, characteristic
13#89430C211yesmaximal, characteristic, complemented
14#89428C211yesmaximal, characteristic, complemented
15#89427C211yesmaximal, characteristic, complemented
16C22×C4×D7C2211yescharacteristic
17C22×Dic14C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×C7×C4⋊C4C2211yescharacteristic
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×C4⋊Dic7C2211yescharacteristic
29C2×C4⋊Dic7C2211yescharacteristic
30C2×C4⋊Dic7C2211yescharacteristic
31C2×Dic7⋊C4C2211yescharacteristic
32C2×Dic7⋊C4C2211yescharacteristic
33C2×Dic7⋊C4C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C2×C4×Dic7C2211yescharacteristic
40C2×C4×Dic7C2211yescharacteristic
41#6459—24no—
42#6452—24no—
43C7×#95C2211yescharacteristic
44C7×#95C2211yescharacteristic
45C7×#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53#2720—71nomaximal, Sylow
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
67C22×Dic7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C2×C4×D7—22no—
70C2×C4×D7—41no—
71C2×C4×D7—41no—
72C2×Dic14—22no—
73C2×Dic14—41no—
74C2×Dic14—41no—
75C7×C4⋊C4—22no—
76C7×C4⋊C4—22no—
77C7×C4⋊C4—41no—
78C7×C4⋊C4—41no—
79(C2×D7)⋊C4—22no—
80(C2×D7)⋊C4—22no—
81(C2×D7)⋊C4—22no—
82(C2×D7)⋊C4—41no—
83(C2×D7)⋊C4—41no—
84(C2×D7)⋊C4Q814yescomplemented
85C4⋊Dic7—22no—
86C4⋊Dic7—22no—
87C4⋊Dic7—41no—
88Dic7⋊C4—22no—
89Dic7⋊C4—22no—
90Dic7⋊C4—22no—
91Dic7⋊C4—22no—
92Dic7⋊C4—41no—
93Dic7⋊C4—41no—
94Dic7⋊C4—41no—
95Dic7⋊C4D414yes—
96C4×Dic7—22no—
97C4×Dic7—22no—
98C2×#125—71no—
99C2×C22⋊Q8—71no—
100#400—71no—
101#399—71no—
102#398D711yescharacteristic, complemented
103#396—71no—
104#395—71no—
105#395—71no—
106#389—71no—
107#388—71no—
108#388—71no—
109#387—71no—
110#385—71no—
111#385—71no—
112#384—71no—
113C23×C7C2411yescharacteristic
114C22×D7—21no—
115C22×D7—21no—
116C22×D7—21no—
117C22×D7—21no—
118C22×D7—21no—
119C22×D7—21no—
120C22×D7C2×Q812yes—
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—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×C7C2×D412yes—
141C2×C4×C7C2×Q812yes—
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—21no—
154C2×Dic7—21no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—21no—
158C2×Dic7—21no—
159C2×Dic7C2×D412yes—
160C2×Dic7C2×D412yes—
161C2×Dic7C2×Q812yes—
162C2×Dic7#4112yes—
163C4×D7—44no—
164Dic14—44no—
165C22×Q8—71no—
166C23×C4—71no—
167#125—144no—
168C22⋊Q8—144no—
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⋊C4—71no—
176C2×C4⋊C4—71no—
177C2×C4⋊C4—71no—
178C2×C4⋊C4—71no—
179C2×C4⋊C4—71no—
180C2×C4⋊C4C2×D711yescharacteristic
181C2×C4⋊C4C2×D711yescharacteristic
182C2×C4⋊C4C2×D711yescharacteristic
183C2×C4⋊C4C2×D711yescharacteristic
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×C42—71no—
191C2×C42—71no—
192#95—71no—
193#95—71no—
194#95—71no—
195#95—71no—
196#95—71no—
197#95—71no—
198#95—71no—
199#95C2×D711yescharacteristic
200#95C2×D711yescharacteristic
201#95C2×D711yescharacteristic
202C22×C7C22×D411yescharacteristic
203C22×C7C22×Q811yescharacteristic
204C22×C7C2×#4111yescharacteristic
205C22×C7H(5,𝔽2)11yescharacteristic
206C22×C7H(5,𝔽2)11yescharacteristic
207C22×C72−1+411yescharacteristic
208C22×C72−1+411yescharacteristic
209C2×D7—22no—
210C2×D7—22no—
211C2×D7—22no—
212C2×D7—41no—
213C2×D7—41no—
214C2×D7—41no—
215C2×D7—41no—
216C4×C7—22no—
217C4×C7—22no—
218C4×C7—41no—
219C4×C7—41no—
220C4×C7—41no—
221C4×C7—41no—
222C4×C7—41no—
223Dic7—22no—
224Dic7—22no—
225Dic7—22no—
226Dic7—22no—
227Dic7—41no—
228Dic7—41no—
229Dic7—41no—
230C24—71no—
231C2×Q8—142no—
232C2×Q8—281no—
233C2×Q8—281no—
234C22×C4—71no—
235C22×C4—71no—
236C22×C4—71no—
237C22×C4—71no—
238C22×C4—71no—
239C22×C4—71no—
240C22×C4—71no—
241C22×C4—142no—
242C22×C4—281no—
243C22×C4—281no—
244C22×C4C22×D711yescharacteristic
245C22×C4C22×D711yescharacteristic
246C22×C4C22×D711yescharacteristic
247C22×C4C22×D711yescharacteristic
248C22×C4C22×D711yescharacteristic
249C22×C4C22×D711yescharacteristic
250C22×C4C22×D711yescharacteristic
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—74no—
256C4⋊C4—142no—
257C4⋊C4—142no—
258C4⋊C4—142no—
259C4⋊C4—142no—
260C4⋊C4—142no—
261C4⋊C4—142no—
262C4⋊C4—281no—
263C4⋊C4—281no—
264C4⋊C4—281no—
265C4⋊C4—281no—
266C22⋊C4—74no—
267C22⋊C4—142no—
268C22⋊C4—142no—
269C22⋊C4—142no—
270C22⋊C4—281no—
271C22⋊C4—281no—
272C42—142no—
273C42—142no—
274C2×C7#53511yescharacteristic
275C2×C7#53911yescharacteristic
276C2×C7#54411yescharacteristic
277C2×C7#54711yescharacteristic
278C2×C7D4×Q811yescharacteristic
279C2×C7#55411yescharacteristic
280C2×C7#56311yescharacteristic
281D7—42no—
282C23—72no—
283C23—141no—
284C23—141no—
285C23—141no—
286C23—141no—
287C23—141no—
288C23—141no—
289C23C23×D711yescharacteristic
290Q8—284no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
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—72no—
311C2×C4—72no—
312C2×C4—72no—
313C2×C4—72no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—141no—
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—284no—
332C2×C4D4×D712yes—
333C2×C4Q8×D712yes—
334C7#272011yescharacteristic, complemented, Sylow
335C22—142no—
336C22—142no—
337C22—142no—
338C22—281no—
339C22—281no—
340C22—281no—
341C22—281no—
342C22C2×D4×D711yescharacteristic
343C22#654811yescharacteristic
344C22C2×Q8×D711yescharacteristic
345C22#655111yescharacteristic
346C22D7×#4111yescharacteristic
347C22#655311yescharacteristic
348C22#655411yescharacteristic
349C4—22no—
350C4—22no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—142no—
357C4—142no—
358C4—142no—
359C4—142no—
360C4—281no—
361C4—281no—
362C4—281no—
363C2—282no—
364C2#9029811yescharacteristic
365C2#9032711yescharacteristic
366C2#9033711yescharacteristic
367C2#9034611yescharacteristic
368C2#9038011yescharacteristic
369C2#9038311yescharacteristic
370C2#9041811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#398C2
prime radicalC23×D7C23
hypercenter#398D7
hypocenterC7#2720
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#6452C2semidirect
3. C7×#398C2semidirect
7. #89751C2semidirect
8. #89748C2semidirect
9. #89745C2semidirect
13. #89430C2semidirect
14. #89428C2semidirect
15. #89427C2semidirect
84. (C2×D7)⋊C4Q8semidirect
102. #398D7semidirect
334. C7#2720semidirect

Maximal subgroups

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

subgroupindex
C2×#64592
C2×#64522
C7×#3982
#897662
#897642
#897612
#897512
#897482
#897452
#894422
#894412
#894402
#894302
#894282
#894272
#27207