Finite Group Databasebeta

← #11735466

Subgroups — #11735466

Order 896.

492 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6452C211yesmaximal, characteristic, complemented
2C2×#6440C211yesmaximal, characteristic
3C7×#399C211yesmaximal, characteristic
4#89753C211yesmaximal, characteristic, complemented
5#89748C211yesmaximal, characteristic, complemented
6#89745C211yesmaximal, characteristic, complemented
7#89744C211yesmaximal, characteristic, complemented
8#89744C211yesmaximal, characteristic, complemented
9#89721C211yesmaximal, characteristic
10#89720C211yesmaximal, characteristic
11#89720C211yesmaximal, characteristic
12#89714C211yesmaximal, characteristic
13C22⋊C4×Dic7C211yesmaximal, characteristic
14#89425C211yesmaximal, characteristic, complemented
15#89419C211yesmaximal, characteristic, complemented
16C23×Dic7C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20C2×C7×C22⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C22⋊Dic7C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C4⋊Dic7C2211yescharacteristic
27C2×C4⋊Dic7C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic
29C2×Dic7⋊C4C2211yescharacteristic
30C2×Dic7⋊C4C2211yescharacteristic
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
41C2×C4×Dic7C2211yescharacteristic
42C2×C4×Dic7C2211yescharacteristic
43#6452—24no—
44#6440—24no—
45C7×#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53#2441—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C22×C4×C7C2311yescharacteristic
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×Dic7—22no—
62C22×Dic7—22no—
63C22×Dic7Q814yes—
64C22×Dic7C2311yescharacteristic
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C22×Dic7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C7×C4⋊C4—22no—
73C7×C4⋊C4—22no—
74C7×C4⋊C4—41no—
75C7×C22⋊C4—22no—
76C7×C22⋊C4—22no—
77C7×C22⋊C4—41no—
78C22⋊Dic7—22no—
79C22⋊Dic7—22no—
80C22⋊Dic7—41no—
81C4⋊Dic7—22no—
82C4⋊Dic7—22no—
83C4⋊Dic7—22no—
84Dic7⋊C4—22no—
85Dic7⋊C4—22no—
86Dic7⋊C4—22no—
87Dic7⋊C4—22no—
88Dic7⋊C4—22no—
89Dic7⋊C4—22no—
90Dic7⋊C4—41no—
91Dic7⋊C4—41no—
92Dic7⋊C4—41no—
93Dic7⋊C4D414yes—
94C4×Dic7—22no—
95C4×Dic7—22no—
96C4×Dic7—22no—
97C4×Dic7—22no—
98C2×#125—71no—
99C2×#117—71no—
100#399—71no—
101#399D711yescharacteristic
102#396—71no—
103#396—71no—
104#389—71no—
105#389—71no—
106#389—71no—
107#389—71no—
108#387—71no—
109#387—71no—
110#384—71no—
111#380—71no—
112C4×C22⋊C4—71no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7C2×Q812yes—
120C23×C7C2411yescharacteristic
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×C7C2×D412yes—
138C2×C4×C7#4112yes—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7—28no—
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×Dic7C2×D412yes—
159C2×Dic7C2×D412yes—
160C2×Dic7C2×Q812yes—
161C2×Dic7C2×Q812yes—
162C2×Dic7#4112yes—
163C2×Dic7#4112yes—
164C23×C4—71no—
165#125—144no—
166#117—144no—
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⋊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×C22⋊C4—71no—
184C2×C22⋊C4—71no—
185C2×C22⋊C4—71no—
186C2×C22⋊C4C2×D711yescharacteristic
187C2×C22⋊C4C2×D711yescharacteristic
188C2×C22⋊C4C2×D711yescharacteristic
189C2×C42—71no—
190C2×C42—71no—
191C2×C42—71no—
192C2×C42—71no—
193#95—71no—
194#95—71no—
195#95—71no—
196#95—71no—
197#95—71no—
198#95—71no—
199#95—71no—
200#95C2×D711yescharacteristic
201C22×C7—22no—
202C22×C7—22no—
203C22×C7—22no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7—41no—
208C22×C7C22×D411yescharacteristic
209C22×C7C22×Q811yescharacteristic
210C22×C7C2×#4111yescharacteristic
211C22×C7C2×#4111yescharacteristic
212C22×C7C2×#4111yescharacteristic
213C22×C7H(5,𝔽2)11yescharacteristic
214C22×C72−1+411yescharacteristic
215C4×C7—22no—
216C4×C7—22no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7—41no—
220C4×C7—41no—
221Dic7—22no—
222Dic7—22no—
223Dic7—22no—
224Dic7—41no—
225Dic7—41no—
226Dic7—41no—
227Dic7—41no—
228Dic7C22⋊Q814yes—
229C24C22×D711yescharacteristic
230C22×C4—71no—
231C22×C4—74no—
232C22×C4—71no—
233C22×C4—71no—
234C22×C4—71no—
235C22×C4—71no—
236C22×C4—71no—
237C22×C4—71no—
238C22×C4—71no—
239C22×C4—142no—
240C22×C4—142no—
241C22×C4C22×D711yescharacteristic
242C22×C4C22×D711yescharacteristic
243C22×C4C22×D711yescharacteristic
244C22×C4C22×D711yescharacteristic
245C22×C4C22×D711yescharacteristic
246C22×C4C22×D711yescharacteristic
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—41no—
250C4⋊C4—74no—
251C4⋊C4—142no—
252C4⋊C4—142no—
253C4⋊C4—142no—
254C4⋊C4—142no—
255C4⋊C4—142no—
256C4⋊C4—142no—
257C4⋊C4—142no—
258C4⋊C4—142no—
259C4⋊C4—142no—
260C4⋊C4—281no—
261C4⋊C4—281no—
262C4⋊C4—281no—
263C22⋊C4—22no—
264C22⋊C4—22no—
265C22⋊C4—41no—
266C22⋊C4—142no—
267C22⋊C4—142no—
268C22⋊C4—281no—
269C42—142no—
270C42—142no—
271C42—142no—
272C42—142no—
273C2×C7—42no—
274C2×C7C2×C22⋊Q811yescharacteristic
275C2×C7#53311yescharacteristic
276C2×C7#54411yescharacteristic
277C2×C7#54711yescharacteristic
278C2×C7#55011yescharacteristic
279C2×C7#55211yescharacteristic
280C2×C7#55311yescharacteristic
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23Q8×D712yes—
288C23C23×D711yescharacteristic
289C2×C4—21no—
290C2×C4—21no—
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—72no—
306C2×C4—72no—
307C2×C4—72no—
308C2×C4—72no—
309C2×C4—72no—
310C2×C4—72no—
311C2×C4—148no—
312C2×C4—141no—
313C2×C4—141no—
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×C4D4×D712yes—
331C2×C4#115912yes—
332C7#244111yescharacteristic, complemented, Sylow
333C22—22no—
334C22—22no—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22C2×D4×D711yescharacteristic
341C22C2×#115911yescharacteristic
342C22#654811yescharacteristic
343C22C2×Q8×D711yescharacteristic
344C22D7×#4111yescharacteristic
345C22D7×#4111yescharacteristic
346C22#655411yescharacteristic
347C4—22no—
348C4—22no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—74no—
354C4—142no—
355C4—142no—
356C4—142no—
357C4—281no—
358C4—281no—
359C4—281no—
360C4—281no—
361C2—42no—
362C2#9031311yescharacteristic
363C2#9031411yescharacteristic
364C2D7×C22⋊Q811yescharacteristic
365C2#9034111yescharacteristic
366C2#9034611yescharacteristic
367C2#9035211yescharacteristic
368C2#9040911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#399C2
prime radicalC24×C7C23
hypercenter#399D7
hypocenterC7#2441
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#6452C2semidirect
4. #89753C2semidirect
5. #89748C2semidirect
6. #89745C2semidirect
7. #89744C2semidirect
8. #89744C2semidirect
14. #89425C2semidirect
15. #89419C2semidirect
332. C7#2441semidirect

Maximal subgroups

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

subgroupindex
C2×#64522
C2×#64402
C7×#3992
#897532
#897482
#897452
#897442
#897212
#897202
#897142
C22⋊C4×Dic72
#894252
#894192
#24417