Finite Group Databasebeta

← #11732902

Subgroups — #11732902

Order 896.

568 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×(C22⋊D7)⋊C4C211yesmaximal, characteristic
2C7×#388C211yesmaximal, characteristic, complemented
3#89761C211yesmaximal, characteristic, complemented
4#89749C211yesmaximal, characteristic, complemented
5#89728C211yesmaximal, characteristic
6#89728C211yesmaximal, characteristic
7#89723C211yesmaximal, characteristic, complemented
8#89715C211yesmaximal, characteristic, complemented
9#89714C211yesmaximal, characteristic, complemented
10C4×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
11#89705C211yesmaximal, characteristic, complemented
12#89440C211yesmaximal, characteristic, complemented
13D7×#95C211yesmaximal, characteristic, complemented
14#89424C211yesmaximal, characteristic, complemented
15#89423C211yesmaximal, characteristic, complemented
16C22×C22⋊D7C2211yescharacteristic
17C23×Dic7C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
19C22×C4×D7C2211yescharacteristic
20C22×C4×D7C2211yescharacteristic
21C2×C7×C4⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C42×C7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic
28C2×(C2×D7)⋊C4C2211yescharacteristic
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×C4⋊Dic7C2211yescharacteristic
32C2×Dic7⋊C4C2211yescharacteristic
33C2×Dic7⋊C4C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×C4×Dic7C2211yescharacteristic
36C2×C4×Dic7C2211yescharacteristic
37(C22⋊D7)⋊C4C418yescomplemented
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
51C7⋊#95C2211yescharacteristic
52#2318—71nomaximal, Sylow
53C24×C7C2311yescharacteristic
54C23×D7C2311yescharacteristic
55C22×C4×C7C2311yescharacteristic
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C2×C22⋊D7—22no—
62C2×C22⋊D7—22no—
63C2×C22⋊D7C2×C414yes—
64C22×Dic7—41no—
65C22×Dic7—41no—
66C22×Dic7C2×C414yescomplemented
67C22×Dic7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C2×C4×D7—22no—
75C2×C4×D7—22no—
76C2×C4×D7—41no—
77C2×C4×D7—41no—
78C2×C4×D7—41no—
79C2×C4×D7—41no—
80C2×C4×D7—41no—
81C2×C4×D7—41no—
82C2×C4×D7C2×C414yes—
83C7×C4⋊C4—41no—
84C7×C22⋊C4—41no—
85C7×C22⋊C4—41no—
86C7×C22⋊C4C2×C414yescomplemented
87C42×C7—41no—
88C22⋊Dic7—41no—
89C22⋊Dic7—41no—
90(C2×D7)⋊C4—22no—
91(C2×D7)⋊C4—22no—
92(C2×D7)⋊C4C2×C414yes—
93C4⋊Dic7—41no—
94Dic7⋊C4—41no—
95Dic7⋊C4—41no—
96Dic7⋊C4C2×C414yescomplemented
97C4×Dic7—41no—
98C4×Dic7C2×C414yescomplemented
99C2×C4×D4—71no—
100#388—71no—
101#388D711yescharacteristic, complemented
102#387—71no—
103#387—71no—
104#386—71no—
105#386—71no—
106#385—71no—
107#385—71no—
108#383—71no—
109#382—71no—
110#381—71no—
111#380—71no—
112C4×C22⋊C4—71no—
113C2×#95—71no—
114C23×C7—21no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7C22×C412yes—
121C23×C7C2411yescharacteristic
122C22×D7—21no—
123C22×D7—21no—
124C22×D7—21no—
125C22×D7—21no—
126C22×D7C22×C412yes—
127C22×D7#4112yes—
128C22×D7#4112yes—
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—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7C22×C412yes—
146C2×C4×C7C22×C412yes—
147C22⋊D7—28no—
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—
164C2×Dic7—21no—
165C2×Dic7—21no—
166C2×Dic7—44no—
167C2×Dic7C22×C412yes—
168C2×Dic7C22×C412yes—
169C2×Dic7C22×C412yes—
170C4×D7—44no—
171C4×D7—44no—
172C4×D7—44no—
173C22×D4—71no—
174C23×C4—71no—
175C23×C4—71no—
176C23×C4—71no—
177C23×C4—71no—
178C4×D4—78no—
179C2×C4⋊C4—71no—
180C2×C4⋊C4—71no—
181C2×C4⋊C4—71no—
182C2×C4⋊C4—71no—
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⋊C4C2×D711yescharacteristic
190C2×C22⋊C4C2×D711yescharacteristic
191C2×C22⋊C4C2×D711yescharacteristic
192C2×C42—71no—
193C2×C42—71no—
194C2×C42C2×D711yescharacteristic
195#95—71no—
196#95—71no—
197#95—71no—
198#95—71no—
199#95—71no—
200#95—71no—
201#95—71no—
202#95—71no—
203#95—71no—
204#95—71no—
205#95—71no—
206#95—71no—
207#95C2×D711yescharacteristic
208#95C2×D711yescharacteristic
209C22×C7—22no—
210C22×C7—22no—
211C22×C7—22no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7C23×C411yescharacteristic
217C22×C7C2×#4111yescharacteristic
218C22×C7C2×#4111yescharacteristic
219C22×C7H(5,𝔽2)11yescharacteristic
220C22×C7H(5,𝔽2)11yescharacteristic
221C22×C7H(5,𝔽2)11yescharacteristic
222C22×C72−1+411yescharacteristic
223C2×D7—22no—
224C2×D7—22no—
225C2×D7—22no—
226C2×D7—22no—
227C2×D7—22no—
228C2×D7—22no—
229C2×D7#11714yes—
230C4×C7—22no—
231C4×C7—22no—
232C4×C7—41no—
233C4×C7—41no—
234C4×C7—41no—
235C4×C7—41no—
236Dic7—22no—
237Dic7—22no—
238Dic7—22no—
239Dic7—41no—
240Dic7—41no—
241Dic7—41no—
242Dic7—41no—
243C24—71no—
244C24C22×D711yescharacteristic
245C2×D4—74no—
246C2×D4—142no—
247C2×D4—142no—
248C22×C4—71no—
249C22×C4—71no—
250C22×C4—71no—
251C22×C4—74no—
252C22×C4—74no—
253C22×C4—71no—
254C22×C4—71no—
255C22×C4—71no—
256C22×C4—71no—
257C22×C4—142no—
258C22×C4—142no—
259C22×C4—281no—
260C22×C4—281no—
261C22×C4—281no—
262C22×C4—281no—
263C22×C4—281no—
264C22×C4—281no—
265C22×C4—281no—
266C22×C4—281no—
267C22×C4C22×D711yescharacteristic
268C22×C4C22×D711yescharacteristic
269C22×C4C22×D711yescharacteristic
270C22×C4C22×D711yescharacteristic
271C22×C4C22×D711yescharacteristic
272C22×C4C22×D711yescharacteristic
273C4⋊C4—41no—
274C4⋊C4—74no—
275C4⋊C4—281no—
276C4⋊C4—281no—
277C4⋊C4—281no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—74no—
281C22⋊C4—142no—
282C22⋊C4—142no—
283C22⋊C4—281no—
284C22⋊C4—281no—
285C22⋊C4C4×D714yescomplemented
286C42—41no—
287C42—74no—
288C42—281no—
289C2×C7—42no—
290C2×C7C2×#11711yescharacteristic
291C2×C7#51811yescharacteristic
292C2×C7#52011yescharacteristic
293C2×C7#53811yescharacteristic
294C2×C7#53811yescharacteristic
295C2×C7#53911yescharacteristic
296C2×C7#53911yescharacteristic
297D7—24no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—72no—
305C23—72no—
306C23—72no—
307C23—141no—
308C23—141no—
309C23—141no—
310C23—141no—
311C23C2×C4×D712yes—
312C23C23×D711yescharacteristic
313D4—148no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—72no—
331C2×C4—72no—
332C2×C4—72no—
333C2×C4—141no—
334C2×C4—141no—
335C2×C4—141no—
336C2×C4—141no—
337C2×C4—141no—
338C2×C4—141no—
339C2×C4—141no—
340C2×C4—141no—
341C2×C4—141no—
342C2×C4—141no—
343C2×C4—141no—
344C2×C4—141no—
345C2×C4—141no—
346C2×C4—141no—
347C2×C4—141no—
348C2×C4—141no—
349C2×C4—141no—
350C2×C4—141no—
351C2×C4—284no—
352C2×C4—284no—
353C2×C4—284no—
354C2×C4—284no—
355C2×C4C2×C4×D712yes—
356C2×C4C2×C4×D712yes—
357C7#231811yescharacteristic, complemented, Sylow
358C22—22no—
359C22—22no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—74no—
366C22—142no—
367C22—142no—
368C22—142no—
369C22—142no—
370C22—142no—
371C22—142no—
372C22C22×C4×D711yescharacteristic
373C22#654811yescharacteristic
374C22#654811yescharacteristic
375C22D7×#4111yescharacteristic
376C22D7×#4111yescharacteristic
377C22#655311yescharacteristic
378C22#655411yescharacteristic
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—142no—
386C4—142no—
387C4—142no—
388C4—281no—
389C4—281no—
390C4—281no—
391C4—281no—
392C2—42no—
393C2—144no—
394C2D7×#11711yescharacteristic
395C2#9023611yescharacteristic
396C2#9023711yescharacteristic
397C2#9030411yescharacteristic
398C2#9035111yescharacteristic
399C2#9036911yescharacteristic
400C2#9039811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutatorC22×C7C23×C4
FittingC7×#388C2
socle, cosocleC23×C7C24
prime radicalC22×C22⋊D7C22
hypercenter#388D7
hypocenterC7#2318
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C7×#388C2semidirect
3. #89761C2semidirect
4. #89749C2semidirect
7. #89723C2semidirect
8. #89715C2semidirect
9. #89714C2semidirect
10. C4×(C2×D7)⋊C4C2semidirect
11. #89705C2semidirect
12. #89440C2semidirect
13. D7×#95C2semidirect
14. #89424C2semidirect
15. #89423C2semidirect
37. (C22⋊D7)⋊C4C4semidirect
66. C22×Dic7C2×C4semidirect
86. C7×C22⋊C4C2×C4semidirect
96. Dic7⋊C4C2×C4semidirect
98. C4×Dic7C2×C4semidirect
101. #388D7semidirect
285. C22⋊C4C4×D7semidirect
357. C7#2318semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D7)⋊C42
C7×#3882
#897612
#897492
#897282
#897232
#897152
#897142
C4×(C2×D7)⋊C42
#897052
#894402
D7×#952
#894242
#894232
#23187