Finite Group Databasebeta

← #11731840

Subgroups — #11731840

Order 896.

652 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊D7C211yesmaximal, characteristic
2C2×#6449C211yesmaximal, characteristic
3C2×#6440C211yesmaximal, characteristic, complemented
4C7×#385C211yesmaximal, characteristic, complemented
5C2×C7⋊#95C211yesmaximal, characteristic, complemented
6#89762C211yesmaximal, characteristic, complemented
7#89761C211yesmaximal, characteristic, complemented
8C4⋊C4×Dic7C211yesmaximal, characteristic, complemented
9#89745C211yesmaximal, characteristic, complemented
10#89730C211yesmaximal, characteristic
11C22⋊C4×Dic7C211yesmaximal, characteristic, complemented
12#89441C211yesmaximal, characteristic, complemented
13#89438C211yesmaximal, characteristic, complemented
14#89426C211yesmaximal, characteristic, complemented
15#89426C211yesmaximal, characteristic, complemented
16C23×C4×C7C2211yescharacteristic
17C22×C22⋊D7C2211yescharacteristic
18C23×Dic7C2211yescharacteristic
19C23×Dic7C2211yescharacteristic
20C22×C4×D7C2211yescharacteristic
21C2×C7×C4⋊C4C2211yescharacteristic
22C2×C7×C4⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C4×C22⋊D7—24no—
27C2×(C2×D7)⋊C4C2211yescharacteristic
28C2×(C2×D7)⋊C4C2211yescharacteristic
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×C4⋊Dic7C2211yescharacteristic
33C2×C4⋊Dic7C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×C4×Dic7C2211yescharacteristic
38C2×C4×Dic7C2211yescharacteristic
39C2×C4×Dic7C2211yescharacteristic
40C2×C4×Dic7C2211yescharacteristic
41C2×C4×Dic7C2211yescharacteristic
42#6449C418yescomplemented
43#6440—24no—
44C7×#95C2211yescharacteristic
45C7×#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53C7⋊#95C2211yescharacteristic
54#2359—71nomaximal, Sylow
55C24×C7C2311yescharacteristic
56C23×D7C2311yescharacteristic
57C22×C4×C7—22no—
58C22×C4×C7—22no—
59C22×C4×C7—41no—
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C2×C22⋊D7—22no—
67C2×C22⋊D7—22no—
68C2×C22⋊D7C2×C414yes—
69C22×Dic7—22no—
70C22×Dic7—22no—
71C22×Dic7—22no—
72C22×Dic7—22no—
73C22×Dic7—41no—
74C22×Dic7C2×C414yescomplemented
75C22×Dic7C2311yescharacteristic
76C22×Dic7C2311yescharacteristic
77C22×Dic7C2311yescharacteristic
78C22×Dic7C2311yescharacteristic
79C22×Dic7C2311yescharacteristic
80C22×Dic7C2311yescharacteristic
81C22×Dic7C2311yescharacteristic
82C2×C4×D7—22no—
83C2×C4×D7—22no—
84C2×C4×D7—41no—
85C7×C4⋊C4—22no—
86C7×C4⋊C4—41no—
87C7×C22⋊C4—22no—
88C7×C22⋊C4C2×C414yescomplemented
89C22⋊Dic7D414yes—
90(C2×D7)⋊C4—22no—
91(C2×D7)⋊C4—22no—
92(C2×D7)⋊C4—41no—
93(C2×D7)⋊C4C2×C414yescomplemented
94(C2×D7)⋊C4C2×C414yescomplemented
95C4⋊Dic7C2×C414yescomplemented
96C4⋊Dic7C2×C414yescomplemented
97Dic7⋊C4—22no—
98Dic7⋊C4—22no—
99Dic7⋊C4—22no—
100C4×Dic7—22no—
101C4×Dic7—22no—
102C4×Dic7—22no—
103C4×Dic7—22no—
104C4×Dic7—41no—
105C2×#123—71no—
106C2×C4×D4—71no—
107C2×#117—71no—
108#389—71no—
109#388—71no—
110#388—71no—
111#387—71no—
112#387—71no—
113#386—71no—
114#385—71no—
115#385D711yescharacteristic, complemented
116C4×C4⋊C4—71no—
117C4×C22⋊C4—71no—
118C4×C22⋊C4—71no—
119C2×#95—71no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7C22×C412yes—
125C23×C7C2×D412yes—
126C23×C7#4112yes—
127C23×C7C2411yescharacteristic
128C22×D7—21no—
129C22×D7—21no—
130C22×D7—21no—
131C22×D7—21no—
132C22×D7—21no—
133C22×D7—21no—
134C22×D7C22×C412yes—
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×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—44no—
151C2×C4×C7C22×C412yes—
152C2×C4×C7C22×C412yes—
153C2×C4×C7#4112yes—
154C22⋊D7—28no—
155C2×Dic7—28no—
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—21no—
167C2×Dic7—21no—
168C2×Dic7—21no—
169C2×Dic7—21no—
170C2×Dic7—21no—
171C2×Dic7—44no—
172C2×Dic7C22×C412yes—
173C2×Dic7C22×C412yes—
174C2×Dic7C22×C412yes—
175C2×Dic7C2×D412yes—
176C2×Dic7C2×D412yes—
177C2×Dic7#4112yes—
178C4×D7—44no—
179C22×D4—71no—
180C23×C4—71no—
181C23×C4—71no—
182C23×C4—71no—
183C23×C4C2×D711yescharacteristic
184#123—78no—
185C4×D4—144no—
186#117—144no—
187C2×C4⋊C4—71no—
188C2×C4⋊C4—71no—
189C2×C4⋊C4—71no—
190C2×C4⋊C4—71no—
191C2×C4⋊C4—71no—
192C2×C4⋊C4C2×D711yescharacteristic
193C2×C4⋊C4C2×D711yescharacteristic
194C2×C22⋊C4—71no—
195C2×C22⋊C4—71no—
196C2×C22⋊C4—71no—
197C2×C22⋊C4—71no—
198C2×C22⋊C4—71no—
199C2×C22⋊C4—71no—
200C2×C22⋊C4C2×D711yescharacteristic
201C2×C22⋊C4C2×D711yescharacteristic
202C2×C42—71no—
203C2×C42—71no—
204C2×C42—71no—
205C2×C42—71no—
206C2×C42—71no—
207#95—71no—
208#95—71no—
209#95—71no—
210#95—71no—
211#95—71no—
212#95—71no—
213#95—71no—
214#95—71no—
215#95C2×D711yescharacteristic
216#95C2×D711yescharacteristic
217C22×C7—22no—
218C22×C7—22no—
219C22×C7—22no—
220C22×C7—22no—
221C22×C7—22no—
222C22×C7—22no—
223C22×C7C4×D414yes—
224C22×C7C23×C411yescharacteristic
225C22×C7C22×D411yescharacteristic
226C22×C7C2×#4111yescharacteristic
227C22×C7C2×#4111yescharacteristic
228C22×C7C2×#4111yescharacteristic
229C22×C7H(5,𝔽2)11yescharacteristic
230C22×C72−1+411yescharacteristic
231C2×D7—22no—
232C2×D7—22no—
233C2×D7—22no—
234C2×D7—41no—
235C2×D7—41no—
236C2×D7—41no—
237C2×D7—41no—
238C4×C7—22no—
239C4×C7—22no—
240C4×C7—22no—
241C4×C7—41no—
242C4×C7—41no—
243C4×C7—41no—
244Dic7—22no—
245Dic7—22no—
246Dic7—22no—
247Dic7—22no—
248Dic7—22no—
249Dic7—22no—
250Dic7—41no—
251C24—71no—
252C24C22×D711yescharacteristic
253C2×D4—74no—
254C2×D4—142no—
255C2×D4—142no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—41no—
259C22×C4—74no—
260C22×C4—71no—
261C22×C4—71no—
262C22×C4—71no—
263C22×C4—71no—
264C22×C4—71no—
265C22×C4—71no—
266C22×C4—71no—
267C22×C4—142no—
268C22×C4—142no—
269C22×C4—142no—
270C22×C4—142no—
271C22×C4—142no—
272C22×C4—142no—
273C22×C4—281no—
274C22×C4—281no—
275C22×C4C22×D711yescharacteristic
276C22×C4C22×D711yescharacteristic
277C22×C4C22×D711yescharacteristic
278C22×C4C22×D711yescharacteristic
279C22×C4C22×D711yescharacteristic
280C22×C4C22×D711yescharacteristic
281C4⋊C4—22no—
282C4⋊C4—41no—
283C4⋊C4—74no—
284C4⋊C4—74no—
285C4⋊C4—142no—
286C4⋊C4—142no—
287C4⋊C4—142no—
288C22⋊C4—22no—
289C22⋊C4—74no—
290C22⋊C4—74no—
291C22⋊C4—74no—
292C22⋊C4—142no—
293C22⋊C4—142no—
294C22⋊C4—281no—
295C22⋊C4C4×D714yescomplemented
296C42—142no—
297C42—142no—
298C42—142no—
299C42—142no—
300C42—281no—
301C2×C7—24no—
302C2×C7C2×C4×D411yescharacteristic
303C2×C7C4×#4111yescharacteristic
304C2×C7#52011yescharacteristic
305C2×C7#54611yescharacteristic
306C2×C7#54711yescharacteristic
307C2×C7#55111yescharacteristic
308C2×C7#55211yescharacteristic
309D7—42no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—72no—
315C23—141no—
316C23—141no—
317C23—141no—
318C23—141no—
319C23—141no—
320C23—141no—
321C23C2×C4×D712yes—
322C23D4×D712yes—
323C23#116112yes—
324C23C23×D711yescharacteristic
325D4—148no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—44no—
342C2×C4—72no—
343C2×C4—72no—
344C2×C4—72no—
345C2×C4—72no—
346C2×C4—72no—
347C2×C4—72no—
348C2×C4—148no—
349C2×C4—141no—
350C2×C4—141no—
351C2×C4—141no—
352C2×C4—141no—
353C2×C4—141no—
354C2×C4—141no—
355C2×C4—141no—
356C2×C4—141no—
357C2×C4—141no—
358C2×C4—141no—
359C2×C4—141no—
360C2×C4—141no—
361C2×C4—141no—
362C2×C4—141no—
363C2×C4—141no—
364C2×C4—284no—
365C2×C4—284no—
366C2×C4C2×C4×D712yes—
367C2×C4C2×C4×D712yes—
368C2×C4#115912yes—
369C7#235911yescharacteristic, complemented, Sylow
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—142no—
377C22—142no—
378C22—142no—
379C22—281no—
380C22—281no—
381C22—281no—
382C22—281no—
383C22#645614yes—
384C22C22×C4×D711yescharacteristic
385C22C2×D4×D711yescharacteristic
386C22C2×#115911yescharacteristic
387C22#654811yescharacteristic
388C22C2×#116111yescharacteristic
389C22D7×#4111yescharacteristic
390C22#655411yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—142no—
398C4—142no—
399C4—142no—
400C4—142no—
401C4—142no—
402C4—142no—
403C4—281no—
404C2—24no—
405C2—282no—
406C2C2×#645611yescharacteristic
407C2C4×#115911yescharacteristic
408C2#9023711yescharacteristic
409C2#9028211yescharacteristic
410C2#9032811yescharacteristic
411C2#9033311yescharacteristic
412C2#9034611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutatorC22×C7C23×C4
FittingC7×#385C2
socle, cosocleC23×C7C24
prime radicalC22×C22⋊D7C22
hypercenter#385D7
hypocenterC7#2359
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. C2×#6440C2semidirect
4. C7×#385C2semidirect
5. C2×C7⋊#95C2semidirect
6. #89762C2semidirect
7. #89761C2semidirect
8. C4⋊C4×Dic7C2semidirect
9. #89745C2semidirect
11. C22⋊C4×Dic7C2semidirect
12. #89441C2semidirect
13. #89438C2semidirect
14. #89426C2semidirect
15. #89426C2semidirect
42. #6449C4semidirect
74. C22×Dic7C2×C4semidirect
88. C7×C22⋊C4C2×C4semidirect
93. (C2×D7)⋊C4C2×C4semidirect
94. (C2×D7)⋊C4C2×C4semidirect
95. C4⋊Dic7C2×C4semidirect
96. C4⋊Dic7C2×C4semidirect
115. #385D7semidirect
295. C22⋊C4C4×D7semidirect
369. C7#2359semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D72
C2×#64492
C2×#64402
C7×#3852
C2×C7⋊#952
#897622
#897612
C4⋊C4×Dic72
#897452
#897302
C22⋊C4×Dic72
#894412
#894382
#894262
#23597