Finite Group Databasebeta

← #11732752

Subgroups — #11732752

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×#6457C211yesmaximal, characteristic
2C2×#6456C211yesmaximal, characteristic
3C2×#6435C211yesmaximal, characteristic, complemented
4C7×#387C211yesmaximal, characteristic, complemented
5C4⋊C4×Dic7C211yesmaximal, characteristic, complemented
6#89744C211yesmaximal, characteristic, complemented
7C4×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
8#89441C211yesmaximal, characteristic, complemented
9#89441C211yesmaximal, characteristic, complemented
10#89440C211yesmaximal, characteristic, complemented
11#89440C211yesmaximal, characteristic, complemented
12#89438C211yesmaximal, characteristic, complemented
13#89437C211yesmaximal, characteristic
14D7×#95C211yesmaximal, characteristic, complemented
15#89424C211yesmaximal, characteristic, complemented
16C22×D28C2211yescharacteristic
17C22×C4×D7C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
19C22×C4×D7C2211yescharacteristic
20C22×C4×D7C2211yescharacteristic
21C2×C7×C4⋊C4C2211yescharacteristic
22C2×C7×C4⋊C4C2211yescharacteristic
23C2×C42×C7C2211yescharacteristic
24C2×(C2×D7)⋊C4C2211yescharacteristic
25C2×(C2×D7)⋊C4C2211yescharacteristic
26C2×(C2×D7)⋊C4C2211yescharacteristic
27C2×(C2×D7)⋊C4C2211yescharacteristic
28C2×(C2×D7)⋊C4C2211yescharacteristic
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×C4⋊Dic7C2211yescharacteristic
33C2×Dic7⋊C4C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×C4×Dic7C2211yescharacteristic
38C2×C4×Dic7C2211yescharacteristic
39C2×C4×Dic7C2211yescharacteristic
40C2×C4×Dic7C2211yescharacteristic
41#6457C418yescomplemented
42#6456—24no—
43#6435—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
55C23×D7C2311yescharacteristic
56C23×D7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C22×Dic7C2311yescharacteristic
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C22×Dic7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C2×D28—22no—
71C2×D28—22no—
72C2×D28C2×C414yes—
73C2×C4×D7—22no—
74C2×C4×D7—22no—
75C2×C4×D7—22no—
76C2×C4×D7—22no—
77C2×C4×D7—22no—
78C2×C4×D7—22no—
79C2×C4×D7—22no—
80C2×C4×D7—22no—
81C2×C4×D7—41no—
82C2×C4×D7—41no—
83C2×C4×D7—41no—
84C2×C4×D7C2×C414yescomplemented
85C7×C4⋊C4—22no—
86C7×C4⋊C4C2×C414yescomplemented
87C42×C7—22no—
88(C2×D7)⋊C4—22no—
89(C2×D7)⋊C4—22no—
90(C2×D7)⋊C4—22no—
91(C2×D7)⋊C4—41no—
92(C2×D7)⋊C4C2×C414yescomplemented
93(C2×D7)⋊C4C2×C414yescomplemented
94(C2×D7)⋊C4C2×C414yescomplemented
95(C2×D7)⋊C4D414yes—
96C4⋊Dic7—22no—
97Dic7⋊C4—22no—
98Dic7⋊C4—22no—
99Dic7⋊C4—41no—
100Dic7⋊C4C2×C414yescomplemented
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#387D711yescharacteristic, complemented
113#386—71no—
114#385—71no—
115#385—71no—
116C4×C4⋊C4—71no—
117C4×C22⋊C4—71no—
118C4×C22⋊C4—71no—
119C2×#95—71no—
120C23×C7C2411yescharacteristic
121C22×D7—21no—
122C22×D7—21no—
123C22×D7—21no—
124C22×D7—21no—
125C22×D7—21no—
126C22×D7—21no—
127C22×D7—21no—
128C22×D7—21no—
129C22×D7—21no—
130C22×D7—21no—
131C22×D7C22×C412yes—
132C22×D7C22×C412yes—
133C22×D7C2×D412yes—
134C22×D7#4112yes—
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—21no—
151C2×C4×C7—21no—
152C2×C4×C7C22×C412yes—
153C2×C4×C7C22×C412yes—
154C2×C4×C7C22×C412yes—
155C2×C4×C7C2×D412yes—
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×Dic7C22×C412yes—
170C2×Dic7C22×C412yes—
171C2×Dic7C2×D412yes—
172C2×Dic7#4112yes—
173C2×Dic7#4112yes—
174D28—28no—
175C4×D7—28no—
176C4×D7—44no—
177C4×D7—44no—
178C4×D7—44no—
179C22×D4—71no—
180C23×C4—71no—
181C23×C4—71no—
182C23×C4—71no—
183C23×C4—71no—
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⋊C4—71no—
201C2×C22⋊C4—71no—
202C2×C42—71no—
203C2×C42—71no—
204C2×C42—71no—
205C2×C42—71no—
206C2×C42C2×D711yescharacteristic
207#95—71no—
208#95—71no—
209#95—71no—
210#95—71no—
211#95—71no—
212#95—71no—
213#95C2×D711yescharacteristic
214#95C2×D711yescharacteristic
215#95C2×D711yescharacteristic
216#95C2×D711yescharacteristic
217C22×C7C23×C411yescharacteristic
218C22×C7C22×D411yescharacteristic
219C22×C7C2×#4111yescharacteristic
220C22×C7C2×#4111yescharacteristic
221C22×C7C2×#4111yescharacteristic
222C22×C7H(5,𝔽2)11yescharacteristic
223C22×C72−1+411yescharacteristic
224C2×D7—22no—
225C2×D7—22no—
226C2×D7—22no—
227C2×D7—22no—
228C2×D7—22no—
229C2×D7—22no—
230C2×D7—22no—
231C2×D7—22no—
232C2×D7—22no—
233C2×D7—41no—
234C2×D7—41no—
235C2×D7—41no—
236C2×D7—41no—
237C2×D7C4×D414yes—
238C4×C7—22no—
239C4×C7—22no—
240C4×C7—22no—
241C4×C7—22no—
242C4×C7—41no—
243C4×C7—41no—
244C4×C7—41no—
245Dic7—22no—
246Dic7—22no—
247Dic7—22no—
248Dic7—22no—
249Dic7—22no—
250Dic7—41no—
251C24—71no—
252C24—71no—
253C2×D4—74no—
254C2×D4—142no—
255C2×D4—142no—
256C22×C4—71no—
257C22×C4—71no—
258C22×C4—71no—
259C22×C4—71no—
260C22×C4—71no—
261C22×C4—74no—
262C22×C4—71no—
263C22×C4—142no—
264C22×C4—142no—
265C22×C4—142no—
266C22×C4—142no—
267C22×C4—142no—
268C22×C4—142no—
269C22×C4—142no—
270C22×C4—142no—
271C22×C4—281no—
272C22×C4—281no—
273C22×C4—281no—
274C22×C4C22×D711yescharacteristic
275C22×C4C22×D711yescharacteristic
276C22×C4C22×D711yescharacteristic
277C22×C4C22×D711yescharacteristic
278C22×C4C22×D711yescharacteristic
279C22×C4C22×D711yescharacteristic
280C22×C4C22×D711yescharacteristic
281C4⋊C4—22no—
282C4⋊C4—74no—
283C4⋊C4—142no—
284C4⋊C4—142no—
285C4⋊C4—142no—
286C4⋊C4—281no—
287C4⋊C4C4×D714yescomplemented
288C22⋊C4—74no—
289C22⋊C4—74no—
290C22⋊C4—74no—
291C22⋊C4—74no—
292C22⋊C4—142no—
293C22⋊C4—142no—
294C22⋊C4—142no—
295C22⋊C4—281no—
296C42—22no—
297C42—142no—
298C42—142no—
299C42—142no—
300C42—281no—
301C2×C7C2×C4×D411yescharacteristic
302C2×C7C4×#4111yescharacteristic
303C2×C7#52011yescharacteristic
304C2×C7#54611yescharacteristic
305C2×C7#54711yescharacteristic
306C2×C7#55111yescharacteristic
307C2×C7#55211yescharacteristic
308D7—24no—
309D7—42no—
310C23—72no—
311C23—72no—
312C23—72no—
313C23—72no—
314C23—141no—
315C23—141no—
316C23—141no—
317C23—141no—
318C23—141no—
319C23—141no—
320C23—141no—
321C23—141no—
322C23—141no—
323C23—141no—
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—21no—
342C2×C4—21no—
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—284no—
363C2×C4—284no—
364C2×C4—284no—
365C2×C4C2×C4×D712yes—
366C2×C4C2×C4×D712yes—
367C2×C4C2×C4×D712yes—
368C2×C4D4×D712yes—
369C7#235911yescharacteristic, complemented, Sylow
370C22—74no—
371C22—142no—
372C22—142no—
373C22—142no—
374C22—142no—
375C22—142no—
376C22—142no—
377C22—142no—
378C22—142no—
379C22—142no—
380C22—281no—
381C22—281no—
382C22—281no—
383C22—281no—
384C22C22×C4×D711yescharacteristic
385C22C2×D4×D711yescharacteristic
386C22#655111yescharacteristic
387C22D7×#4111yescharacteristic
388C22D7×#4111yescharacteristic
389C22D7×#4111yescharacteristic
390C22#655311yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—142no—
399C4—142no—
400C4—142no—
401C4—142no—
402C4—142no—
403C4—281no—
404C2—144no—
405C2—282no—
406C2#9021311yescharacteristic
407C2C4×D4×D711yescharacteristic
408C2#9026511yescharacteristic
409C2#9032211yescharacteristic
410C2#9034511yescharacteristic
411C2#9038411yescharacteristic
412C2#9039611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutatorC22×C7C23×C4
FittingC7×#387C2
socle, cosocleC23×C7C24
prime radicalC22×D28C22
hypercenter#387D7
hypocenterC7#2359
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
3. C2×#6435C2semidirect
4. C7×#387C2semidirect
5. C4⋊C4×Dic7C2semidirect
6. #89744C2semidirect
7. C4×(C2×D7)⋊C4C2semidirect
8. #89441C2semidirect
9. #89441C2semidirect
10. #89440C2semidirect
11. #89440C2semidirect
12. #89438C2semidirect
14. D7×#95C2semidirect
15. #89424C2semidirect
41. #6457C4semidirect
84. C2×C4×D7C2×C4semidirect
86. C7×C4⋊C4C2×C4semidirect
92. (C2×D7)⋊C4C2×C4semidirect
93. (C2×D7)⋊C4C2×C4semidirect
94. (C2×D7)⋊C4C2×C4semidirect
100. Dic7⋊C4C2×C4semidirect
112. #387D7semidirect
287. C4⋊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×#64572
C2×#64562
C2×#64352
C7×#3872
C4⋊C4×Dic72
#897442
C4×(C2×D7)⋊C42
#894412
#894402
#894382
#894372
D7×#952
#894242
#23597