Finite Group Databasebeta

← #11738043

Subgroups — #11738043

Order 896.

484 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6481C211yesmaximal, characteristic, complemented
2C2×#6478C211yesmaximal, characteristic, complemented
3#90359C211yesmaximal, characteristic
4C4×Q8×D7C211yesmaximal, characteristic, complemented
5C7×#464C211yesmaximal, characteristic, complemented
6#89954C211yesmaximal, characteristic, complemented
7#89935C211yesmaximal, characteristic, complemented
8#89826C211yesmaximal, characteristic, complemented
9#89800C211yesmaximal, characteristic, complemented
10#89617C211yesmaximal, characteristic, complemented
11#89609C211yesmaximal, characteristic, complemented
12#89575C211yesmaximal, characteristic, complemented
13#89565C211yesmaximal, characteristic, complemented
14#89543C211yesmaximal, characteristic, complemented
15#89534C211yesmaximal, characteristic, complemented
16C2×#1161C2211yescharacteristic
17C2×Q8×D7C2211yescharacteristic
18C2×#1159C2211yescharacteristic
19C2×C7×Q16C2211yescharacteristic
20C2×C7×SD16C2211yescharacteristic
21C7×#124C2211yescharacteristic
22C4×C7×Q8C2211yescharacteristic
23Q8×Dic7C2211yescharacteristic
24#6507C2211yescharacteristic
25C2×#1144C2211yescharacteristic
26C2×#1144C2211yescharacteristic
27C2×#1143C2211yescharacteristic
28#6503C2211yescharacteristic
29C2×#1142C2211yescharacteristic
30#6481—24no—
31#6478—24no—
32C2×Dic28C2211yescharacteristic
33C2×#1132C2211yescharacteristic
34C2×#1131C2211yescharacteristic
35C2×#1131C2211yescharacteristic
36D7×C4⋊C4—21no—
37D7×C4⋊C4C2211yescharacteristic
38#6450—21no—
39#6450C2211yescharacteristic
40#6447—21no—
41#6447—21no—
42(C22⋊D7)⋊C4—21no—
43(C22⋊D7)⋊C4—21no—
44#6438C2211yescharacteristic
45C42×D7—21no—
46C42×D7C2211yescharacteristic
47C4×Dic14—21no—
48C4×Dic14C2211yescharacteristic
49C7×C4⋊C8C2211yescharacteristic
50C7×#103C2211yescharacteristic
51C7×#102C2211yescharacteristic
52#6409C2211yescharacteristic
53#6406C2211yescharacteristic
54#6395C2211yescharacteristic
55#6390C2211yescharacteristic
56Dic7⋊C8C2211yescharacteristic
57Dic7⋊C8C2211yescharacteristic
58#6384C2211yescharacteristic
59#6383C2211yescharacteristic
60C4⋊(C7⋊C8)C2211yescharacteristic
61#3037—71nomaximal, Sylow
62C2×C7×Q8C2311yescharacteristic
63C2×C7×Q8C2311yescharacteristic
64C2×C7×D4C2311yescharacteristic
65C2×C22⋊D7—21no—
66C2×C22⋊D7—21no—
67C2×C22⋊D7—21no—
68C22×Dic7—21no—
69#1161—22no—
70#1161—22no—
71Q8×D7—22no—
72Q8×D7D414yescomplemented
73#1159—22no—
74#1159—22no—
75C2×D28—21no—
76C2×D28C2311yescharacteristic
77C2×C4×D7—21no—
78C2×C4×D7—21no—
79C2×C4×D7—21no—
80C2×C4×D7D411yescharacteristic
81C2×C4×D7D411yescharacteristic
82C2×C4×D7C2311yescharacteristic
83C2×Dic14—21no—
84C2×Dic14C2311yescharacteristic
85C2×Dic14C2311yescharacteristic
86C7×Q16—22no—
87C7×SD16—22no—
88C2×C7×C8C2311yescharacteristic
89C2×C7×C8C2311yescharacteristic
90C7×C4⋊C4—21no—
91C7×C4⋊C4C2311yescharacteristic
92C7×C22⋊C4—21no—
93C7×C22⋊C4—21no—
94C42×C7—21no—
95C42×C7C2311yescharacteristic
96#1144—22no—
97#1144—22no—
98#1143—22no—
99#1142—22no—
100(C2×D7)⋊C4—21no—
101(C2×D7)⋊C4—21no—
102C4⋊Dic7—21no—
103C4⋊Dic7C2311yescharacteristic
104Dic7⋊C4—21no—
105Dic7⋊C4—21no—
106Dic7⋊C4—21no—
107Dic7⋊C4—21no—
108Dic7⋊C4—21no—
109C4×Dic7—21no—
110C4×Dic7—21no—
111C4×Dic7—21no—
112C4×Dic7D411yescharacteristic
113C4×Dic7D411yescharacteristic
114C4×Dic7C2311yescharacteristic
115C2×C7⋊C8C2311yescharacteristic
116C2×C7⋊C8C2311yescharacteristic
117Dic28—22no—
118#1132—22no—
119#1131—22no—
120#1131—22no—
121C2×#137—71no—
122C2×#137—71no—
123#532—71no—
124C2×C4×Q8—71no—
125#464—71no—
126#464—71no—
127#464—71no—
128#464D711yescharacteristic, complemented
129#462—71no—
130#462—71no—
131#460—71no—
132#460—71no—
133#423—71no—
134#417—71no—
135#417—71no—
136C23×C7—21no—
137C22×D7—21no—
138C22×D7C2×D411yescharacteristic
139C7×Q8—21no—
140C7×Q8—21no—
141C7×Q8—21no—
142C7×Q8C2×D412yes—
143C7×D4—21no—
144C7×D4—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7C2×D411yescharacteristic
149C2×C4×C7C2×D411yescharacteristic
150C2×C4×C7C2411yescharacteristic
151C22⋊D7—42no—
152C22⋊D7—42no—
153C22⋊D7—42no—
154C2×Dic7—21no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—42no—
158C2×Dic7C2×D411yescharacteristic
159C2×Dic7C2×D411yescharacteristic
160C2×Dic7C2×D411yescharacteristic
161D28—21no—
162D28—21no—
163D28—42no—
164C4×D7—24no—
165C4×D7—22no—
166C4×D7—41no—
167C4×D7—42no—
168C4×D7—42no—
169C4×D7C2×D412yes—
170C4×D7#4112yes—
171Dic14—21no—
172Dic14—24no—
173Dic14—21no—
174Dic14—21no—
175Dic14C2×D412yes—
176C7×C8—21no—
177C7×C8—21no—
178C7⋊C8—21no—
179C7⋊C8—21no—
180C2×#41—71no—
181C2×#41—71no—
182C22×Q8—71no—
183#137—144no—
184#137—144no—
185C2×Q16—71no—
186C2×Q16—71no—
187C2×Q16—71no—
188C2×Q16C2×D711yescharacteristic
189C2×SD16—71no—
190C2×SD16—71no—
191C2×SD16—71no—
192C2×SD16C2×D711yescharacteristic
193C2×M16—71no—
194C2×M16—71no—
195C4⋊Q8—71no—
196#127—71no—
197#124—71no—
198#124C2×D711yescharacteristic
199#121—141no—
200#121—141no—
201C4×Q8—71no—
202C4×Q8—71no—
203C4×Q8—71no—
204C4×Q8—141no—
205C4×Q8—141no—
206C4×Q8C2×D711yescharacteristic
207C4×D4—141no—
208C4×D4—141no—
209C2×C4⋊C4—71no—
210C2×C4⋊C4—141no—
211C2×C42—71no—
212C2×C42—141no—
213C4⋊C8—71no—
214C4⋊C8—71no—
215C4⋊C8—71no—
216C4⋊C8C2×D711yescharacteristic
217#103—71no—
218#103—71no—
219#103—71no—
220#103C2×D711yescharacteristic
221#102—71no—
222#102—71no—
223#102—71no—
224#102C2×D711yescharacteristic
225C22×C7—41no—
226C22×C7—41no—
227C22×C7—41no—
228C22×C7C22×D411yescharacteristic
229C2×D7—21no—
230C2×D7—21no—
231C2×D7—41no—
232C2×D7—41no—
233C2×D7—41no—
234C2×D7#12112yes—
235C4×C7—22no—
236C4×C7—21no—
237C4×C7—21no—
238C4×C7—41no—
239C4×C7—41no—
240C4×C7C22×D411yescharacteristic
241C4×C7C2×#4111yescharacteristic
242Dic7—22no—
243Dic7—21no—
244Dic7—41no—
245Dic7—41no—
246Dic7#12112yes—
247Dic7#13712yescomplemented
248#41—142no—
249#41—142no—
250#41—142no—
251#41—142no—
252C2×Q8—74no—
253C2×Q8—71no—
254C2×Q8—71no—
255C2×Q8—141no—
256C2×Q8—142no—
257C2×Q8C22×D711yescharacteristic
258C2×Q8C22×D711yescharacteristic
259C2×D4—71no—
260C2×D4—141no—
261C2×D4—141no—
262C2×D4—141no—
263C2×D4—141no—
264C2×D4C22×D711yescharacteristic
265C22×C4—71no—
266C22×C4—71no—
267C22×C4—71no—
268C22×C4—141no—
269C22×C4—141no—
270C22×C4—141no—
271C22×C4—141no—
272Q16—22no—
273Q16—142no—
274Q16—142no—
275Q16—142no—
276SD16—22no—
277SD16—142no—
278SD16—142no—
279SD16—142no—
280M16—142no—
281M16—142no—
282C2×C8—71no—
283C2×C8—71no—
284C2×C8C22×D711yescharacteristic
285C2×C8C22×D711yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—71no—
288C4⋊C4—141no—
289C4⋊C4—141no—
290C4⋊C4—141no—
291C4⋊C4—141no—
292C4⋊C4—141no—
293C4⋊C4—141no—
294C4⋊C4C22×D711yescharacteristic
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—141no—
298C22⋊C4—141no—
299C42—21no—
300C42—71no—
301C42—71no—
302C42—71no—
303C42—141no—
304C42—141no—
305C42—141no—
306C42C22×D711yescharacteristic
307C2×C7—81no—
308C2×C7C2×#12111yescharacteristic
309C2×C7C2×#13711yescharacteristic
310C2×C7#57511yescharacteristic
311D7—22no—
312D7—81no—
313C23—21no—
314C23—71no—
315C23—141no—
316Q8—21no—
317Q8—21no—
318Q8—21no—
319Q8—72no—
320Q8—144no—
321Q8—141no—
322Q8—141no—
323Q8—141no—
324Q8D4×D712yes—
325D4—21no—
326D4—21no—
327D4—141no—
328D4—141no—
329D4—282no—
330D4—282no—
331D4—282no—
332D4—282no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—71no—
337C2×C4—71no—
338C2×C4—72no—
339C2×C4—72no—
340C2×C4—71no—
341C2×C4—142no—
342C2×C4—144no—
343C2×C4—141no—
344C2×C4—141no—
345C2×C4—141no—
346C2×C4—281no—
347C2×C4—282no—
348C2×C4—282no—
349C2×C4—282no—
350C2×C4D4×D711yescharacteristic
351C2×C4D4×D711yescharacteristic
352C2×C4C23×D711yescharacteristic
353C8—21no—
354C8—21no—
355C8—141no—
356C8—141no—
357C7#303711yescharacteristic, complemented, Sylow
358C22—41no—
359C22—41no—
360C22—41no—
361C22—72no—
362C22—141no—
363C22—141no—
364C22—281no—
365C22—281no—
366C22—281no—
367C22C2×D4×D711yescharacteristic
368C4—21no—
369C4—21no—
370C4—22no—
371C4—41no—
372C4—41no—
373C4—72no—
374C4—72no—
375C4—141no—
376C4—142no—
377C4—281no—
378C4—281no—
379C4C2×D4×D711yescharacteristic
380C4D7×#4111yescharacteristic
381C2—81no—
382C2—142no—
383C2—561no—
384C2D7×#12111yescharacteristic
385C2#9045911yescharacteristic
386C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#464C2
socleC22×C7C22×D4
prime radical#90359C2
hypercenter#464D7
hypocenterC7#3037
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#6481C2semidirect
2. C2×#6478C2semidirect
4. C4×Q8×D7C2semidirect
5. C7×#464C2semidirect
6. #89954C2semidirect
7. #89935C2semidirect
8. #89826C2semidirect
9. #89800C2semidirect
10. #89617C2semidirect
11. #89609C2semidirect
12. #89575C2semidirect
13. #89565C2semidirect
14. #89543C2semidirect
15. #89534C2semidirect
72. Q8×D7D4semidirect
128. #464D7semidirect
247. Dic7#137semidirect
357. C7#3037semidirect

Maximal subgroups

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

subgroupindex
C2×#64812
C2×#64782
#903592
C4×Q8×D72
C7×#4642
#899542
#899352
#898262
#898002
#896172
#896092
#895752
#895652
#895432
#895342
#30377