Finite Group Databasebeta

← #20286439

Subgroups — #20286439

Order 6784.

494 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133238C211yesmaximal, characteristic, complemented
2#16811292C211yesmaximal, characteristic, complemented
3#16811272C211yesmaximal, characteristic, complemented
4#16810994C211yesmaximal, characteristic, complemented
5#16810852C211yesmaximal, characteristic, complemented
6#16810849C211yesmaximal, characteristic, complemented
7#16810831C211yesmaximal, characteristic, complemented
8#16810816C211yesmaximal, characteristic, complemented
9#16810785C211yesmaximal, characteristic, complemented
10#16810784C211yesmaximal, characteristic, complemented
11#16810768C211yesmaximal, characteristic, complemented
12#16810765C211yesmaximal, characteristic, complemented
13SD16×Dic53C211yesmaximal, characteristic, complemented
14C8×C22⋊D53C211yesmaximal, characteristic, complemented
15C2×#135×C53C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719947C2211yescharacteristic
18#13133238—24no—
19C2×#11719925C2211yescharacteristic
20C2×#11719913C2211yescharacteristic
21C2×#11719923C2211yescharacteristic
22C2×#11719924C2211yescharacteristic
23C2×#11719922C2211yescharacteristic
24C2×C8×D53C2211yescharacteristic
25#13133215C2211yescharacteristic
26#13133213—21no—
27#13133213C2211yescharacteristic
28#13133200—21no—
29#13133198—21no—
30#13133196C2211yescharacteristic
31#13133193C2211yescharacteristic
32#13133191—21no—
33#13133191C2211yescharacteristic
34#13133189—21no—
35#13133188—21no—
36#13133188C2211yescharacteristic
37Q8×Dic53C2211yescharacteristic
38D4×Dic53C2211yescharacteristic
39C4×C22⋊D53—21no—
40C4×C22⋊D53C2211yescharacteristic
41C2×#11719919—21no—
42#13133149C2211yescharacteristic
43#13133146C2211yescharacteristic
44#13133142C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133141C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133138C2211yescharacteristic
49#13133138C2211yescharacteristic
50C22⋊#88322C2211yescharacteristic
51#13133129C2211yescharacteristic
52C8×Dic53C2211yescharacteristic
53C2×#41×C53C2211yescharacteristic
54C2×#41×C53C2211yescharacteristic
55#135×C53—24no—
56C2×Q16×C53C2211yescharacteristic
57C2×SD16×C53C2211yescharacteristic
58C2×SD16×C53C2211yescharacteristic
59C2×D8×C53C2211yescharacteristic
60C22×C8×C53C2211yescharacteristic
61#41×C53—22no—
62#41×C53—22no—
63#41×C53—41no—
64#41×C53—41no—
65C2×Q8×C53C2311yescharacteristic
66C2×Q8×C53C2311yescharacteristic
67C2×D4×C53—21no—
68C2×D4×C53—21no—
69C2×D4×C53C2311yescharacteristic
70C2×D4×C53C2311yescharacteristic
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53C2311yescharacteristic
74C2×C22⋊D53—21no—
75C2×C22⋊D53—21no—
76C2×C22⋊D53—21no—
77C2×C22⋊D53—21no—
78C2×C22⋊D53—21no—
79C2×C22⋊D53D411yescharacteristic
80C22×Dic53—21no—
81C22×Dic53—21no—
82#11719949—22no—
83#11719949—41no—
84#11719947—22no—
85#11719947—41no—
86C2×D212—21no—
87C2×D212C2311yescharacteristic
88C2×C4×D53—21no—
89C2×C4×D53C2311yescharacteristic
90C2×Dic106C2311yescharacteristic
91Q16×C53—22no—
92SD16×C53—22no—
93SD16×C53D414yescomplemented
94D8×C53—22no—
95C2×C8×C53—22no—
96C2×C8×C53C2311yescharacteristic
97C2×C8×C53C2311yescharacteristic
98C22⋊Dic53—21no—
99C22⋊Dic53—21no—
100C22⋊Dic53—21no—
101C22⋊Dic53D411yescharacteristic
102#11719925—22no—
103#11719924—41no—
104#11719923—41no—
105#11719922—22no—
106#11719921—21no—
107#11719921—21no—
108#11719921D411yescharacteristic
109C4⋊Dic53—21no—
110C4⋊Dic53C2311yescharacteristic
111C4⋊Dic53C2311yescharacteristic
112#11719919—21no—
113#11719919—21no—
114#11719919—21no—
115#11719919—21no—
116#11719919D411yescharacteristic
117C4×Dic53—21no—
118C4×Dic53C2311yescharacteristic
119C2×#88322C2311yescharacteristic
120C2×#88322C2311yescharacteristic
121#11719913—22no—
122C8×D53—22no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53C2×D411yescharacteristic
126C22×D53—21no—
127C22×D53C2×D411yescharacteristic
128Q8×C53—21no—
129Q8×C53—21no—
130Q8×C53—21no—
131Q8×C53C2×D412yes—
132D4×C53—21no—
133D4×C53—21no—
134D4×C53—21no—
135D4×C53—42no—
136D4×C53—42no—
137D4×C53C2×D412yes—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—42no—
143C2×C4×C53—42no—
144C2×C4×C53C2×D411yescharacteristic
145C2×C4×C53C2411yescharacteristic
146C22⋊D53—22no—
147C22⋊D53—42no—
148C22⋊D53—42no—
149C22⋊D53—81no—
150C22⋊D53—81no—
151C22⋊D53—81no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—21no—
155C2×Dic53—42no—
156C2×Dic53—42no—
157C2×Dic53C2×D411yescharacteristic
158C2×Dic53C2×D411yescharacteristic
159C2×Dic53C2×D411yescharacteristic
160D212—21no—
161D212—21no—
162D212—42no—
163C4×D53—21no—
164C4×D53—21no—
165C4×D53—42no—
166Dic106—21no—
167Dic106—21no—
168C8×C53—21no—
169C8×C53C2×D412yes—
170#88322—21no—
171#88322—21no—
172C22×C53—22no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53C22×D411yescharacteristic
182C2×D53—21no—
183C2×D53—21no—
184C2×D53—21no—
185C2×D53—41no—
186C2×D53—41no—
187C2×D53—41no—
188C4×C53—22no—
189C4×C53—21no—
190C4×C53—41no—
191C4×C53C22×D411yescharacteristic
192C4×C53H(5,𝔽2)11yescharacteristic
193Dic53—21no—
194Dic53—21no—
195Dic53—41no—
196Dic53—41no—
197Dic53—41no—
198Dic53#13512yescomplemented
199#3284—531nomaximal, Sylow
200C2×C53—41no—
201C2×C53—42no—
202C2×C53—81no—
203C2×C53D4211yescharacteristic
204C2×C53C2×#13511yescharacteristic
205C2×C53#57711yescharacteristic
206D53—41no—
207D53—81no—
208C2×#135—531no—
209C2×#135D5311yescharacteristic, complemented
210#550—531no—
211#547—531no—
212#492—531no—
213#465—531no—
214#465—531no—
215#462—531no—
216#459—531no—
217#452—531no—
218#452—531no—
219#449—531no—
220#449—531no—
221C4×SD16—531no—
222C8×D4—531no—
223C53#328411yescharacteristic, complemented, Sylow
224C2×#41—531no—
225C2×#41—531no—
226C2×#41C2×D5311yescharacteristic
227C2×#41C2×D5311yescharacteristic
228#135—24no—
229#135—1064no—
230C2×Q16—531no—
231C2×Q16C2×D5311yescharacteristic
232C2×SD16—531no—
233C2×SD16—531no—
234C2×SD16—531no—
235C2×SD16C2×D5311yescharacteristic
236C2×SD16C2×D5311yescharacteristic
237C2×D8—531no—
238C2×D8C2×D5311yescharacteristic
239C22×C8—531no—
240C22×C8C2×D5311yescharacteristic
241C4⋊Q8—531no—
242#127—531no—
243#127—1061no—
244#123—1061no—
245#123—1061no—
246C22⋊Q8—531no—
247C22⋊Q8—531no—
248#121—531no—
249#121—531no—
250#121—1061no—
251#121—1061no—
252#121—1061no—
253C4×Q8—531no—
254C4×D4—531no—
255C4×D4—531no—
256C4×D4—1061no—
257C2×C4⋊C4—1061no—
258#106—531no—
259C4⋊C8—531no—
260#103—531no—
261#103—531no—
262#103—531no—
263#102—531no—
264#102—531no—
265#102—531no—
266C22⋊C8—531no—
267C22⋊C8—531no—
268C4×C8—531no—
269#41—22no—
270#41—22no—
271#41—41no—
272#41—41no—
273#41—1062no—
274#41—1062no—
275#41—2121no—
276#41—2121no—
277C2×Q8—531no—
278C2×Q8C22×D5311yescharacteristic
279C2×Q8C22×D5311yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—531no—
283C2×D4—531no—
284C2×D4—1061no—
285C2×D4—1061no—
286C2×D4—1061no—
287C2×D4—1061no—
288C2×D4—1061no—
289C2×D4—1061no—
290C2×D4C22×D5311yescharacteristic
291C2×D4C22×D5311yescharacteristic
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—531no—
295C22×C4—1061no—
296C22×C4—1061no—
297C22×C4—1061no—
298C22×C4C22×D5311yescharacteristic
299Q16—22no—
300Q16—1062no—
301SD16—22no—
302SD16—1062no—
303SD16—2121no—
304SD16—2121no—
305SD16C22⋊D5314yescomplemented
306D8—22no—
307D8—1062no—
308C2×C8—22no—
309C2×C8—531no—
310C2×C8—531no—
311C2×C8—1062no—
312C2×C8C22×D5311yescharacteristic
313C2×C8C22×D5311yescharacteristic
314C4⋊C4—531no—
315C4⋊C4—531no—
316C4⋊C4—531no—
317C4⋊C4—1061no—
318C4⋊C4—1061no—
319C4⋊C4—1061no—
320C4⋊C4—1061no—
321C4⋊C4—1061no—
322C22⋊C4—531no—
323C22⋊C4—531no—
324C22⋊C4—1061no—
325C22⋊C4—1061no—
326C22⋊C4—1061no—
327C22⋊C4—1061no—
328C22⋊C4—1061no—
329C42—531no—
330C42—1061no—
331C23—21no—
332C23—21no—
333C23—531no—
334C23—1061no—
335C23D4×D5311yescharacteristic
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—1061no—
340Q8—1061no—
341Q8C2×C22⋊D5312yes—
342D4—21no—
343D4—21no—
344D4—21no—
345D4—42no—
346D4—42no—
347D4—1062no—
348D4—1061no—
349D4—1061no—
350D4—4241no—
351D4—4241no—
352D4—4241no—
353D4—2122no—
354D4—2122no—
355D4—2122no—
356D4C2×C22⋊D5312yes—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—531no—
364C2×C4—531no—
365C2×C4—531no—
366C2×C4—1061no—
367C2×C4—1061no—
368C2×C4—1061no—
369C2×C4—1061no—
370C2×C4—1061no—
371C2×C4—2122no—
372C2×C4—2122no—
373C2×C4—2122no—
374C2×C4D4×D5311yescharacteristic
375C2×C4C23×D5311yescharacteristic
376C8—21no—
377C8—1061no—
378C8—1061no—
379C8C2×C22⋊D5312yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—1061no—
390C22—1061no—
391C22—1061no—
392C22—2121no—
393C22—2121no—
394C22—2121no—
395C22C2×D4×D5311yescharacteristic
396C4—21no—
397C4—22no—
398C4—41no—
399C4—532no—
400C4—1061no—
401C4—1061no—
402C4—2121no—
403C4—2121no—
404C4—2121no—
405C4C22×C22⋊D5311yescharacteristic
406C4#1313326211yescharacteristic
407C2—41no—
408C2—42no—
409C2—81no—
410C2—4241no—
411C2—2121no—
412C2D4×C22⋊D5311yescharacteristic
413C2#135×D5311yescharacteristic
414C2#1681143011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×#135×C53C2
socleC22×C53C22×D4
hypercenterC2×#135D53
hypocenterC53#3284
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#13133238C2semidirect
2. #16811292C2semidirect
3. #16811272C2semidirect
4. #16810994C2semidirect
5. #16810852C2semidirect
6. #16810849C2semidirect
7. #16810831C2semidirect
8. #16810816C2semidirect
9. #16810785C2semidirect
10. #16810784C2semidirect
11. #16810768C2semidirect
12. #16810765C2semidirect
13. SD16×Dic53C2semidirect
14. C8×C22⋊D53C2semidirect
15. C2×#135×C53C2semidirect
93. SD16×C53D4semidirect
198. Dic53#135semidirect
209. C2×#135D53semidirect
223. C53#3284semidirect
305. SD16C22⋊D53semidirect