Finite Group Databasebeta

← #20278564

Subgroups — #20278564

Order 6784.

620 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133215C211yesmaximal, characteristic, complemented
2C2×#13133195C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D53C211yesmaximal, characteristic
4#16810546C211yesmaximal, characteristic, complemented
5#16810545C211yesmaximal, characteristic, complemented
6#16810535C211yesmaximal, characteristic, complemented
7#16810534C211yesmaximal, characteristic, complemented
8#16810523C211yesmaximal, characteristic
9#16810523C211yesmaximal, characteristic
10#16810518C211yesmaximal, characteristic, complemented
11#16810518C211yesmaximal, characteristic, complemented
12#16810496C211yesmaximal, characteristic, complemented
13#16810494C211yesmaximal, characteristic, complemented
14C4⋊C4×Dic53C211yesmaximal, characteristic, complemented
15C2×C22⋊Q8×C53C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C22×C4×D53C2211yescharacteristic
19#13133215—24no—
20#13133195—24no—
21C4×C22⋊D53—24no—
22C2×C4⋊Dic53C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×#11719919C2211yescharacteristic
29C2×#11719919C2211yescharacteristic
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×C22⋊Dic53C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×C4×Dic53C2211yescharacteristic
39C2×C4×Dic53C2211yescharacteristic
40C2×C4×Dic53C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47C22×Q8×C53C2211yescharacteristic
48C23×C4×C53C2211yescharacteristic
49C22⋊Q8×C53—24no—
50C2×C4⋊C4×C53C2211yescharacteristic
51C2×C4⋊C4×C53C2211yescharacteristic
52C2×C4⋊C4×C53C2211yescharacteristic
53C2×C22⋊C4×C53C2211yescharacteristic
54C2×C22⋊C4×C53C2211yescharacteristic
55C24×C53C2311yescharacteristic
56C23×D53C2311yescharacteristic
57C2×Q8×C53—22no—
58C2×Q8×C53—22no—
59C2×Q8×C53—41no—
60C22×C4×C53—22no—
61C22×C4×C53—22no—
62C22×C4×C53—41no—
63C22×C4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C22×C4×C53C2311yescharacteristic
69C2×C22⋊D53—22no—
70C2×C22⋊D53—22no—
71C2×C22⋊D53Q814yescomplemented
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C22×Dic53C2311yescharacteristic
76C22×Dic53C2311yescharacteristic
77C22×Dic53C2311yescharacteristic
78C22×Dic53C2311yescharacteristic
79C2×C4×D53—22no—
80C2×C4×D53—22no—
81C2×C4×D53—41no—
82C2×Dic106—22no—
83C2×Dic106—22no—
84C2×Dic106—41no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53D414yescomplemented
88C22⋊C4×C53—22no—
89C22⋊C4×C53—22no—
90C22⋊Dic53—22no—
91C22⋊Dic53—22no—
92C22⋊Dic53—41no—
93#11719921—22no—
94#11719921—22no—
95#11719921—22no—
96#11719921—22no—
97#11719921—41no—
98C4⋊Dic53—22no—
99#11719919—22no—
100#11719919—22no—
101#11719919—22no—
102#11719919—22no—
103#11719919—22no—
104#11719919—22no—
105#11719919D414yes—
106C4×Dic53—22no—
107C4×Dic53—22no—
108C4×Dic53—22no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53C2×Q812yes—
116C23×C53C2411yescharacteristic
117C22×D53—21no—
118C22×D53—21no—
119C22×D53—21no—
120C22×D53—21no—
121C22×D53—21no—
122C22×D53—21no—
123C22×D53C2×Q812yes—
124Q8×C53—44no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—44no—
140C2×C4×C53C2×D412yes—
141C2×C4×C53C2×D412yes—
142C2×C4×C53C2×D412yes—
143C2×C4×C53C2×D412yes—
144C22⋊D53—28no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—21no—
155C2×Dic53—21no—
156C2×Dic53—21no—
157C2×Dic53—21no—
158C2×Dic53—21no—
159C2×Dic53—21no—
160C2×Dic53—21no—
161C2×Dic53C2×D412yes—
162C2×Dic53C2×D412yes—
163C2×Dic53C2×Q812yes—
164C2×Dic53#4112yes—
165C2×Dic53#4112yes—
166C4×D53—44no—
167Dic106—44no—
168C22×C53—22no—
169C22×C53—22no—
170C22×C53—22no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53C22×D411yescharacteristic
176C22×C53C22×D411yescharacteristic
177C22×C53C22×Q811yescharacteristic
178C22×C53C2×#4111yescharacteristic
179C22×C53C2×#4111yescharacteristic
180C22×C53H(5,𝔽2)11yescharacteristic
181C22×C532−1+411yescharacteristic
182C2×D53—22no—
183C2×D53—22no—
184C2×D53—22no—
185C2×D53—41no—
186C2×D53—41no—
187C2×D53—41no—
188C2×D53—41no—
189C4×C53—22no—
190C4×C53—22no—
191C4×C53—22no—
192C4×C53—22no—
193C4×C53—41no—
194C4×C53—41no—
195Dic53—22no—
196Dic53—22no—
197Dic53—41no—
198Dic53—41no—
199Dic53—41no—
200Dic53—41no—
201Dic53C22⋊Q814yescomplemented
202#2452—531nomaximal, Sylow
203C2×C53—42no—
204C2×C53C2×C22⋊Q811yescharacteristic
205C2×C53#53211yescharacteristic
206C2×C53#54211yescharacteristic
207C2×C53D4211yescharacteristic
208C2×C53#54811yescharacteristic
209C2×C53D4×Q811yescharacteristic
210C2×C53#55411yescharacteristic
211D53—42no—
212C2×C4⋊Q8—531no—
213C2×C22⋊Q8—531no—
214C2×C22⋊Q8D5311yescharacteristic, complemented
215C2×C4×D4—531no—
216#396—531no—
217#396—531no—
218#393—531no—
219#393—531no—
220#390—531no—
221#390—531no—
222#389—531no—
223#389—531no—
224#385—531no—
225#385—531no—
226C4×C4⋊C4—531no—
227C53#245211yescharacteristic, complemented, Sylow
228C22×Q8—531no—
229C22×Q8C2×D5311yescharacteristic
230C22×D4—531no—
231C23×C4—531no—
232C23×C4C2×D5311yescharacteristic
233C4⋊Q8—1064no—
234C22⋊Q8—24no—
235C22⋊Q8—1064no—
236C4×D4—1064no—
237C2×C4⋊C4—531no—
238C2×C4⋊C4—531no—
239C2×C4⋊C4—531no—
240C2×C4⋊C4—531no—
241C2×C4⋊C4—531no—
242C2×C4⋊C4—531no—
243C2×C4⋊C4—531no—
244C2×C4⋊C4—531no—
245C2×C4⋊C4C2×D5311yescharacteristic
246C2×C4⋊C4C2×D5311yescharacteristic
247C2×C4⋊C4C2×D5311yescharacteristic
248C2×C22⋊C4—531no—
249C2×C22⋊C4—531no—
250C2×C22⋊C4—531no—
251C2×C22⋊C4—531no—
252C2×C22⋊C4—531no—
253C2×C22⋊C4—531no—
254C2×C22⋊C4—531no—
255C2×C22⋊C4—531no—
256C2×C22⋊C4C2×D5311yescharacteristic
257C2×C22⋊C4C2×D5311yescharacteristic
258C2×C42—531no—
259C2×C42—531no—
260C2×C42—531no—
261#95—531no—
262#95—531no—
263#95—531no—
264#95—531no—
265#95—531no—
266#95—531no—
267C24—531no—
268C24C22×D5311yescharacteristic
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—1062no—
273C2×Q8—1062no—
274C2×Q8—2121no—
275C2×D4—534no—
276C2×D4—1062no—
277C2×D4—1062no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—41no—
281C22×C4—531no—
282C22×C4—531no—
283C22×C4—531no—
284C22×C4—531no—
285C22×C4—531no—
286C22×C4—531no—
287C22×C4—531no—
288C22×C4—1062no—
289C22×C4—1062no—
290C22×C4—2121no—
291C22×C4C22×D5311yescharacteristic
292C22×C4C22×D5311yescharacteristic
293C22×C4C22×D5311yescharacteristic
294C22×C4C22×D5311yescharacteristic
295C22×C4C22×D5311yescharacteristic
296C22×C4C22×D5311yescharacteristic
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—534no—
300C4⋊C4—1062no—
301C4⋊C4—1062no—
302C4⋊C4—1062no—
303C4⋊C4—1062no—
304C4⋊C4—1062no—
305C4⋊C4—1062no—
306C4⋊C4—1062no—
307C4⋊C4C22⋊D5314yescomplemented
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—1062no—
311C22⋊C4—1062no—
312C22⋊C4—1062no—
313C22⋊C4—1062no—
314C22⋊C4—1062no—
315C22⋊C4—1062no—
316C22⋊C4—2121no—
317C22⋊C4—2121no—
318C42—1062no—
319C42—1062no—
320C42—1062no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—532no—
328C23—1061no—
329C23—1061no—
330C23—1061no—
331C23—1061no—
332C23—1061no—
333C23—1061no—
334C23Q8×D5312yes—
335C23C23×D5311yescharacteristic
336Q8—44no—
337Q8—2124no—
338D4—1068no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—44no—
354C2×C4—532no—
355C2×C4—532no—
356C2×C4—532no—
357C2×C4—532no—
358C2×C4—532no—
359C2×C4—1061no—
360C2×C4—1061no—
361C2×C4—1061no—
362C2×C4—1061no—
363C2×C4—1061no—
364C2×C4—1061no—
365C2×C4—1061no—
366C2×C4—1061no—
367C2×C4—1061no—
368C2×C4—1061no—
369C2×C4—1061no—
370C2×C4—1061no—
371C2×C4—1061no—
372C2×C4—1061no—
373C2×C4—1061no—
374C2×C4—1061no—
375C2×C4—2124no—
376C2×C4D4×D5312yes—
377C2×C4C2×C22⋊D5312yes—
378C2×C4C2×C22⋊D5312yes—
379C2×C4C2×C22⋊D5312yes—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—1062no—
388C22—1062no—
389C22—1062no—
390C22—2121no—
391C22—2121no—
392C22—2121no—
393C22—2121no—
394C22C2×D4×D5311yescharacteristic
395C22C22×C22⋊D5311yescharacteristic
396C22C2×Q8×D5311yescharacteristic
397C22#41×D5311yescharacteristic
398C22#41×D5311yescharacteristic
399C22#1313326211yescharacteristic
400C22#1313326511yescharacteristic
401C4—22no—
402C4—22no—
403C4—22no—
404C4—22no—
405C4—41no—
406C4—41no—
407C4—534no—
408C4—1062no—
409C4—1062no—
410C4—2121no—
411C4—2121no—
412C4—2121no—
413C4—2121no—
414C2—42no—
415C2—2122no—
416C2C22⋊Q8×D5311yescharacteristic
417C2#1681116911yescharacteristic
418C2#1681124311yescharacteristic
419C2D4×C22⋊D5311yescharacteristic
420C2#1681128311yescharacteristic
421C2Q8×C22⋊D5311yescharacteristic
422C2#1681132711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×C22⋊Q8×C53C2
hypercenterC2×C22⋊Q8D53
hypocenterC53#2452
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#13133215C2semidirect
2. C2×#13133195C2semidirect
4. #16810546C2semidirect
5. #16810545C2semidirect
6. #16810535C2semidirect
7. #16810534C2semidirect
10. #16810518C2semidirect
11. #16810518C2semidirect
12. #16810496C2semidirect
13. #16810494C2semidirect
14. C4⋊C4×Dic53C2semidirect
15. C2×C22⋊Q8×C53C2semidirect
71. C2×C22⋊D53Q8semidirect
87. C4⋊C4×C53D4semidirect
201. Dic53C22⋊Q8semidirect
214. C2×C22⋊Q8D53semidirect
227. C53#2452semidirect
307. C4⋊C4C22⋊D53semidirect