Finite Group Databasebeta

← #20286826

Subgroups — #20286826

Order 6784.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811330C211yesmaximal, characteristic, complemented
2#16810982C211yesmaximal, characteristic, complemented
3#16810951C211yesmaximal, characteristic, complemented
4#16810939C211yesmaximal, characteristic, complemented
5#16810908C211yesmaximal, characteristic, complemented
6#16810879C211yesmaximal, characteristic, complemented
7#16810850C211yesmaximal, characteristic, complemented
8#16810842C211yesmaximal, characteristic, complemented
9#16810785C211yesmaximal, characteristic, complemented
10#16810779C211yesmaximal, characteristic, complemented
11#16810728C211yesmaximal, characteristic, complemented
12#16810695C211yesmaximal, characteristic, complemented
13#16810624C211yesmaximal, characteristic, complemented
14C2×#13133141C211yesmaximal, characteristic, complemented
15C53×#548C211yesmaximal, characteristic
16C2×#11719925C2211yescharacteristic
17C2×#11719925C2211yescharacteristic
18C2×#11719923C2211yescharacteristic
19C2×#11719917C2211yescharacteristic
20C22×#88322C2211yescharacteristic
21#13133217C2211yescharacteristic
22#13133208—21no—
23#13133196C2211yescharacteristic
24#13133196C2211yescharacteristic
25#13133194—21no—
26#13133194—21no—
27C4×Dic106C2211yescharacteristic
28D4×Dic53—21no—
29C2×C4⋊Dic53C2211yescharacteristic
30C2×C4⋊Dic53C2211yescharacteristic
31#13133148C2211yescharacteristic
32#13133148C2211yescharacteristic
33#13133148C2211yescharacteristic
34C4⋊#88322C2211yescharacteristic
35#13133141C2211yescharacteristic
36#13133141C2211yescharacteristic
37#13133141C2211yescharacteristic
38#13133141C2211yescharacteristic
39#13133141C2211yescharacteristic
40#13133141C2211yescharacteristic
41#13133140C2211yescharacteristic
42#13133138C2211yescharacteristic
43#13133138C2211yescharacteristic
44#13133138C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46C22⋊#88322C2211yescharacteristic
47#13133128C2211yescharacteristic
48C2×#41×C53C2211yescharacteristic
49C22×Q8×C53C2211yescharacteristic
50#124×C53—21no—
51#124×C53C2211yescharacteristic
52C22⋊Q8×C53—21no—
53C22⋊Q8×C53C2211yescharacteristic
54#121×C53—21no—
55#121×C53C2211yescharacteristic
56C4×Q8×C53C2211yescharacteristic
57C4×D4×C53—21no—
58C4×D4×C53C2211yescharacteristic
59#41×C53—22no—
60#41×C53—41no—
61C2×Q8×C53—21no—
62C2×Q8×C53—22no—
63C2×Q8×C53—41no—
64C2×Q8×C53C2311yescharacteristic
65C2×Q8×C53C2311yescharacteristic
66C2×D4×C53—21no—
67C2×D4×C53—21no—
68C2×D4×C53D411yescharacteristic
69C2×D4×C53C2311yescharacteristic
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53C2311yescharacteristic
73C22×C4×C53C2311yescharacteristic
74C22×Dic53—21no—
75C22×Dic53—21no—
76C2×Dic106—21no—
77C2×Dic106C2311yescharacteristic
78C4⋊C4×C53—21no—
79C4⋊C4×C53—21no—
80C4⋊C4×C53D411yescharacteristic
81C4⋊C4×C53C2311yescharacteristic
82C22⋊C4×C53—21no—
83C22⋊C4×C53—21no—
84C22⋊C4×C53—21no—
85C22⋊C4×C53—21no—
86C22⋊C4×C53D411yescharacteristic
87C22⋊C4×C53D411yescharacteristic
88C42×C53—21no—
89C42×C53C2311yescharacteristic
90C22⋊Dic53—21no—
91C22⋊Dic53—21no—
92#11719925—22no—
93#11719925—41no—
94#11719923—41no—
95C4⋊Dic53—21no—
96C4⋊Dic53—21no—
97C4⋊Dic53—21no—
98C4⋊Dic53C2311yescharacteristic
99C4⋊Dic53C2311yescharacteristic
100C4⋊Dic53C2311yescharacteristic
101#11719919—21no—
102#11719919—21no—
103#11719919—21no—
104C4×Dic53—21no—
105#11719917—41no—
106C2×#88322—41no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C23×C53—21no—
112C23×C53C2×D411yescharacteristic
113C23×C53C2×D411yescharacteristic
114Q8×C53—21no—
115Q8×C53—21no—
116Q8×C53—21no—
117Q8×C53—42no—
118Q8×C53#4112yes—
119D4×C53—22no—
120D4×C53—21no—
121D4×C53—21no—
122D4×C53—42no—
123D4×C53—42no—
124C2×C4×C53—21no—
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—42no—
132C2×C4×C53—42no—
133C2×C4×C53C2×D411yescharacteristic
134C2×C4×C53C2×D411yescharacteristic
135C2×C4×C53C2×D411yescharacteristic
136C2×C4×C53C2×D411yescharacteristic
137C2×C4×C53C2411yescharacteristic
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—81no—
143C2×Dic53—81no—
144Dic106—21no—
145Dic106—21no—
146Dic106—81no—
147#88322—21no—
148#88322—21no—
149#88322—21no—
150#88322—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53#13712yes—
160C22×C53C22×D411yescharacteristic
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—22no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53C2×#4111yescharacteristic
169C4×C53H(5,𝔽2)11yescharacteristic
170Dic53—41no—
171Dic53—41no—
172Dic53—41no—
173Dic53—41no—
174#3316—531nomaximal, Sylow
175C2×C53—22no—
176C2×C53—41no—
177C2×C53—81no—
178C2×C53#54611yescharacteristic
179C2×C53C2×#13711yescharacteristic
180C2×C53#57711yescharacteristic
181#554—531no—
182#548D5311yescharacteristic
183#489—531no—
184#483—531no—
185#481—531no—
186#475—531no—
187#470—531no—
188#465—531no—
189#464—531no—
190#452—531no—
191#451—531no—
192#441—531no—
193#435—531no—
194#417—531no—
195C2×#103—531no—
196C53#331611yescharacteristic, complemented, Sylow
197C2×#41C2×D5311yescharacteristic
198C22×Q8C2×D5311yescharacteristic
199C2×Q16—531no—
200C2×Q16—531no—
201C2×SD16—531no—
202C2×M16—531no—
203C22×C8—531no—
204C4⋊Q8—531no—
205#125—1061no—
206#124—21no—
207#124C2×D5311yescharacteristic
208C22⋊Q8—21no—
209C22⋊Q8—531no—
210C22⋊Q8—531no—
211C22⋊Q8—1061no—
212C22⋊Q8—1061no—
213C22⋊Q8C2×D5311yescharacteristic
214#121—21no—
215#121C2×D5311yescharacteristic
216C4×Q8—531no—
217C4×Q8C2×D5311yescharacteristic
218C4×D4—21no—
219C4×D4—1061no—
220C4×D4C2×D5311yescharacteristic
221C2×C4⋊C4—531no—
222C2×C4⋊C4—531no—
223#106—531no—
224#106—531no—
225#106—531no—
226C4⋊C8—531no—
227#103—531no—
228#103—531no—
229#103—531no—
230#103—531no—
231#103—531no—
232#103—531no—
233#103—531no—
234#102—531no—
235#102—531no—
236#102—531no—
237C22⋊C8—531no—
238C22⋊C8—531no—
239#97—531no—
240#41—22no—
241#41—41no—
242C2×Q8—21no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—531no—
246C2×Q8—1061no—
247C2×Q8C22×D5311yescharacteristic
248C2×Q8C22×D5311yescharacteristic
249C2×D4—21no—
250C2×D4—21no—
251C2×D4C22⋊D5311yescharacteristic
252C2×D4C22×D5311yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4C22×D5311yescharacteristic
258C22×C4C22×D5311yescharacteristic
259Q16—1062no—
260Q16—2121no—
261SD16—2121no—
262M16—2121no—
263C2×C8—531no—
264C2×C8—531no—
265C2×C8—531no—
266C2×C8—531no—
267C2×C8—2121no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—531no—
271C4⋊C4—531no—
272C4⋊C4—531no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4C22⋊D5311yescharacteristic
280C4⋊C4C22×D5311yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4C22⋊D5311yescharacteristic
288C22⋊C4C22⋊D5311yescharacteristic
289C42—21no—
290C42—1061no—
291C42C22×D5311yescharacteristic
292C23—21no—
293C23C2×C22⋊D5311yescharacteristic
294C23C2×C22⋊D5311yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—42no—
299Q8—1061no—
300Q8—1061no—
301Q8—4241no—
302Q8#1171994712yes—
303D4—22no—
304D4—21no—
305D4—21no—
306D4—42no—
307D4—42no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—4241no—
322C2×C4—4241no—
323C2×C4C2×C22⋊D5311yescharacteristic
324C2×C4C2×C22⋊D5311yescharacteristic
325C2×C4C2×C22⋊D5311yescharacteristic
326C2×C4C2×C22⋊D5311yescharacteristic
327C2×C4C23×D5311yescharacteristic
328C8—1061no—
329C8—1061no—
330C8—1061no—
331C8—1061no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22#1313324712yes—
341C22C22×C22⋊D5311yescharacteristic
342C4—21no—
343C4—21no—
344C4—22no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—2121no—
350C4—2121no—
351C4—2121no—
352C4—2121no—
353C4C2×#1171994711yescharacteristic
354C4#1313326211yescharacteristic
355C2—22no—
356C2—41no—
357C2—81no—
358C2#1681126711yescharacteristic
359C2C2×#1313324711yescharacteristic
360C2#1681143111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#548C2
socleC22×C53C22×D4
hypercenter#548D53
hypocenterC53#3316
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #16811330C2semidirect
2. #16810982C2semidirect
3. #16810951C2semidirect
4. #16810939C2semidirect
5. #16810908C2semidirect
6. #16810879C2semidirect
7. #16810850C2semidirect
8. #16810842C2semidirect
9. #16810785C2semidirect
10. #16810779C2semidirect
11. #16810728C2semidirect
12. #16810695C2semidirect
13. #16810624C2semidirect
14. C2×#13133141C2semidirect
196. C53#3316semidirect