Finite Group Databasebeta

← #20286126

Subgroups — #20286126

Order 6784.

358 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811228C211yesmaximal, characteristic, complemented
2#16810930C211yesmaximal, characteristic, complemented
3#16810923C211yesmaximal, characteristic, complemented
4#16810910C211yesmaximal, characteristic, complemented
5#16810900C211yesmaximal, characteristic, complemented
6#16810874C211yesmaximal, characteristic, complemented
7#16810871C211yesmaximal, characteristic, complemented
8#16810857C211yesmaximal, characteristic, complemented
9#16810854C211yesmaximal, characteristic, complemented
10#16810835C211yesmaximal, characteristic, complemented
11#16810835C211yesmaximal, characteristic, complemented
12#16810823C211yesmaximal, characteristic, complemented
13#16810818C211yesmaximal, characteristic, complemented
14#16810650C211yesmaximal, characteristic, complemented
15C53×#542C211yesmaximal, characteristic, complemented
16C2×#11719924C2211yescharacteristic
17C2×#11719924C2211yescharacteristic
18C2×#11719922C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20C2×#11719917C2211yescharacteristic
21C22×#88322C2211yescharacteristic
22#13133214C2211yescharacteristic
23#13133209C2211yescharacteristic
24#13133202—21no—
25#13133202—21no—
26#13133191C2211yescharacteristic
27#13133191C2211yescharacteristic
28#13133190—21no—
29#13133190—21no—
30C4×D212C2211yescharacteristic
31C4×D212C2211yescharacteristic
32#13133150C2211yescharacteristic
33#13133150C2211yescharacteristic
34#13133148C2211yescharacteristic
35#13133148C2211yescharacteristic
36C4⋊#88322C2211yescharacteristic
37C4⋊#88322C2211yescharacteristic
38C4⋊#88322C2211yescharacteristic
39C4⋊#88322C2211yescharacteristic
40#13133141C2211yescharacteristic
41#13133141C2211yescharacteristic
42#13133138C2211yescharacteristic
43#13133138C2211yescharacteristic
44#13133137C2211yescharacteristic
45#13133137C2211yescharacteristic
46#13133137C2211yescharacteristic
47#13133137C2211yescharacteristic
48C4⋊Q8×C53C2211yescharacteristic
49#126×C53—21no—
50#124×C53—21no—
51#124×C53C2211yescharacteristic
52#123×C53—21no—
53C22⋊Q8×C53—21no—
54C22⋊Q8×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic
56C4×Q8×C53C2211yescharacteristic
57C4×D4×C53C2211yescharacteristic
58#117×C53C2211yescharacteristic
59C2×Q8×C53—21no—
60C2×Q8×C53C2311yescharacteristic
61C2×D4×C53—21no—
62C2×D4×C53C2311yescharacteristic
63C22×C4×C53—21no—
64C22×C4×C53C2311yescharacteristic
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×D212—21no—
68C2×D212—21no—
69C2×D212C2311yescharacteristic
70C2×D212C2311yescharacteristic
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C4⋊C4×C53—21no—
74C4⋊C4×C53—21no—
75C4⋊C4×C53—21no—
76C4⋊C4×C53D411yescharacteristic
77C4⋊C4×C53D411yescharacteristic
78C4⋊C4×C53C2311yescharacteristic
79C4⋊C4×C53C2311yescharacteristic
80C22⋊C4×C53—21no—
81C22⋊C4×C53—21no—
82C22⋊C4×C53—21no—
83C22⋊C4×C53—21no—
84C22⋊C4×C53—21no—
85C22⋊C4×C53D411yescharacteristic
86C22⋊C4×C53D411yescharacteristic
87C42×C53—21no—
88C42×C53C2311yescharacteristic
89C42×C53C2311yescharacteristic
90#11719924—41no—
91#11719924—41no—
92#11719922—41no—
93#11719922—41no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98C4⋊Dic53C2311yescharacteristic
99C4⋊Dic53C2311yescharacteristic
100#11719919—21no—
101#11719919—21no—
102#11719917—41no—
103C2×#88322—41no—
104C2×#88322C2311yescharacteristic
105C2×#88322C2311yescharacteristic
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108C23×C53—21no—
109C23×C53C2×D411yescharacteristic
110C22×D53—21no—
111C22×D53—21no—
112Q8×C53—21no—
113Q8×C53—21no—
114Q8×C53—81no—
115D4×C53—21no—
116D4×C53—21no—
117D4×C53—81no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—81no—
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2×D411yescharacteristic
127C2×C4×C53C2×D411yescharacteristic
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2411yescharacteristic
130C22⋊D53—81no—
131C22⋊D53—81no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134D212—21no—
135D212—21no—
136D212—21no—
137D212—21no—
138D212—81no—
139D212—81no—
140C4×D53—81no—
141C4×D53—81no—
142#88322—21no—
143#88322—21no—
144#88322—21no—
145#88322—21no—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53C22×D411yescharacteristic
153C2×D53—41no—
154C2×D53—41no—
155C2×D53—41no—
156C2×D53—41no—
157C2×D53—41no—
158C2×D53—41no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—41no—
165C4×C53—41no—
166C4×C53—41no—
167C4×C53H(5,𝔽2)11yescharacteristic
168C4×C532−1+411yescharacteristic
169Dic53—41no—
170Dic53—41no—
171#3257—531nomaximal, Sylow
172C2×C53—41no—
173C2×C53—81no—
174C2×C53#53711yescharacteristic
175C2×C53#57611yescharacteristic
176C2×C53#57711yescharacteristic
177D53—81no—
178D53—81no—
179#542D5311yescharacteristic, complemented
180#540—531no—
181#479—531no—
182#478—531no—
183#476—531no—
184#474—531no—
185#469—531no—
186#468—531no—
187#466—531no—
188#465—531no—
189#463—531no—
190#463—531no—
191#460—531no—
192#459—531no—
193#424—531no—
194C53#325711yescharacteristic, complemented, Sylow
195C2×SD16—531no—
196C2×SD16—531no—
197C2×D8—531no—
198C2×D8—531no—
199C2×M16—531no—
200C22×C8—531no—
201C4⋊Q8C2×D5311yescharacteristic
202#127—531no—
203#126—21no—
204#125—531no—
205#124—21no—
206#124C2×D5311yescharacteristic
207#123—21no—
208#123—1061no—
209#123—1061no—
210C22⋊Q8—21no—
211C22⋊Q8C2×D5311yescharacteristic
212#121—531no—
213#121—531no—
214#121—1061no—
215#121—1061no—
216#121C2×D5311yescharacteristic
217C4×Q8C2×D5311yescharacteristic
218C4×D4—531no—
219C4×D4—531no—
220C4×D4C2×D5311yescharacteristic
221#117C2×D5311yescharacteristic
222#107—531no—
223#107—531no—
224#106—531no—
225#106—531no—
226C4⋊C8—531no—
227C4⋊C8—531no—
228C4⋊C8—531no—
229C4⋊C8—531no—
230#103—531no—
231#103—531no—
232#102—531no—
233#102—531no—
234#102—531no—
235#102—531no—
236#102—531no—
237#102—531no—
238C2×Q8—21no—
239C2×Q8C22×D5311yescharacteristic
240C2×D4—21no—
241C2×D4—531no—
242C2×D4—531no—
243C2×D4—1061no—
244C2×D4—1061no—
245C2×D4—1061no—
246C2×D4—1061no—
247C2×D4C22×D5311yescharacteristic
248C22×C4—21no—
249C22×C4—1061no—
250C22×C4—1061no—
251C22×C4C22×D5311yescharacteristic
252SD16—2121no—
253SD16—2121no—
254D8—2121no—
255D8—2121no—
256M16—2121no—
257C2×C8—531no—
258C2×C8—531no—
259C2×C8—531no—
260C2×C8—531no—
261C2×C8—2121no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—531no—
266C4⋊C4—531no—
267C4⋊C4—1061no—
268C4⋊C4—1061no—
269C4⋊C4C22⋊D5311yescharacteristic
270C4⋊C4C22⋊D5311yescharacteristic
271C4⋊C4C22×D5311yescharacteristic
272C4⋊C4C22×D5311yescharacteristic
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4C22⋊D5311yescharacteristic
283C22⋊C4C22⋊D5311yescharacteristic
284C42—21no—
285C42C22×D5311yescharacteristic
286C42C22×D5311yescharacteristic
287C23—21no—
288C23—1061no—
289C23—1061no—
290C23C2×C22⋊D5311yescharacteristic
291Q8—21no—
292Q8—21no—
293Q8—81no—
294D4—21no—
295D4—21no—
296D4—81no—
297D4—1061no—
298D4—1061no—
299D4—1061no—
300D4—1061no—
301D4—4241no—
302D4—4241no—
303D4—4241no—
304D4—4241no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—4241no—
314C2×C4—4241no—
315C2×C4C2×C22⋊D5311yescharacteristic
316C2×C4C2×C22⋊D5311yescharacteristic
317C2×C4C2×C22⋊D5311yescharacteristic
318C2×C4C2×C22⋊D5311yescharacteristic
319C2×C4C2×C22⋊D5311yescharacteristic
320C2×C4C23×D5311yescharacteristic
321C8—1061no—
322C8—1061no—
323C8—1061no—
324C8—1061no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—2121no—
332C22—2121no—
333C22—2121no—
334C22—2121no—
335C22—2121no—
336C22—2121no—
337C22C22×C22⋊D5311yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—2121no—
347C4—2121no—
348C4#1313326311yescharacteristic
349C4#1313326411yescharacteristic
350C2—41no—
351C2—81no—
352C2—4241no—
353C2—4241no—
354C2#1681120111yescharacteristic
355C2#1681142611yescharacteristic
356C2#1681143111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#542C2
socleC22×C53C22×D4
hypercenter#542D53
hypocenterC53#3257
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #16811228C2semidirect
2. #16810930C2semidirect
3. #16810923C2semidirect
4. #16810910C2semidirect
5. #16810900C2semidirect
6. #16810874C2semidirect
7. #16810871C2semidirect
8. #16810857C2semidirect
9. #16810854C2semidirect
10. #16810835C2semidirect
11. #16810835C2semidirect
12. #16810823C2semidirect
13. #16810818C2semidirect
14. #16810650C2semidirect
15. C53×#542C2semidirect
179. #542D53semidirect
194. C53#3257semidirect