Finite Group Databasebeta

← #20286766

Subgroups — #20286766

Order 6784.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811313C211yesmaximal, characteristic
2#16810991C211yesmaximal, characteristic, complemented
3#16810961C211yesmaximal, characteristic, complemented
4#16810951C211yesmaximal, characteristic, complemented
5#16810930C211yesmaximal, characteristic, complemented
6#16810879C211yesmaximal, characteristic, complemented
7#16810862C211yesmaximal, characteristic, complemented
8#16810830C211yesmaximal, characteristic, complemented
9#16810779C211yesmaximal, characteristic, complemented
10#16810779C211yesmaximal, characteristic, complemented
11#16810728C211yesmaximal, characteristic, complemented
12#16810695C211yesmaximal, characteristic
13#16810632C211yesmaximal, characteristic, complemented
14C2×#13133141C211yesmaximal, characteristic, complemented
15D4×Q8×C53C211yesmaximal, characteristic
16C2×#11719925C2211yescharacteristic
17C2×#11719925C2211yescharacteristic
18C2×#11719925C2211yescharacteristic
19C2×#11719917C2211yescharacteristic
20C22×#88322C2211yescharacteristic
21#13133210—21no—
22#13133209C2211yescharacteristic
23#13133207—21no—
24#13133198—21no—
25#13133196C2211yescharacteristic
26#13133196C2211yescharacteristic
27C4×Dic106C2211yescharacteristic
28#13133176C2211yescharacteristic
29#13133173—21no—
30C2×C4⋊Dic53C2211yescharacteristic
31#13133150C2211yescharacteristic
32#13133148C2211yescharacteristic
33#13133148C2211yescharacteristic
34C4⋊#88322C2211yescharacteristic
35#13133141C2211yescharacteristic
36#13133141C2211yescharacteristic
37#13133141C2211yescharacteristic
38#13133141C2211yescharacteristic
39#13133141C2211yescharacteristic
40#13133141C2211yescharacteristic
41#13133141C2211yescharacteristic
42#13133141C2211yescharacteristic
43#13133141C2211yescharacteristic
44#13133140C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46C22⋊#88322C2211yescharacteristic
47#13133128C2211yescharacteristic
48C22×Q8×C53C2211yescharacteristic
49C22×Q8×C53C2211yescharacteristic
50C4⋊Q8×C53—21no—
51C4⋊Q8×C53C2211yescharacteristic
52C22⋊Q8×C53—21no—
53C22⋊Q8×C53—21no—
54C22⋊Q8×C53C2211yescharacteristic
55C22⋊Q8×C53C2211yescharacteristic
56C4×Q8×C53C2211yescharacteristic
57C4×D4×C53—21no—
58C4×D4×C53C2211yescharacteristic
59C2×Q8×C53—21no—
60C2×Q8×C53—21no—
61C2×Q8×C53—22no—
62C2×Q8×C53—22no—
63C2×Q8×C53—41no—
64C2×Q8×C53—41no—
65C2×Q8×C53C2311yescharacteristic
66C2×Q8×C53C2311yescharacteristic
67C2×Q8×C53C2311yescharacteristic
68C2×D4×C53D411yescharacteristic
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53C2311yescharacteristic
72C22×C4×C53C2311yescharacteristic
73C22×Dic53—21no—
74C2×Dic106C2311yescharacteristic
75C4⋊C4×C53—21no—
76C4⋊C4×C53—21no—
77C4⋊C4×C53—21no—
78C4⋊C4×C53—21no—
79C4⋊C4×C53—21no—
80C4⋊C4×C53D411yescharacteristic
81C4⋊C4×C53C2311yescharacteristic
82C22⋊C4×C53—21no—
83C22⋊C4×C53—21no—
84C22⋊C4×C53D411yescharacteristic
85C22⋊C4×C53D411yescharacteristic
86C42×C53—21no—
87C42×C53C2311yescharacteristic
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90C22⋊Dic53—21no—
91#11719925—22no—
92#11719925—41no—
93#11719925—41no—
94C4⋊Dic53—21no—
95C4⋊Dic53C2311yescharacteristic
96C4⋊Dic53C2311yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103C4×Dic53—21no—
104C4×Dic53—21no—
105#11719917—41no—
106C2×#88322—41no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C23×C53C2×D411yescharacteristic
112C23×C53C2×D411yescharacteristic
113Q8×C53—21no—
114Q8×C53—21no—
115Q8×C53—21no—
116Q8×C53—21no—
117Q8×C53—21no—
118Q8×C53—42no—
119Q8×C53—42no—
120Q8×C53#4112yes—
121D4×C53—22no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
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—42no—
131C2×C4×C53—42no—
132C2×C4×C53C2×D411yescharacteristic
133C2×C4×C53C2×D411yescharacteristic
134C2×C4×C53C2×D411yescharacteristic
135C2×C4×C53C2×D411yescharacteristic
136C2×C4×C53C2411yescharacteristic
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—42no—
142Dic106—21no—
143Dic106—21no—
144#88322—21no—
145#88322—21no—
146#88322—21no—
147#88322—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53#13712yes—
154C22×C53C22×D411yescharacteristic
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—22no—
158C4×C53—21no—
159C4×C53—21no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53C2×#4111yescharacteristic
164C4×C53H(5,𝔽2)11yescharacteristic
165Dic53—41no—
166Dic53—41no—
167Dic53—41no—
168Dic53—41no—
169#3312—531nomaximal, Sylow
170C2×C53—22no—
171C2×C53—41no—
172C2×C53#54611yescharacteristic
173C2×C53C2×#13711yescharacteristic
174C2×C53#57811yescharacteristic
175#552—531no—
176D4×Q8D5311yescharacteristic
177#491—531no—
178#485—531no—
179#483—531no—
180#479—531no—
181#470—531no—
182#467—531no—
183#462—531no—
184#451—531no—
185#451—531no—
186#441—531no—
187#435—531no—
188#419—531no—
189C2×#103—531no—
190C53#331211yescharacteristic, complemented, Sylow
191C22×Q8C2×D5311yescharacteristic
192C22×Q8C2×D5311yescharacteristic
193C2×Q16—531no—
194C2×Q16—531no—
195C2×Q16—531no—
196C2×M16—531no—
197C22×C8—531no—
198C4⋊Q8—21no—
199C4⋊Q8C2×D5311yescharacteristic
200#126—1061no—
201#125—531no—
202#125—1061no—
203#123—1061no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206C22⋊Q8—531no—
207C22⋊Q8—531no—
208C22⋊Q8C2×D5311yescharacteristic
209C22⋊Q8C2×D5311yescharacteristic
210C4×Q8—531no—
211C4×Q8C2×D5311yescharacteristic
212C4×D4—21no—
213C4×D4C2×D5311yescharacteristic
214#117—531no—
215#117—1061no—
216C2×C4⋊C4—531no—
217#107—531no—
218#106—531no—
219#106—531no—
220C4⋊C8—531no—
221#103—531no—
222#103—531no—
223#103—531no—
224#103—531no—
225#103—531no—
226#103—531no—
227#103—531no—
228#103—531no—
229#103—531no—
230#103—531no—
231C22⋊C8—531no—
232C22⋊C8—531no—
233#97—531no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—531no—
241C2×Q8C22×D5311yescharacteristic
242C2×Q8C22×D5311yescharacteristic
243C2×Q8C22×D5311yescharacteristic
244C2×D4C22⋊D5311yescharacteristic
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—1061no—
248C22×C4C22×D5311yescharacteristic
249C22×C4C22×D5311yescharacteristic
250Q16—1062no—
251Q16—2121no—
252Q16—2121no—
253M16—2121no—
254C2×C8—531no—
255C2×C8—531no—
256C2×C8—531no—
257C2×C8—531no—
258C2×C8—2121no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—531no—
265C4⋊C4—531no—
266C4⋊C4—531no—
267C4⋊C4—1061no—
268C4⋊C4—1061no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4C22⋊D5311yescharacteristic
274C4⋊C4C22×D5311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—1061no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C22⋊C4C22⋊D5311yescharacteristic
281C22⋊C4C22⋊D5311yescharacteristic
282C42—21no—
283C42—1061no—
284C42—1061no—
285C42C22×D5311yescharacteristic
286C23C2×C22⋊D5311yescharacteristic
287C23C2×C22⋊D5311yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—42no—
294Q8—42no—
295Q8—1061no—
296Q8—1061no—
297Q8#1171994712yes—
298D4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—2122no—
314C2×C4C2×C22⋊D5311yescharacteristic
315C2×C4C2×C22⋊D5311yescharacteristic
316C2×C4C2×C22⋊D5311yescharacteristic
317C2×C4C2×C22⋊D5311yescharacteristic
318C2×C4C23×D5311yescharacteristic
319C8—1061no—
320C8—1061no—
321C8—1061no—
322C8—1061no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22#1313324812yes—
329C22C22×C22⋊D5311yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—21no—
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—2121no—
339C4—2121no—
340C4—2121no—
341C4—2121no—
342C4C2×#1171994711yescharacteristic
343C4#1313326211yescharacteristic
344C2—22no—
345C2—41no—
346C2#1681126711yescharacteristic
347C2C2×#1313324811yescharacteristic
348C2#1681143811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingD4×Q8×C53C2
socleC22×C53C22×D4
hypercenterD4×Q8D53
hypocenterC53#3312
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
2. #16810991C2semidirect
3. #16810961C2semidirect
4. #16810951C2semidirect
5. #16810930C2semidirect
6. #16810879C2semidirect
7. #16810862C2semidirect
8. #16810830C2semidirect
9. #16810779C2semidirect
10. #16810779C2semidirect
11. #16810728C2semidirect
13. #16810632C2semidirect
14. C2×#13133141C2semidirect
190. C53#3312semidirect