Finite Group Databasebeta

← #27278411

Subgroups — #27278411

Order 13568.

392 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#20283381C211yesmaximal, characteristic
2#20273491C211yesmaximal, characteristic, complemented
3#20273470C211yesmaximal, characteristic, complemented
4#20273097C211yesmaximal, characteristic
5#20272618C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7C53×#1806C211yesmaximal, characteristic
8#16810684—22no—
9#16810680—21no—
10#16810676—21no—
11#16810676C2211yescharacteristic
12#16810647C2211yescharacteristic
13C2×C4⋊#88322C2211yescharacteristic
14#16810590—21no—
15#16810579—21no—
16C2×C4×#88322C2211yescharacteristic
17#16810375—21no—
18#16810375—21no—
19#16810373—22no—
20Q8⋊#88322C411yescharacteristic, complemented
21Q8⋊#88322C411yescharacteristic, complemented
22D4⋊#88322C411yescharacteristic, complemented
23D4⋊#88322C411yescharacteristic, complemented
24C4×#41×C53C2211yescharacteristic
25C53×#423C2211yescharacteristic
26C2×#104×C53—21no—
27C4×C4⋊C4×C53C2211yescharacteristic
28C53×#339—21no—
29C2×#11719917—21no—
30C22×#88322—21no—
31C22×#88322—21no—
32C4⋊#88322—21no—
33C4⋊#88322—21no—
34C4⋊#88322C2×C411yescharacteristic
35C4⋊#88322C2×C411yescharacteristic
36C4⋊#88322D412yes—
37C4⋊#88322D411yescharacteristic
38C4⋊#88322D411yescharacteristic
39C22⋊#88322—22no—
40C22⋊#88322—22no—
41C22⋊#88322—21no—
42#13133128—22no—
43C4×#88322—22no—
44C4×#88322—21no—
45C4×#88322—21no—
46C4×#88322C2×C411yescharacteristic
47C4×#88322C2×C411yescharacteristic
48C2×#41×C53D411yescharacteristic
49C2×M16×C53—21no—
50C4×Q8×C53C2×C411yescharacteristic
51C4×Q8×C53Q811yescharacteristic
52C4×D4×C53—21no—
53C4×D4×C53—21no—
54C4×D4×C53C2×C411yescharacteristic
55C4×D4×C53Q811yescharacteristic
56#117×C53—21no—
57#117×C53D411yescharacteristic
58C2×C4⋊C4×C53—21no—
59C2×C42×C53—21no—
60C2×C42×C53—21no—
61C2×C42×C53C2311yescharacteristic
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64#104×C53—42no—
65#95×C53—21no—
66#41×C53—21no—
67#41×C53—21no—
68#41×C53—21no—
69#41×C53—21no—
70C2×Q8×C53C4⋊C411yescharacteristic
71C2×D4×C53—21no—
72C2×D4×C53C4⋊C411yescharacteristic
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—41no—
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53#4111yescharacteristic
79C22×C4×C53#4111yescharacteristic
80M16×C53—41no—
81M16×C53—41no—
82C2×C8×C53—41no—
83C4⋊C4×C53—21no—
84C4⋊C4×C53—21no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—41no—
87C4⋊C4×C53C4⋊C411yescharacteristic
88C4⋊C4×C53C4⋊C411yescharacteristic
89C22⋊C4×C53—21no—
90C22⋊C4×C53—21no—
91C22⋊C4×C53—21no—
92C42×C53—21no—
93C42×C53—21no—
94C42×C53—21no—
95C42×C53—41no—
96C42×C53—41no—
97C42×C53C22×C411yescharacteristic
98C42×C53C2×D411yescharacteristic
99C42×C53C2×D411yescharacteristic
100C42×C53C2×Q811yescharacteristic
101#11719917—41no—
102#11719917—41no—
103C2×#88322—21no—
104C2×#88322—21no—
105C2×#88322—21no—
106C2×#88322—21no—
107C2×#88322—21no—
108C2×#88322—22no—
109C2×#88322—21no—
110C2×#88322—42no—
111C2×#88322—41no—
112C2×#88322—41no—
113C23×C53—21no—
114C23×C53#12311yescharacteristic
115Q8×C53—21no—
116Q8×C53—21no—
117D4×C53—21no—
118D4×C53—21no—
119D4×C53—41no—
120D4×C53—41no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—42no—
132C2×C4×C53—81no—
133C2×C4×C53—81no—
134C2×C4×C53—81no—
135C2×C4×C53C2×C4⋊C411yescharacteristic
136C2×C4×C53C4×D411yescharacteristic
137C2×C4×C53C4×D411yescharacteristic
138C2×C4×C53#12011yescharacteristic
139C2×C4×C53C22⋊Q811yescharacteristic
140C2×C4×C53C22⋊Q811yescharacteristic
141C2×C4×C53#13511yescharacteristic
142C2×C4×C53#13511yescharacteristic
143C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C53#13711yescharacteristic
145C8×C53—81no—
146#88322—22no—
147#88322—41no—
148#88322—41no—
149#88322—41no—
150#88322—41no—
151#88322—41no—
152#88322—42no—
153#8905—531nomaximal, Sylow
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53#38511yescharacteristic
160C22×C53#48211yescharacteristic
161C22×C53#48511yescharacteristic
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—41no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—82no—
169C4×C53#42711yescharacteristic
170C4×C53#42811yescharacteristic
171C4×C53#44311yescharacteristic
172C4×C53#44411yescharacteristic
173C4×C53#44911yescharacteristic
174C4×C53#45211yescharacteristic
175#2925—531no—
176#1806D5311yescharacteristic
177#1571—531no—
178#1566—531no—
179#1476—531no—
180#1303—531no—
181#1279—531no—
182C2×C53—21no—
183C2×C53—21no—
184C2×C53—81no—
185C2×C53#185411yescharacteristic
186C2×C53#186111yescharacteristic
187C2×C53#186611yescharacteristic
188C4×#41C2×D5311yescharacteristic
189#433—1062no—
190#432—1061no—
191#431—531no—
192#431—1061no—
193#423—531no—
194#423C2×D5311yescharacteristic
195C2×C4⋊C8—531no—
196C2×#104—21no—
197#408—1061no—
198#405—1061no—
199C2×C4×C8—531no—
200C4×C4⋊C4C2×D5311yescharacteristic
201#339—21no—
202#335—1061no—
203#335—1061no—
204#334—1062no—
205Q8⋊C8—531no—
206Q8⋊C8—531no—
207D4⋊C8—531no—
208D4⋊C8—531no—
209C53#890511yescharacteristic, complemented, Sylow
210C2×#41C22⋊D5311yescharacteristic
211C2×M16—21no—
212C2×M16—1061no—
213C22×C8—1061no—
214C22×C8—1061no—
215C4×Q8Dic10611yescharacteristic
216C4×Q8C4×D5311yescharacteristic
217C4×D4—21no—
218C4×D4—21no—
219C4×D4Dic10611yescharacteristic
220C4×D4C4×D5311yescharacteristic
221#117—21no—
222#117C22⋊D5311yescharacteristic
223C2×C4⋊C4—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42C22×D5311yescharacteristic
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—531no—
230C4⋊C8—531no—
231C4⋊C8—532no—
232C4⋊C8—531no—
233C4⋊C8—531no—
234C4⋊C8—1061no—
235C4⋊C8—1061no—
236#104—42no—
237C22⋊C8—1062no—
238C22⋊C8—1062no—
239C22⋊C8—1061no—
240#97—1062no—
241C4×C8—531no—
242C4×C8—531no—
243C4×C8—1062no—
244C4×C8—1061no—
245C4×C8—1061no—
246#95—21no—
247#41—21no—
248#41—21no—
249#41—21no—
250#41—21no—
251C2×Q8#1171991911yescharacteristic
252C2×D4—21no—
253C2×D4#1171991911yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—41no—
258C22×C4#1171994711yescharacteristic
259C22×C4#1171994711yescharacteristic
260C22×C4C2×C22⋊D5311yescharacteristic
261M16—41no—
262M16—41no—
263M16—2121no—
264M16—2121no—
265C2×C8—41no—
266C2×C8—1061no—
267C2×C8—1061no—
268C2×C8—1061no—
269C2×C8—1061no—
270C2×C8—1061no—
271C2×C8—1062no—
272C2×C8—1061no—
273C2×C8—2122no—
274C2×C8—2121no—
275C2×C8—2121no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—22no—
279C4⋊C4—41no—
280C4⋊C4#1171991911yescharacteristic
281C4⋊C4#1171991911yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—41no—
289C42—41no—
290C42C2×Dic10611yescharacteristic
291C42C2×C4×D5311yescharacteristic
292C42D4×D5311yescharacteristic
293C42D4×D5311yescharacteristic
294C23—21no—
295C23#1313319811yescharacteristic
296Q8—21no—
297Q8—21no—
298D4—21no—
299D4—21no—
300D4—41no—
301D4—41no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4C2×#1171991911yescharacteristic
317C2×C4#1313317711yescharacteristic
318C2×C4#1313317711yescharacteristic
319C2×C4#1313318511yescharacteristic
320C2×C4#1313319411yescharacteristic
321C2×C4#1313319411yescharacteristic
322C2×C4#1313323711yescharacteristic
323C2×C4#1313323711yescharacteristic
324C2×C4#1313324311yescharacteristic
325C2×C4#1313324811yescharacteristic
326C8—81no—
327C8—1062no—
328C8—2121no—
329C8—2121no—
330C8—2121no—
331C8—2122no—
332C8—2121no—
333C8—2121no—
334C22—21no—
335C22—21no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22#1681049511yescharacteristic
340C22#1681094411yescharacteristic
341C22#1681096111yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—82no—
349C4#1681065911yescharacteristic
350C4#1681066311yescharacteristic
351C4#1681073711yescharacteristic
352C4#1681074211yescharacteristic
353C4#1681076611yescharacteristic
354C4#1681078311yescharacteristic
355C2—21no—
356C2—21no—
357C2—81no—
358C2#2027491211yescharacteristic
359C2#2027495311yescharacteristic
360C2#2027498811yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1806C2
socleC22×C53#385
prime radicalC2×#41×C53D4
cosocleC2×C42×C53C23
hypercenter#1806D53
hypocenterC53#8905
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #20273491C2semidirect
3. #20273470C2semidirect
5. #20272618C2semidirect
6. #20272556C2semidirect
20. Q8⋊#88322C4semidirect
21. Q8⋊#88322C4semidirect
22. D4⋊#88322C4semidirect
23. D4⋊#88322C4semidirect
209. C53#8905semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202833812
#202734912
#202734702
#202730972
#202726182
#202725562
C53×#18062
#890553