Finite Group Databasebeta

← #27278404

Subgroups — #27278404

Order 13568.

438 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283283C211yesmaximal, characteristic
2#20273487C211yesmaximal, characteristic, complemented
3#20273470C211yesmaximal, characteristic, complemented
4#20273097C211yesmaximal, characteristic
5#20272618C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7C53×#1806C211yesmaximal, characteristic
8C2×#13133224—21no—
9#16810676C2211yescharacteristic
10#16810650—22no—
11#16810650—21no—
12#16810647—21no—
13#16810647C2211yescharacteristic
14C2×C4⋊#88322—21no—
15C2×C4⋊#88322C2211yescharacteristic
16C2×C4×#88322C2211yescharacteristic
17#16810373—22no—
18#16810373—21no—
19#16810373—21no—
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—
29#13133224—42no—
30#13133224—42no—
31C2×#11719917—22no—
32C2×#11719917—21no—
33C22×#88322—22no—
34C22×#88322—21no—
35C22×#88322—21no—
36C4⋊#88322—22no—
37C4⋊#88322—22no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322C2×C411yescharacteristic
41C4⋊#88322C2×C411yescharacteristic
42C4⋊#88322D411yescharacteristic
43C4⋊#88322D411yescharacteristic
44C4⋊#88322D412yes—
45C22⋊#88322—21no—
46C4×#88322—21no—
47C4×#88322—21no—
48C4×#88322C2×C411yescharacteristic
49C4×#88322C2×C411yescharacteristic
50C2×#41×C53D411yescharacteristic
51C2×M16×C53—21no—
52C4×Q8×C53C2×C411yescharacteristic
53C4×Q8×C53Q811yescharacteristic
54C4×D4×C53—21no—
55C4×D4×C53—21no—
56C4×D4×C53C2×C411yescharacteristic
57C4×D4×C53Q811yescharacteristic
58#117×C53—21no—
59#117×C53D411yescharacteristic
60C2×C4⋊C4×C53—21no—
61C2×C42×C53—21no—
62C2×C42×C53—21no—
63C2×C42×C53C2311yescharacteristic
64C4⋊C8×C53—21no—
65C4⋊C8×C53—21no—
66#104×C53—42no—
67#95×C53—21no—
68#41×C53—21no—
69#41×C53—21no—
70#41×C53SD1612yes—
71#41×C53SD1612yes—
72C2×Q8×C53C4⋊C411yescharacteristic
73C2×D4×C53—21no—
74C2×D4×C53C4⋊C411yescharacteristic
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—41no—
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53#4111yescharacteristic
81C22×C4×C53#4111yescharacteristic
82M16×C53—41no—
83M16×C53—41no—
84C2×C8×C53—41no—
85C4⋊C4×C53—21no—
86C4⋊C4×C53—21no—
87C4⋊C4×C53—22no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53C4⋊C411yescharacteristic
90C4⋊C4×C53C4⋊C411yescharacteristic
91C22⋊C4×C53—21no—
92C22⋊C4×C53—21no—
93C22⋊C4×C53—21no—
94C42×C53—21no—
95C42×C53—21no—
96C42×C53—21no—
97C42×C53—41no—
98C42×C53—41no—
99C42×C53C22×C411yescharacteristic
100C42×C53C2×D411yescharacteristic
101C42×C53C2×D411yescharacteristic
102C42×C53C2×Q811yescharacteristic
103#11719917—44no—
104#11719917—41no—
105#11719917—41no—
106C2×#88322—21no—
107C2×#88322—21no—
108C2×#88322—21no—
109C2×#88322—21no—
110C2×#88322—21no—
111C2×#88322—22no—
112C2×#88322—21no—
113C2×#88322—42no—
114C2×#88322—44no—
115C2×#88322—41no—
116C2×#88322—41no—
117C23×C53—21no—
118C23×C53#12311yescharacteristic
119Q8×C53—21no—
120Q8×C53#10612yes—
121D4×C53—22no—
122D4×C53—22no—
123D4×C53—21no—
124D4×C53#10612yes—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—22no—
129C2×C4×C53—22no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—42no—
136C2×C4×C53—81no—
137C2×C4×C53—81no—
138C2×C4×C53—81no—
139C2×C4×C53C2×C4⋊C411yescharacteristic
140C2×C4×C53C4×D411yescharacteristic
141C2×C4×C53C4×D411yescharacteristic
142C2×C4×C53#12011yescharacteristic
143C2×C4×C53C22⋊Q811yescharacteristic
144C2×C4×C53C22⋊Q811yescharacteristic
145C2×C4×C53C2×SD1611yescharacteristic
146C2×C4×C53C2×SD1611yescharacteristic
147C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C53#13711yescharacteristic
149C8×C53—81no—
150#88322—22no—
151#88322—41no—
152#88322—41no—
153#88322—41no—
154#88322—41no—
155#88322—42no—
156#88322—41no—
157#8904—531nomaximal, Sylow
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—22no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53#38511yescharacteristic
164C22×C53#48111yescharacteristic
165C22×C53#48311yescharacteristic
166C4×C53—22no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53—82no—
173C4×C53C2×#10611yescharacteristic
174C4×C53#42811yescharacteristic
175C4×C53#44311yescharacteristic
176C4×C53#44411yescharacteristic
177C4×C53#44811yescharacteristic
178C4×C53#45011yescharacteristic
179#2906—531no—
180#1806D5311yescharacteristic
181#1570—531no—
182#1566—531no—
183#1476—531no—
184#1303—531no—
185#1279—531no—
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—42no—
189C2×C53#185311yescharacteristic
190C2×C53#186111yescharacteristic
191C2×C53#186611yescharacteristic
192C2×#131—1061no—
193C4×#41C2×D5311yescharacteristic
194#431—531no—
195#424—1062no—
196#424—1061no—
197#423—531no—
198#423—1061no—
199#423C2×D5311yescharacteristic
200C2×C4⋊C8—531no—
201C2×C4⋊C8—1061no—
202C2×#104—21no—
203C2×C4×C8—531no—
204C4×C4⋊C4C2×D5311yescharacteristic
205#339—21no—
206#334—1062no—
207#334—1061no—
208#334—1061no—
209Q8⋊C8—531no—
210Q8⋊C8—531no—
211D4⋊C8—531no—
212D4⋊C8—531no—
213C53#890411yescharacteristic, complemented, Sylow
214C2×#41C22⋊D5311yescharacteristic
215#131—2122no—
216#131—2122no—
217C2×M16—21no—
218C2×M16—1062no—
219C2×M16—1061no—
220C22×C8—1062no—
221C22×C8—1061no—
222C22×C8—1061no—
223C4×Q8Dic10611yescharacteristic
224C4×Q8C4×D5311yescharacteristic
225C4×D4—21no—
226C4×D4—21no—
227C4×D4Dic10611yescharacteristic
228C4×D4C4×D5311yescharacteristic
229#117—21no—
230#117C22⋊D5311yescharacteristic
231C2×C4⋊C4—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42C22×D5311yescharacteristic
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—531no—
238C4⋊C8—531no—
239C4⋊C8—531no—
240C4⋊C8—531no—
241C4⋊C8—532no—
242C4⋊C8—1062no—
243C4⋊C8—1062no—
244C4⋊C8—1061no—
245C4⋊C8—1061no—
246#104—42no—
247C22⋊C8—1061no—
248C4×C8—531no—
249C4×C8—531no—
250C4×C8—1061no—
251C4×C8—1061no—
252#95—21no—
253#41—21no—
254#41—21no—
255#41#1171992312yes—
256#41#1171992412yes—
257C2×Q8#1171991911yescharacteristic
258C2×D4—21no—
259C2×D4#1171991911yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4#1171994711yescharacteristic
265C22×C4#1171994711yescharacteristic
266C22×C4C2×C22⋊D5311yescharacteristic
267M16—41no—
268M16—41no—
269M16—2124no—
270M16—2121no—
271M16—2121no—
272C2×C8—41no—
273C2×C8—1061no—
274C2×C8—1061no—
275C2×C8—1061no—
276C2×C8—1061no—
277C2×C8—1061no—
278C2×C8—1061no—
279C2×C8—1062no—
280C2×C8—2122no—
281C2×C8—2124no—
282C2×C8—2121no—
283C2×C8—2121no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—22no—
287C4⋊C4—41no—
288C4⋊C4#1171991911yescharacteristic
289C4⋊C4#1171991911yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—41no—
297C42—41no—
298C42C2×Dic10611yescharacteristic
299C42C2×C4×D5311yescharacteristic
300C42D4×D5311yescharacteristic
301C42D4×D5311yescharacteristic
302C23—21no—
303C23#1313319811yescharacteristic
304Q8—21no—
305Q8#1313314812yes—
306D4—22no—
307D4—22no—
308D4—21no—
309D4#1313314812yes—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4C2×#1171991911yescharacteristic
325C2×C4#1313317711yescharacteristic
326C2×C4#1313317711yescharacteristic
327C2×C4#1313318511yescharacteristic
328C2×C4#1313319411yescharacteristic
329C2×C4#1313319411yescharacteristic
330C2×C4C2×#1171992411yescharacteristic
331C2×C4C2×#1171992311yescharacteristic
332C2×C4#1313324211yescharacteristic
333C2×C4#1313324711yescharacteristic
334C8—81no—
335C8—1062no—
336C8—2121no—
337C8—2121no—
338C8—2122no—
339C8—2121no—
340C8—2121no—
341C8—2121no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22#1681049511yescharacteristic
348C22#1681093911yescharacteristic
349C22#1681095111yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—82no—
357C4C2×#1313314811yescharacteristic
358C4#1681066611yescharacteristic
359C4#1681073711yescharacteristic
360C4#1681074211yescharacteristic
361C4#1681076111yescharacteristic
362C4#1681077311yescharacteristic
363C2—21no—
364C2—21no—
365C2—42no—
366C2#2027490711yescharacteristic
367C2#2027495311yescharacteristic
368C2#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#8904
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. #20273487C2semidirect
3. #20273470C2semidirect
5. #20272618C2semidirect
6. #20272556C2semidirect
20. Q8⋊#88322C4semidirect
21. Q8⋊#88322C4semidirect
22. D4⋊#88322C4semidirect
23. D4⋊#88322C4semidirect
213. C53#8904semidirect

Maximal subgroups

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

subgroupindex
#202832832
#202734872
#202734702
#202730972
#202726182
#202725562
C53×#18062
#890453