Finite Group Databasebeta

← #27286629

Subgroups — #27286629

Order 13568.

710 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810997C211yesmaximal, characteristic, complemented
2C2×#16810818C211yesmaximal, characteristic
3#20275510C211yesmaximal, characteristic, complemented
4C2×#16810369C211yesmaximal, characteristic, complemented
5C2×#16810366C211yesmaximal, characteristic, complemented
6#20272549C211yesmaximal, characteristic, complemented
7C2×C53×#474C211yesmaximal, characteristic, complemented
8C22×#11719924—22no—
9C22×#11719922—21no—
10C2×#13133214C2211yescharacteristic
11#16810997—22no—
12#16810997—22no—
13#16810818—22no—
14#16810818—22no—
15C2×C4⋊#88322C2211yescharacteristic
16C2×C4⋊#88322C2211yescharacteristic
17C2×#13133137—21no—
18C2×C22⋊#88322—22no—
19C2×C4×#88322C2211yescharacteristic
20#16810369—22no—
21#16810366—22no—
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×C4×D4×C53C2211yescharacteristic
24C53×#474—22no—
25C53×#474—22no—
26C2×#107×C53—21no—
27C2×C4⋊C8×C53C2211yescharacteristic
28C2×#102×C53—21no—
29C22×D212—21no—
30C22×D212—21no—
31C22×D212—21no—
32C2×#11719924—28no—
33C2×#11719924—44no—
34C2×#11719924—44no—
35C2×#11719922—24no—
36C2×#11719922—42no—
37C2×#11719922—42no—
38C22×#88322—21no—
39C22×#88322—21no—
40C22×#88322D412yes—
41#13133214—21no—
42#13133214—21no—
43#13133214—21no—
44#13133214D412yes—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C4⋊#88322D412yes—
49#13133137—24no—
50C22⋊#88322—44no—
51C4×#88322—21no—
52C4×#88322—21no—
53C22×Q8×C53—21no—
54C22×D4×C53D411yescharacteristic
55C23×C4×C53—21no—
56C22×C8×C53—21no—
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53—21no—
59C4⋊Q8×C53—21no—
60C4⋊Q8×C53D412yes—
61C4×D4×C53—21no—
62C4×D4×C53—21no—
63C4×D4×C53—21no—
64C4×D4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53D411yescharacteristic
68C2×C22⋊C4×C53—21no—
69C2×C42×C53C2311yescharacteristic
70#107×C53—24no—
71C4⋊C8×C53—21no—
72C4⋊C8×C53D412yes—
73#102×C53—24no—
74C24×C53—21no—
75C23×D53—41no—
76C2×Q8×C53—22no—
77C2×Q8×C53—41no—
78C2×Q8×C53—42no—
79C2×Q8×C53—42no—
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—21no—
85C2×D4×C53SD1612yescomplemented
86C22×C4×C53—21no—
87C22×C4×C53—41no—
88C22×C4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53—42no—
91C22×C4×C53—42no—
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53#4111yescharacteristic
95C2×D212—22no—
96C2×D212—22no—
97C2×D212—22no—
98C2×D212—42no—
99C2×D212—41no—
100C2×D212—41no—
101C2×D212—41no—
102C2×D212—42no—
103C2×D212—41no—
104C2×D212—41no—
105C2×D212—41no—
106C2×D212—41no—
107C2×D212—82no—
108C2×C8×C53—22no—
109C2×C8×C53—41no—
110C2×C8×C53—41no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—41no—
116C4⋊C4×C53—41no—
117C22⋊C4×C53—41no—
118C22⋊C4×C53—41no—
119C42×C53C2×D411yescharacteristic
120C42×C53C2×D411yescharacteristic
121C42×C53C2×D411yescharacteristic
122C42×C53C2×D411yescharacteristic
123#11719924—416no—
124#11719922—48no—
125C2×#88322—22no—
126C2×#88322—22no—
127C2×#88322—22no—
128C2×#88322—41no—
129C2×#88322—41no—
130C2×#88322—41no—
131C2×#88322—42no—
132C2×#88322D814yescomplemented
133C23×C53—22no—
134C23×C53—41no—
135C23×C53—42no—
136C23×C53—41no—
137C23×C53—42no—
138C23×C53#12111yescharacteristic
139C22×D53—42no—
140C22×D53—82no—
141C22×D53—81no—
142C22×D53—81no—
143C22×D53—82no—
144Q8×C53—44no—
145Q8×C53—81no—
146Q8×C53—81no—
147D4×C53—22no—
148D4×C53—22no—
149D4×C53—41no—
150D4×C53—41no—
151C2×C4×C53—22no—
152C2×C4×C53—41no—
153C2×C4×C53—42no—
154C2×C4×C53—42no—
155C2×C4×C53—82no—
156C2×C4×C53—82no—
157C2×C4×C53—81no—
158C2×C4×C53—81no—
159C2×C4×C53#12011yescharacteristic
160C2×C4×C53#12011yescharacteristic
161C2×C4×C53#12011yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#12711yescharacteristic
165C2×C4×C53C2×D812yes—
166C2×C4×C53C2×SD1611yescharacteristic
167C2×C4×C53#13511yescharacteristic
168C2×C4×C53AGL(1,ℤ/8)12yes—
169C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C53#13711yescharacteristic
171D212—44no—
172D212—42no—
173D212—42no—
174D212—82no—
175D212—82no—
176D212—81no—
177D212—81no—
178D212—81no—
179D212—81no—
180C8×C53—42no—
181#88322—24no—
182#88322—42no—
183#88322—81no—
184#10266—531nomaximal, Sylow
185C22×C53—44no—
186C22×C53—42no—
187C22×C53—82no—
188C22×C53—81no—
189C22×C53—81no—
190C22×C53#39211yescharacteristic
191C22×C53#45912yes—
192C22×C53#46511yescharacteristic
193C22×C53#46811yescharacteristic
194C22×C53#47112yes—
195C2×D53—82no—
196C2×D53—82no—
197C2×D53—82no—
198C2×D53—162no—
199C2×D53—162no—
200C4×C53—21no—
201C4×C53—21no—
202C4×C53—81no—
203C4×C53—82no—
204C4×C53#44712yes—
205C4×C53#44811yescharacteristic
206C4×C53#44911yescharacteristic
207C4×C53#45011yescharacteristic
208C4×C53#45211yescharacteristic
209C4×C53#49311yescharacteristic
210C4×C53#49611yescharacteristic
211C2×#492—531no—
212C2×#474D5311yescharacteristic, complemented
213C2×#459—531no—
214#1954—531no—
215C2×#332—531no—
216C2×#331—531no—
217#1277—531no—
218C2×C53—82no—
219C2×C53#166512yes—
220C2×C53#168512yes—
221C2×C53#200011yescharacteristic
222C2×C53#201111yescharacteristic
223C2×C53#201911yescharacteristic
224D53—162no—
225C22×SD16—1062no—
226C22×D8—1061no—
227C2×C4⋊Q8C2×D5311yescharacteristic
228C2×#127—531no—
229C2×C4×D4C2×D5311yescharacteristic
230#492—1062no—
231#492—1062no—
232#474—22no—
233#474—22no—
234#459—1062no—
235#459—1062no—
236C2×#107—21no—
237C2×C4⋊C8—531no—
238C2×C4⋊C8—531no—
239C2×C4⋊C8C2×D5311yescharacteristic
240C2×#102—21no—
241C2×#102—1061no—
242C2×C22⋊C8—1062no—
243C2×C4×C8—531no—
244#332—1062no—
245#331—1062no—
246C53#1026611yescharacteristic, complemented, Sylow
247C22×Q8—21no—
248C22×D4—1061no—
249C22×D4—1061no—
250C22×D4—1061no—
251C22×D4C22⋊D5311yescharacteristic
252C23×C4—21no—
253C2×SD16—1068no—
254C2×SD16—2124no—
255C2×SD16—2124no—
256C2×D8—1064no—
257C2×D8—2122no—
258C2×D8—2122no—
259C22×C8—21no—
260C22×C8—532no—
261C22×C8—1061no—
262C22×C8—1061no—
263C4⋊Q8—21no—
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4⋊Q8C22⋊D5312yes—
267#127—532no—
268#127—1061no—
269#127—1061no—
270#127—1061no—
271C4×D4—21no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4C22⋊D5311yescharacteristic
278C2×C22⋊C4—21no—
279C2×C42C22×D5311yescharacteristic
280#107—24no—
281C4⋊C8—21no—
282C4⋊C8—532no—
283C4⋊C8—1061no—
284C4⋊C8—1061no—
285C4⋊C8—1061no—
286C4⋊C8C22⋊D5312yes—
287#102—24no—
288#102—1064no—
289C22⋊C8—2124no—
290C4×C8—1061no—
291C4×C8—1061no—
292C24—21no—
293C24—2121no—
294C2×Q8—22no—
295C2×Q8—41no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—1062no—
304C2×D4—1062no—
305C2×D4—1062no—
306C2×D4—4242no—
307C2×D4—2121no—
308C2×D4—2121no—
309C2×D4—2121no—
310C2×D4—2122no—
311C2×D4—2121no—
312C2×D4—2121no—
313C2×D4—2122no—
314C2×D4—2121no—
315C2×D4—2121no—
316C2×D4#1171992412yescomplemented
317C22×C4—21no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4D4×D5311yescharacteristic
324C22×C4#1171994711yescharacteristic
325C22×C4C2×C22⋊D5311yescharacteristic
326SD16—21216no—
327D8—2128no—
328C2×C8—22no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—534no—
332C2×C8—1062no—
333C2×C8—1062no—
334C2×C8—1062no—
335C2×C8—2121no—
336C2×C8—2122no—
337C2×C8—2121no—
338C2×C8—2121no—
339C4⋊C4—21no—
340C4⋊C4—22no—
341C4⋊C4—21no—
342C4⋊C4—22no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C42D4×D5311yescharacteristic
348C42D4×D5311yescharacteristic
349C42C2×C22⋊D5311yescharacteristic
350C42C2×C22⋊D5311yescharacteristic
351C23—22no—
352C23—41no—
353C23—42no—
354C23—41no—
355C23—42no—
356C23—4242no—
357C23—4241no—
358C23—4241no—
359C23—4242no—
360C23—2122no—
361C23#1313318911yescharacteristic
362Q8—44no—
363Q8—81no—
364Q8—81no—
365D4—22no—
366D4—22no—
367D4—41no—
368D4—41no—
369D4—4242no—
370D4—4242no—
371D4—4241no—
372D4—4241no—
373D4—4241no—
374D4—4241no—
375D4—2124no—
376D4—2122no—
377D4—2122no—
378C2×C4—22no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4#1313318511yescharacteristic
387C2×C4#1313318511yescharacteristic
388C2×C4#1313318611yescharacteristic
389C2×C4#1313318811yescharacteristic
390C2×C4#1313318911yescharacteristic
391C2×C4#1313321311yescharacteristic
392C2×C4D8×D5312yes—
393C2×C4C2×#1171992411yescharacteristic
394C2×C4#1313323711yescharacteristic
395C2×C4#1313324112yes—
396C2×C4#1313324211yescharacteristic
397C2×C4#1313324811yescharacteristic
398C8—42no—
399C8—1064no—
400C8—4241no—
401C8—2122no—
402C22—42no—
403C22—44no—
404C22—82no—
405C22—81no—
406C22—81no—
407C22—8482no—
408C22—8482no—
409C22—4242no—
410C22—4242no—
411C22—4242no—
412C22#1681053111yescharacteristic
413C22#1681081612yes—
414C22#1681085411yescharacteristic
415C22#1681087111yescharacteristic
416C22#1681088812yes—
417C4—21no—
418C4—21no—
419C4—81no—
420C4—82no—
421C4#1681075512yes—
422C4#1681076211yescharacteristic
423C4#1681076811yescharacteristic
424C4#1681077311yescharacteristic
425C4#1681078311yescharacteristic
426C4#1681099811yescharacteristic
427C4#1681101111yescharacteristic
428C2—82no—
429C2—8482no—
430C2#2027405812yes—
431C2#2027418212yes—
432C2#2027571611yescharacteristic
433C2#2027578411yescharacteristic
434C2#2027582611yescharacteristic
subgroupG/H
centerC23#13133189
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#474C2
socleC23×C53#121
prime radicalC2×#16810818C2
hypercenterC2×#474D53
hypocenterC53#10266
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×#16810997C2semidirect
3. #20275510C2semidirect
4. C2×#16810369C2semidirect
5. C2×#16810366C2semidirect
6. #20272549C2semidirect
7. C2×C53×#474C2semidirect
85. C2×D4×C53SD16semidirect
132. C2×#88322D8semidirect
212. C2×#474D53semidirect
246. C53#10266semidirect
316. C2×D4#11719924semidirect

Maximal subgroups

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

subgroupindex
C2×#168109972
C2×#168108182
#202755102
C2×#168103692
C2×#168103662
#202725492
C2×C53×#4742
#1026653