Finite Group Databasebeta

← #27300076

Subgroups — #27300076

Order 13568.

544 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133204C211yesmaximal, characteristic
2C2×#16810417C212yesmaximal
3#20274703C211yesmaximal, characteristic
4#20272839C212yesmaximal
5C53×#1799C211yesmaximal, characteristic
6C2×#13133204C2211yescharacteristic
7C2×C4×Dic106C411yescharacteristic, complemented
8C2×#13133132C2212yes—
9#16810528C411yescharacteristic, complemented
10#16810526C2211yescharacteristic
11#16810517—21no—
12#16810517—21no—
13#16810488C411yescharacteristic, complemented
14#16810483—21no—
15C4×C22⋊Dic53—21no—
16C4×C22⋊Dic53—21no—
17C42×Dic53C411yescharacteristic, complemented
18#16810417—24no—
19#16810390—21no—
20C2×C4×D4×C53C2211yescharacteristic
21C2×#99×C53C2212yes—
22C53×#380—21no—
23C53×#342—21no—
24C22×Dic106C2×C411yescharacteristic
25#13133204—21no—
26#13133204—21no—
27#13133204D412yes—
28#13133204Q812yes—
29C4×Dic106—41no—
30C4×Dic106—41no—
31C2×C4⋊Dic53—41no—
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×#11719919—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—22no—
39C2×C4×Dic53—21no—
40C2×C4×Dic53—21no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53C2×C411yescharacteristic
44#13133132—22no—
45#13133132—22no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—41no—
51C22×D4×C53C2311yescharacteristic
52C23×C4×C53D411yescharacteristic
53C23×C4×C53D411yescharacteristic
54C4×D4×C53—41no—
55C4×D4×C53—41no—
56C2×C4⋊C4×C53—41no—
57C2×C4⋊C4×C53C2×C411yescharacteristic
58C2×C22⋊C4×C53—22no—
59C2×C22⋊C4×C53D411yescharacteristic
60C2×C22⋊C4×C53D411yescharacteristic
61C2×C42×C53C2×C411yescharacteristic
62#99×C53—22no—
63#99×C53—22no—
64#95×C53—41no—
65C24×C53C2×D411yescharacteristic
66C24×C53C2×D411yescharacteristic
67C2×D4×C53—41no—
68C2×D4×C53—41no—
69C2×D4×C53C2×D411yescharacteristic
70C2×D4×C53C2×Q811yescharacteristic
71C2×D4×C53#4111yescharacteristic
72C2×D4×C53#4111yescharacteristic
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—42no—
78C22×C4×C53—42no—
79C22×C4×C53—42no—
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22⋊C411yescharacteristic
83C22×C4×C53C22⋊C411yescharacteristic
84C22×C4×C53C22×C411yescharacteristic
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C22×Dic53—21no—
89C22×Dic53—42no—
90C2×Dic106—21no—
91C2×Dic106—41no—
92C2×Dic106—41no—
93C2×Dic106C4⋊C412yescomplemented
94C4⋊C4×C53—41no—
95C4⋊C4×C53—82no—
96C22⋊C4×C53—41no—
97C22⋊C4×C53—41no—
98C22⋊C4×C53—42no—
99C22⋊C4×C53—42no—
100C42×C53—21no—
101C42×C53—21no—
102C22⋊Dic53—22no—
103C22⋊Dic53—22no—
104C22⋊Dic53—22no—
105C22⋊Dic53—22no—
106C22⋊Dic53—42no—
107C22⋊Dic53—42no—
108C22⋊Dic53—42no—
109C22⋊Dic53—42no—
110C4⋊Dic53—41no—
111C4⋊Dic53—82no—
112#11719919—42no—
113C4×Dic53—21no—
114C4×Dic53—42no—
115C4×Dic53—42no—
116C4×Dic53—41no—
117C4×Dic53—41no—
118C4×Dic53—41no—
119C4×Dic53C4⋊C412yescomplemented
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53C2×C22⋊C411yescharacteristic
129C23×C53#12111yescharacteristic
130C23×C53#12111yescharacteristic
131C23×C53C22⋊Q811yescharacteristic
132C23×C53C22⋊Q811yescharacteristic
133D4×C53—41no—
134D4×C53—41no—
135D4×C53—41no—
136D4×C53—41no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—22no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—42no—
144C2×C4×C53—42no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—81no—
149C2×C4×C53—81no—
150C2×C4×C53—82no—
151C2×C4×C53—82no—
152C2×C4×C53—82no—
153C2×C4×C53#9911yescharacteristic
154C2×C4×C53#9911yescharacteristic
155C2×C4×C53C2×C4⋊C411yescharacteristic
156C2×C4×C53#11711yescharacteristic
157C2×Dic53—22no—
158C2×Dic53—22no—
159C2×Dic53—22no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—41no—
164C2×Dic53—41no—
165C2×Dic53—41no—
166C2×Dic53—41no—
167C2×Dic53—41no—
168C2×Dic53—41no—
169C2×Dic53—82no—
170C2×Dic53—82no—
171C2×Dic53—82no—
172Dic106—42no—
173Dic106—42no—
174#12798—531nomaximal, Sylow
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—41no—
184C22×C53—41no—
185C22×C53—41no—
186C22×C53—41no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53—81no—
190C22×C53—81no—
191C22×C53#38011yescharacteristic
192C22×C53C2×#9911yescharacteristic
193C22×C53#41011yescharacteristic
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—41no—
198C4×C53—41no—
199C4×C53—41no—
200C4×C53—162no—
201C4×C53#35412yescomplemented
202Dic53—42no—
203Dic53—42no—
204Dic53—42no—
205Dic53—81no—
206Dic53—162no—
207C4×#124—531no—
208C2×#354—532no—
209#1799—531no—
210#1799D5311yescharacteristic
211#1393—532no—
212C2×C53—21no—
213C2×C53—21no—
214C2×C53—41no—
215C2×C53—41no—
216C2×C53—41no—
217C2×C53—41no—
218C2×C53#179911yescharacteristic
219C2×C53C2×#35411yescharacteristic
220C2×C53#212611yescharacteristic
221C2×#124—531no—
222C2×C4×Q8—531no—
223C2×C4×D4C2×D5311yescharacteristic
224C2×#99—532no—
225C2×#99C2×D5312yes—
226#391—531no—
227#390—531no—
228#388—1061no—
229#388—1061no—
230#382—531no—
231#380—21no—
232#380—1061no—
233C4×C22⋊C4—1061no—
234C4×C22⋊C4—1061no—
235C43—531no—
236#354—1064no—
237#342—21no—
238#342—1061no—
239C53#1279811yescharacteristic, complemented, Sylow
240C22×Q8—531no—
241C22×D4C22×D5311yescharacteristic
242C23×C4D21211yescharacteristic
243C23×C4C22⋊D5311yescharacteristic
244#124—532no—
245#124—532no—
246#124—1061no—
247#124—1061no—
248C4×Q8—2121no—
249C4×Q8—2121no—
250C4×D4—41no—
251C4×D4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—531no—
254C2×C4⋊C4—1061no—
255C2×C4⋊C4—2121no—
256C2×C4⋊C4C4×D5311yescharacteristic
257C2×C22⋊C4—22no—
258C2×C22⋊C4—1061no—
259C2×C22⋊C4—1061no—
260C2×C22⋊C4—1061no—
261C2×C22⋊C4—1061no—
262C2×C22⋊C4—1062no—
263C2×C22⋊C4D21211yescharacteristic
264C2×C22⋊C4C22⋊D5311yescharacteristic
265C2×C42—531no—
266C2×C42—531no—
267C2×C42—1061no—
268C2×C42—1061no—
269C2×C42—1061no—
270C2×C42C4×D5311yescharacteristic
271#99—22no—
272#99—22no—
273#99—1062no—
274#99—1062no—
275#95—41no—
276#95—1061no—
277#95—1061no—
278#95—1061no—
279#95—1061no—
280#95—2121no—
281C24C2×D21211yescharacteristic
282C24C2×C22⋊D5311yescharacteristic
283C2×Q8—532no—
284C2×Q8—1061no—
285C2×Q8—2121no—
286C2×Q8—2121no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4D4×D5311yescharacteristic
290C2×D4#1171994711yescharacteristic
291C2×D4Q8×D5311yescharacteristic
292C2×D4#1171994911yescharacteristic
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—1061no—
301C22×C4—1061no—
302C22×C4—1061no—
303C22×C4—1061no—
304C22×C4—2122no—
305C22×C4#1171992111yescharacteristic
306C22×C4#1171992111yescharacteristic
307C22×C4#1171992111yescharacteristic
308C22×C4#1171992111yescharacteristic
309C22×C4C2×C4×D5311yescharacteristic
310C4⋊C4—41no—
311C4⋊C4—82no—
312C4⋊C4—4242no—
313C4⋊C4—2121no—
314C4⋊C4—2122no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—1062no—
320C22⋊C4—1062no—
321C22⋊C4—1062no—
322C22⋊C4—1062no—
323C22⋊C4—2122no—
324C22⋊C4—2122no—
325C22⋊C4—2122no—
326C22⋊C4—2122no—
327C42—21no—
328C42—21no—
329C42—532no—
330C42—1061no—
331C42—2121no—
332C42—2122no—
333C42—2122no—
334C42—2121no—
335C42—2121no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23C2×#1171992111yescharacteristic
345C23#1313318811yescharacteristic
346C23#1313319211yescharacteristic
347C23#1313319311yescharacteristic
348C23#1313319711yescharacteristic
349Q8—2122no—
350Q8—2122no—
351D4—41no—
352D4—41no—
353D4—41no—
354D4—41no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—1062no—
372C2×C4—1062no—
373C2×C4—1062no—
374C2×C4—4242no—
375C2×C4—4242no—
376C2×C4—4242no—
377C2×C4—2121no—
378C2×C4—2121no—
379C2×C4—2121no—
380C2×C4—2121no—
381C2×C4—2121no—
382C2×C4—2121no—
383C2×C4—2121no—
384C2×C4—2121no—
385C2×C4—2121no—
386C2×C4#1313313111yescharacteristic
387C2×C4#1313313111yescharacteristic
388C2×C4C4⋊C4×D5311yescharacteristic
389C2×C4#1313317511yescharacteristic
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—81no—
405C22—81no—
406C22#1681048011yescharacteristic
407C22C2×#1313313111yescharacteristic
408C22#1681059611yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—162no—
416C4—8482no—
417C4—4241no—
418C4—2122no—
419C4—2122no—
420C4—2122no—
421C4#1681041712yescomplemented
422C2—21no—
423C2—21no—
424C2—41no—
425C2—41no—
426C2—41no—
427C2—41no—
428C2#2027470011yescharacteristic
429C2C2×#1681041711yescharacteristic
430C2#2027637811yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#1799C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenter#1799D53
hypocenterC53#12798
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
7. C2×C4×Dic106C4semidirect
9. #16810528C4semidirect
13. #16810488C4semidirect
17. C42×Dic53C4semidirect
93. C2×Dic106C4⋊C4semidirect
119. C4×Dic53C4⋊C4semidirect
201. C4×C53#354semidirect
239. C53#12798semidirect
421. C4#16810417semidirect

Maximal subgroups

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

subgroupindex
C4×#131332042
C2×#168104172
#202747032
#202728392
C53×#17992
#1279853