Finite Group Databasebeta

← #27275529

Subgroups — #27275529

Order 13568.

684 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C2×#16811235C211yesmaximal, characteristic
2#20273111C212yesmaximal, complemented
3#20273109C212yesmaximal, complemented
4#20273029C211yesmaximal, characteristic
5C53×#1458C211yesmaximal, characteristic
6#16811235—24no—
7#16811235—24no—
8C2×#13133217C411yescharacteristic
9C2×#13133210—21no—
10C2×#13133210—21no—
11C2×#13133209C411yescharacteristic
12C2×#13133207—21no—
13C2×#13133207—21no—
14C2×#13133196C411yescharacteristic
15C2×#13133196C411yescharacteristic
16C2×C4×Dic106C2211yescharacteristic
17C2×C4×Dic106C2211yescharacteristic
18#16810676C2212yes—
19#16810355—24no—
20#16810353—24no—
21#16810349—21no—
22#16810347—21no—
23C2×#117×C53C2211yescharacteristic
24C53×#431C2212yes—
25C53×#324—21no—
26C53×#323—21no—
27C22×Dic106C2×C411yescharacteristic
28C22×Dic106C2×C411yescharacteristic
29#13133217—22no—
30#13133217—22no—
31#13133210—28no—
32#13133210—28no—
33#13133209—22no—
34#13133209—22no—
35#13133207—28no—
36#13133207—28no—
37#13133196—22no—
38#13133196—22no—
39#13133196—22no—
40#13133196—22no—
41C4×Dic106—22no—
42C4×Dic106—22no—
43C4×Dic106D414yes—
44C4×Dic106D414yes—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53—21no—
47C2×C4⋊Dic53C2×C411yescharacteristic
48C2×C4⋊Dic53C2×C411yescharacteristic
49C2×#11719919—21no—
50C2×#11719919—21no—
51C2×#11719919—21no—
52C2×#11719919—21no—
53C2×#11719919—21no—
54C2×#11719919—21no—
55C2×C22⋊Dic53—21no—
56C2×C22⋊Dic53—21no—
57C2×C4×Dic53—21no—
58C2×C4×Dic53—21no—
59C4⋊#88322—22no—
60C22⋊#88322—22no—
61C4×#88322—22no—
62C23×C4×C53C2×C411yescharacteristic
63#117×C53—22no—
64#117×C53—22no—
65C2×C4⋊C4×C53D411yescharacteristic
66C2×C4⋊C4×C53D411yescharacteristic
67C2×C22⋊C4×C53D411yescharacteristic
68C2×C22⋊C4×C53D411yescharacteristic
69C2×C42×C53C2×C411yescharacteristic
70C2×C42×C53C2311yescharacteristic
71C4⋊C8×C53—22no—
72C22⋊C8×C53—22no—
73C4×C8×C53—22no—
74C24×C53C22⋊C411yescharacteristic
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—22no—
78C22×C4×C53—22no—
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22×C411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C22×Dic53—21no—
89C2×Dic106—22no—
90C2×Dic106—22no—
91C2×Dic106—22no—
92C2×Dic106—22no—
93C2×Dic106—21no—
94C2×Dic106—21no—
95C2×Dic106C22⋊C412yes—
96C2×Dic106C22⋊C412yes—
97C2×C8×C53—42no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—22no—
100C22⋊C4×C53—22no—
101C22⋊C4×C53—22no—
102C42×C53—21no—
103C42×C53—21no—
104C42×C53C2×D412yes—
105C42×C53C2×D412yes—
106C22⋊Dic53—24no—
107C22⋊Dic53—24no—
108C4⋊Dic53—24no—
109C4⋊Dic53—24no—
110C4⋊Dic53—21no—
111C4⋊Dic53—21no—
112C4⋊Dic53C22⋊C412yes—
113C4⋊Dic53C22⋊C412yes—
114#11719919—24no—
115#11719919—24no—
116#11719919—24no—
117#11719919—24no—
118#11719919—24no—
119#11719919—24no—
120C4×Dic53—24no—
121C4×Dic53—24no—
122C2×#88322—42no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53C2×C22⋊C411yescharacteristic
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—42no—
137C2×C4×C53—42no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53C2×C22⋊C411yescharacteristic
141C2×C4×C53C2×C22⋊C411yescharacteristic
142C2×C4×C53#12012yes—
143C2×C4×C53#12012yes—
144C2×C4×C53#13112yes—
145C2×C4×C53#13112yes—
146C2×Dic53—22no—
147C2×Dic53—22no—
148C2×Dic53—22no—
149C2×Dic53—22no—
150C2×Dic53—42no—
151C2×Dic53—42no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156Dic106—24no—
157Dic106—24no—
158Dic106—42no—
159Dic106—42no—
160C8×C53—82no—
161#88322—82no—
162#8327—531nomaximal, Sylow
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—81no—
169C22×C53—81no—
170C22×C53#37911yescharacteristic
171C22×C53#40811yescharacteristic
172C22×C53#40811yescharacteristic
173C4×C53—22no—
174C4×C53—22no—
175C4×C53—41no—
176C4×C53—41no—
177C4×C53—41no—
178C4×C53#40812yes—
179C4×C53#40812yes—
180Dic53—42no—
181Dic53—42no—
182Dic53—42no—
183Dic53—42no—
184C2×#541—531no—
185#1479—532no—
186#1479—532no—
187#1458—531no—
188#1458D5311yescharacteristic
189C2×C53—21no—
190C2×C53—21no—
191C2×C53—81no—
192C2×C53#178011yescharacteristic
193C2×C53#179611yescharacteristic
194C2×C53#179611yescharacteristic
195#541—1064no—
196#541—1064no—
197C2×C4⋊Q8—531no—
198C2×#126—1061no—
199C2×#126—1061no—
200C2×#125—531no—
201C2×#125—1061no—
202C2×#125—1061no—
203C2×C22⋊Q8—531no—
204C2×C22⋊Q8—531no—
205C2×C4×Q8—531no—
206C2×C4×Q8—531no—
207C2×#117C2×D5311yescharacteristic
208#431—532no—
209#431C2×D5312yes—
210Q8⋊C8—1064no—
211Q8⋊C8—1064no—
212#324—21no—
213#324—1061no—
214#323—21no—
215#323—1061no—
216C53#832711yescharacteristic, complemented, Sylow
217C22×Q8—531no—
218C22×Q8—531no—
219C23×C4C4×D5311yescharacteristic
220C4⋊Q8—1062no—
221C4⋊Q8—1062no—
222#126—1068no—
223#126—1068no—
224#125—1062no—
225#125—1062no—
226#125—1068no—
227#125—1068no—
228C22⋊Q8—1062no—
229C22⋊Q8—1062no—
230C22⋊Q8—1062no—
231C22⋊Q8—1062no—
232C4×Q8—534no—
233C4×Q8—534no—
234C4×Q8—1062no—
235C4×Q8—1062no—
236#117—22no—
237#117—22no—
238C2×C4⋊C4—531no—
239C2×C4⋊C4—531no—
240C2×C4⋊C4—1061no—
241C2×C4⋊C4—1061no—
242C2×C4⋊C4—1061no—
243C2×C4⋊C4—1061no—
244C2×C4⋊C4—1061no—
245C2×C4⋊C4—1061no—
246C2×C4⋊C4—1061no—
247C2×C4⋊C4—1061no—
248C2×C4⋊C4D21211yescharacteristic
249C2×C4⋊C4C22⋊D5311yescharacteristic
250C2×C22⋊C4—1061no—
251C2×C22⋊C4—1061no—
252C2×C22⋊C4D21211yescharacteristic
253C2×C22⋊C4C22⋊D5311yescharacteristic
254C2×C42—1061no—
255C2×C42—1061no—
256C2×C42C4×D5311yescharacteristic
257C2×C42C22×D5311yescharacteristic
258C4⋊C8—22no—
259C4⋊C8—1062no—
260C22⋊C8—22no—
261C22⋊C8—1062no—
262C4×C8—22no—
263C4×C8—1062no—
264C24#1171992111yescharacteristic
265C2×Q8—532no—
266C2×Q8—532no—
267C2×Q8—1061no—
268C2×Q8—1061no—
269C2×Q8—1062no—
270C2×Q8—1062no—
271C2×Q8—1062no—
272C2×Q8—1062no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—1061no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4#1171992111yescharacteristic
282C22×C4#1171992111yescharacteristic
283C22×C4#1171992111yescharacteristic
284C22×C4C2×C4×D5311yescharacteristic
285C22×C4C2×D21211yescharacteristic
286C22×C4C2×C22⋊D5311yescharacteristic
287C2×C8—42no—
288C2×C8—2122no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—532no—
292C4⋊C4—532no—
293C4⋊C4—1064no—
294C4⋊C4—1064no—
295C4⋊C4—1064no—
296C4⋊C4—1064no—
297C4⋊C4—1064no—
298C4⋊C4—1064no—
299C4⋊C4—1061no—
300C4⋊C4—1061no—
301C4⋊C4—1064no—
302C4⋊C4—1064no—
303C22⋊C4—22no—
304C22⋊C4—22no—
305C22⋊C4—1064no—
306C22⋊C4—1064no—
307C42—21no—
308C42—21no—
309C42—1064no—
310C42—1064no—
311C42D4×D5312yes—
312C42D4×D5312yes—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23C2×#1171992111yescharacteristic
319Q8—1064no—
320Q8—1064no—
321Q8—2122no—
322Q8—2122no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—1062no—
335C2×C4—1062no—
336C2×C4—1062no—
337C2×C4—1062no—
338C2×C4—2122no—
339C2×C4—2122no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4—2121no—
343C2×C4—2121no—
344C2×C4C22⋊C4×D5311yescharacteristic
345C2×C4C22⋊C4×D5311yescharacteristic
346C2×C4#1313318512yes—
347C2×C4#1313318712yes—
348C2×C4#1313322312yes—
349C2×C4#1313322512yes—
350C8—82no—
351C8—4242no—
352C22—21no—
353C22—21no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—81no—
358C22—81no—
359C22#1681047811yescharacteristic
360C22#1681058911yescharacteristic
361C22#1681059111yescharacteristic
362C4—22no—
363C4—22no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—2122no—
368C4—2122no—
369C4—2122no—
370C4—2122no—
371C4#1681058812yes—
372C4#1681058812yes—
373C2—21no—
374C2—21no—
375C2—81no—
376C2#2027462611yescharacteristic
377C2#2027468811yescharacteristic
378C2#2027469111yescharacteristic
subgroupG/H
centerC22#16810478
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1458C2
socleC22×C53#379
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1458D53
hypocenterC53#8327
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
2. #20273111C2semidirect
3. #20273109C2semidirect
216. C53#8327semidirect

Maximal subgroups

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

subgroupindex
C2×#168112352
#202731112
#202731092
#202730292
C53×#14582
#832753