Finite Group Databasebeta

← #27276632

Subgroups — #27276632

Order 13568.

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

normal subgroups only

This group has 656 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
1C2×#16811235C211yesmaximal, characteristic
2#20273440C212yesmaximal, complemented
3#20273341C212yesmaximal, complemented
4#20273049C211yesmaximal, characteristic
5C53×#1464C211yesmaximal, characteristic
6#16811235—24no—
7#16811235—24no—
8C2×#13133217C2211yescharacteristic
9C2×#13133210—21no—
10C2×#13133210—21no—
11C2×#13133209C2211yescharacteristic
12C2×#13133207—21no—
13C2×#13133207—21no—
14C2×#13133196C411yescharacteristic
15C2×#13133196C411yescharacteristic
16C2×C4×Dic106C411yescharacteristic
17C2×C4×Dic106C411yescharacteristic
18#16810680C2212yes—
19#16810371—24no—
20#16810364—24no—
21#16810349—21no—
22#16810347—21no—
23C2×#117×C53C2211yescharacteristic
24C53×#432C2212yes—
25C53×#324—21no—
26C53×#323—21no—
27C22×Dic106C2×C411yescharacteristic
28C22×Dic106C2×C411yescharacteristic
29#13133217—22no—
30#13133217D414yes—
31#13133210—28no—
32#13133210—28no—
33#13133209—22no—
34#13133209D414yes—
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×Dic106—22no—
44C4×Dic106—22no—
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—
61#13133128—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—
73#97×C53—22no—
74C24×C53C22⋊C411yescharacteristic
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—42no—
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22×C411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C2×Dic106—21no—
89C2×Dic106—21no—
90C2×Dic106—42no—
91C2×Dic106—42no—
92C2×Dic106C22⋊C412yes—
93C2×Dic106C22⋊C412yes—
94C2×C8×C53—42no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—22no—
97C22⋊C4×C53—22no—
98C22⋊C4×C53—22no—
99C42×C53—21no—
100C42×C53—21no—
101C42×C53C2×D412yes—
102C42×C53C2×D412yes—
103C22⋊Dic53—24no—
104C22⋊Dic53—24no—
105C4⋊Dic53—21no—
106C4⋊Dic53—21no—
107C4⋊Dic53—24no—
108C4⋊Dic53—24no—
109C4⋊Dic53C22⋊C412yes—
110C4⋊Dic53C22⋊C412yes—
111#11719919—24no—
112#11719919—24no—
113#11719919—24no—
114#11719919—24no—
115#11719919—24no—
116#11719919—24no—
117C4×Dic53—24no—
118C4×Dic53—24no—
119C2×#88322—42no—
120C23×C53—21no—
121C23×C53—22no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53C2×C22⋊C411yescharacteristic
125C2×C4×C53—22no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—22no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53C2×C22⋊C411yescharacteristic
137C2×C4×C53C2×C22⋊C411yescharacteristic
138C2×C4×C53#12012yes—
139C2×C4×C53#12012yes—
140C2×Dic53—22no—
141C2×Dic53—22no—
142C2×Dic53—22no—
143C2×Dic53—22no—
144C2×Dic53—42no—
145C2×Dic53—42no—
146C2×Dic53—42no—
147C2×Dic53—42no—
148Dic106—42no—
149Dic106—42no—
150Dic106—42no—
151Dic106—42no—
152C8×C53—82no—
153#88322—82no—
154#8527—531nomaximal, Sylow
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—82no—
161C22×C53#37911yescharacteristic
162C22×C53#41312yes—
163C4×C53—22no—
164C4×C53—22no—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—41no—
170Dic53—42no—
171Dic53—42no—
172Dic53—42no—
173Dic53—42no—
174C2×#541—531no—
175#1558—532no—
176#1534—532no—
177#1464—531no—
178#1464D5311yescharacteristic
179C2×C53—22no—
180C2×C53—81no—
181C2×C53#178511yescharacteristic
182C2×C53#179611yescharacteristic
183C2×C53#179611yescharacteristic
184#541—1064no—
185#541—1064no—
186C2×C4⋊Q8—531no—
187C2×#126—1061no—
188C2×#126—1061no—
189C2×#125—531no—
190C2×#125—1061no—
191C2×#125—1061no—
192C2×C22⋊Q8—531no—
193C2×C22⋊Q8—531no—
194C2×C4×Q8—531no—
195C2×C4×Q8—531no—
196C2×#117C2×D5311yescharacteristic
197#432—532no—
198#432C2×D5312yes—
199#333—1064no—
200#330—1064no—
201#324—21no—
202#324—1061no—
203#323—21no—
204#323—1061no—
205C53#852711yescharacteristic, complemented, Sylow
206C22×Q8—531no—
207C22×Q8—531no—
208C23×C4C4×D5311yescharacteristic
209C4⋊Q8—534no—
210C4⋊Q8—1062no—
211#126—1068no—
212#126—1068no—
213#125—534no—
214#125—1068no—
215#125—1068no—
216#125—1062no—
217C22⋊Q8—1062no—
218C22⋊Q8—1062no—
219C22⋊Q8—1062no—
220C22⋊Q8—1062no—
221C4×Q8—1062no—
222C4×Q8—1062no—
223C4×Q8—1062no—
224C4×Q8—1062no—
225#117—22no—
226#117—22no—
227C2×C4⋊C4—531no—
228C2×C4⋊C4—531no—
229C2×C4⋊C4—1061no—
230C2×C4⋊C4—1061no—
231C2×C4⋊C4—1061no—
232C2×C4⋊C4—1061no—
233C2×C4⋊C4—1061no—
234C2×C4⋊C4—1061no—
235C2×C4⋊C4—1061no—
236C2×C4⋊C4—1061no—
237C2×C4⋊C4D21211yescharacteristic
238C2×C4⋊C4C22⋊D5311yescharacteristic
239C2×C22⋊C4—1061no—
240C2×C22⋊C4—1061no—
241C2×C22⋊C4D21211yescharacteristic
242C2×C22⋊C4C22⋊D5311yescharacteristic
243C2×C42—1061no—
244C2×C42—1061no—
245C2×C42C4×D5311yescharacteristic
246C2×C42C22×D5311yescharacteristic
247C4⋊C8—22no—
248C4⋊C8—1062no—
249C22⋊C8—22no—
250C22⋊C8—1062no—
251#97—22no—
252#97—1062no—
253C24#1171992111yescharacteristic
254C2×Q8—532no—
255C2×Q8—532no—
256C2×Q8—1061no—
257C2×Q8—1061no—
258C2×Q8—2122no—
259C2×Q8—2122no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—42no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4#1171992111yescharacteristic
268C22×C4#1171992111yescharacteristic
269C22×C4#1171992111yescharacteristic
270C22×C4C2×C4×D5311yescharacteristic
271C22×C4C2×D21211yescharacteristic
272C22×C4C2×C22⋊D5311yescharacteristic
273C2×C8—42no—
274C2×C8—2122no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—532no—
278C4⋊C4—532no—
279C4⋊C4—1064no—
280C4⋊C4—1064no—
281C4⋊C4—1064no—
282C4⋊C4—1064no—
283C4⋊C4—1061no—
284C4⋊C4—1061no—
285C4⋊C4—1064no—
286C4⋊C4—1064no—
287C4⋊C4—1064no—
288C4⋊C4—1064no—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—1064no—
292C22⋊C4—1064no—
293C42—21no—
294C42—21no—
295C42—1064no—
296C42—1064no—
297C42D4×D5312yes—
298C42D4×D5312yes—
299C23—21no—
300C23—22no—
301C23—41no—
302C23—41no—
303C23C2×#1171992111yescharacteristic
304Q8—2122no—
305Q8—2122no—
306Q8—2122no—
307Q8—2122no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—1062no—
320C2×C4—1062no—
321C2×C4—1062no—
322C2×C4—1062no—
323C2×C4—2122no—
324C2×C4—2122no—
325C2×C4—2122no—
326C2×C4—2122no—
327C2×C4C22⋊C4×D5311yescharacteristic
328C2×C4C22⋊C4×D5311yescharacteristic
329C2×C4#1313318512yes—
330C2×C4#1313318712yes—
331C8—82no—
332C8—4242no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—82no—
339C22#1681047811yescharacteristic
340C22#1681060512yes—
341C4—22no—
342C4—22no—
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—2122no—
349C4—2122no—
350C4—2122no—
351C4—2122no—
352C2—22no—
353C2—81no—
354C2#2027464111yescharacteristic
355C2#2027468811yescharacteristic
356C2#2027469111yescharacteristic
subgroupG/H
centerC22#16810478
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1464C2
socleC22×C53#379
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1464D53
hypocenterC53#8527
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #20273440C2semidirect
3. #20273341C2semidirect
205. C53#8527semidirect

Maximal subgroups

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

subgroupindex
C2×#168112352
#202734402
#202733412
#202730492
C53×#14642
#852753