Finite Group Databasebeta

← #28299469

Subgroups — #28299469

Order 13568.

444 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 118 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
1#20290631C211yesmaximal, characteristic
2C4×#13133284C211yesmaximal, characteristic
3#20289535C212yesmaximal
4#20289534C212yesmaximal
5#20277190C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×#13133284—21no—
8C2×C4×#88324—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×Q8×Dic53C411yescharacteristic
13#16810530C2211yescharacteristic
14#16810521C411yescharacteristic
15#16810519C2211yescharacteristic
16#16810508—21no—
17#16810506—21no—
18C4⋊C4×Dic53C411yescharacteristic
19#16810474—21no—
20#16810468—21no—
21C42×Dic53C2211yescharacteristic
22C53×#391C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133284—24no—
26#13133284—42no—
27C4×#88324—42no—
28Q8×Dic53—22no—
29Q8×Dic53C814yescomplemented
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53C2×C411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53C2311yescharacteristic
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48C22×Q8×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C42×C53C2×C411yescharacteristic
51#95×C53—21no—
52#95×C53—21no—
53C2×Q8×C53—21no—
54C2×Q8×C53—41no—
55C2×Q8×C53—41no—
56C2×Q8×C53C2×C812yes—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22×C411yescharacteristic
62C22×Dic53—21no—
63C22×Dic53—21no—
64C22×Dic53C22⋊C411yescharacteristic
65C22×Dic53C22⋊C411yescharacteristic
66C22×Dic53C2×D411yescharacteristic
67C22×Dic53C2×D411yescharacteristic
68C2×#88324—24no—
69C2×#88324—42no—
70C2×#88324—42no—
71C2×#88324—42no—
72C2×#88324—42no—
73C2×#88324—42no—
74C4⋊C4×C53—41no—
75C42×C53—41no—
76C4⋊Dic53—21no—
77C4⋊Dic53—24no—
78C4⋊Dic53—41no—
79C4⋊Dic53C2×C812yes—
80#11719919—42no—
81#11719919—41no—
82#11719919—41no—
83C4×Dic53—24no—
84C4×Dic53—22no—
85C4×Dic53—21no—
86C4×Dic53—21no—
87C4×Dic53—41no—
88C4×Dic53—41no—
89C4×Dic53—42no—
90C4×Dic53C2×C812yes—
91C4×Dic53M1612yes—
92C4×Dic53SD1612yes—
93C4×Dic53Q1612yes—
94C23×C53C2×C22⋊C411yescharacteristic
95Q8×C53—42no—
96Q8×C53—42no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53#10312yes—
111C2×C4×C53C22×C811yescharacteristic
112C2×C4×C53C2×M1611yescharacteristic
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—22no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53C22⋊C812yes—
126C2×Dic53C22⋊C812yes—
127C2×Dic53#10312yes—
128C2×Dic53#10412yes—
129C2×Dic53C2×SD1611yescharacteristic
130C2×Dic53C2×Q1611yescharacteristic
131C2×Dic53AGL(1,ℤ/8)11yescharacteristic
132C2×Dic53#13711yescharacteristic
133#88324—44no—
134#88324—82no—
135#7577—531nomaximal, Sylow
136C22×C53C2×C22⋊C811yescharacteristic
137C22×C53#41011yescharacteristic
138C22×C53#41311yescharacteristic
139C22×C53C2×#10311yescharacteristic
140C22×C53#41711yescharacteristic
141C22×C53C2×#10411yescharacteristic
142C22×C53#42111yescharacteristic
143C4×C53—22no—
144C4×C53—21no—
145C4×C53—21no—
146C4×C53—42no—
147C4×C53—41no—
148C4×C53—81no—
149Dic53—22no—
150Dic53—21no—
151Dic53—21no—
152Dic53—21no—
153Dic53—21no—
154Dic53—42no—
155Dic53—81no—
156Dic53Q8⋊C814yes—
157#2295—531no—
158#1831—531no—
159C4×C4⋊C8—531no—
160#1272—532no—
161#1271—532no—
162C2×C53#145811yescharacteristic
163C2×C53C2×Q8⋊C811yescharacteristic
164C2×C53#147711yescharacteristic
165C2×C53#150911yescharacteristic
166C2×C53#151811yescharacteristic
167C2×C53#152911yescharacteristic
168C2×C53#155011yescharacteristic
169C2×C4×Q8—531no—
170C2×C4⋊C8—1061no—
171C2×C4⋊C8—1061no—
172C2×C4×C8—1061no—
173#391—531no—
174#391#615711yescharacteristic
175#389—531no—
176#389—531no—
177#387—1061no—
178#387—1061no—
179C4×C4⋊C4—531no—
180C4×C4⋊C4—1061no—
181#376—1061no—
182C43—531no—
183#336—532no—
184#336—532no—
185#336—1061no—
186C53#757711yescharacteristic, complemented, Sylow
187C22×Q8C2×#615711yescharacteristic
188C22×C8—1062no—
189C22×C8—1062no—
190C4×Q8—534no—
191C4×Q8—1062no—
192C2×C4⋊C4—531no—
193C2×C4⋊C4—531no—
194C2×C4⋊C4—1061no—
195C2×C4⋊C4—1061no—
196C2×C4⋊C4—1061no—
197C2×C4⋊C4C2×#615711yescharacteristic
198C2×C42—531no—
199C2×C42—531no—
200C2×C42—531no—
201C2×C42—531no—
202C2×C42—531no—
203C2×C42—531no—
204C2×C42—1061no—
205C2×C42—1061no—
206C2×C42C2×#615711yescharacteristic
207C4⋊C8—1064no—
208C4⋊C8—2122no—
209C4×C8—2122no—
210#95—21no—
211#95—21no—
212#95—1061no—
213#95—1061no—
214#95—1061no—
215#95—1061no—
216#95—1061no—
217C2×Q8—21no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8D53⋊C812yes—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—531no—
224C22×C4—531no—
225C22×C4—531no—
226C22×C4—531no—
227C22×C4—1061no—
228C22×C4—1061no—
229C22×C4C22⋊#615711yescharacteristic
230C22×C4C22⋊#615711yescharacteristic
231C22×C4C22×#615711yescharacteristic
232C2×C8—1064no—
233C2×C8—2122no—
234C2×C8—2122no—
235C2×C8—2122no—
236C2×C8—2122no—
237C2×C8—2122no—
238C4⋊C4—41no—
239C4⋊C4—532no—
240C4⋊C4—1064no—
241C4⋊C4—1061no—
242C4⋊C4—2121no—
243C4⋊C4—2122no—
244C4⋊C4—2121no—
245C4⋊C4—2121no—
246C42—41no—
247C42—532no—
248C42—532no—
249C42—532no—
250C42—532no—
251C42—1061no—
252C42—1062no—
253C42—1061no—
254C42—1064no—
255C42—2121no—
256C42—2121no—
257C42—2122no—
258C23C2×C22⋊#615711yescharacteristic
259Q8—42no—
260Q8—42no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—532no—
275C2×C4—532no—
276C2×C4—532no—
277C2×C4—532no—
278C2×C4—531no—
279C2×C4—531no—
280C2×C4—531no—
281C2×C4—531no—
282C2×C4—1061no—
283C2×C4—1061no—
284C2×C4—1061no—
285C2×C4—1061no—
286C2×C4—1061no—
287C2×C4—1061no—
288C2×C4—1062no—
289C2×C4—2121no—
290C2×C4—2121no—
291C2×C4—2121no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4#1313328012yes—
295C2×C4C2×D53⋊C811yescharacteristic
296C2×C4D53⋊M1611yescharacteristic
297C8—4242no—
298C8—2124no—
299C22D53⋊(C22⋊C8)11yescharacteristic
300C22#1681160311yescharacteristic
301C22#1681161011yescharacteristic
302C22C2×#1313328011yescharacteristic
303C22#1681161411yescharacteristic
304C22D53⋊#10411yescharacteristic
305C22#1681162211yescharacteristic
306C4—21no—
307C4—21no—
308C4—22no—
309C4—41no—
310C4—42no—
311C4—81no—
312C4—534no—
313C4—1061no—
314C4—1061no—
315C4—1061no—
316C4—1061no—
317C4—1062no—
318C4—4241no—
319C4—2122no—
320C2#2028994711yescharacteristic
321C2D53⋊(Q8⋊C8)11yescharacteristic
322C2#2029001311yescharacteristic
323C2#2029011611yescharacteristic
324C2#2029014911yescharacteristic
325C2#2029018311yescharacteristic
326C2#2029023311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#391C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#391#6157
hypocenterC53#7577
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
29. Q8×Dic53C8semidirect
186. C53#7577semidirect

Maximal subgroups

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

subgroupindex
#202906312
C4×#131332842
#202895352
#202895342
#202771902
#757753