Finite Group Databasebeta

← #27276358

Subgroups — #27276358

Order 13568.

656 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 139 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#20273380C211yesmaximal, characteristic, complemented
2#20273353C211yesmaximal, characteristic, complemented
3#20273149C212yesmaximal
4#20273071C212yesmaximal, complemented
5C53×#3504C211yesmaximal, characteristic
6#16810680C2212yes—
7#16810676C2212yes—
8#16810369—22no—
9#16810365—21no—
10#16810365—21no—
11Q8⋊#88322—22no—
12D4⋊#88322—22no—
13#16810349—24no—
14C53×#557—22no—
15C53×#554—22no—
16C53×#554—24no—
17C53×#544—22no—
18C53×#544C412yes—
19C53×#533—21no—
20C53×#533C2211yescharacteristic
21C2×#125×C53—21no—
22C2×#125×C53C2211yescharacteristic
23C53×#520C412yes—
24C4×#41×C53C2211yescharacteristic
25C4⋊#88322—22no—
26C4⋊#88322—22no—
27C22⋊#88322—22no—
28C22⋊#88322—22no—
29#13133128—22no—
30C4×#88322—22no—
31C2×#41×C53C2×C411yescharacteristic
32C4⋊Q8×C53—24no—
33C4⋊Q8×C53—22no—
34C4⋊Q8×C53—22no—
35C4⋊Q8×C53D412yes—
36#125×C53—24no—
37#125×C53—24no—
38#125×C53—22no—
39#125×C53—22no—
40#125×C53—21no—
41#125×C53—21no—
42#125×C53—22no—
43#125×C53D411yescharacteristic
44#125×C53D411yescharacteristic
45C22⋊Q8×C53—24no—
46C22⋊Q8×C53—24no—
47C22⋊Q8×C53—24no—
48C22⋊Q8×C53—24no—
49C22⋊Q8×C53—22no—
50C4×Q8×C53—24no—
51C4×Q8×C53—22no—
52C4×Q8×C53—22no—
53C4×Q8×C53D412yes—
54C4×D4×C53—22no—
55C4×D4×C53—24no—
56C4×D4×C53—22no—
57C4×D4×C53D412yes—
58#117×C53—22no—
59#117×C53—21no—
60#117×C53C2×C411yescharacteristic
61#117×C53C2×C412yes—
62C2×C4⋊C4×C53—24no—
63C2×C4⋊C4×C53—22no—
64C2×C4⋊C4×C53C2×C412yes—
65C2×C4⋊C4×C53D414yes—
66C2×C42×C53—21no—
67C2×C42×C53C2311yescharacteristic
68#41×C53—22no—
69#41×C53—22no—
70C2×Q8×C53—24no—
71C2×Q8×C53—22no—
72C2×Q8×C53C22⋊C411yescharacteristic
73C2×D4×C53—21no—
74C2×D4×C53C22⋊C411yescharacteristic
75C22×C4×C53—21no—
76C22×C4×C53—24no—
77C22×C4×C53—22no—
78C22×C4×C53C22⋊C414yes—
79C22×C4×C53C22×C411yescharacteristic
80C22×C4×C53C2×D412yes—
81C4⋊C4×C53—24no—
82C4⋊C4×C53—24no—
83C4⋊C4×C53—24no—
84C4⋊C4×C53—24no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—22no—
88C4⋊C4×C53—24no—
89C4⋊C4×C53—24no—
90C4⋊C4×C53—22no—
91C4⋊C4×C53—22no—
92C4⋊C4×C53—22no—
93C4⋊C4×C53C22⋊C412yes—
94C4⋊C4×C53C22⋊C412yes—
95C4⋊C4×C53C22⋊C412yes—
96C22⋊C4×C53—24no—
97C22⋊C4×C53—22no—
98C22⋊C4×C53—21no—
99C22⋊C4×C53—22no—
100C42×C53—24no—
101C42×C53—21no—
102C42×C53—22no—
103C42×C53—22no—
104C42×C53C2×D411yescharacteristic
105C42×C53C2×D411yescharacteristic
106C42×C53C2×D412yes—
107C2×#88322—42no—
108C2×#88322—42no—
109C23×C53—21no—
110C23×C53C2×C22⋊C411yescharacteristic
111Q8×C53—21no—
112Q8×C53—82no—
113Q8×C53—84no—
114Q8×C53#10112yes—
115D4×C53—24no—
116D4×C53—21no—
117D4×C53#10112yes—
118C2×C4×C53—24no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—22no—
123C2×C4×C53—24no—
124C2×C4×C53—22no—
125C2×C4×C53—22no—
126C2×C4×C53—42no—
127C2×C4×C53—44no—
128C2×C4×C53—82no—
129C2×C4×C53—84no—
130C2×C4×C53C2×C22⋊C411yescharacteristic
131C2×C4×C53C2×C22⋊C411yescharacteristic
132C2×C4×C53#12012yes—
133C2×C4×C53#12012yes—
134#88322—82no—
135#88322—82no—
136#8478—531nomaximal, Sylow
137C22×C53—22no—
138C22×C53—21no—
139C22×C53—22no—
140C22×C53—42no—
141C22×C53#37911yescharacteristic
142C4×C53—22no—
143C4×C53—22no—
144C4×C53—21no—
145C4×C53—22no—
146C4×C53—44no—
147C4×C53—42no—
148C4×C53—42no—
149C4×C53—42no—
150C4×C53C2×#10111yescharacteristic
151C4×C53C2×#10111yescharacteristic
152#3504D5311yescharacteristic
153#1544—531no—
154#1537—531no—
155#1488—532no—
156#1470—532no—
157C2×C53—41no—
158C2×C53—42no—
159C2×C53#178411yescharacteristic
160C2×C53#179412yes—
161#557—22no—
162#554—22no—
163#554—24no—
164#544—22no—
165#544Dic5312yes—
166#533—21no—
167#533C2×D5311yescharacteristic
168C2×#125—21no—
169C2×#125C2×D5311yescharacteristic
170#520Dic5312yes—
171C4×#41C2×D5311yescharacteristic
172#432—532no—
173#431—532no—
174#332—1062no—
175#330—1061no—
176#330—1061no—
177Q8⋊C8—1062no—
178D4⋊C8—1062no—
179#324—1064no—
180C53#847811yescharacteristic, complemented, Sylow
181C2×#41C2×Dic5311yescharacteristic
182C4⋊Q8—24no—
183C4⋊Q8—22no—
184C4⋊Q8—22no—
185C4⋊Q8C22⋊D5312yes—
186#125—24no—
187#125—24no—
188#125—22no—
189#125—22no—
190#125—21no—
191#125—21no—
192#125—22no—
193#125C22⋊D5311yescharacteristic
194#125C22⋊D5311yescharacteristic
195C22⋊Q8—24no—
196C22⋊Q8—24no—
197C22⋊Q8—24no—
198C22⋊Q8—22no—
199C22⋊Q8—24no—
200C4×Q8—24no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8C22⋊D5312yes—
204C4×D4—22no—
205C4×D4—22no—
206C4×D4—24no—
207C4×D4C22⋊D5312yes—
208#117—22no—
209#117—21no—
210#117C2×Dic5311yescharacteristic
211#117C2×Dic5312yes—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4C2×Dic5312yes—
215C2×C4⋊C4C22⋊D5314yes—
216C2×C42—21no—
217C2×C42C22×D5311yescharacteristic
218C4⋊C8—1062no—
219C4⋊C8—1062no—
220C22⋊C8—1062no—
221C22⋊C8—1062no—
222#97—1062no—
223C4×C8—1062no—
224#41—22no—
225#41—22no—
226C2×Q8—24no—
227C2×Q8—22no—
228C2×Q8C22⋊Dic5311yescharacteristic
229C2×D4—21no—
230C2×D4C22⋊Dic5311yescharacteristic
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4C22⋊Dic5314yes—
235C22×C4C22×Dic5311yescharacteristic
236C22×C4C2×C22⋊D5312yes—
237C2×C8—2122no—
238C2×C8—2122no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4C22⋊Dic5312yes—
252C4⋊C4C22⋊Dic5312yes—
253C4⋊C4C22⋊Dic5312yes—
254C22⋊C4—24no—
255C22⋊C4—22no—
256C22⋊C4—21no—
257C22⋊C4—22no—
258C42—24no—
259C42—21no—
260C42—22no—
261C42—22no—
262C42C2×C22⋊D5312yes—
263C42C2×C22⋊D5311yescharacteristic
264C42C2×C22⋊D5311yescharacteristic
265C23—21no—
266C23C2×C22⋊Dic5311yescharacteristic
267Q8—21no—
268Q8—82no—
269Q8—84no—
270Q8#1313313612yes—
271D4—24no—
272D4—21no—
273D4#1313313612yes—
274C2×C4—24no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—82no—
285C2×C4—84no—
286C2×C4C2×C22⋊Dic5311yescharacteristic
287C2×C4C2×C22⋊Dic5311yescharacteristic
288C2×C4#1313318612yes—
289C2×C4#1313318612yes—
290C8—4242no—
291C8—4242no—
292C22—21no—
293C22—22no—
294C22—22no—
295C22—42no—
296C22#1681047911yescharacteristic
297C4—21no—
298C4—22no—
299C4—22no—
300C4—22no—
301C4—44no—
302C4—42no—
303C4—42no—
304C4—42no—
305C4C2×#1313313611yescharacteristic
306C4C2×#1313313611yescharacteristic
307C2—41no—
308C2—42no—
309C2#2027463911yescharacteristic
310C2#2027467912yes—
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3504C2
socleC22×C53#379
prime radicalC2×#41×C53C2×C4
cosocleC2×C42×C53C23
hypercenter#3504D53
hypocenterC53#8478
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3504

Complements

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

NH ≅ G/Ntype
1. #20273380C2semidirect
2. #20273353C2semidirect
4. #20273071C2semidirect
180. C53#8478semidirect

Maximal subgroups

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

subgroupindex
#202733802
#202733532
#202731492
#202730712
C53×#35042
#847853