Finite Group Databasebeta

← #27276238

Subgroups — #27276238

Order 13568.

746 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 746 conjugacy classes of subgroups — too many to draw. Only its 143 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#20275514C211yesmaximal, characteristic
2#20273415C212yesmaximal, complemented
3#20273330C212yesmaximal, complemented
4#20273056C211yesmaximal, characteristic
5C53×#3501C211yesmaximal, characteristic
6#16810680C2212yes—
7C2×C4⋊#88322C2211yescharacteristic
8C2×C22⋊#88322—22no—
9C2×#13133128C2211yescharacteristic
10#16810369—22no—
11#16810369—22no—
12#16810365—24no—
13#16810347—22no—
14C53×#554—22no—
15C53×#554—24no—
16D4×Q8×C53—22no—
17C53×#544C412yes—
18C53×#543—22no—
19C53×#533C2212yes—
20C2×C22⋊Q8×C53—22no—
21C53×#518C412yes—
22C2×C4×D4×C53C2211yescharacteristic
23C22×#88322—21no—
24C22×#88322—21no—
25C4⋊#88322—22no—
26C4⋊#88322—22no—
27C22⋊#88322—22no—
28C22⋊#88322—44no—
29#13133128—22no—
30#13133128—22no—
31C22×Q8×C53—22no—
32C22×D4×C53C2×C411yescharacteristic
33C23×C4×C53D412yes—
34C4⋊Q8×C53—24no—
35C4⋊Q8×C53D412yes—
36C4⋊Q8×C53D412yes—
37#125×C53—24no—
38#125×C53D414yes—
39C22⋊Q8×C53—24no—
40C22⋊Q8×C53—24no—
41C22⋊Q8×C53—28no—
42C22⋊Q8×C53—28no—
43C22⋊Q8×C53—24no—
44C22⋊Q8×C53—22no—
45C22⋊Q8×C53—22no—
46C4×Q8×C53—22no—
47C4×Q8×C53—22no—
48C4×D4×C53—24no—
49C4×D4×C53—24no—
50C4×D4×C53—22no—
51C4×D4×C53—22no—
52#117×C53C2×C414yes—
53C2×C4⋊C4×C53—22no—
54C2×C4⋊C4×C53—24no—
55C2×C4⋊C4×C53C2×C411yescharacteristic
56C2×C22⋊C4×C53—24no—
57C2×C22⋊C4×C53D412yes—
58C2×C42×C53C2311yescharacteristic
59C24×C53C22⋊C412yes—
60C2×Q8×C53—24no—
61C2×Q8×C53—22no—
62C2×Q8×C53—22no—
63C2×Q8×C53—82no—
64C2×Q8×C53—82no—
65C2×D4×C53—24no—
66C2×D4×C53—22no—
67C22×C4×C53—24no—
68C22×C4×C53—24no—
69C22×C4×C53—22no—
70C22×C4×C53—22no—
71C22×C4×C53—42no—
72C22×C4×C53—42no—
73C22×C4×C53C22⋊C412yes—
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C4⋊C4×C53—24no—
78C4⋊C4×C53—28no—
79C4⋊C4×C53—22no—
80C4⋊C4×C53—24no—
81C4⋊C4×C53—22no—
82C4⋊C4×C53—24no—
83C4⋊C4×C53—22no—
84C4⋊C4×C53C22⋊C414yes—
85C4⋊C4×C53C22⋊C414yes—
86C22⋊C4×C53—28no—
87C22⋊C4×C53—24no—
88C22⋊C4×C53—24no—
89C42×C53—24no—
90C42×C53C2×D412yes—
91C42×C53C2×D412yes—
92C2×#88322—22no—
93C2×#88322—41no—
94C2×#88322—41no—
95C2×#88322—42no—
96C2×#88322—42no—
97C2×#88322—41no—
98C23×C53—22no—
99C23×C53—22no—
100C23×C53—42no—
101C23×C53—42no—
102C23×C53#10114yes—
103C23×C53C2×C22⋊C411yescharacteristic
104Q8×C53—42no—
105Q8×C53—42no—
106Q8×C53—82no—
107Q8×C53—84no—
108D4×C53—28no—
109C2×C4×C53—22no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—22no—
113C2×C4×C53—24no—
114C2×C4×C53—24no—
115C2×C4×C53—41no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—82no—
120C2×C4×C53—84no—
121C2×C4×C53—84no—
122C2×C4×C53C2×C22⋊C412yes—
123C2×C4×C53#12011yescharacteristic
124C2×C4×C53#12012yes—
125C2×C4×C53#12011yescharacteristic
126#88322—42no—
127#88322—82no—
128#88322—81no—
129#8457—531nomaximal, Sylow
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—24no—
133C22×C53—44no—
134C22×C53—42no—
135C22×C53—42no—
136C22×C53—82no—
137C22×C53#37911yescharacteristic
138C22×C53C2×#10112yes—
139C4×C53—21no—
140C4×C53—22no—
141C4×C53—22no—
142C4×C53—42no—
143C4×C53—44no—
144C4×C53—44no—
145C4×C53#45311yescharacteristic
146C4×C53#45511yescharacteristic
147#3501D5311yescharacteristic
148#1955—531no—
149#1552—532no—
150#1531—532no—
151#1466—531no—
152C2×C53—22no—
153C2×C53—44no—
154C2×C53#178411yescharacteristic
155C2×C53#179211yescharacteristic
156C2×C53#179411yescharacteristic
157#554—22no—
158#554—24no—
159D4×Q8—22no—
160#544Dic5312yes—
161#543—22no—
162#533C2×D5312yes—
163C2×C22⋊Q8—22no—
164#518Dic5312yes—
165C2×C4×D4C2×D5311yescharacteristic
166#432—532no—
167C2×C4⋊C8—531no—
168C2×C22⋊C8—1062no—
169C2×#97—531no—
170#332—1062no—
171#332—1062no—
172#330—1064no—
173#323—1062no—
174C53#845711yescharacteristic, complemented, Sylow
175C22×Q8—22no—
176C22×D4C2×Dic5311yescharacteristic
177C23×C4C22⋊D5312yes—
178C22×C8—1061no—
179C22×C8—1061no—
180C4⋊Q8—24no—
181C4⋊Q8C22⋊D5312yes—
182C4⋊Q8C22⋊D5312yes—
183#125—24no—
184#125C22⋊D5314yes—
185C22⋊Q8—24no—
186C22⋊Q8—24no—
187C22⋊Q8—28no—
188C22⋊Q8—28no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C22⋊Q8—24no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4—22no—
197C4×D4—22no—
198#117C2×Dic5314yes—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4C2×Dic5311yescharacteristic
202C2×C22⋊C4—24no—
203C2×C22⋊C4C22⋊D5312yes—
204C2×C42C22×D5311yescharacteristic
205C4⋊C8—1062no—
206C4⋊C8—1062no—
207C22⋊C8—1062no—
208C22⋊C8—2124no—
209#97—1062no—
210#97—1062no—
211C24C22⋊Dic5312yes—
212C2×Q8—24no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—82no—
216C2×Q8—82no—
217C2×D4—24no—
218C2×D4—22no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4C22⋊Dic5312yes—
226C22×C4C22×Dic5311yescharacteristic
227C22×C4C2×C22⋊D5311yescharacteristic
228C22×C4C2×C22⋊D5311yescharacteristic
229C2×C8—1062no—
230C2×C8—2122no—
231C2×C8—2122no—
232C2×C8—2121no—
233C2×C8—2121no—
234C2×C8—2121no—
235C4⋊C4—24no—
236C4⋊C4—28no—
237C4⋊C4—22no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—24no—
241C4⋊C4—22no—
242C4⋊C4C22⋊Dic5314yes—
243C4⋊C4C22⋊Dic5314yes—
244C22⋊C4—28no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C42—24no—
248C42C2×C22⋊D5312yes—
249C42C2×C22⋊D5312yes—
250C23—22no—
251C23—22no—
252C23—42no—
253C23—42no—
254C23#1313313614yes—
255C23C2×C22⋊Dic5311yescharacteristic
256Q8—42no—
257Q8—42no—
258Q8—82no—
259Q8—84no—
260D4—28no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—22no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—82no—
272C2×C4—84no—
273C2×C4—84no—
274C2×C4C2×C22⋊Dic5312yes—
275C2×C4#1313318612yes—
276C2×C4#1313318611yescharacteristic
277C2×C4#1313318611yescharacteristic
278C8—4242no—
279C8—4241no—
280C8—2122no—
281C22—21no—
282C22—21no—
283C22—24no—
284C22—44no—
285C22—42no—
286C22—42no—
287C22—82no—
288C22#1681047911yescharacteristic
289C22C2×#1313313612yes—
290C4—22no—
291C4—21no—
292C4—22no—
293C4—42no—
294C4—44no—
295C4—44no—
296C4#1681079111yescharacteristic
297C4#1681080111yescharacteristic
298C2—22no—
299C2—44no—
300C2#2027463911yescharacteristic
301C2#2027466911yescharacteristic
302C2#2027467911yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3501C2
socleC22×C53#379
prime radicalC22×D4×C53C2×C4
cosocleC2×C42×C53C23
hypercenter#3501D53
hypocenterC53#8457
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
2. #20273415C2semidirect
3. #20273330C2semidirect
174. C53#8457semidirect

Maximal subgroups

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

subgroupindex
#202755142
#202734152
#202733302
#202730562
C53×#35012
#845753