Finite Group Databasebeta

← #27293471

Subgroups — #27293471

Order 13568.

598 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 95 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×#16810528C211yesmaximal, characteristic
2#20273045C211yesmaximal, characteristic
3#20272648C214yesmaximal, complemented
4C53×#1463C211yesmaximal, characteristic
5C23×Dic106C411yescharacteristic
6C22×C4×Dic53C411yescharacteristic
7#16810587—21no—
8#16810528—21no—
9#16810528—21no—
10#16810528C412yes—
11#16810528C2212yes—
12C2×#13133124—21no—
13C2×#13133124—21no—
14#16810349—21no—
15#16810349C2212yes—
16C22×C4⋊C4×C53C2211yescharacteristic
17C53×#407—21no—
18C53×#324—21no—
19C53×#324C2212yes—
20C22×Dic106—21no—
21C22×Dic106—24no—
22C22×Dic106—41no—
23C22×Dic106—41no—
24C22×Dic106—42no—
25C22×Dic106C2×C412yes—
26C23×Dic53—21no—
27C23×Dic53—21no—
28C2×#11719917—41no—
29C2×C4×Dic53—21no—
30C2×C4×Dic53—42no—
31C2×C4×Dic53C2×C412yes—
32C22⋊#88322—22no—
33C22⋊#88322—41no—
34#13133124—22no—
35#13133124—22no—
36#13133124—22no—
37#13133124—22no—
38C23×C4×C53C2×C411yescharacteristic
39C23×C4×C53D411yescharacteristic
40C23×C4×C53D411yescharacteristic
41C2×M16×C53—41no—
42C2×C4⋊C4×C53—41no—
43C2×C4⋊C4×C53—41no—
44C2×C4⋊C4×C53C2×C411yescharacteristic
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C4⋊C4×C53D411yescharacteristic
47C2×C4⋊C4×C53C2311yescharacteristic
48C22⋊C8×C53—22no—
49C22⋊C8×C53—41no—
50C24×C53C22⋊C411yescharacteristic
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—42no—
60C22×C4×C53—42no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×Dic53—24no—
68C22×Dic53—24no—
69C22×Dic53—22no—
70C22×Dic53—22no—
71C22×Dic53—42no—
72C22×Dic53—42no—
73C22×Dic53—42no—
74C22×Dic53—41no—
75C22×Dic53—41no—
76C2×Dic106—24no—
77C2×Dic106—22no—
78C2×Dic106—42no—
79C2×Dic106—48no—
80C2×Dic106—48no—
81C2×Dic106—42no—
82C2×Dic106—42no—
83C2×Dic106—42no—
84C2×Dic106M1614yescomplemented
85M16×C53—82no—
86C2×C8×C53—42no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C4×Dic53—42no—
92C4×Dic53—42no—
93#11719917—82no—
94C2×#88322—42no—
95C23×C53—22no—
96C23×C53—22no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53#9911yescharacteristic
100C23×C53#9911yescharacteristic
101C23×C53C2×C22⋊C411yescharacteristic
102C2×C4×C53—24no—
103C2×C4×C53—24no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—81no—
119C2×C4×C53—81no—
120C2×C4×C53#10012yes—
121C2×C4×C53C2×M1612yes—
122C2×Dic53—24no—
123C2×Dic53—24no—
124C2×Dic53—44no—
125C2×Dic53—44no—
126C2×Dic53—48no—
127C2×Dic53—42no—
128C2×Dic53—42no—
129C2×Dic53—42no—
130C2×Dic53—42no—
131Dic106—48no—
132Dic106—44no—
133Dic106—44no—
134C8×C53—82no—
135#88322—82no—
136#11506—531nomaximal, Sylow
137C22×C53—24no—
138C22×C53—24no—
139C22×C53—21no—
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—21no—
145C22×C53#35412yes—
146C22×C53#35612yes—
147C22×C53#40711yescharacteristic
148C22×C53C2×#9911yescharacteristic
149C22×C53C2×#10011yescharacteristic
150C4×C53—22no—
151C4×C53—22no—
152C4×C53—41no—
153C4×C53—41no—
154C4×C53—41no—
155C4×C53—41no—
156Dic53—44no—
157Dic53—44no—
158C2×#391—531no—
159#1463—531no—
160#1463D5311yescharacteristic
161#1319—534no—
162C2×C53—21no—
163C2×C53—22no—
164C2×C53—22no—
165C2×C53—21no—
166C2×C53#146211yescharacteristic
167C2×C53C2×#35411yescharacteristic
168C2×C53C2×#35611yescharacteristic
169C23×Q8—531no—
170C22×C4⋊C4C2×D5311yescharacteristic
171C22×C42—531no—
172#407—21no—
173#407—1061no—
174#391—532no—
175#391—532no—
176#391—1061no—
177#391—1061no—
178C2×#95—1061no—
179C2×#95—1061no—
180#324—21no—
181#324—532no—
182#324—1061no—
183#324C2×D5312yes—
184C53#1150611yescharacteristic, complemented, Sylow
185C22×Q8—532no—
186C22×Q8—1064no—
187C22×Q8—1061no—
188C22×Q8—2122no—
189C22×Q8—2121no—
190C22×Q8—2121no—
191C23×C4—1061no—
192C23×C4—1061no—
193C23×C4C4×D5311yescharacteristic
194C23×C4D21211yescharacteristic
195C23×C4C22⋊D5311yescharacteristic
196C2×M16—41no—
197C2×M16—2121no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4C4×D5311yescharacteristic
201C2×C4⋊C4D21211yescharacteristic
202C2×C4⋊C4C22⋊D5311yescharacteristic
203C2×C4⋊C4C22×D5311yescharacteristic
204C2×C42—532no—
205C2×C42—1061no—
206C2×C42—2122no—
207C22⋊C8—22no—
208C22⋊C8—41no—
209C22⋊C8—1062no—
210C22⋊C8—2121no—
211#95—1062no—
212#95—1062no—
213#95—1062no—
214#95—1062no—
215C24#1171992111yescharacteristic
216C2×Q8—534no—
217C2×Q8—1064no—
218C2×Q8—1062no—
219C2×Q8—2128no—
220C2×Q8—2128no—
221C2×Q8—2122no—
222C2×Q8—2122no—
223C2×Q8—2122no—
224C2×Q8—2122no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—1064no—
236C22×C4—1064no—
237C22×C4—1062no—
238C22×C4—1062no—
239C22×C4—2122no—
240C22×C4—2122no—
241C22×C4—2122no—
242C22×C4—2121no—
243C22×C4—2121no—
244C22×C4#1171992111yescharacteristic
245C22×C4#1171992111yescharacteristic
246C22×C4#1171992111yescharacteristic
247C22×C4C2×C4×D5311yescharacteristic
248C22×C4C2×D21211yescharacteristic
249C22×C4C2×C22⋊D5311yescharacteristic
250M16—82no—
251M16—4242no—
252C2×C8—42no—
253C2×C8—2122no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C42—2122no—
259C42—2122no—
260C23—21no—
261C23—21no—
262C23—22no—
263C23—22no—
264C23#1313313111yescharacteristic
265C23#1313313111yescharacteristic
266C23C2×#1171992111yescharacteristic
267Q8—2124no—
268Q8—2128no—
269Q8—2124no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—1064no—
289C2×C4—1064no—
290C2×C4—2128no—
291C2×C4—2122no—
292C2×C4—2124no—
293C2×C4—2124no—
294C2×C4—2122no—
295C2×C4—2122no—
296C2×C4—2122no—
297C2×C4#1313313312yes—
298C2×C4M16×D5312yes—
299C8—82no—
300C8—4242no—
301C22—24no—
302C22—24no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22#1681041712yes—
310C22#1681042112yes—
311C22#1681058611yescharacteristic
312C22C2×#1313313111yescharacteristic
313C22C2×#1313313311yescharacteristic
314C4—22no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—2124no—
321C4—2124no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—22no—
326C2#2027304111yescharacteristic
327C2C2×#1681041711yescharacteristic
328C2C2×#1681042111yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1463C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1463D53
hypocenterC53#11506
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
3. #20272648C2semidirect
84. C2×Dic106M16semidirect
184. C53#11506semidirect

Maximal subgroups

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

subgroupindex
C2×#168105282
#202730452
#202726482
C53×#14632
#1150653