Finite Group Databasebeta

← #27294396

Subgroups — #27294396

Order 13568.

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

normal subgroups only

This group has 452 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
1#20277983C211yesmaximal, characteristic
2#20273029C211yesmaximal, characteristic
3#20272744C212yesmaximal, complemented
4#20272741C212yesmaximal, complemented
5C53×#1742C211yesmaximal, characteristic
6C2×C4×Dic106C411yescharacteristic
7#16810676—21no—
8#16810562C2211yescharacteristic
9#16810559C411yescharacteristic
10#16810556—21no—
11#16810546—21no—
12#16810543C2211yescharacteristic
13#16810535C411yescharacteristic
14C4×#11719919—21no—
15C4×C4⋊Dic53C411yescharacteristic
16C4×C22⋊Dic53—21no—
17#16810349C2212yes—
18#16810347—21no—
19C22×M16×C53—21no—
20C2×#117×C53C2211yescharacteristic
21C53×#343—21no—
22C53×#343C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C4×Dic106—41no—
25C4×Dic106—41no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×C4×Dic53—21no—
34C2×C4×Dic53—21no—
35C4⋊#88322—41no—
36C22⋊#88322—22no—
37C4×#88322—41no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C23×C4×C53D411yescharacteristic
45C2×M16×C53—24no—
46C2×M16×C53—22no—
47C2×M16×C53—41no—
48C2×M16×C53—42no—
49C22×C8×C53—41no—
50#117×C53—41no—
51#117×C53—41no—
52C2×C4⋊C4×C53D411yescharacteristic
53C2×C4⋊C4×C53C2311yescharacteristic
54C2×C22⋊C4×C53C2×C411yescharacteristic
55C2×C22⋊C4×C53D411yescharacteristic
56C2×C42×C53C2×C411yescharacteristic
57C2×C42×C53D411yescharacteristic
58C24×C53C22⋊C411yescharacteristic
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—22no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53SD1612yes—
66C22×C4×C53Q1612yes—
67C22×C4×C53C22×C411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C2×Dic106—41no—
75C2×Dic106—42no—
76M16×C53—24no—
77M16×C53—44no—
78M16×C53—42no—
79C2×C8×C53—42no—
80C2×C8×C53—44no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C22⋊C4×C53—41no—
84C22⋊C4×C53—41no—
85C42×C53—21no—
86C42×C53—21no—
87C42×C53—21no—
88C42×C53—21no—
89C22⋊Dic53—42no—
90C22⋊Dic53—42no—
91C4⋊Dic53—42no—
92C4⋊Dic53—41no—
93#11719919—42no—
94#11719919—42no—
95#11719919—42no—
96C4×Dic53—41no—
97C4×Dic53—41no—
98C4×Dic53—42no—
99C2×#88322—42no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53#10312yes—
105C23×C53C2×C22⋊C411yescharacteristic
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53#9911yescharacteristic
121C2×C4×C53#9911yescharacteristic
122C2×C4×C53#10312yes—
123C2×C4×C53#10412yes—
124C2×C4×C53C2×SD1611yescharacteristic
125C2×C4×C53C2×Q1611yescharacteristic
126C2×Dic53—22no—
127C2×Dic53—22no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138Dic106—42no—
139Dic106—42no—
140C8×C53—44no—
141#88322—82no—
142#11721—531nomaximal, Sylow
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—22no—
146C22×C53—42no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—81no—
150C22×C53C2×#9911yescharacteristic
151C22×C53C2×#10311yescharacteristic
152C22×C53C2×#10411yescharacteristic
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—41no—
156C4×C53—41no—
157C4×C53—41no—
158C4×C53—41no—
159C4×C53#32812yes—
160C4×C53#32812yes—
161Dic53—42no—
162Dic53—42no—
163Dic53—81no—
164Dic53—81no—
165#2396—531no—
166#1742D5311yescharacteristic
167#1458—531no—
168#1354—532no—
169#1353—532no—
170C2×C53—21no—
171C2×C53—21no—
172C2×C53—42no—
173C2×C53C2×#32811yescharacteristic
174C2×C53#212011yescharacteristic
175C2×C53#213211yescharacteristic
176C22×M16—21no—
177C2×C4×Q8—531no—
178C2×#117C2×D5311yescharacteristic
179#431—1061no—
180#400—531no—
181#399—531no—
182#398—1061no—
183#396—1061no—
184#395—531no—
185#393—531no—
186C4×C4⋊C4—531no—
187C4×C4⋊C4—1061no—
188C4×C22⋊C4—1061no—
189#343—21no—
190#343C2×D5312yes—
191#324—532no—
192#323—1061no—
193C53#1172111yescharacteristic, complemented, Sylow
194C22×Q8—531no—
195C23×C4C22⋊D5311yescharacteristic
196C2×M16—24no—
197C2×M16—22no—
198C2×M16—41no—
199C2×M16—42no—
200C22×C8—41no—
201C4×Q8—2121no—
202C4×Q8—2121no—
203#117—41no—
204#117—41no—
205C2×C4⋊C4—531no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4C22⋊D5311yescharacteristic
211C2×C4⋊C4C22×D5311yescharacteristic
212C2×C22⋊C4—1061no—
213C2×C22⋊C4—1061no—
214C2×C22⋊C4C4×D5311yescharacteristic
215C2×C22⋊C4D21211yescharacteristic
216C2×C42—1061no—
217C2×C42—1061no—
218C2×C42C4×D5311yescharacteristic
219C2×C42D21211yescharacteristic
220C4⋊C8—2121no—
221C22⋊C8—1062no—
222C4×C8—2121no—
223#95—531no—
224#95—531no—
225#95—1061no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229C24#1171992111yescharacteristic
230C2×Q8—2121no—
231C2×Q8—2122no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—1061no—
236C22×C4—1061no—
237C22×C4—1061no—
238C22×C4—1061no—
239C22×C4#1171992111yescharacteristic
240C22×C4#1171992111yescharacteristic
241C22×C4#1171992111yescharacteristic
242C22×C4#1171992312yes—
243C22×C4#1171992512yes—
244C22×C4C2×C4×D5311yescharacteristic
245C22×C4C2×D21211yescharacteristic
246C22×C4C2×C22⋊D5311yescharacteristic
247M16—24no—
248M16—44no—
249M16—42no—
250C2×C8—42no—
251C2×C8—44no—
252C2×C8—2122no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—2121no—
256C4⋊C4—2122no—
257C4⋊C4—2122no—
258C4⋊C4—2122no—
259C4⋊C4—2122no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—2122no—
263C22⋊C4—2122no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—2121no—
269C42—2121no—
270C42—2122no—
271C23—21no—
272C23—21no—
273C23—41no—
274C23—41no—
275C23#1313314012yes—
276C23C2×#1171992111yescharacteristic
277Q8—2122no—
278Q8—2122no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—1062no—
294C2×C4—1062no—
295C2×C4—2121no—
296C2×C4—2121no—
297C2×C4—2121no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4#1313313111yescharacteristic
306C2×C4#1313313111yescharacteristic
307C2×C4#1313314012yes—
308C2×C4#1313314512yes—
309C2×C4C2×#1171992311yescharacteristic
310C2×C4C2×#1171992511yescharacteristic
311C8—44no—
312C8—4242no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22—81no—
320C22C2×#1313313111yescharacteristic
321C22C2×#1313314011yescharacteristic
322C22C2×#1313314511yescharacteristic
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—4241no—
330C4—4241no—
331C4—2122no—
332C4—2122no—
333C4#1681035912yes—
334C4#1681035912yes—
335C2—21no—
336C2—21no—
337C2—42no—
338C2C2×#1681035911yescharacteristic
339C2#2027635711yescharacteristic
340C2#2027639811yescharacteristic
subgroupG/H
centerC2×C4#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1742C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1742D53
hypocenterC53#11721
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #1742

Complements

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

NH ≅ G/Ntype
3. #20272744C2semidirect
4. #20272741C2semidirect
193. C53#11721semidirect

Maximal subgroups

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

subgroupindex
#202779832
#202730292
#202727442
#202727412
C53×#17422
#1172153