Finite Group Databasebeta

← #27299600

Subgroups — #27299600

Order 13568.

1018 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C22×#13133215C211yesmaximal, characteristic
2C2×#16810421C214yesmaximal, complemented
3#20274639C211yesmaximal, characteristic
4C53×#1784C211yesmaximal, characteristic
5C23×Dic106C411yescharacteristic
6C2×#13133215—28no—
7C2×#13133215—21no—
8C2×#13133215—21no—
9C2×#13133215C412yes—
10C2×#13133215C2212yes—
11C22×#11719919—21no—
12C22×#11719919—21no—
13C22×C4×Dic53C411yescharacteristic
14C2×#13133136—21no—
15C2×#13133136C2212yes—
16#16810587—21no—
17#16810421—28no—
18C23×Q8×C53C2211yescharacteristic
19C2×#101×C53—21no—
20C2×#101×C53C2212yes—
21C53×#407—21no—
22C22×Dic106—21no—
23C22×Dic106—24no—
24C22×Dic106—42no—
25C22×Dic106—42no—
26C22×Dic106C2×C412yes—
27C23×Dic53—21no—
28C23×Dic53—21no—
29C2×#11719917—22no—
30#13133215—28no—
31#13133215—28no—
32#13133215—24no—
33#13133215—24no—
34#13133215—24no—
35#13133215D418yes—
36C2×#11719919—28no—
37C2×#11719919—28no—
38C2×#11719919—22no—
39C2×#11719919—22no—
40C2×#11719919—22no—
41C2×#11719919—22no—
42C2×C4×Dic53—24no—
43C2×C4×Dic53—21no—
44C2×C4×Dic53C2×C412yes—
45#13133136—24no—
46#13133136—42no—
47C22⋊#88322—42no—
48C22×Q8×C53—24no—
49C22×Q8×C53—42no—
50C22×Q8×C53—42no—
51C22×Q8×C53C2×C411yescharacteristic
52C22×Q8×C53D411yescharacteristic
53C22×Q8×C53D411yescharacteristic
54C22×Q8×C53C2311yescharacteristic
55C23×C4×C53C2×C411yescharacteristic
56C23×C4×C53D411yescharacteristic
57C23×C4×C53D411yescharacteristic
58C2×M16×C53—22no—
59#101×C53—24no—
60#101×C53—42no—
61C22⋊C8×C53—42no—
62C24×C53C22⋊C411yescharacteristic
63C2×Q8×C53—22no—
64C2×Q8×C53—22no—
65C2×Q8×C53—22no—
66C2×Q8×C53—22no—
67C2×Q8×C53—22no—
68C2×Q8×C53—41no—
69C2×Q8×C53—41no—
70C2×Q8×C53—41no—
71C2×Q8×C53—41no—
72C2×Q8×C53—41no—
73C2×Q8×C53—41no—
74C2×Q8×C53—416no—
75C2×Q8×C53C2×D414yes—
76C22×C4×C53—22no—
77C22×C4×C53—22no—
78C22×C4×C53—22no—
79C22×C4×C53—22no—
80C22×C4×C53—22no—
81C22×C4×C53—22no—
82C22×C4×C53—22no—
83C22×C4×C53—41no—
84C22×C4×C53—41no—
85C22×C4×C53C22⋊C411yescharacteristic
86C22×C4×C53C22⋊C411yescharacteristic
87C22×C4×C53C22⋊C411yescharacteristic
88C22×C4×C53C22×C411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×Dic53—24no—
92C22×Dic53—28no—
93C22×Dic53—22no—
94C22×Dic53—24no—
95C22×Dic53—22no—
96C22×Dic53—42no—
97C22×Dic53—42no—
98C2×Dic106—24no—
99C2×Dic106—22no—
100C2×Dic106—42no—
101C2×Dic106—416no—
102C2×Dic106—42no—
103C2×Dic106—42no—
104C2×Dic106—42no—
105C2×Dic106C22⋊C414yes—
106M16×C53—42no—
107M16×C53—42no—
108C2×C8×C53—42no—
109#11719919—28no—
110#11719919—28no—
111#11719919—28no—
112#11719919—28no—
113C4×Dic53—24no—
114C4×Dic53—22no—
115C4×Dic53C22⋊C414yes—
116#11719917—42no—
117#11719917—42no—
118C2×#88322—42no—
119C23×C53—24no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53#9912yes—
123C23×C53C2×C22⋊C411yescharacteristic
124Q8×C53—44no—
125Q8×C53—44no—
126Q8×C53—44no—
127Q8×C53—44no—
128C2×C4×C53—28no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—44no—
144C2×C4×C53—44no—
145C2×C4×C53#9912yes—
146C2×C4×C53C2×C22⋊C412yes—
147C2×C4×C53#12012yes—
148C2×C4×C53#12012yes—
149C2×Dic53—216no—
150C2×Dic53—28no—
151C2×Dic53—24no—
152C2×Dic53—24no—
153C2×Dic53—48no—
154C2×Dic53—42no—
155C2×Dic53—42no—
156Dic106—48no—
157Dic106—44no—
158Dic106—44no—
159C8×C53—82no—
160#88322—82no—
161#12700—531nomaximal, Sylow
162C22×C53—24no—
163C22×C53—24no—
164C22×C53—22no—
165C22×C53—22no—
166C22×C53—22no—
167C22×C53#35614yescomplemented
168C22×C53#37911yescharacteristic
169C22×C53C2×#9911yescharacteristic
170C22×C53C2×#9911yescharacteristic
171C4×C53—22no—
172C4×C53—22no—
173C4×C53—22no—
174C4×C53—22no—
175C4×C53—41no—
176C4×C53—41no—
177Dic53—28no—
178Dic53—44no—
179C22×C4⋊Q8—531no—
180C2×#356—534no—
181#1784—531no—
182#1784D5311yescharacteristic
183C2×C53—21no—
184C2×C53—24no—
185C2×C53—21no—
186C2×C53#178111yescharacteristic
187C2×C53C2×#35612yes—
188C23×Q8—531no—
189C23×Q8C2×D5311yescharacteristic
190C2×C4⋊Q8—532no—
191C2×C4⋊Q8—532no—
192C2×C4⋊Q8—1068no—
193C2×C4⋊Q8—1061no—
194C2×C4⋊Q8—1061no—
195C22×C4⋊C4—1061no—
196C22×C4⋊C4—1061no—
197C22×C42—531no—
198C2×#101—21no—
199C2×#101—532no—
200C2×#101—1061no—
201C2×#101C2×D5312yes—
202#407—21no—
203#407—1061no—
204#356—1068no—
205C53#1270011yescharacteristic, complemented, Sylow
206C22×Q8—24no—
207C22×Q8—42no—
208C22×Q8—42no—
209C22×Q8—532no—
210C22×Q8—1064no—
211C22×Q8—1061no—
212C22×Q8—2122no—
213C22×Q8—2122no—
214C22×Q8C4×D5311yescharacteristic
215C22×Q8D21211yescharacteristic
216C22×Q8C22⋊D5311yescharacteristic
217C22×Q8C22×D5311yescharacteristic
218C23×C4—1061no—
219C23×C4—1061no—
220C23×C4C4×D5311yescharacteristic
221C23×C4D21211yescharacteristic
222C23×C4C22⋊D5311yescharacteristic
223C2×M16—22no—
224C2×M16—1062no—
225C4⋊Q8—538no—
226C4⋊Q8—1064no—
227C4⋊Q8—1064no—
228C4⋊Q8—1068no—
229C4⋊Q8—1068no—
230C4⋊Q8—1064no—
231C2×C4⋊C4—1068no—
232C2×C4⋊C4—1068no—
233C2×C4⋊C4—1062no—
234C2×C4⋊C4—1062no—
235C2×C4⋊C4—1062no—
236C2×C4⋊C4—1062no—
237C2×C42—532no—
238C2×C42—1064no—
239C2×C42—1061no—
240#101—24no—
241#101—42no—
242#101—1064no—
243#101—2122no—
244C22⋊C8—42no—
245C22⋊C8—2122no—
246C24#1171992111yescharacteristic
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—416no—
259C2×Q8—534no—
260C2×Q8—1064no—
261C2×Q8—1062no—
262C2×Q8—21216no—
263C2×Q8—2122no—
264C2×Q8—2122no—
265C2×Q8—2122no—
266C2×Q8—2122no—
267C2×Q8D4×D5314yes—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—1064no—
278C22×C4—1068no—
279C22×C4—1062no—
280C22×C4—1064no—
281C22×C4—1062no—
282C22×C4—2122no—
283C22×C4—2122no—
284C22×C4#1171992111yescharacteristic
285C22×C4#1171992111yescharacteristic
286C22×C4#1171992111yescharacteristic
287C22×C4C2×C4×D5311yescharacteristic
288C22×C4C2×D21211yescharacteristic
289C22×C4C2×C22⋊D5311yescharacteristic
290M16—42no—
291M16—42no—
292M16—2122no—
293M16—2122no—
294C2×C8—42no—
295C2×C8—2122no—
296C4⋊C4—1068no—
297C4⋊C4—1068no—
298C4⋊C4—1068no—
299C4⋊C4—1068no—
300C42—534no—
301C42—1064no—
302C42—1062no—
303C23—21no—
304C23—24no—
305C23—21no—
306C23#1313313112yes—
307C23C2×#1171992111yescharacteristic
308Q8—44no—
309Q8—44no—
310Q8—44no—
311Q8—44no—
312Q8—2124no—
313Q8—2128no—
314Q8—2124no—
315C2×C4—28no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—10616no—
333C2×C4—1068no—
334C2×C4—1064no—
335C2×C4—1064no—
336C2×C4—2128no—
337C2×C4—2122no—
338C2×C4—2122no—
339C2×C4#1313313112yes—
340C2×C4C22⋊C4×D5312yes—
341C2×C4#1313318512yes—
342C2×C4#1313318712yes—
343C8—82no—
344C8—4242no—
345C22—24no—
346C22—24no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22#1681042114yescomplemented
351C22#1681047811yescharacteristic
352C22C2×#1313313111yescharacteristic
353C22C2×#1313313111yescharacteristic
354C4—22no—
355C4—22no—
356C4—22no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—1068no—
361C4—2124no—
362C2—21no—
363C2—21no—
364C2—24no—
365C2#2027462911yescharacteristic
366C2C2×#1681042112yes—
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC53×#1784C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC22×Q8×C53C23
hypercenter#1784D53
hypocenterC53#12700
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ #1784

Complements

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

NH ≅ G/Ntype
2. C2×#16810421C2semidirect
167. C22×C53#356semidirect
205. C53#12700semidirect
350. C22#16810421semidirect

Maximal subgroups

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

subgroupindex
C22×#131332152
C2×#168104212
#202746392
C53×#17842
#1270053