Finite Group Databasebeta

← #28299852

Subgroups — #28299852

Order 13568.

562 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C4×C22⋊#88324C211yesmaximal, characteristic
2#20290378C211yesmaximal, characteristic
3#20289534C212yesmaximal, complemented
4#20289533C212yesmaximal, complemented
5#20283065C211yesmaximal, characteristic
6C22×#11719940—21no—
7C2×C22⋊#88324—21no—
8C2×C4×#88324—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133216C411yescharacteristic
13C2×#13133194—21no—
14C22×C4×Dic53C2211yescharacteristic
15#16810527—21no—
16#16810522C2211yescharacteristic
17#16810520—21no—
18#16810519C2211yescharacteristic
19#16810492C411yescharacteristic
20#16810483C411yescharacteristic
21#16810481—21no—
22C53×#380C411yescharacteristic
23C2×#11719940—24no—
24C2×#11719940—42no—
25C2×#11719940—42no—
26C22×#88324—22no—
27C22×#88324—22no—
28C22⋊#88324—24no—
29C4×#88324—42no—
30C22×Dic106—21no—
31C23×Dic53D411yescharacteristic
32C23×Dic53D411yescharacteristic
33#13133216—42no—
34#13133194—42no—
35#13133194—42no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×#11719919—21no—
40C2×#11719919—21no—
41C2×#11719919—21no—
42C2×#11719919—21no—
43C2×C22⋊Dic53—21no—
44C2×C4×Dic53—42no—
45C2×C4×Dic53C2×C411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48C2×C4×Dic53C2311yescharacteristic
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52C23×C4×C53C2×C411yescharacteristic
53C2×C4⋊C4×C53C2×C411yescharacteristic
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C22⋊C4×C53—21no—
56#95×C53—21no—
57C24×C53C22⋊C411yescharacteristic
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—41no—
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22×C411yescharacteristic
65C22×Dic53—22no—
66C22×Dic53—22no—
67C22×Dic53—22no—
68C22×Dic53—22no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—41no—
72C22×Dic53C22⋊C411yescharacteristic
73C22×Dic53C22⋊C411yescharacteristic
74C22×Dic53D812yes—
75C22×Dic53SD1612yes—
76C22×Dic53C2×D411yescharacteristic
77C22×Dic53C2×D411yescharacteristic
78C2×Dic106—42no—
79C2×Dic106—42no—
80C2×Dic106—42no—
81#11719940—48no—
82C2×#88324—24no—
83C2×#88324—24no—
84C2×#88324—42no—
85C2×#88324—42no—
86C2×#88324—42no—
87C2×#88324—42no—
88C4⋊C4×C53—22no—
89C4⋊C4×C53—41no—
90C22⋊C4×C53—42no—
91C22⋊Dic53—42no—
92C4⋊Dic53—22no—
93C4⋊Dic53—41no—
94C4⋊Dic53—41no—
95C4⋊Dic53—41no—
96#11719919—42no—
97#11719919—42no—
98#11719919—42no—
99#11719919—41no—
100#11719919—41no—
101C4×Dic53—22no—
102C4×Dic53—22no—
103C4×Dic53—41no—
104C4×Dic53—41no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53#10212yes—
112C23×C53C2×C22⋊C411yescharacteristic
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—44no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53#10112yes—
125C2×C4×C53#13112yes—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—24no—
133C2×Dic53—24no—
134C2×Dic53—44no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53#9912yes—
142C2×Dic53#10212yes—
143C2×Dic53#10412yes—
144C2×Dic53#10412yes—
145C2×Dic53C2×D811yescharacteristic
146C2×Dic53C2×SD1611yescharacteristic
147C2×Dic53#13712yes—
148Dic106—84no—
149#88324—44no—
150#88324—44no—
151#7685—531nomaximal, Sylow
152C22×C53—22no—
153C22×C53—22no—
154C22×C53—22no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53#40811yescharacteristic
160C22×C53C2×#9911yescharacteristic
161C22×C53C2×#10111yescharacteristic
162C22×C53C2×#10211yescharacteristic
163C22×C53#41911yescharacteristic
164C22×C53C2×#10411yescharacteristic
165C22×C53C2×#10411yescharacteristic
166C4×C53—22no—
167C4×C53—22no—
168C4×C53—81no—
169C4×C53—81no—
170Dic53—21no—
171Dic53—21no—
172Dic53—22no—
173Dic53—22no—
174Dic53—81no—
175Dic53—81no—
176Dic53#32714yes—
177#2857—531no—
178C4×C22⋊C8—531no—
179#1728—531no—
180#1271—532no—
181#1271—532no—
182C2×C53—42no—
183C2×C53#147211yescharacteristic
184C2×C53#148511yescharacteristic
185C2×C53#149211yescharacteristic
186C2×C53C2×#32711yescharacteristic
187C2×C53#150811yescharacteristic
188C2×C53#152611yescharacteristic
189C2×C53#154611yescharacteristic
190C22×M16—1061no—
191C2×C4⋊Q8—531no—
192C2×C22⋊Q8—1061no—
193C22×C42—531no—
194C2×C22⋊C8—1061no—
195C2×C4×C8—1061no—
196#391—1061no—
197#389—531no—
198#389—531no—
199#389—1061no—
200#384—531no—
201#380—531no—
202#380—1061no—
203#380#615711yescharacteristic
204#336—532no—
205#336—532no—
206#336—1061no—
207C53#768511yescharacteristic, complemented, Sylow
208C22×Q8—1061no—
209C23×C4—531no—
210C23×C4—531no—
211C23×C4C2×#615711yescharacteristic
212C2×M16—1064no—
213C2×M16—2122no—
214C2×M16—2122no—
215C22×C8—1062no—
216C22×C8—1062no—
217C4⋊Q8—2122no—
218C22⋊Q8—2122no—
219C22⋊Q8—2122no—
220C2×C4⋊C4—531no—
221C2×C4⋊C4—531no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4—1061no—
227C2×C4⋊C4C2×#615711yescharacteristic
228C2×C4⋊C4C2×#615711yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—1061no—
231C2×C42—531no—
232C2×C42—531no—
233C2×C42—531no—
234C2×C42—531no—
235C2×C42—2122no—
236C22⋊C8—1064no—
237C4×C8—2122no—
238#95—21no—
239#95—1061no—
240#95—1061no—
241#95—1061no—
242C24C22⋊#615711yescharacteristic
243C2×Q8—2122no—
244C2×Q8—2122no—
245C2×Q8—2122no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—531no—
252C22×C4—532no—
253C22×C4—532no—
254C22×C4—531no—
255C22×C4—531no—
256C22×C4—531no—
257C22×C4—1062no—
258C22×C4—1062no—
259C22×C4—1062no—
260C22×C4—1062no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—2121no—
264C22×C4C22⋊#615711yescharacteristic
265C22×C4C22×#615711yescharacteristic
266M16—2128no—
267C2×C8—1064no—
268C2×C8—1064no—
269C2×C8—2122no—
270C2×C8—2122no—
271C2×C8—2122no—
272C2×C8—2122no—
273C4⋊C4—22no—
274C4⋊C4—41no—
275C4⋊C4—1062no—
276C4⋊C4—2121no—
277C4⋊C4—2122no—
278C4⋊C4—2121no—
279C4⋊C4—2121no—
280C4⋊C4—2121no—
281C4⋊C4—2121no—
282C4⋊C4—2122no—
283C4⋊C4—2122no—
284C22⋊C4—42no—
285C22⋊C4—2122no—
286C42—1062no—
287C42—1062no—
288C42—2121no—
289C42—2121no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23#1313327912yes—
297C23C2×C22⋊#615711yescharacteristic
298Q8—4244no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—44no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—532no—
311C2×C4—532no—
312C2×C4—532no—
313C2×C4—531no—
314C2×C4—531no—
315C2×C4—532no—
316C2×C4—532no—
317C2×C4—1064no—
318C2×C4—1064no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—2124no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4#1313327812yes—
333C2×C4#1313330412yes—
334C8—2124no—
335C8—2124no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22#1681159611yescharacteristic
344C22D53⋊#9911yescharacteristic
345C22C2×#1313327811yescharacteristic
346C22C2×#1313327911yescharacteristic
347C22#1681161611yescharacteristic
348C22D53⋊#10411yescharacteristic
349C22D53⋊#10411yescharacteristic
350C4—22no—
351C4—22no—
352C4—81no—
353C4—81no—
354C4—534no—
355C4—1061no—
356C4—1061no—
357C4—1062no—
358C4—1062no—
359C4—4241no—
360C4—4241no—
361C2—42no—
362C2#2028999411yescharacteristic
363C2#2029003711yescharacteristic
364C2#2029006011yescharacteristic
365C2D53⋊#32711yescharacteristic
366C2#2029011311yescharacteristic
367C2#2029017211yescharacteristic
368C2#2029022411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#380C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#380#6157
hypocenterC53#7685
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #20289534C2semidirect
4. #20289533C2semidirect
207. C53#7685semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊#883242
#202903782
#202895342
#202895332
#202830652
#768553