Finite Group Databasebeta

← #27300353

Subgroups — #27300353

Order 13568.

642 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 99 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#20277307C211yesmaximal, characteristic
2#20274633C211yesmaximal, characteristic
3#20272839C214yesmaximal, complemented
4C53×#1782C211yesmaximal, characteristic
5C2×#13133176C411yescharacteristic, complemented
6C22×C4⋊Dic53C411yescharacteristic, complemented
7C2×#13133132—21no—
8C2×#13133132C2212yes—
9#16810530C412yescomplemented
10#16810526C2212yes—
11#16810503—21no—
12#16810503—21no—
13#16810486—21no—
14#16810483—21no—
15#16810483—21no—
16C22×#41×C53C2211yescharacteristic
17C2×#99×C53—21no—
18C2×#99×C53C2212yes—
19C53×#381—21no—
20C23×Dic53—21no—
21#13133176—42no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—42no—
25C2×C4⋊Dic53C2×C412yes—
26C2×C22⋊Dic53—21no—
27C2×C22⋊Dic53—22no—
28C2×C22⋊Dic53—22no—
29C2×C22⋊Dic53—22no—
30C2×C4×Dic53C2×C412yes—
31#13133132—24no—
32#13133132—42no—
33#13133124—22no—
34#13133124—22no—
35#13133124—42no—
36C2×#41×C53—24no—
37C2×#41×C53—42no—
38C2×#41×C53—42no—
39C22×Q8×C53C2×C411yescharacteristic
40C22×D4×C53D411yescharacteristic
41C22×D4×C53D411yescharacteristic
42C22×D4×C53C2311yescharacteristic
43C23×C4×C53C2×C411yescharacteristic
44C23×C4×C53D411yescharacteristic
45C23×C4×C53D411yescharacteristic
46C2×C22⋊C4×C53—22no—
47#99×C53—24no—
48#99×C53—42no—
49#95×C53—42no—
50C24×C53C22⋊C411yescharacteristic
51C24×C53C2×D411yescharacteristic
52C24×C53C2×D411yescharacteristic
53#41×C53—416no—
54C2×Q8×C53—22no—
55C2×Q8×C53—41no—
56C2×Q8×C53—41no—
57C2×D4×C53—22no—
58C2×D4×C53—22no—
59C2×D4×C53—22no—
60C2×D4×C53—22no—
61C2×D4×C53—41no—
62C2×D4×C53—41no—
63C2×D4×C53—41no—
64C2×D4×C53—41no—
65C2×D4×C53#4114yes—
66C22×C4×C53—22no—
67C22×C4×C53—22no—
68C22×C4×C53—22no—
69C22×C4×C53—22no—
70C22×C4×C53—22no—
71C22×C4×C53—22no—
72C22×C4×C53—22no—
73C22×C4×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53—42no—
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22×C411yescharacteristic
80C22×Dic53—22no—
81C22×Dic53—22no—
82C22×Dic53—42no—
83C22×Dic53—42no—
84C22×Dic53—82no—
85C22⋊C4×C53—42no—
86C22⋊C4×C53—42no—
87C22⋊Dic53—42no—
88C22⋊Dic53—42no—
89C22⋊Dic53—42no—
90C22⋊Dic53—82no—
91C22⋊Dic53—82no—
92C4⋊Dic53—42no—
93C4⋊Dic53—42no—
94C4⋊Dic53—42no—
95C4×Dic53—42no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53#9912yes—
109C23×C53C2×C22⋊C411yescharacteristic
110C23×C53#12312yes—
111C23×C53#12312yes—
112Q8×C53—44no—
113D4×C53—44no—
114D4×C53—44no—
115D4×C53—44no—
116C2×C4×C53—28no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—44no—
128C2×C4×C53—44no—
129C2×C4×C53—82no—
130C2×C4×C53—82no—
131C2×C4×C53—82no—
132C2×C4×C53#9912yes—
133C2×C4×C53#11712yes—
134C2×Dic53—42no—
135C2×Dic53—42no—
136C2×Dic53—42no—
137C2×Dic53—42no—
138C2×Dic53—42no—
139C2×Dic53—42no—
140C2×Dic53—84no—
141C2×Dic53—82no—
142C2×Dic53—82no—
143C2×Dic53—82no—
144#12859—531nomaximal, Sylow
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—22no—
148C22×C53—22no—
149C22×C53—22no—
150C22×C53—42no—
151C22×C53—42no—
152C22×C53—42no—
153C22×C53—42no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—81no—
161C22×C53—81no—
162C22×C53#38111yescharacteristic
163C22×C53C2×#9911yescharacteristic
164C22×C53C2×#9911yescharacteristic
165C4×C53—22no—
166C4×C53—22no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—162no—
170Dic53—82no—
171Dic53—82no—
172Dic53—162no—
173#2311—531no—
174#1782—531no—
175#1782D5311yescharacteristic
176#1393—534no—
177C2×C53—21no—
178C2×C53—21no—
179C2×C53—42no—
180C2×C53—42no—
181C2×C53—42no—
182C2×C53#181911yescharacteristic
183C2×C53#212811yescharacteristic
184C2×C53#212911yescharacteristic
185C22×#41C2×D5311yescharacteristic
186C2×#117—531no—
187C22×C4⋊C4—531no—
188C2×#99—21no—
189C2×#99—532no—
190C2×#99—1061no—
191C2×#99C2×D5312yes—
192#391—532no—
193#390—532no—
194#386—1061no—
195#386—1061no—
196#381—21no—
197#381—1061no—
198#380—1061no—
199#380—1061no—
200C53#1285911yescharacteristic, complemented, Sylow
201C2×#41—24no—
202C2×#41—42no—
203C2×#41—42no—
204C22×Q8C4×D5311yescharacteristic
205C22×D4D21211yescharacteristic
206C22×D4C22⋊D5311yescharacteristic
207C22×D4C22×D5311yescharacteristic
208C23×C4—1061no—
209C23×C4C4×D5311yescharacteristic
210C23×C4D21211yescharacteristic
211C23×C4C22⋊D5311yescharacteristic
212#117—2122no—
213C2×C4⋊C4—532no—
214C2×C4⋊C4—1061no—
215C2×C4⋊C4—1061no—
216C2×C4⋊C4—2122no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—1062no—
219C2×C22⋊C4—1062no—
220C2×C22⋊C4—1061no—
221C2×C22⋊C4—1062no—
222C2×C42—532no—
223#99—24no—
224#99—42no—
225#99—1064no—
226#99—2122no—
227#95—42no—
228#95—1062no—
229#95—1062no—
230#95—2122no—
231C24#1171992111yescharacteristic
232C24C2×D21211yescharacteristic
233C24C2×C22⋊D5311yescharacteristic
234#41—416no—
235C2×Q8—22no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4#1171994714yes—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—1062no—
258C22×C4—1062no—
259C22×C4—4242no—
260C22×C4—2122no—
261C22×C4—2122no—
262C22×C4#1171992111yescharacteristic
263C22×C4#1171992111yescharacteristic
264C22×C4#1171992111yescharacteristic
265C22×C4C2×C4×D5311yescharacteristic
266C4⋊C4—2122no—
267C4⋊C4—2122no—
268C4⋊C4—2122no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—4242no—
272C22⋊C4—4242no—
273C22⋊C4—2122no—
274C22⋊C4—2122no—
275C22⋊C4—2122no—
276C42—2122no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23#1313313112yes—
290C23C2×#1171992111yescharacteristic
291C23#1313319812yes—
292C23#1313320112yes—
293Q8—44no—
294D4—44no—
295D4—44no—
296D4—44no—
297C2×C4—28no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—4242no—
314C2×C4—4242no—
315C2×C4—4242no—
316C2×C4—4244no—
317C2×C4—2122no—
318C2×C4—2122no—
319C2×C4—2122no—
320C2×C4—2122no—
321C2×C4—2122no—
322C2×C4—2122no—
323C2×C4#1313313112yes—
324C2×C4#1313317312yes—
325C22—21no—
326C22—21no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—81no—
341C22—81no—
342C22#1681048411yescharacteristic
343C22C2×#1313313111yescharacteristic
344C22C2×#1313313111yescharacteristic
345C4—22no—
346C4—22no—
347C4—41no—
348C4—41no—
349C4—162no—
350C4—8482no—
351C4—4242no—
352C4—4242no—
353C2—21no—
354C2—21no—
355C2—42no—
356C2—42no—
357C2—42no—
358C2#2027478711yescharacteristic
359C2#2027638611yescharacteristic
360C2#2027638911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#1782C2
socleC22×C53C2×#99
prime radicalC22×#41×C53C22
cosocleC22×D4×C53C23
hypercenter#1782D53
hypocenterC53#12859
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #1782

Complements

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

NH ≅ G/Ntype
3. #20272839C2semidirect
5. C2×#13133176C4semidirect
6. C22×C4⋊Dic53C4semidirect
9. #16810530C4semidirect
200. C53#12859semidirect

Maximal subgroups

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

subgroupindex
#202773072
#202746332
#202728392
C53×#17822
#1285953