Finite Group Databasebeta

← #27300365

Subgroups — #27300365

Order 13568.

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

normal subgroups only

This group has 614 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#20274643C211yesmaximal, characteristic
3#20272841C214yesmaximal, complemented
4C53×#1785C211yesmaximal, characteristic
5C2×#13133176C411yescharacteristic
6C22×C4⋊Dic53C411yescharacteristic
7C2×#13133136C2212yes—
8C2×#13133134—21no—
9#16810587—21no—
10#16810530C2212yes—
11#16810526C412yes—
12#16810503—21no—
13#16810503—21no—
14#16810483—21no—
15#16810483—21no—
16C22×#41×C53C2211yescharacteristic
17C2×#101×C53C2212yes—
18C2×#100×C53—21no—
19C53×#407—21no—
20C23×Dic53—21no—
21C2×#11719917—22no—
22#13133176—42no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—42no—
26C2×C4⋊Dic53C2×C412yes—
27C2×C22⋊Dic53—21no—
28C2×C22⋊Dic53—22no—
29C2×C22⋊Dic53—22no—
30C2×C4×Dic53C2×C412yes—
31#13133136—24no—
32#13133134—42no—
33C22⋊#88322—42no—
34#13133124—22no—
35#13133124—22no—
36C2×#41×C53—24no—
37C2×#41×C53—42no—
38C2×#41×C53—42no—
39C22×Q8×C53C2311yescharacteristic
40C22×D4×C53C2×C411yescharacteristic
41C22×D4×C53D411yescharacteristic
42C22×D4×C53D411yescharacteristic
43C23×C4×C53C2×C411yescharacteristic
44C23×C4×C53D411yescharacteristic
45C23×C4×C53D411yescharacteristic
46C2×M16×C53—22no—
47#101×C53—24no—
48#100×C53—42no—
49C22⋊C8×C53—42no—
50C24×C53C22⋊C411yescharacteristic
51C24×C53C22⋊C411yescharacteristic
52C24×C53C22⋊C411yescharacteristic
53#41×C53—416no—
54C2×Q8×C53—41no—
55C2×Q8×C53—41no—
56C2×Q8×C53#4114yes—
57C2×D4×C53—22no—
58C2×D4×C53—22no—
59C2×D4×C53—22no—
60C2×D4×C53—22no—
61C2×D4×C53—22no—
62C2×D4×C53—41no—
63C2×D4×C53—41no—
64C2×D4×C53—41no—
65C2×D4×C53—41no—
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×C53C22⋊C411yescharacteristic
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×Dic53—22no—
80C22×Dic53—22no—
81C22×Dic53—42no—
82C22×Dic53—82no—
83M16×C53—42no—
84M16×C53—42no—
85C2×C8×C53—42no—
86C22⋊Dic53—42no—
87C22⋊Dic53—82no—
88C22⋊Dic53—82no—
89C4⋊Dic53—42no—
90C4⋊Dic53—42no—
91C4⋊Dic53—42no—
92C4×Dic53—42no—
93#11719917—42no—
94#11719917—42no—
95C2×#88322—42no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53#9912yes—
111C23×C53C2×C22⋊C411yescharacteristic
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—44no—
126C2×C4×C53—44no—
127C2×C4×C53#9912yes—
128C2×C4×C53#11712yes—
129C2×C4×C53#12312yes—
130C2×C4×C53#12312yes—
131C2×Dic53—42no—
132C2×Dic53—42no—
133C2×Dic53—42no—
134C2×Dic53—42no—
135C2×Dic53—42no—
136C2×Dic53—42no—
137C2×Dic53—84no—
138C8×C53—82no—
139#88322—82no—
140#12863—531nomaximal, Sylow
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—22no—
144C22×C53—42no—
145C22×C53—42no—
146C22×C53—42no—
147C22×C53—42no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—81no—
157C22×C53—81no—
158C22×C53#38111yescharacteristic
159C22×C53C2×#9911yescharacteristic
160C22×C53C2×#9911yescharacteristic
161C4×C53—22no—
162C4×C53—22no—
163C4×C53—22no—
164C4×C53—22no—
165Dic53—82no—
166Dic53—82no—
167#2311—531no—
168#1785—531no—
169#1785D5311yescharacteristic
170#1394—534no—
171C2×C53—21no—
172C2×C53—21no—
173C2×C53—42no—
174C2×C53—81no—
175C2×C53—81no—
176C2×C53#181911yescharacteristic
177C2×C53#213411yescharacteristic
178C2×C53#213511yescharacteristic
179C22×#41C2×D5311yescharacteristic
180C2×#117—531no—
181C22×C4⋊C4—531no—
182C2×#101—532no—
183C2×#101C2×D5312yes—
184C2×#100—21no—
185C2×#100—1061no—
186#407—21no—
187#407—1061no—
188#391—532no—
189#390—532no—
190#386—1061no—
191#386—1061no—
192#380—1061no—
193#380—1061no—
194C53#1286311yescharacteristic, complemented, Sylow
195C2×#41—24no—
196C2×#41—42no—
197C2×#41—42no—
198C22×Q8C22×D5311yescharacteristic
199C22×D4C4×D5311yescharacteristic
200C22×D4D21211yescharacteristic
201C22×D4C22⋊D5311yescharacteristic
202C23×C4—1061no—
203C23×C4C4×D5311yescharacteristic
204C23×C4D21211yescharacteristic
205C23×C4C22⋊D5311yescharacteristic
206C2×M16—22no—
207C2×M16—1062no—
208#117—2122no—
209C2×C4⋊C4—532no—
210C2×C4⋊C4—1061no—
211C2×C4⋊C4—1061no—
212C2×C4⋊C4—2122no—
213C2×C22⋊C4—1062no—
214C2×C22⋊C4—1062no—
215C2×C22⋊C4—1061no—
216C2×C42—532no—
217#101—24no—
218#101—1064no—
219#100—42no—
220#100—2122no—
221C22⋊C8—42no—
222C22⋊C8—2122no—
223#95—1062no—
224#95—1062no—
225C24#1171992111yescharacteristic
226C24#1171992111yescharacteristic
227C24#1171992111yescharacteristic
228#41—416no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8#1171994714yes—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—1062no—
251C22×C4—1062no—
252C22×C4—4242no—
253C22×C4—2122no—
254C22×C4#1171992111yescharacteristic
255C22×C4C2×C4×D5311yescharacteristic
256C22×C4C2×D21211yescharacteristic
257C22×C4C2×C22⋊D5311yescharacteristic
258M16—42no—
259M16—42no—
260M16—2122no—
261M16—2122no—
262C2×C8—42no—
263C2×C8—2122no—
264C4⋊C4—2122no—
265C4⋊C4—2122no—
266C4⋊C4—2122no—
267C22⋊C4—4242no—
268C22⋊C4—4242no—
269C22⋊C4—2122no—
270C42—2122no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23#1313313112yes—
286C23C2×#1171992111yescharacteristic
287Q8—44no—
288D4—44no—
289D4—44no—
290D4—44no—
291C2×C4—28no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—4244no—
303C2×C4—2122no—
304C2×C4—2122no—
305C2×C4—2122no—
306C2×C4—2122no—
307C2×C4—2122no—
308C2×C4—2122no—
309C2×C4#1313313112yes—
310C2×C4#1313317312yes—
311C2×C4#1313319812yes—
312C2×C4#1313320112yes—
313C8—82no—
314C8—4242no—
315C22—21no—
316C22—21no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22—81no—
332C22#1681048411yescharacteristic
333C22C2×#1313313111yescharacteristic
334C22C2×#1313313111yescharacteristic
335C4—22no—
336C4—22no—
337C4—22no—
338C4—22no—
339C4—4242no—
340C4—4242no—
341C2—21no—
342C2—21no—
343C2—42no—
344C2—81no—
345C2—81no—
346C2#2027478711yescharacteristic
347C2#2027640611yescharacteristic
348C2#2027640911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC53×#1785C2
socleC22×C53C2×#99
prime radicalC22×#41×C53C22
cosocleC22×Q8×C53C23
hypercenter#1785D53
hypocenterC53#12863
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #1785

Complements

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

NH ≅ G/Ntype
3. #20272841C2semidirect
194. C53#12863semidirect

Maximal subgroups

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

subgroupindex
#202773072
#202746432
#202728412
C53×#17852
#1286353