Finite Group Databasebeta

← #27297593

Subgroups — #27297593

Order 13568.

368 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 71 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#20276987C211yesmaximal, characteristic, complemented
2#20276765C211yesmaximal, characteristic, complemented
3#20274204C211yesmaximal, characteristic, complemented
4#20272758C211yesmaximal, characteristic, complemented
5#20272685C211yesmaximal, characteristic, complemented
6#20272673C211yesmaximal, characteristic, complemented
7C53×#3277C211yesmaximal, characteristic
8C2×#13133163—21no—
9C2×#13133163—21no—
10C2×#13133161—21no—
11#16811003C2211yescharacteristic
12#16810903—21no—
13C4⋊#11719908C2211yescharacteristic
14#16810426—21no—
15#16810399C2211yescharacteristic
16#16810374C2211yescharacteristic
17#16810370—21no—
18C2×#137×C53—21no—
19C53×#548—21no—
20C53×#495C2211yescharacteristic
21C53×#468—21no—
22C53×#464—21no—
23C53×#451—21no—
24C53×#449—21no—
25C4×Q16×C53C2211yescharacteristic
26C53×#436C2211yescharacteristic
27C2×Dic212—21no—
28C2×Dic212—21no—
29#13133217—21no—
30#13133163—42no—
31#13133163—42no—
32#13133161—42no—
33C2×#11719908—21no—
34C2×#11719908—21no—
35#13133150—41no—
36C4⋊#88322—21no—
37#13133138—41no—
38C2×#41×C53—21no—
39C22×Q8×C53—21no—
40#137×C53—42no—
41#137×C53—42no—
42C2×Q16×C53—21no—
43C2×Q16×C53D411yescharacteristic
44C2×Q16×C53D411yescharacteristic
45C2×SD16×C53—21no—
46C2×SD16×C53—21no—
47C2×D8×C53D411yescharacteristic
48C2×M16×C53—21no—
49#124×C53—21no—
50#124×C53—41no—
51C22⋊Q8×C53—21no—
52C22⋊Q8×C53—41no—
53#121×C53—21no—
54#121×C53—41no—
55C4×Q8×C53—21no—
56C4×D4×C53—41no—
57C4×D4×C53D411yescharacteristic
58#107×C53D411yescharacteristic
59C4⋊C8×C53D411yescharacteristic
60#103×C53—21no—
61#103×C53—21no—
62#102×C53—21no—
63C22⋊C8×C53—21no—
64C4×C8×C53C2311yescharacteristic
65#41×C53—42no—
66#41×C53—81no—
67C2×Q8×C53—21no—
68C2×Q8×C53—21no—
69C2×Q8×C53—41no—
70C2×Q8×C53—42no—
71C2×Q8×C53—81no—
72C2×D4×C53—21no—
73C2×D4×C53—41no—
74C2×D4×C53—41no—
75C2×D4×C53D811yescharacteristic
76C22×C4×C53—21no—
77C22×C4×C53—41no—
78C22×C4×C53—41no—
79C2×Dic106—41no—
80Q16×C53—21no—
81Q16×C53—41no—
82Q16×C53—42no—
83Q16×C53D812yes—
84SD16×C53—42no—
85SD16×C53—41no—
86SD16×C53—41no—
87D8×C53—41no—
88M16×C53—41no—
89M16×C53—41no—
90C2×C8×C53—21no—
91C2×C8×C53C2×D411yescharacteristic
92C2×C8×C53C2×D411yescharacteristic
93C4⋊C4×C53—21no—
94C4⋊C4×C53—41no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53D811yescharacteristic
97C22⋊C4×C53—21no—
98C22⋊C4×C53—41no—
99C22⋊C4×C53—41no—
100C22⋊C4×C53—41no—
101C22⋊C4×C53—41no—
102C42×C53—41no—
103C42×C53C2×D411yescharacteristic
104C4⋊Dic53—41no—
105C4⋊Dic53—41no—
106C2×#88322—41no—
107Dic212—41no—
108Dic212—41no—
109Dic212—41no—
110Dic212—41no—
111#11719908—22no—
112#11719908—41no—
113C23×C53—21no—
114C23×C53—41no—
115Q8×C53—22no—
116Q8×C53—41no—
117Q8×C53—41no—
118Q8×C53—41no—
119Q8×C53—82no—
120D4×C53—41no—
121D4×C53—41no—
122D4×C53—41no—
123D4×C53—82no—
124D4×C53—82no—
125C2×C4×C53—21no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—82no—
132C2×C4×C53—82no—
133C2×C4×C53#12011yescharacteristic
134C2×C4×C53C2×D811yescharacteristic
135C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
136C2×Dic53—81no—
137Dic106—81no—
138Dic106—81no—
139C8×C53—21no—
140C8×C53—41no—
141C8×C53C2×D811yescharacteristic
142C8×C53AGL(1,ℤ/8)11yescharacteristic
143#88322—81no—
144#12327—531nomaximal, Sylow
145C22×C53—41no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53—81no—
149C22×C53—81no—
150C22×C53—81no—
151C22×C53#44711yescharacteristic
152C4×C53—21no—
153C4×C53—21no—
154C4×C53—41no—
155C4×C53—42no—
156C4×C53—81no—
157C4×C53—81no—
158C4×C53#44711yescharacteristic
159C4×C53#45311yescharacteristic
160Dic53—161no—
161#3277D5311yescharacteristic
162#2251—531no—
163#2209—531no—
164#1689—531no—
165#1360—531no—
166#1334—531no—
167#1330—531no—
168C2×C53—81no—
169C2×C53—161no—
170C2×C53#161911yescharacteristic
171C2×C53#219111yescharacteristic
172C2×C53#219511yescharacteristic
173C2×#137—21no—
174#548—21no—
175C2×Q32—1061no—
176C2×Q32—1061no—
177C2×SD32—1061no—
178#495C2×D5311yescharacteristic
179#494—531no—
180#474—1061no—
181#468—21no—
182#464—21no—
183#451—21no—
184#449—21no—
185C4×Q16C2×D5311yescharacteristic
186#436C2×D5311yescharacteristic
187C4⋊C16—531no—
188#358—1061no—
189#346—531no—
190#334—531no—
191#332—1061no—
192C53#1232711yescharacteristic, complemented, Sylow
193C2×#41—21no—
194C22×Q8—21no—
195#137—42no—
196#137—42no—
197C2×Q16—21no—
198C2×Q16—1061no—
199C2×Q16—1061no—
200C2×Q16C22⋊D5311yescharacteristic
201C2×Q16C22⋊D5311yescharacteristic
202C2×SD16—21no—
203C2×SD16—21no—
204C2×D8C22⋊D5311yescharacteristic
205C2×M16—21no—
206C4⋊Q8—1061no—
207#124—21no—
208#124—41no—
209C22⋊Q8—21no—
210C22⋊Q8—41no—
211#121—21no—
212#121—41no—
213C4×Q8—21no—
214C4×D4—41no—
215C4×D4C22⋊D5311yescharacteristic
216Q32—2122no—
217Q32—2122no—
218SD32—2122no—
219C2×C16—1061no—
220C2×C16—1061no—
221#107—2121no—
222#107C22⋊D5311yescharacteristic
223C4⋊C8—1061no—
224C4⋊C8C22⋊D5311yescharacteristic
225#103—21no—
226#103—21no—
227#102—21no—
228#102—2121no—
229C22⋊C8—21no—
230C4×C8C22×D5311yescharacteristic
231#41—42no—
232#41—81no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—41no—
236C2×Q8—42no—
237C2×Q8—81no—
238C2×Q8—2121no—
239C2×D4—21no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4#1171992211yescharacteristic
243C22×C4—21no—
244C22×C4—41no—
245C22×C4—41no—
246Q16—21no—
247Q16—41no—
248Q16—42no—
249Q16—2121no—
250Q16—2121no—
251Q16—2121no—
252Q16—2121no—
253Q16#1171992212yes—
254SD16—42no—
255SD16—41no—
256SD16—41no—
257D8—41no—
258M16—41no—
259M16—41no—
260C2×C8—21no—
261C2×C8—2121no—
262C2×C8C2×C22⋊D5311yescharacteristic
263C2×C8C2×C22⋊D5311yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—2121no—
268C4⋊C4—2121no—
269C4⋊C4#1171992211yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C42—41no—
276C42C2×C22⋊D5311yescharacteristic
277C16—1062no—
278C16—2121no—
279C23—21no—
280C23—41no—
281Q8—22no—
282Q8—41no—
283Q8—41no—
284Q8—41no—
285Q8—82no—
286Q8—4241no—
287Q8—4241no—
288D4—41no—
289D4—41no—
290D4—41no—
291D4—82no—
292D4—82no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—4241no—
302C2×C4#1313318611yescharacteristic
303C2×C4C2×#1171992211yescharacteristic
304C2×C4#1313324311yescharacteristic
305C8—21no—
306C8—41no—
307C8—4241no—
308C8C2×#1171992211yescharacteristic
309C8#1313324311yescharacteristic
310C22—41no—
311C22—41no—
312C22—41no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22#1681075711yescharacteristic
317C4—21no—
318C4—21no—
319C4—41no—
320C4—42no—
321C4—81no—
322C4—81no—
323C4—8481no—
324C4#1681075711yescharacteristic
325C4#1681079111yescharacteristic
326C2—81no—
327C2—161no—
328C2#2027376911yescharacteristic
329C2#2027666611yescharacteristic
330C2#2027669311yescharacteristic
subgroupG/H
centerC22#16810757
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
Fitting, prime radicalC53×#3277C2
socleC22×C53#447
hypercenter#3277D53
hypocenterC53#12327
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #436 ◁ #3277

Complements

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

NH ≅ G/Ntype
1. #20276987C2semidirect
2. #20276765C2semidirect
3. #20274204C2semidirect
4. #20272758C2semidirect
5. #20272685C2semidirect
6. #20272673C2semidirect
192. C53#12327semidirect

Maximal subgroups

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

subgroupindex
#202769872
#202767652
#202742042
#202727582
#202726852
#202726732
C53×#32772
#1232753