Finite Group Databasebeta

← #27296206

Subgroups — #27296206

Order 13568.

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

normal subgroups only

This group has 476 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#20276631C212yesmaximal, complemented
2#20275725C211yesmaximal, characteristic, complemented
3#20272724C212yesmaximal, complemented
4#20272639C211yesmaximal, characteristic, complemented
5C53×#3186C211yesmaximal, characteristic
6C22×Dic212C2211yescharacteristic, complemented
7C2×#13133196—21no—
8C2×#13133161—22no—
9C2×#13133142—21no—
10#16810426—22no—
11C22⋊#11719908C2212yescomplemented
12#16810386C2211yescharacteristic, complemented
13C2×#121×C53—21no—
14C53×#480—22no—
15C53×#466—21no—
16C53×#466C2212yes—
17C53×#447—22no—
18C2×C22⋊C8×C53C2211yescharacteristic
19C22×Dic106—21no—
20C2×Dic212—22no—
21C2×Dic212—42no—
22#13133196—42no—
23#13133196—42no—
24C2×C4⋊Dic53—21no—
25C2×#11719919—41no—
26C2×C22⋊Dic53—41no—
27#13133161—44no—
28C2×#11719908—22no—
29#13133142—42no—
30C22×D4×C53—21no—
31C22×D4×C53—41no—
32C23×C4×C53D411yescharacteristic
33C2×D8×C53—21no—
34C2×D8×C53D412yes—
35C22×C8×C53D411yescharacteristic
36C22×C8×C53C2311yescharacteristic
37#121×C53—22no—
38#121×C53—22no—
39#121×C53—41no—
40#121×C53—41no—
41C2×C4⋊C4×C53—21no—
42C2×C22⋊C4×C53—41no—
43#107×C53—21no—
44#107×C53D412yes—
45#102×C53—22no—
46#102×C53—22no—
47C22⋊C8×C53—22no—
48C24×C53—41no—
49C24×C53D811yescharacteristic
50C2×D4×C53—22no—
51C2×D4×C53—41no—
52C2×D4×C53—42no—
53C2×D4×C53—42no—
54C2×D4×C53—81no—
55C2×D4×C53—81no—
56C2×D4×C53—82no—
57C2×D4×C53—82no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—41no—
61C22×C4×C53—41no—
62C22×C4×C53—41no—
63C22×C4×C53D811yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×Dic53—41no—
66C22×Dic53—41no—
67C2×Dic106—22no—
68C2×Dic106—41no—
69C2×Dic106—41no—
70C2×Dic106—41no—
71C2×Dic106—41no—
72C2×Dic106—41no—
73D8×C53—42no—
74D8×C53—81no—
75C2×C8×C53—21no—
76C2×C8×C53—41no—
77C2×C8×C53—41no—
78C2×C8×C53C2×D412yes—
79C4⋊C4×C53—22no—
80C4⋊C4×C53—41no—
81C22⋊C4×C53—42no—
82C22⋊C4×C53—42no—
83C22⋊Dic53—44no—
84C4⋊Dic53—22no—
85C4⋊Dic53—41no—
86#11719919—44no—
87Dic212—24no—
88Dic212—42no—
89#11719908—42no—
90C23×C53—21no—
91C23×C53—21no—
92C23×C53—41no—
93C23×C53—41no—
94C23×C53—42no—
95C23×C53—82no—
96C23×C53—82no—
97C23×C53—82no—
98C23×C53SD3212yescomplemented
99C23×C53C2×D811yescharacteristic
100D4×C53—42no—
101D4×C53—42no—
102D4×C53—82no—
103D4×C53—84no—
104D4×C53—82no—
105D4×C53—82no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—42no—
113C2×C4×C53—81no—
114C2×C4×C53—82no—
115C2×C4×C53#12011yescharacteristic
116C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
117C2×Dic53—42no—
118C2×Dic53—42no—
119C2×Dic53—81no—
120C2×Dic53—81no—
121C2×Dic53—81no—
122C2×Dic53—81no—
123Dic106—42no—
124Dic106—42no—
125Dic106—81no—
126Dic106—81no—
127C8×C53—22no—
128C8×C53—41no—
129#12105—531nomaximal, Sylow
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—22no—
133C22×C53—42no—
134C22×C53—41no—
135C22×C53—41no—
136C22×C53—81no—
137C22×C53—82no—
138C22×C53—82no—
139C22×C53—82no—
140C22×C53—164no—
141C22×C53#44711yescharacteristic
142C22×C53C2×SD3211yescharacteristic
143C22×C53#50911yescharacteristic
144C4×C53—21no—
145C4×C53—41no—
146C4×C53—82no—
147C4×C53#45311yescharacteristic
148C4×C53B(4,𝔽2)11yescharacteristic
149Dic53—82no—
150Dic53—82no—
151#3186D5311yescharacteristic
152#2185—532no—
153#2001—531no—
154#1347—532no—
155#1314—531no—
156C2×C53—21no—
157C2×C53—21no—
158C2×C53—42no—
159C2×C53—162no—
160C2×C53#159511yescharacteristic
161C2×C53#218711yescharacteristic
162C2×C53#219111yescharacteristic
163C22×Q16—531no—
164C2×C22⋊Q8—1061no—
165C2×#121—21no—
166C2×SD32—1062no—
167#480—22no—
168#466—21no—
169#466C2×D5312yes—
170#447—22no—
171C2×#103—1061no—
172C2×C22⋊C8C2×D5311yescharacteristic
173#358—1062no—
174C22⋊C16—532no—
175#340—531no—
176C53#1210511yescharacteristic, complemented, Sylow
177C22×Q8—1061no—
178C22×D4—21no—
179C22×D4—41no—
180C23×C4C22⋊D5311yescharacteristic
181C2×Q16—1062no—
182C2×Q16—2122no—
183C2×D8—21no—
184C2×D8C22⋊D5312yes—
185C22×C8C22⋊D5311yescharacteristic
186C22×C8C22×D5311yescharacteristic
187C22⋊Q8—2122no—
188C22⋊Q8—2122no—
189#121—22no—
190#121—22no—
191#121—41no—
192#121—41no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—1061no—
195C2×C4⋊C4—2121no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—2121no—
198SD32—2124no—
199C2×C16—1062no—
200#107—21no—
201#107C22⋊D5312yes—
202#103—2122no—
203#102—22no—
204#102—22no—
205C22⋊C8—22no—
206C24—41no—
207C24#1171992211yescharacteristic
208C2×Q8—1062no—
209C2×Q8—2121no—
210C2×Q8—2121no—
211C2×Q8—2121no—
212C2×Q8—2121no—
213C2×Q8—2121no—
214C2×D4—22no—
215C2×D4—41no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—81no—
219C2×D4—81no—
220C2×D4—82no—
221C2×D4—82no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—2121no—
228C22×C4—2121no—
229C22×C4#1171992211yescharacteristic
230C22×C4C2×C22⋊D5311yescharacteristic
231Q16—1064no—
232Q16—2122no—
233D8—42no—
234D8—81no—
235C2×C8—21no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8C2×C22⋊D5312yes—
239C4⋊C4—22no—
240C4⋊C4—41no—
241C4⋊C4—1062no—
242C4⋊C4—2121no—
243C4⋊C4—2124no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—2124no—
247C16—2122no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—42no—
253C23—82no—
254C23—82no—
255C23—82no—
256C23#1313316112yescomplemented
257C23C2×#1171992211yescharacteristic
258Q8—4241no—
259Q8—4241no—
260Q8—2122no—
261Q8—2122no—
262D4—42no—
263D4—42no—
264D4—82no—
265D4—84no—
266D4—82no—
267D4—82no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—81no—
276C2×C4—82no—
277C2×C4—4241no—
278C2×C4—4241no—
279C2×C4—4241no—
280C2×C4—4241no—
281C2×C4—2122no—
282C2×C4—2122no—
283C2×C4#1313318611yescharacteristic
284C2×C4#1313324311yescharacteristic
285C8—22no—
286C8—41no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22—164no—
298C22#1681075711yescharacteristic
299C22C2×#1313316111yescharacteristic
300C22#1681105911yescharacteristic
301C4—21no—
302C4—41no—
303C4—82no—
304C4—4242no—
305C4—4242no—
306C4#1681079111yescharacteristic
307C4#1681081011yescharacteristic
308C2—21no—
309C2—21no—
310C2—42no—
311C2—162no—
312C2#2027363311yescharacteristic
313C2#2027664411yescharacteristic
314C2#2027666611yescharacteristic
subgroupG/H
centerC22#16810757
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
Fitting, prime radicalC53×#3186C2
socleC22×C53#447
hypercenter#3186D53
hypocenterC53#12105
LCSG ▷ C22×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #3186

Complements

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

NH ≅ G/Ntype
1. #20276631C2semidirect
2. #20275725C2semidirect
3. #20272724C2semidirect
4. #20272639C2semidirect
6. C22×Dic212C22semidirect
11. C22⋊#11719908C22semidirect
12. #16810386C22semidirect
98. C23×C53SD32semidirect
176. C53#12105semidirect
256. C23#13133161semidirect

Maximal subgroups

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

subgroupindex
#202766312
#202757252
#202727242
#202726392
C53×#31862
#1210553