Finite Group Databasebeta

← #28308130

Subgroups — #28308130

Order 13568.

360 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 88 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#20291077C211yesmaximal, characteristic, complemented
2#20291073C211yesmaximal, characteristic, complemented
3#20289638C212yesmaximal, complemented
4#20289637C212yesmaximal, complemented
5#20283425C211yesmaximal, characteristic
6#16811544—21no—
7#16811544—21no—
8C22⋊#11719910C2212yes—
9C22⋊#11719910C2212yes—
10#16811512—21no—
11#16811509—21no—
12C2×#13133223C411yescharacteristic
13#16811169C411yescharacteristic
14#16810699—21no—
15#16810699—21no—
16C8×C22⋊D53—21no—
17C8×C22⋊D53—21no—
18#16810646C411yescharacteristic
19#16810591C2211yescharacteristic
20#16810590C2211yescharacteristic
21C2×C8×Dic53C2211yescharacteristic
22C2×#131×C53C411yescharacteristic
23C2×#11719910—22no—
24C2×#11719910—22no—
25C2×#11719945C2×C411yescharacteristic
26#13133223—42no—
27C2×#11719917C2×C411yescharacteristic
28C2×#11719912—21no—
29C22×#88322C2311yescharacteristic
30C2×C8×D53—21no—
31#13133215C811yescharacteristic
32#13133213C811yescharacteristic
33#13133206C811yescharacteristic
34#13133204C811yescharacteristic
35#13133189—21no—
36#13133189—21no—
37C4×C22⋊D53—21no—
38C4×C22⋊D53—21no—
39C2×C4×Dic53C2×C411yescharacteristic
40#13133146—21no—
41#13133146—21no—
42C22⋊#88322—21no—
43C22⋊#88322—21no—
44#13133129—21no—
45#13133129—21no—
46C8×Dic53D411yescharacteristic
47C8×Dic53D411yescharacteristic
48C8×Dic53D411yescharacteristic
49C8×Dic53D411yescharacteristic
50C2×#41×C53C2×C411yescharacteristic
51#131×C53—42no—
52C2×M16×C53—21no—
53C2×M16×C53C2×C411yescharacteristic
54C22×C8×C53—21no—
55C22×C8×C53C2×C411yescharacteristic
56#41×C53—41no—
57#41×C53—41no—
58C2×Q8×C53C2×C811yescharacteristic
59C2×D4×C53—21no—
60C2×D4×C53C2×C811yescharacteristic
61C22×C4×C53—21no—
62C22×C4×C53C22×C411yescharacteristic
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C22×Dic53M1611yescharacteristic
67C22×Dic53M1611yescharacteristic
68#11719945—41no—
69#11719945—41no—
70C2×D212C2×C811yescharacteristic
71C2×C4×D53—21no—
72C2×Dic106C2×C811yescharacteristic
73M16×C53—42no—
74M16×C53—41no—
75C2×C8×C53—22no—
76C2×C8×C53—21no—
77C2×C8×C53—42no—
78C2×C8×C53C22⋊C411yescharacteristic
79C2×C8×C53C22⋊C411yescharacteristic
80C22⋊Dic53—21no—
81C22⋊Dic53—21no—
82#11719921—21no—
83#11719921—21no—
84#11719919—21no—
85#11719919—21no—
86C4×Dic53C22⋊C411yescharacteristic
87C4×Dic53C22⋊C411yescharacteristic
88C4×Dic53C2×C811yescharacteristic
89C4×Dic53C2×C811yescharacteristic
90#11719917—41no—
91C2×#88322—21no—
92C2×#88322—41no—
93C2×#88322C2×D411yescharacteristic
94C2×#88322C2×D411yescharacteristic
95#11719912—42no—
96C8×D53—42no—
97#11719910—42no—
98#11719910—42no—
99C23×C53—21no—
100C23×C53C2×M1611yescharacteristic
101C22×D53—21no—
102Q8×C53—41no—
103D4×C53—41no—
104D4×C53—42no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53C2×C22⋊C411yescharacteristic
111C2×C4×C53C22×C811yescharacteristic
112C22⋊D53—42no—
113C22⋊D53—42no—
114C22⋊D53—42no—
115C2×Dic53—22no—
116C2×Dic53—21no—
117C2×Dic53—41no—
118C2×Dic53C22⋊C811yescharacteristic
119C2×Dic53C22⋊C811yescharacteristic
120C2×Dic53C22⋊C811yescharacteristic
121C2×Dic53C22⋊C811yescharacteristic
122D212—41no—
123C4×D53—41no—
124C4×D53—41no—
125Dic106—41no—
126C8×C53—21no—
127C8×C53—41no—
128C8×C53#10012yes—
129#88322—21no—
130#88322—21no—
131#88322—41no—
132#11492—531nomaximal, Sylow
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—41no—
137C22×C53—41no—
138C22×C53—41no—
139C22×C53C2×C22⋊C811yescharacteristic
140C2×D53—41no—
141C2×D53—41no—
142C2×D53—41no—
143C4×C53—21no—
144C4×C53—41no—
145C4×C53#41011yescharacteristic
146C4×C53C2×#10011yescharacteristic
147Dic53—21no—
148Dic53—21no—
149Dic53—21no—
150Dic53—41no—
151Dic53#35012yes—
152#2931—531no—
153#2163—531no—
154#2162—531no—
155#1315—532no—
156#1315—532no—
157C2×C53—41no—
158C2×C53—81no—
159C2×C53#146011yescharacteristic
160C2×C53C2×#35011yescharacteristic
161C2×C53#211711yescharacteristic
162D53—81no—
163C2×#131—531no—
164C2×#131#615711yescharacteristic
165#532—531no—
166#436—1061no—
167#436—1061no—
168C8×D4—1061no—
169C8×D4—1061no—
170#423—531no—
171#408—531no—
172#408—531no—
173C2×C4×C8—531no—
174C4⋊C16—1061no—
175C4⋊C16—1061no—
176C22⋊C16—532no—
177C22⋊C16—532no—
178#346—1061no—
179C4×C16—1061no—
180C53#1149211yescharacteristic, complemented, Sylow
181C2×#41—531no—
182C2×#41C2×#615711yescharacteristic
183#131—42no—
184#131—2122no—
185C2×M16—21no—
186C2×M16—531no—
187C2×M16—1061no—
188C2×M16C2×#615711yescharacteristic
189C22×C8—21no—
190C22×C8—531no—
191C22×C8—1061no—
192C22×C8C2×#615711yescharacteristic
193C4⋊Q8—531no—
194#127—531no—
195#124—531no—
196#124—531no—
197#121—1061no—
198#121—1061no—
199C4×D4—1061no—
200C4×D4—1061no—
201C2×C42—531no—
202C2×C16—1062no—
203C2×C16—1062no—
204C4⋊C8—1061no—
205C4⋊C8—1061no—
206C22⋊C8—1061no—
207C22⋊C8—1061no—
208C22⋊C8—1061no—
209C22⋊C8—1061no—
210C4×C8—531no—
211C4×C8—531no—
212C4×C8—531no—
213C4×C8—531no—
214#41—41no—
215#41—41no—
216#41—2121no—
217#41—2121no—
218C2×Q8—531no—
219C2×Q8D53⋊C811yescharacteristic
220C2×D4—21no—
221C2×D4—531no—
222C2×D4—1061no—
223C2×D4—1061no—
224C2×D4—1061no—
225C2×D4D53⋊C811yescharacteristic
226C22×C4—21no—
227C22×C4—531no—
228C22×C4—531no—
229C22×C4—1061no—
230C22×C4C22×#615711yescharacteristic
231M16—42no—
232M16—41no—
233M16—2122no—
234M16—2121no—
235C2×C8—22no—
236C2×C8—21no—
237C2×C8—42no—
238C2×C8—531no—
239C2×C8—531no—
240C2×C8—1061no—
241C2×C8—2122no—
242C2×C8—2121no—
243C2×C8C22⋊#615711yescharacteristic
244C2×C8C22⋊#615711yescharacteristic
245C4⋊C4—1061no—
246C4⋊C4—1061no—
247C22⋊C4—1061no—
248C22⋊C4—1061no—
249C22⋊C4—1061no—
250C22⋊C4—1061no—
251C42—531no—
252C42—531no—
253C42—531no—
254C42—531no—
255C16—2122no—
256C16—2122no—
257C23—21no—
258C23—1061no—
259C23D53⋊M1611yescharacteristic
260Q8—41no—
261Q8—2121no—
262D4—41no—
263D4—42no—
264D4—2121no—
265D4—2122no—
266D4—2122no—
267D4—2122no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—531no—
274C2×C4—531no—
275C2×C4—531no—
276C2×C4—531no—
277C2×C4—1062no—
278C2×C4—1061no—
279C2×C4—2121no—
280C2×C4—2121no—
281C2×C4—2121no—
282C2×C4C2×C22⋊#615711yescharacteristic
283C2×C4C2×D53⋊C811yescharacteristic
284C8—21no—
285C8—41no—
286C8—1061no—
287C8—1061no—
288C8—2121no—
289C8#1313327712yes—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—2121no—
297C22—2121no—
298C22—2121no—
299C22D53⋊(C22⋊C8)11yescharacteristic
300C4—21no—
301C4—41no—
302C4—532no—
303C4—1061no—
304C4—1061no—
305C4—1061no—
306C4—2121no—
307C4#1681160211yescharacteristic
308C4C2×#1313327711yescharacteristic
309C2—41no—
310C2—81no—
311C2—4241no—
312C2#2028995411yescharacteristic
313C2D53⋊#35011yescharacteristic
314C2#2029095811yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC2×#131×C53C4
socleC22×C53C2×C22⋊C8
prime radical#16811169C4
cosocleC22×#88322C23
hypercenterC2×#131#6157
hypocenterC53#11492
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
1. #20291077C2semidirect
2. #20291073C2semidirect
3. #20289638C2semidirect
4. #20289637C2semidirect
180. C53#11492semidirect

Maximal subgroups

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

subgroupindex
#202910772
#202910732
#202896382
#202896372
#202834252
#1149253