Finite Group Databasebeta

← #28298227

Subgroups — #28298227

Order 13568.

412 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 78 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#20289553C211yesmaximal, characteristic
2#20289553C211yesmaximal, characteristic
3#20272980C211yesmaximal, characteristic
4C2×C22⋊#88324—21no—
5C2×C22⋊#88324—21no—
6C22×C4⋊Dic53—21no—
7C22×C4⋊Dic53C411yescharacteristic
8C2×#13133124—21no—
9C2×#13133124C2211yescharacteristic
10C2×#95×C53C411yescharacteristic
11C22×#88324—41no—
12C22×#88324—41no—
13C22⋊#88324—22no—
14C22⋊#88324—22no—
15C22⋊#88324—41no—
16C22⋊#88324—41no—
17C23×Dic53—21no—
18C23×Dic53D411yescharacteristic
19C23×Dic53D411yescharacteristic
20C2×C4⋊Dic53—42no—
21C2×C4⋊Dic53—42no—
22C2×C4⋊Dic53—42no—
23C2×C4⋊Dic53—41no—
24C2×C4⋊Dic53—41no—
25C2×C4⋊Dic53—41no—
26#13133124—41no—
27#13133124—42no—
28C23×C4×C53—21no—
29C23×C4×C53C2×C411yescharacteristic
30#95×C53—21no—
31#95×C53C812yescomplemented
32C24×C53C22⋊C411yescharacteristic
33C22×C4×C53—22no—
34C22×C4×C53—22no—
35C22×C4×C53—22no—
36C22×C4×C53—21no—
37C22×C4×C53—21no—
38C22×C4×C53—41no—
39C22×C4×C53—41no—
40C22×C4×C53—41no—
41C22×C4×C53—41no—
42C22×C4×C53—41no—
43C22×C4×C53—41no—
44C22×C4×C53C2×C811yescharacteristic
45C22×C4×C53M1611yescharacteristic
46C22×Dic53—21no—
47C22×Dic53—21no—
48C22×Dic53—21no—
49C22×Dic53—21no—
50C22×Dic53—22no—
51C22×Dic53—41no—
52C22×Dic53—41no—
53C22×Dic53—41no—
54C22×Dic53—41no—
55C22×Dic53—41no—
56C22×Dic53—41no—
57C22×Dic53—41no—
58C22×Dic53—41no—
59C22×Dic53—41no—
60C22×Dic53—41no—
61C22×Dic53D811yescharacteristic
62C22×Dic53SD1611yescharacteristic
63C22×Dic53SD1611yescharacteristic
64C22×Dic53Q1611yescharacteristic
65C2×#88324—42no—
66C2×#88324—42no—
67C2×#88324—81no—
68C2×#88324—81no—
69C2×#88324—81no—
70C2×#88324—81no—
71C4⋊Dic53—44no—
72C4⋊Dic53—44no—
73C4⋊Dic53—42no—
74C4⋊Dic53—42no—
75C23×C53—21no—
76C23×C53—21no—
77C23×C53—21no—
78C23×C53—21no—
79C23×C53C22⋊C811yescharacteristic
80C23×C53#9911yescharacteristic
81C23×C53#10111yescharacteristic
82C23×C53#10211yescharacteristic
83C23×C53#10311yescharacteristic
84C23×C53#10411yescharacteristic
85C23×C53#10411yescharacteristic
86C2×C4×C53—24no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—42no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×Dic53—21no—
101C2×Dic53—21no—
102C2×Dic53—21no—
103C2×Dic53—21no—
104C2×Dic53—41no—
105C2×Dic53—41no—
106C2×Dic53—41no—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53—41no—
110C2×Dic53—41no—
111C2×Dic53—41no—
112C2×Dic53—42no—
113C2×Dic53—42no—
114C2×Dic53—42no—
115C2×Dic53—81no—
116C2×Dic53—81no—
117C2×Dic53—81no—
118C2×Dic53—81no—
119C2×Dic53—81no—
120C2×Dic53—81no—
121#88324—82no—
122#88324—82no—
123#7123—531nomaximal, Sylow
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—21no—
136C22×C53#32411yescharacteristic
137C22×C53D4⋊C811yescharacteristic
138C22×C53Q8⋊C811yescharacteristic
139C22×C53#32711yescharacteristic
140C22×C53#32811yescharacteristic
141C22×C53#33011yescharacteristic
142C22×C53#33211yescharacteristic
143C22×C53AΣL(1,𝔽16)11yescharacteristic
144C22×C53#35211yescharacteristic
145C22×C53#35311yescharacteristic
146C22×C53#35611yescharacteristic
147C4×C53—44no—
148C4×C53—41no—
149C4×C53—41no—
150Dic53—41no—
151Dic53—41no—
152Dic53—41no—
153Dic53—41no—
154Dic53—82no—
155#1444—531no—
156#1280—531no—
157#1280—531no—
158C2×C53—21no—
159C2×C53—21no—
160C2×C53—21no—
161C2×C53—21no—
162C2×C53#127111yescharacteristic
163C2×C53#132511yescharacteristic
164C2×C53#132611yescharacteristic
165C2×C53#134311yescharacteristic
166C2×C53#134611yescharacteristic
167C2×C53#135211yescharacteristic
168C2×C53#135411yescharacteristic
169C22×C4⋊C4—531no—
170C22×C4⋊C4—1061no—
171C2×C22⋊C8—1061no—
172C2×C22⋊C8—1061no—
173C2×#95—531no—
174C2×#95—1061no—
175C2×#95#615711yescharacteristic
176C53#712311yescharacteristic, complemented, Sylow
177C23×C4—21no—
178C23×C4—531no—
179C23×C4—531no—
180C23×C4—1061no—
181C23×C4C2×#615711yescharacteristic
182C22×C8—2121no—
183C22×C8—2121no—
184C2×C4⋊C4—2121no—
185C2×C4⋊C4—2122no—
186C2×C4⋊C4—2121no—
187C2×C4⋊C4—2121no—
188C2×C4⋊C4—2122no—
189C2×C4⋊C4—2122no—
190C22⋊C8—1062no—
191C22⋊C8—1062no—
192C22⋊C8—2121no—
193C22⋊C8—2121no—
194#95—21no—
195#95—2121no—
196#95—2122no—
197#95#8832412yescomplemented
198C24C22⋊#615711yescharacteristic
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—531no—
211C22×C4—531no—
212C22×C4—531no—
213C22×C4—531no—
214C22×C4—1061no—
215C22×C4—1061no—
216C22×C4—1061no—
217C22×C4—1061no—
218C22×C4—1062no—
219C22×C4—2121no—
220C22×C4—2121no—
221C22×C4—2121no—
222C22×C4—2121no—
223C22×C4—2121no—
224C22×C4—2121no—
225C22×C4—2121no—
226C22×C4—2121no—
227C22×C4—2121no—
228C22×C4—2121no—
229C22×C4C2×#8832411yescharacteristic
230C22×C4#1171994011yescharacteristic
231C2×C8—4241no—
232C2×C8—4241no—
233C2×C8—4241no—
234C2×C8—4241no—
235C2×C8—2122no—
236C2×C8—2122no—
237C4⋊C4—2122no—
238C4⋊C4—2124no—
239C4⋊C4—2124no—
240C4⋊C4—2122no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23C22⋊#8832411yescharacteristic
246C23#1313327611yescharacteristic
247C23#1313327811yescharacteristic
248C23#1313327911yescharacteristic
249C23#1313328011yescharacteristic
250C23#1313328111yescharacteristic
251C23#1313328211yescharacteristic
252C2×C4—24no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—1061no—
267C2×C4—1061no—
268C2×C4—1061no—
269C2×C4—1061no—
270C2×C4—4241no—
271C2×C4—4241no—
272C2×C4—4241no—
273C2×C4—4241no—
274C2×C4—4241no—
275C2×C4—4241no—
276C2×C4—2122no—
277C2×C4—2122no—
278C2×C4—2122no—
279C2×C4—2121no—
280C2×C4—2121no—
281C2×C4—2121no—
282C2×C4—2121no—
283C2×C4—2121no—
284C2×C4—2121no—
285C2×C4—2121no—
286C2×C4—2121no—
287C8—4242no—
288C8—4242no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22#1681146811yescharacteristic
302C22D4⋊#8832411yescharacteristic
303C22Q8⋊#8832411yescharacteristic
304C22#1681147511yescharacteristic
305C22#1681147611yescharacteristic
306C22#1681148011yescharacteristic
307C22#1681148211yescharacteristic
308C22#1681152411yescharacteristic
309C22#1681152611yescharacteristic
310C22#1681152811yescharacteristic
311C22#1681153411yescharacteristic
312C4—44no—
313C4—41no—
314C4—41no—
315C4—4242no—
316C4—2121no—
317C4—2121no—
318C4—2121no—
319C4—2121no—
320C2—21no—
321C2—21no—
322C2—21no—
323C2—21no—
324C2#2028953311yescharacteristic
325C2#2028965711yescharacteristic
326C2#2028965911yescharacteristic
327C2#2028969411yescharacteristic
328C2#2028970011yescharacteristic
329C2#2028971111yescharacteristic
330C2#2028971511yescharacteristic
subgroupG/H
centerC23#13133276
commutatorC22×C4×C53C2×C8
FrattiniC23×C4C2×#6157
FittingC2×#95×C53C4
socleC23×C53#99
prime radicalC24×C53C22⋊C4
cosocleC2×#13133124C22
hypercenterC2×#95#6157
hypocenterC53#7123
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
31. #95×C53C8semidirect
176. C53#7123semidirect
197. #95#88324semidirect

Maximal subgroups

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

subgroupindex
#202895532
#202729802
#712353