Finite Group Databasebeta

← #28299596

Subgroups — #28299596

Order 13568.

516 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 102 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#20290571C211yesmaximal, characteristic
2#20290441C211yesmaximal, characteristic
3#20289533C212yesmaximal, complemented
4#20289533C212yesmaximal, complemented
5#20277199C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×C22⋊#88324—21no—
8C2×C22⋊#88324—21no—
9C2×#13133272—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133176C411yescharacteristic
13C22×C4×Dic53C2211yescharacteristic
14#16810522C411yescharacteristic
15#16810522C411yescharacteristic
16#16810522C2211yescharacteristic
17#16810522C2211yescharacteristic
18#16810498—21no—
19#16810498—21no—
20C4×C4⋊Dic53—21no—
21C4×C4⋊Dic53—21no—
22C2×#117×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133284—42no—
26C22⋊#88324—24no—
27C22⋊#88324—42no—
28#13133272—42no—
29C23×Dic53D411yescharacteristic
30C23×Dic53D411yescharacteristic
31#13133176—42no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×C22⋊Dic53—21no—
39C2×C4×Dic53—21no—
40C2×C4×Dic53—42no—
41C2×C4×Dic53C2×C411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53C2311yescharacteristic
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C23×C4×C53C2×C411yescharacteristic
50#117×C53—22no—
51#117×C53—22no—
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C4⋊C4×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53—21no—
55C2×C42×C53—21no—
56C24×C53C22⋊C411yescharacteristic
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22×C411yescharacteristic
65C22×Dic53—22no—
66C22×Dic53—22no—
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—41no—
72C22×Dic53—41no—
73C22×Dic53—41no—
74C22×Dic53C22⋊C411yescharacteristic
75C22×Dic53C22⋊C411yescharacteristic
76C22×Dic53C2×D411yescharacteristic
77C22×Dic53C2×D411yescharacteristic
78C2×#88324—24no—
79C2×#88324—42no—
80C2×#88324—42no—
81C2×#88324—42no—
82C2×#88324—42no—
83C2×#88324—42no—
84C4⋊C4×C53—21no—
85C4⋊C4×C53—21no—
86C4⋊C4×C53M1612yes—
87C4⋊C4×C53M1612yes—
88C22⋊C4×C53—24no—
89C42×C53—24no—
90C22⋊Dic53—42no—
91C4⋊Dic53—42no—
92C4⋊Dic53—42no—
93C4⋊Dic53—42no—
94C4⋊Dic53—41no—
95C4⋊Dic53—41no—
96C4⋊Dic53—41no—
97C4⋊Dic53—41no—
98C4×Dic53—22no—
99C4×Dic53—22no—
100C4×Dic53—41no—
101C4×Dic53—41no—
102C4×Dic53—42no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53#10112yes—
110C23×C53C2×C22⋊C411yescharacteristic
111C2×C4×C53—24no—
112C2×C4×C53—24no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53#9912yes—
124C2×C4×C53C2×M1611yescharacteristic
125C2×C4×C53C2×M1611yescharacteristic
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—44no—
136C2×Dic53—42no—
137C2×Dic53—42no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53#10412yes—
145C2×Dic53#13512yes—
146C2×Dic53AGL(1,ℤ/8)11yescharacteristic
147C2×Dic53#13711yescharacteristic
148#88324—44no—
149#88324—82no—
150#7613—531nomaximal, Sylow
151C22×C53—22no—
152C22×C53—22no—
153C22×C53—22no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53#40711yescharacteristic
159C22×C53C2×#9911yescharacteristic
160C22×C53C2×#10111yescharacteristic
161C22×C53#41611yescharacteristic
162C22×C53#41711yescharacteristic
163C22×C53C2×#10411yescharacteristic
164C22×C53#42111yescharacteristic
165C4×C53—22no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—42no—
169C4×C53—42no—
170Dic53—22no—
171Dic53—22no—
172Dic53—21no—
173Dic53—21no—
174Dic53—41no—
175Dic53—81no—
176Dic53—81no—
177#2296—531no—
178#1797—531no—
179#1749—531no—
180#1271—532no—
181#1271—532no—
182C2×C53—42no—
183C2×C53#146311yescharacteristic
184C2×C53#148811yescharacteristic
185C2×C53#148911yescharacteristic
186C2×C53#150111yescharacteristic
187C2×C53#151311yescharacteristic
188C2×C53#153111yescharacteristic
189C2×C53#155211yescharacteristic
190C2×#117—531no—
191C2×#117#615711yescharacteristic
192C22×C42—531no—
193C2×C4⋊C8—1061no—
194C2×C22⋊C8—1061no—
195C2×C22⋊C8—1061no—
196C2×#97—1061no—
197#389—531no—
198#389—531no—
199#389—531no—
200#389—531no—
201#385—1061no—
202#385—1061no—
203C4×C4⋊C4—1061no—
204C4×C4⋊C4—1061no—
205#336—532no—
206#336—532no—
207C53#761311yescharacteristic, complemented, Sylow
208C23×C4—531no—
209C23×C4—531no—
210C23×C4C2×#615711yescharacteristic
211C22×C8—1062no—
212C22×C8—1062no—
213#117—22no—
214#117—22no—
215#117—2122no—
216C2×C4⋊C4—531no—
217C2×C4⋊C4—531no—
218C2×C4⋊C4—1061no—
219C2×C4⋊C4—1061no—
220C2×C4⋊C4—1061no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4C2×#615711yescharacteristic
223C2×C4⋊C4C2×#615711yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—1061no—
226C2×C42—21no—
227C2×C42—531no—
228C2×C42—531no—
229C2×C42—531no—
230C2×C42—531no—
231C2×C42—1061no—
232C2×C42—2122no—
233C4⋊C8—2122no—
234C22⋊C8—1064no—
235C22⋊C8—2122no—
236#97—2122no—
237#95—1061no—
238#95—1061no—
239#95—1061no—
240#95—1061no—
241C24C22⋊#615711yescharacteristic
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—531no—
249C22×C4—531no—
250C22×C4—531no—
251C22×C4—531no—
252C22×C4—1062no—
253C22×C4—1061no—
254C22×C4—1062no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—1061no—
258C22×C4—2121no—
259C22×C4—2121no—
260C22×C4—2121no—
261C22×C4C22⋊#615711yescharacteristic
262C22×C4C22×#615711yescharacteristic
263C2×C8—1064no—
264C2×C8—2122no—
265C2×C8—2122no—
266C2×C8—2122no—
267C2×C8—2122no—
268C2×C8—2122no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—2121no—
272C4⋊C4—2121no—
273C4⋊C4—2122no—
274C4⋊C4—2121no—
275C4⋊C4—2121no—
276C4⋊C4—2122no—
277C4⋊C4—2122no—
278C4⋊C4#1171994012yes—
279C4⋊C4#1171994012yes—
280C22⋊C4—24no—
281C22⋊C4—2122no—
282C42—24no—
283C42—1062no—
284C42—1062no—
285C42—2121no—
286C42—2121no—
287C42—2122no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23#1313327812yes—
295C23C2×C22⋊#615711yescharacteristic
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—532no—
309C2×C4—531no—
310C2×C4—531no—
311C2×C4—532no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—2124no—
322C2×C4—2122no—
323C2×C4—2122no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4#1313327612yes—
331C2×C4C2×#1171994011yescharacteristic
332C2×C4C2×#1171994011yescharacteristic
333C8—4242no—
334C8—2124no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22#1681159311yescharacteristic
343C22C2×#1313327611yescharacteristic
344C22C2×#1313327811yescharacteristic
345C22#1681161311yescharacteristic
346C22#1681161411yescharacteristic
347C22D53⋊#10411yescharacteristic
348C22#1681162011yescharacteristic
349C4—22no—
350C4—21no—
351C4—21no—
352C4—42no—
353C4—42no—
354C4—1062no—
355C4—1062no—
356C4—1061no—
357C4—1061no—
358C4—4241no—
359C4—4241no—
360C4—2121no—
361C2—42no—
362C2#2028996311yescharacteristic
363C2#2029004611yescharacteristic
364C2#2029005011yescharacteristic
365C2#2029009211yescharacteristic
366C2#2029013211yescharacteristic
367C2#2029018711yescharacteristic
368C2#2029023611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×#117#6157
hypocenterC53#7613
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
3. #20289533C2semidirect
4. #20289533C2semidirect
207. C53#7613semidirect

Maximal subgroups

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

subgroupindex
#202905712
#202904412
#202895332
#202771992
#761353