Finite Group Databasebeta

← #28299528

Subgroups — #28299528

Order 13568.

566 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 110 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#20290607C211yesmaximal, characteristic
2#20290607C211yesmaximal, characteristic
3#20289534C212yesmaximal, complemented
4#20289533C212yesmaximal, complemented
5#389×D53C211yesmaximal, characteristic
6C2×#13133274—21no—
7C2×#13133274—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×C4⋊C4×D53—21no—
13C2×C4⋊C4×D53C411yescharacteristic
14C2×C4⋊C4×D53C411yescharacteristic
15C2×C42×D53C2211yescharacteristic
16#16810522C2211yescharacteristic
17#16810521C411yescharacteristic
18#16810520—21no—
19#16810519C2211yescharacteristic
20#16810518—21no—
21#95×D53—21no—
22C53×#389C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133274—42no—
26#13133274—42no—
27C22×C4×D53—21no—
28C22×C4×D53—21no—
29C22×C4×D53C2×C411yescharacteristic
30C22×C4×D53D411yescharacteristic
31C22×C4×D53D411yescharacteristic
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C4⋊C4×D53—22no—
40C4⋊C4×D53—44no—
41C4⋊C4×D53—42no—
42C4⋊C4×D53C814yescomplemented
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53C2311yescharacteristic
46C42×D53—42no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50C2×C4⋊C4×C53—21no—
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C42×C53C2×C411yescharacteristic
54#95×C53—21no—
55C23×D53C22⋊C411yescharacteristic
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22×C411yescharacteristic
61C22×Dic53—21no—
62C22×Dic53—21no—
63C22×Dic53C22⋊C411yescharacteristic
64C22×Dic53C2×D411yescharacteristic
65C22×Dic53C2×D411yescharacteristic
66C2×C4×D53—22no—
67C2×C4×D53—22no—
68C2×C4×D53—24no—
69C2×C4×D53—22no—
70C2×C4×D53—22no—
71C2×C4×D53—22no—
72C2×C4×D53—22no—
73C2×C4×D53—41no—
74C2×C4×D53—41no—
75C2×C4×D53—42no—
76C2×C4×D53—42no—
77C2×C4×D53—42no—
78C2×C4×D53—42no—
79C2×C4×D53—42no—
80C2×C4×D53C2×C812yes—
81C2×C4×D53M1612yes—
82C2×#88324—42no—
83C2×#88324—42no—
84C2×#88324—42no—
85C2×#88324—42no—
86C2×#88324—42no—
87C2×#88324—42no—
88C4⋊C4×C53—21no—
89C4⋊C4×C53—42no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53C2×C812yes—
92C42×C53—41no—
93C4⋊Dic53—21no—
94C4⋊Dic53—42no—
95C4⋊Dic53—41no—
96C4⋊Dic53C2×C812yes—
97#11719919—24no—
98#11719919—42no—
99#11719919—42no—
100#11719919—42no—
101C4×Dic53—41no—
102C4×Dic53—41no—
103C4×Dic53—41no—
104C23×C53C2×C22⋊C411yescharacteristic
105C22×D53—21no—
106C22×D53—21no—
107C22×D53—21no—
108C22×D53—21no—
109C22×D53C22⋊C812yes—
110C22×D53#9912yes—
111C22×D53#10112yes—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—22no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53C22×C811yescharacteristic
127C2×C4×C53C2×M1611yescharacteristic
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—22no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53C22⋊C812yes—
141C2×Dic53AGL(1,ℤ/8)12yes—
142C2×Dic53#13712yes—
143C4×D53—24no—
144C4×D53—24no—
145C4×D53—44no—
146C4×D53—44no—
147C4×D53—48no—
148C4×D53—84no—
149#88324—82no—
150#88324—82no—
151#7593—531nomaximal, Sylow
152C22×C53C2×C22⋊C811yescharacteristic
153C22×C53C2×#9911yescharacteristic
154C22×C53C2×#10111yescharacteristic
155C22×C53#41811yescharacteristic
156C22×C53#41911yescharacteristic
157C22×C53#42111yescharacteristic
158C22×C53#42111yescharacteristic
159C2×D53—22no—
160C2×D53—22no—
161C2×D53—22no—
162C2×D53—22no—
163C2×D53—22no—
164C2×D53—22no—
165C2×D53#32414yes—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—42no—
169C4×C53—41no—
170C4×C53—41no—
171C4×C53—81no—
172Dic53—22no—
173Dic53—22no—
174Dic53—22no—
175Dic53—42no—
176Dic53—81no—
177C2×#389—531no—
178#1817—531no—
179#1817—531no—
180#1271—532no—
181#1271—532no—
182C2×C53C2×#32411yescharacteristic
183C2×C53#147811yescharacteristic
184C2×C53#147911yescharacteristic
185C2×C53#151011yescharacteristic
186C2×C53#151111yescharacteristic
187C2×C53#153411yescharacteristic
188C2×C53#155611yescharacteristic
189D53—24no—
190C22×C4⋊C4—531no—
191C22×C4⋊C4—531no—
192C22×C4⋊C4—1061no—
193C22×C42—531no—
194C2×C22⋊C8—1061no—
195C2×C22⋊C8—1061no—
196#389—531no—
197#389—531no—
198#389—531no—
199#389—1061no—
200#389—1061no—
201#389#615711yescharacteristic
202C2×#95—1061no—
203#336—532no—
204#336—532no—
205#336—1061no—
206#336—1061no—
207C53#759311yescharacteristic, complemented, Sylow
208C23×C4—531no—
209C23×C4—531no—
210C23×C4—531no—
211C23×C4—1061no—
212C23×C4—1061no—
213C22×C8—1062no—
214C22×C8—1062no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—531no—
217C2×C4⋊C4—534no—
218C2×C4⋊C4—531no—
219C2×C4⋊C4—1062no—
220C2×C4⋊C4—1061no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4—2122no—
226C2×C4⋊C4—2124no—
227C2×C4⋊C4C2×#615711yescharacteristic
228C2×C4⋊C4C2×#615711yescharacteristic
229C2×C42—531no—
230C2×C42—531no—
231C2×C42—531no—
232C2×C42—2122no—
233C2×C42C2×#615711yescharacteristic
234C22⋊C8—2122no—
235C22⋊C8—2122no—
236#95—21no—
237#95—1061no—
238#95—1061no—
239#95—1061no—
240C24—531no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—531no—
244C22×C4—531no—
245C22×C4—531no—
246C22×C4—532no—
247C22×C4—532no—
248C22×C4—1062no—
249C22×C4—1062no—
250C22×C4—1062no—
251C22×C4—1062no—
252C22×C4—1062no—
253C22×C4—1062no—
254C22×C4—1064no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—2121no—
258C22×C4—2121no—
259C22×C4—2122no—
260C22×C4—2122no—
261C22×C4—2122no—
262C22×C4—2122no—
263C22×C4—2122no—
264C22×C4C22⋊#615711yescharacteristic
265C22×C4C22⋊#615711yescharacteristic
266C22×C4C22×#615711yescharacteristic
267C2×C8—2122no—
268C2×C8—2122no—
269C2×C8—2122no—
270C2×C8—2122no—
271C2×C8—2122no—
272C2×C8—2122no—
273C4⋊C4—21no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—532no—
277C4⋊C4—1064no—
278C4⋊C4—1061no—
279C4⋊C4—2121no—
280C4⋊C4—2122no—
281C4⋊C4—2122no—
282C4⋊C4—2122no—
283C4⋊C4—2122no—
284C4⋊C4D53⋊C812yes—
285C42—41no—
286C42—2121no—
287C42—2121no—
288C42—2121no—
289C23—532no—
290C23—532no—
291C23—532no—
292C23—1061no—
293C23—1061no—
294C23—1061no—
295C23—1061no—
296C23C2×C22⋊#615711yescharacteristic
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—532no—
312C2×C4—532no—
313C2×C4—532no—
314C2×C4—1064no—
315C2×C4—1064no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1062no—
323C2×C4—4244no—
324C2×C4—2124no—
325C2×C4—2124no—
326C2×C4—2128no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4C2×D53⋊C811yescharacteristic
333C2×C4D53⋊M1611yescharacteristic
334C8—4242no—
335C8—4242no—
336C22—534no—
337C22—1062no—
338C22—1062no—
339C22—1062no—
340C22—1062no—
341C22—1062no—
342C22—1062no—
343C22D53⋊(C22⋊C8)11yescharacteristic
344C22D53⋊#9911yescharacteristic
345C22D53⋊#10111yescharacteristic
346C22#1681161511yescharacteristic
347C22#1681161611yescharacteristic
348C22#1681162011yescharacteristic
349C22#1681162011yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—42no—
355C4—81no—
356C4—1062no—
357C4—1062no—
358C4—1062no—
359C4—4241no—
360C4—2122no—
361C2—1064no—
362C2D53⋊#32411yescharacteristic
363C2#2029001511yescharacteristic
364C2#2029001811yescharacteristic
365C2#2029012111yescharacteristic
366C2#2029012411yescharacteristic
367C2#2029019811yescharacteristic
368C2#2029024511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#389C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#389#6157
hypocenterC53#7593
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
3. #20289534C2semidirect
4. #20289533C2semidirect
42. C4⋊C4×D53C8semidirect
207. C53#7593semidirect

Maximal subgroups

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

subgroupindex
#202906072
#202895342
#202895332
#389×D532
#759353