Finite Group Databasebeta

← #28299701

Subgroups — #28299701

Order 13568.

478 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 100 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#20290643C211yesmaximal, characteristic, complemented
2#20290632C211yesmaximal, characteristic, complemented
3#20289533C212yesmaximal, complemented
4#20289532C212yesmaximal
5#20277173C211yesmaximal, characteristic
6C2×C4⋊#88324—21no—
7#16811489—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133179C411yescharacteristic
13C2×#13133175—21no—
14#16810525C411yescharacteristic
15#16810524C2211yescharacteristic
16#16810522C2211yescharacteristic
17#16810515—21no—
18#16810513—21no—
19C4⋊C4×Dic53C411yescharacteristic
20#16810470—21no—
21C42×Dic53C2211yescharacteristic
22C53×#389C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C4⋊#88324—42no—
26C22×D212C2×C411yescharacteristic
27C22×C4×D53—21no—
28#13133179—22no—
29#13133179—22no—
30#13133175—42no—
31#13133175—42no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×#11719921—21no—
35C2×#11719921—21no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×C4×Dic53—21no—
40C2×C4×Dic53C2×C411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53C2311yescharacteristic
46#13133124—21no—
47#13133124—21no—
48C2×C4⋊C4×C53—21no—
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C42×C53C2×C411yescharacteristic
52#95×C53—21no—
53C23×D53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22×C411yescharacteristic
59C22×Dic53—21no—
60C22×Dic53C22⋊C411yescharacteristic
61C22×Dic53C22⋊C411yescharacteristic
62C22×Dic53C2×D411yescharacteristic
63C22×Dic53C2×D411yescharacteristic
64C2×D212—21no—
65C2×D212—41no—
66C2×D212—41no—
67C2×D212M1612yes—
68C2×C4×D53—24no—
69C2×C4×D53—42no—
70C2×C4×D53—42no—
71C2×#88324—42no—
72C2×#88324—42no—
73C2×#88324—42no—
74C2×#88324—42no—
75C2×#88324—42no—
76C2×#88324—42no—
77C4⋊C4×C53—21no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53M1612yes—
81C42×C53—41no—
82#11719921—24no—
83#11719921—42no—
84#11719921—42no—
85#11719921—41no—
86#11719921—41no—
87#11719921—42no—
88C4⋊Dic53—41no—
89C4⋊Dic53—41no—
90C4⋊Dic53—41no—
91C4×Dic53—21no—
92C4×Dic53—21no—
93C4×Dic53—22no—
94C4×Dic53—41no—
95C4×Dic53—41no—
96C4×Dic53—41no—
97C4×Dic53—42no—
98C4×Dic53D812yes—
99C4×Dic53SD1612yes—
100C23×C53C2×C22⋊C411yescharacteristic
101C22×D53—22no—
102C22×D53—41no—
103C22×D53—41no—
104C22×D53—41no—
105C22×D53—41no—
106C22×D53—41no—
107C22×D53—41no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—22no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53#10212yes—
122C2×C4×C53C2×M1611yescharacteristic
123C2×C4×C53C2×M1611yescharacteristic
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53#10212yes—
137C2×Dic53C2×D811yescharacteristic
138C2×Dic53C2×SD1611yescharacteristic
139C2×Dic53#13512yes—
140D212—42no—
141D212—42no—
142C4×D53—48no—
143#88324—82no—
144#88324—82no—
145#7642—531nomaximal, Sylow
146C22×C53#40711yescharacteristic
147C22×C53#41011yescharacteristic
148C22×C53#41311yescharacteristic
149C22×C53C2×#10211yescharacteristic
150C22×C53#41611yescharacteristic
151C22×C53#42111yescharacteristic
152C22×C53#42111yescharacteristic
153C2×D53—42no—
154C2×D53—42no—
155C2×D53—42no—
156C2×D53—81no—
157C2×D53—81no—
158C2×D53—81no—
159C2×D53—81no—
160C4×C53—22no—
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—42no—
164C4×C53—41no—
165C4×C53—81no—
166Dic53—22no—
167Dic53—21no—
168Dic53—21no—
169Dic53—22no—
170Dic53—41no—
171Dic53—81no—
172#2293—531no—
173#1839—531no—
174#1831—531no—
175#1271—532no—
176#1270—532no—
177C2×C53#146411yescharacteristic
178C2×C53#148611yescharacteristic
179C2×C53#148811yescharacteristic
180C2×C53#151711yescharacteristic
181C2×C53#152111yescharacteristic
182C2×C53#153511yescharacteristic
183C2×C53#154011yescharacteristic
184D53—82no—
185C2×C4×D4—531no—
186C2×#117—1061no—
187C2×C4⋊C8—1061no—
188#390—531no—
189#390—531no—
190#389—531no—
191#389#615711yescharacteristic
192#388—1061no—
193#388—1061no—
194C4×C4⋊C4—531no—
195C4×C22⋊C4—1061no—
196C43—531no—
197#336—532no—
198#336—532no—
199#336—1061no—
200#336—1061no—
201#336—1061no—
202C53#764211yescharacteristic, complemented, Sylow
203C22×D4—531no—
204C23×C4—1061no—
205C22×C8—1062no—
206C22×C8—1062no—
207C4×D4—1062no—
208C4×D4—1062no—
209#117—2122no—
210#117—2122no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—531no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4C2×#615711yescharacteristic
215C2×C4⋊C4C2×#615711yescharacteristic
216C2×C22⋊C4—1061no—
217C2×C22⋊C4—1061no—
218C2×C22⋊C4—1061no—
219C2×C22⋊C4—1061no—
220C2×C22⋊C4—1061no—
221C2×C42—531no—
222C2×C42—531no—
223C2×C42—531no—
224C2×C42—531no—
225C2×C42—531no—
226C2×C42—531no—
227C2×C42—1061no—
228C2×C42C2×#615711yescharacteristic
229C4⋊C8—2122no—
230#95—21no—
231#95—1061no—
232#95—1061no—
233C24—1061no—
234C2×D4—532no—
235C2×D4—1061no—
236C2×D4—2121no—
237C2×D4—2121no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—531no—
241C22×C4—531no—
242C22×C4—531no—
243C22×C4—531no—
244C22×C4—1064no—
245C22×C4—1061no—
246C22×C4—2122no—
247C22×C4—2122no—
248C22×C4C22⋊#615711yescharacteristic
249C22×C4C22⋊#615711yescharacteristic
250C22×C4C22×#615711yescharacteristic
251C2×C8—2122no—
252C2×C8—2122no—
253C2×C8—2122no—
254C2×C8—2122no—
255C2×C8—2122no—
256C2×C8—2122no—
257C4⋊C4—21no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—2121no—
261C4⋊C4—2121no—
262C4⋊C4—2121no—
263C4⋊C4#1171993612yes—
264C22⋊C4—1064no—
265C22⋊C4—2122no—
266C22⋊C4—2121no—
267C22⋊C4—2121no—
268C22⋊C4—2122no—
269C22⋊C4—2122no—
270C42—41no—
271C42—532no—
272C42—532no—
273C42—1062no—
274C42—1061no—
275C42—1061no—
276C42—2121no—
277C42—2121no—
278C42—2121no—
279C42—2122no—
280C23—1062no—
281C23—2121no—
282C23—2121no—
283C23—2121no—
284C23—2121no—
285C23—2121no—
286C23—2121no—
287C23C2×C22⋊#615711yescharacteristic
288D4—2122no—
289D4—2122no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—532no—
304C2×C4—531no—
305C2×C4—531no—
306C2×C4—532no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—2128no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4#1313327912yes—
321C2×C4D53⋊M1611yescharacteristic
322C2×C4C2×#1171993611yescharacteristic
323C8—4242no—
324C8—4242no—
325C22—4241no—
326C22—4241no—
327C22—4241no—
328C22—4241no—
329C22—2122no—
330C22—2122no—
331C22—2122no—
332C22#1681159411yescharacteristic
333C22#1681160411yescharacteristic
334C22#1681161011yescharacteristic
335C22C2×#1313327911yescharacteristic
336C22#1681161311yescharacteristic
337C22#1681162011yescharacteristic
338C22#1681162211yescharacteristic
339C4—22no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—42no—
344C4—81no—
345C4—1061no—
346C4—1061no—
347C4—1062no—
348C4—1062no—
349C4—4241no—
350C4—2121no—
351C2—4242no—
352C2#2028996911yescharacteristic
353C2#2029004111yescharacteristic
354C2#2029004911yescharacteristic
355C2#2029014611yescharacteristic
356C2#2029015811yescharacteristic
357C2#2029020211yescharacteristic
358C2#2029021311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#389C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C4×Dic53C23
hypercenter#389#6157
hypocenterC53#7642
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
1. #20290643C2semidirect
2. #20290632C2semidirect
3. #20289533C2semidirect
202. C53#7642semidirect

Maximal subgroups

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

subgroupindex
#202906432
#202906322
#202895332
#202895322
#202771732
#764253