Finite Group Databasebeta

← #28299413

Subgroups — #28299413

Order 13568.

538 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C4×#13133284C211yesmaximal, characteristic, complemented
2#20290430C211yesmaximal, characteristic, complemented
3#20289533C212yesmaximal, complemented
4#20289532C212yesmaximal
5#20277173C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×C4×#88324—21no—
8C2×C4×#88324—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—
25#13133284—24no—
26C4×#88324—24no—
27C4×#88324—42no—
28C22×D212C2×C411yescharacteristic
29C22×C4×D53—21no—
30#13133179—22no—
31#13133179C814yescomplemented
32#13133175—42no—
33#13133175—42no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×#11719921—21no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53C2311yescharacteristic
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×D53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22×C411yescharacteristic
61C22×Dic53—21no—
62C22×Dic53C22⋊C411yescharacteristic
63C22×Dic53C22⋊C411yescharacteristic
64C22×Dic53C2×D411yescharacteristic
65C22×Dic53C2×D411yescharacteristic
66C2×D212—21no—
67C2×D212—41no—
68C2×D212—41no—
69C2×D212C2×C812yes—
70C2×C4×D53—24no—
71C2×C4×D53—42no—
72C2×C4×D53—42no—
73C2×#88324—24no—
74C2×#88324—24no—
75C2×#88324—42no—
76C2×#88324—42no—
77C2×#88324—42no—
78C2×#88324—42no—
79C4⋊C4×C53—21no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—42no—
82C4⋊C4×C53C2×C812yes—
83C42×C53—41no—
84#11719921—24no—
85#11719921—42no—
86#11719921—42no—
87#11719921—41no—
88#11719921—41no—
89#11719921—42no—
90C4⋊Dic53—41no—
91C4⋊Dic53—41no—
92C4⋊Dic53—41no—
93C4×Dic53—22no—
94C4×Dic53—22no—
95C4×Dic53—22no—
96C4×Dic53—41no—
97C4×Dic53—42no—
98C4×Dic53C2×C812yes—
99C4×Dic53M1612yes—
100C4×Dic53D812yes—
101C4×Dic53SD1612yes—
102C23×C53C2×C22⋊C411yescharacteristic
103C22×D53—22no—
104C22×D53—41no—
105C22×D53—41no—
106C22×D53—41no—
107C22×D53—41no—
108C22×D53—41no—
109C22×D53—41no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—22no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53#10212yes—
123C2×C4×C53#10412yes—
124C2×C4×C53C22×C811yescharacteristic
125C2×C4×C53C2×M1611yescharacteristic
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53C22⋊C812yes—
136C2×Dic53C22⋊C812yes—
137C2×Dic53#10212yes—
138C2×Dic53#10412yes—
139C2×Dic53C2×D811yescharacteristic
140C2×Dic53C2×SD1611yescharacteristic
141C2×Dic53#13512yes—
142D212—42no—
143D212—42no—
144C4×D53—48no—
145#88324—44no—
146#88324—44no—
147#7560—531nomaximal, Sylow
148C22×C53C2×C22⋊C811yescharacteristic
149C22×C53#41011yescharacteristic
150C22×C53#41311yescharacteristic
151C22×C53C2×#10211yescharacteristic
152C22×C53#41611yescharacteristic
153C22×C53C2×#10411yescharacteristic
154C22×C53C2×#10411yescharacteristic
155C2×D53—42no—
156C2×D53—42no—
157C2×D53—42no—
158C2×D53—81no—
159C2×D53—81no—
160C2×D53—81no—
161C2×D53—81no—
162C4×C53—22no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—22no—
166C4×C53—42no—
167C4×C53—81no—
168Dic53—22no—
169Dic53—22no—
170Dic53—21no—
171Dic53—21no—
172Dic53—81no—
173Dic53D4⋊C814yes—
174#2293—531no—
175C4×C4⋊C8—531no—
176#1744—531no—
177#1271—532no—
178#1270—532no—
179C2×C53#145811yescharacteristic
180C2×C53C2×D4⋊C811yescharacteristic
181C2×C53#147611yescharacteristic
182C2×C53#150011yescharacteristic
183C2×C53#150211yescharacteristic
184C2×C53#152411yescharacteristic
185C2×C53#154011yescharacteristic
186D53—82no—
187C2×C4×D4—531no—
188C2×#117—1061no—
189C2×C4⋊C8—1061no—
190C2×C4×C8—1061no—
191C2×C4×C8—1061no—
192#390—531no—
193#390—531no—
194#389—531no—
195#389#615711yescharacteristic
196#388—1061no—
197#388—1061no—
198C4×C4⋊C4—531no—
199C4×C22⋊C4—1061no—
200C43—531no—
201#336—532no—
202#336—532no—
203#336—1061no—
204C53#756011yescharacteristic, complemented, Sylow
205C22×D4—531no—
206C23×C4—1061no—
207C22×C8—1062no—
208C22×C8—1062no—
209C4×D4—534no—
210C4×D4—1062no—
211#117—2122no—
212#117—2122no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—531no—
215C2×C4⋊C4—1061no—
216C2×C4⋊C4C2×#615711yescharacteristic
217C2×C4⋊C4C2×#615711yescharacteristic
218C2×C22⋊C4—1061no—
219C2×C22⋊C4—1061no—
220C2×C22⋊C4—1061no—
221C2×C22⋊C4—1061no—
222C2×C22⋊C4—1061no—
223C2×C42—531no—
224C2×C42—531no—
225C2×C42—531no—
226C2×C42—531no—
227C2×C42—531no—
228C2×C42—531no—
229C2×C42—1061no—
230C2×C42C2×#615711yescharacteristic
231C4⋊C8—1064no—
232C4×C8—1064no—
233C4×C8—2122no—
234#95—21no—
235#95—1061no—
236#95—1061no—
237C24—1061no—
238C2×D4—532no—
239C2×D4—1061no—
240C2×D4—2121no—
241C2×D4—2121no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—531no—
245C22×C4—531no—
246C22×C4—531no—
247C22×C4—531no—
248C22×C4—1064no—
249C22×C4—1061no—
250C22×C4—2122no—
251C22×C4—2122no—
252C22×C4C22⋊#615711yescharacteristic
253C22×C4C22⋊#615711yescharacteristic
254C22×C4C22×#615711yescharacteristic
255C2×C8—1064no—
256C2×C8—1064no—
257C2×C8—2122no—
258C2×C8—2122no—
259C2×C8—2122no—
260C2×C8—2122no—
261C4⋊C4—21no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—2121no—
265C4⋊C4—2121no—
266C4⋊C4—2121no—
267C4⋊C4D53⋊C812yes—
268C22⋊C4—1064no—
269C22⋊C4—2121no—
270C22⋊C4—2121no—
271C22⋊C4—2122no—
272C22⋊C4—2122no—
273C22⋊C4—2122no—
274C42—41no—
275C42—532no—
276C42—532no—
277C42—532no—
278C42—532no—
279C42—1062no—
280C42—1062no—
281C42—1062no—
282C42—2121no—
283C42—2122no—
284C23—1062no—
285C23—2121no—
286C23—2121no—
287C23—2121no—
288C23—2121no—
289C23—2121no—
290C23—2121no—
291C23C2×C22⋊#615711yescharacteristic
292D4—2122no—
293D4—2122no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—532no—
307C2×C4—532no—
308C2×C4—532no—
309C2×C4—532no—
310C2×C4—531no—
311C2×C4—531no—
312C2×C4—532no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—2128no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4#1313327912yes—
324C2×C4#1313328212yes—
325C2×C4C2×D53⋊C811yescharacteristic
326C2×C4D53⋊M1611yescharacteristic
327C8—2124no—
328C8—2124no—
329C22—4241no—
330C22—4241no—
331C22—4241no—
332C22—4241no—
333C22—2122no—
334C22—2122no—
335C22—2122no—
336C22D53⋊(C22⋊C8)11yescharacteristic
337C22#1681160411yescharacteristic
338C22#1681161011yescharacteristic
339C22C2×#1313327911yescharacteristic
340C22#1681161311yescharacteristic
341C22D53⋊#10411yescharacteristic
342C22C2×#1313328211yescharacteristic
343C4—22no—
344C4—21no—
345C4—21no—
346C4—22no—
347C4—42no—
348C4—81no—
349C4—534no—
350C4—1062no—
351C4—1061no—
352C4—1061no—
353C4—1062no—
354C4—4241no—
355C2—4242no—
356C2#2028994911yescharacteristic
357C2D53⋊(D4⋊C8)11yescharacteristic
358C2#2029000911yescharacteristic
359C2#2029008911yescharacteristic
360C2#2029009611yescharacteristic
361C2#2029016811yescharacteristic
362C2#2029021311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#389C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C4×Dic53C23
hypercenter#389#6157
hypocenterC53#7560
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
1. C4×#13133284C2semidirect
2. #20290430C2semidirect
3. #20289533C2semidirect
31. #13133179C8semidirect
204. C53#7560semidirect

Maximal subgroups

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

subgroupindex
C4×#131332842
#202904302
#202895332
#202895322
#202771732
#756053