Finite Group Databasebeta

← #28299461

Subgroups — #28299461

Order 13568.

520 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 118 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#20290631C211yesmaximal, characteristic, complemented
2C4×#13133284C211yesmaximal, characteristic, complemented
3#20289533C212yesmaximal, complemented
4#20289532C212yesmaximal
5#20277173C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×#13133284—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—
26#13133284—42no—
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—42no—
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—41no—
90#11719921—41no—
91C4⋊Dic53—41no—
92C4⋊Dic53—41no—
93C4⋊Dic53—41no—
94C4×Dic53—22no—
95C4×Dic53—21no—
96C4×Dic53—21no—
97C4×Dic53—41no—
98C4×Dic53—41no—
99C4×Dic53—42no—
100C4×Dic53C2×C812yes—
101C4×Dic53M1612yes—
102C4×Dic53D812yes—
103C4×Dic53SD1612yes—
104C23×C53C2×C22⋊C411yescharacteristic
105C22×D53—22no—
106C22×D53—41no—
107C22×D53—41no—
108C22×D53—41no—
109C22×D53—41no—
110C22×D53—41no—
111C22×D53—41no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—22no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53#10212yes—
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—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53C22⋊C812yes—
138C2×Dic53C22⋊C812yes—
139C2×Dic53#10212yes—
140C2×Dic53#10412yes—
141C2×Dic53C2×D811yescharacteristic
142C2×Dic53C2×SD1611yescharacteristic
143C2×Dic53AGL(1,ℤ/8)11yescharacteristic
144C2×Dic53#13711yescharacteristic
145D212—42no—
146D212—42no—
147C4×D53—48no—
148#88324—44no—
149#88324—82no—
150#7574—531nomaximal, Sylow
151C22×C53C2×C22⋊C811yescharacteristic
152C22×C53#41011yescharacteristic
153C22×C53#41311yescharacteristic
154C22×C53C2×#10211yescharacteristic
155C22×C53#41711yescharacteristic
156C22×C53C2×#10411yescharacteristic
157C22×C53#42111yescharacteristic
158C2×D53—42no—
159C2×D53—42no—
160C2×D53—42no—
161C2×D53—81no—
162C2×D53—81no—
163C2×D53—81no—
164C2×D53—81no—
165C4×C53—22no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—42no—
169C4×C53—41no—
170C4×C53—81no—
171Dic53—22no—
172Dic53—21no—
173Dic53—21no—
174Dic53—21no—
175Dic53—21no—
176Dic53—81no—
177Dic53D4⋊C814yes—
178#2293—531no—
179#1831—531no—
180C4×C4⋊C8—531no—
181#1271—532no—
182#1270—532no—
183C2×C53#145811yescharacteristic
184C2×C53C2×D4⋊C811yescharacteristic
185C2×C53#147711yescharacteristic
186C2×C53#150911yescharacteristic
187C2×C53#151711yescharacteristic
188C2×C53#152911yescharacteristic
189C2×C53#154711yescharacteristic
190D53—82no—
191C2×C4×D4—531no—
192C2×#117—1061no—
193C2×C4⋊C8—1061no—
194C2×C4⋊C8—1061no—
195C2×C4×C8—1061no—
196#390—531no—
197#390—531no—
198#389—531no—
199#389#615711yescharacteristic
200#388—1061no—
201#388—1061no—
202C4×C4⋊C4—531no—
203C4×C22⋊C4—1061no—
204C43—531no—
205#336—532no—
206#336—532no—
207#336—1061no—
208C53#757411yescharacteristic, complemented, Sylow
209C22×D4—531no—
210C23×C4—1061no—
211C22×C8—1062no—
212C22×C8—1062no—
213C4×D4—534no—
214C4×D4—1062no—
215#117—2122no—
216#117—2122no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—531no—
219C2×C4⋊C4—1061no—
220C2×C4⋊C4C2×#615711yescharacteristic
221C2×C4⋊C4C2×#615711yescharacteristic
222C2×C22⋊C4—1061no—
223C2×C22⋊C4—1061no—
224C2×C22⋊C4—1061no—
225C2×C22⋊C4—1061no—
226C2×C22⋊C4—1061no—
227C2×C42—531no—
228C2×C42—531no—
229C2×C42—531no—
230C2×C42—531no—
231C2×C42—531no—
232C2×C42—531no—
233C2×C42—1061no—
234C2×C42C2×#615711yescharacteristic
235C4⋊C8—1064no—
236C4⋊C8—2122no—
237C4×C8—2122no—
238#95—21no—
239#95—1061no—
240#95—1061no—
241C24—1061no—
242C2×D4—532no—
243C2×D4—1061no—
244C2×D4—2121no—
245C2×D4—2121no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—531no—
249C22×C4—531no—
250C22×C4—531no—
251C22×C4—531no—
252C22×C4—1064no—
253C22×C4—1061no—
254C22×C4—2122no—
255C22×C4—2122no—
256C22×C4C22⋊#615711yescharacteristic
257C22×C4C22⋊#615711yescharacteristic
258C22×C4C22×#615711yescharacteristic
259C2×C8—1064no—
260C2×C8—2122no—
261C2×C8—2122no—
262C2×C8—2122no—
263C2×C8—2122no—
264C2×C8—2122no—
265C4⋊C4—21no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—2121no—
269C4⋊C4—2121no—
270C4⋊C4—2121no—
271C4⋊C4D53⋊C812yes—
272C22⋊C4—1064no—
273C22⋊C4—2122no—
274C22⋊C4—2121no—
275C22⋊C4—2121no—
276C22⋊C4—2121no—
277C22⋊C4—2121no—
278C22⋊C4—2122no—
279C42—41no—
280C42—532no—
281C42—532no—
282C42—532no—
283C42—532no—
284C42—1061no—
285C42—1062no—
286C42—1061no—
287C42—2121no—
288C42—2121no—
289C42—2122no—
290C23—1062no—
291C23—2121no—
292C23—2121no—
293C23—2121no—
294C23—2121no—
295C23—2121no—
296C23—2121no—
297C23C2×C22⋊#615711yescharacteristic
298D4—2122no—
299D4—2122no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—532no—
314C2×C4—532no—
315C2×C4—532no—
316C2×C4—532no—
317C2×C4—531no—
318C2×C4—531no—
319C2×C4—531no—
320C2×C4—531no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—2128no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4#1313327912yes—
332C2×C4C2×D53⋊C811yescharacteristic
333C2×C4D53⋊M1611yescharacteristic
334C8—4242no—
335C8—2124no—
336C22—4241no—
337C22—4241no—
338C22—4241no—
339C22—4241no—
340C22—2122no—
341C22—2122no—
342C22—2122no—
343C22D53⋊(C22⋊C8)11yescharacteristic
344C22#1681160411yescharacteristic
345C22#1681161011yescharacteristic
346C22C2×#1313327911yescharacteristic
347C22#1681161411yescharacteristic
348C22D53⋊#10411yescharacteristic
349C22#1681162211yescharacteristic
350C4—21no—
351C4—21no—
352C4—22no—
353C4—41no—
354C4—42no—
355C4—81no—
356C4—534no—
357C4—1061no—
358C4—1061no—
359C4—1061no—
360C4—1061no—
361C4—1062no—
362C4—4241no—
363C2—4242no—
364C2#2028994911yescharacteristic
365C2D53⋊(D4⋊C8)11yescharacteristic
366C2#2029001311yescharacteristic
367C2#2029011811yescharacteristic
368C2#2029014611yescharacteristic
369C2#2029018311yescharacteristic
370C2#2029022711yescharacteristic
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#7574
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. #20290631C2semidirect
2. C4×#13133284C2semidirect
3. #20289533C2semidirect
31. #13133179C8semidirect
208. C53#7574semidirect

Maximal subgroups

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

subgroupindex
#202906312
C4×#131332842
#202895332
#202895322
#202771732
#757453