Finite Group Databasebeta

← #28306339

Subgroups — #28306339

Order 13568.

592 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 114 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#20290572C211yesmaximal, characteristic
2#20289891C211yesmaximal, characteristic
3#20289588C214yesmaximal
4#20283062C211yesmaximal, characteristic
5C2×#13133284—21no—
6C2×#13133274C2212yes—
7C2×(C2×#6157)⋊C4—21no—
8C2×(C2×#6157)⋊C4—21no—
9C2×(C2×#6157)⋊C4C2212yes—
10#16811489—21no—
11C2×#13133193—21no—
12C2×C42×D53C2211yescharacteristic
13#16810530—21no—
14#16810519—21no—
15#16810519C412yescomplemented
16#16810493C411yescharacteristic, complemented
17#16810480—21no—
18#16810480C2212yes—
19C2×C4⋊Q8×C53C411yescharacteristic, complemented
20C23×#6157—22no—
21C22×#88324—22no—
22#13133284—42no—
23#13133274—24no—
24(C2×#6157)⋊C4—22no—
25(C2×#6157)⋊C4—42no—
26(C2×#6157)⋊C4—42no—
27(C2×#6157)⋊C4Q814yescomplemented
28C22×C4×D53D411yescharacteristic
29C22×C4×D53D411yescharacteristic
30C22×C4×D53C2311yescharacteristic
31#13133193—42no—
32#13133193—42no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53C2×C412yes—
36C2×#11719919—22no—
37C2×#11719921—22no—
38C2×C4×Dic53C2×C411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C42×D53—42no—
42#13133124—22no—
43#13133124—22no—
44C22×Q8×C53—21no—
45C4⋊Q8×C53—42no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53C2×C412yes—
48C2×C42×C53C2×C411yescharacteristic
49C23×D53C2×D411yescharacteristic
50C22×#6157—24no—
51C22×#6157—42no—
52C22×#6157—42no—
53C22×#6157—42no—
54C22×#6157—42no—
55C22×#6157—42no—
56C22×#6157—42no—
57C2×Q8×C53—42no—
58C2×Q8×C53—42no—
59C2×Q8×C53—42no—
60C22×C4×C53—22no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C22×Dic53—22no—
65C22×Dic53C22⋊C411yescharacteristic
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C2×D411yescharacteristic
68C2×C4×D53—22no—
69C2×C4×D53—22no—
70C2×C4×D53—41no—
71C2×C4×D53—41no—
72C2×C4×D53—41no—
73C2×C4×D53—41no—
74C2×C4×D53—41no—
75C2×C4×D53D812yes—
76C2×C4×D53SD1612yes—
77C2×C4×D53C2×Q812yes—
78C2×C4×D53#4112yes—
79C2×#88324—42no—
80C2×#88324—42no—
81C2×#88324—42no—
82C4⋊C4×C53—42no—
83C4⋊C4×C53—42no—
84C42×C53—22no—
85#11719921—42no—
86#11719921—42no—
87C4⋊Dic53—42no—
88C4⋊Dic53—42no—
89C4⋊Dic53—42no—
90#11719919—44no—
91C4×Dic53—22no—
92C4×Dic53—41no—
93C4×Dic53—41no—
94C23×C53C2×C22⋊C411yescharacteristic
95C22×D53—21no—
96C22×D53—21no—
97C22×D53—21no—
98C22×D53—21no—
99C22×D53C22⋊Q812yes—
100C22×D53C2×D811yescharacteristic
101C22×D53C2×SD1611yescharacteristic
102C22×D53#13712yes—
103C2×#6157—44no—
104C2×#6157—44no—
105C2×#6157—44no—
106C2×#6157—82no—
107C2×#6157—82no—
108C2×#6157—82no—
109C2×#6157—82no—
110Q8×C53—84no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53#9912yes—
121C2×C4×C53#10212yes—
122C2×C4×C53C4×Q812yes—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—42no—
131C2×Dic53—42no—
132C2×Dic53—42no—
133C2×Dic53#10212yes—
134C2×Dic53#12411yescharacteristic
135C2×Dic53C4⋊Q811yescharacteristic
136C4×D53—42no—
137C4×D53—42no—
138C4×D53—42no—
139C4×D53—42no—
140C4×D53—44no—
141#88324—82no—
142#9912—531nomaximal, Sylow
143C22×C53#39111yescharacteristic
144C22×C53C2×#9911yescharacteristic
145C22×C53#41011yescharacteristic
146C22×C53C2×#10211yescharacteristic
147C22×C53#41911yescharacteristic
148C22×C53#42111yescharacteristic
149C22×C53#42111yescharacteristic
150C2×D53—22no—
151C2×D53—22no—
152C2×D53—22no—
153C2×D53—22no—
154C2×D53—21no—
155C2×D53—21no—
156C2×D53—21no—
157C2×D53—21no—
158C2×D53#47412yes—
159C2×D53#47512yes—
160#6157—84no—
161C4×C53—22no—
162C4×C53—22no—
163C4×C53—22no—
164C4×C53—82no—
165Dic53—21no—
166Dic53—21no—
167Dic53—22no—
168Dic53—41no—
169Dic53—82no—
170#2857—531no—
171#1797—531no—
172#1434—531no—
173#1294—534no—
174C2×C53#151111yescharacteristic
175C2×C53#151511yescharacteristic
176C2×C53#151711yescharacteristic
177C2×C53#151911yescharacteristic
178C2×C53#198811yescharacteristic
179C2×C53#199511yescharacteristic
180C2×C53#199611yescharacteristic
181D53—22no—
182D53—22no—
183C2×C4⋊Q8#615711yescharacteristic, complemented
184C2×C22⋊Q8—1061no—
185C22×C42—531no—
186C2×C4⋊C8—1061no—
187C2×C22⋊C8—532no—
188#391—1061no—
189#389—532no—
190#389—1061no—
191#384—531no—
192#380—532no—
193#380—1061no—
194C2×#95—532no—
195C2×#95—1061no—
196C2×#95—1061no—
197#336—1061no—
198C53#991211yescharacteristic, complemented, Sylow
199C22×Q8—21no—
200C23×C4—531no—
201C23×C4—531no—
202C23×C4—531no—
203C23×C4—1062no—
204C22×C8—1062no—
205C4⋊Q8—42no—
206C22⋊Q8—2122no—
207C22⋊Q8—2122no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—532no—
210C2×C4⋊C4—1062no—
211C2×C4⋊C4—1061no—
212C2×C4⋊C4—1061no—
213C2×C4⋊C4C2×#615712yes—
214C2×C22⋊C4—1062no—
215C2×C42—531no—
216C2×C42—531no—
217C2×C42—531no—
218C2×C42—2122no—
219C2×C42C2×#615711yescharacteristic
220C4⋊C8—2122no—
221C22⋊C8—1064no—
222#95—534no—
223#95—1062no—
224#95—1062no—
225#95—1062no—
226#95—2122no—
227#95—2122no—
228C24—531no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C22×C4—22no—
233C22×C4—531no—
234C22×C4—531no—
235C22×C4—532no—
236C22×C4—532no—
237C22×C4—532no—
238C22×C4—532no—
239C22×C4—531no—
240C22×C4—1062no—
241C22×C4—1062no—
242C22×C4—1064no—
243C22×C4—1062no—
244C22×C4—2121no—
245C22×C4—2121no—
246C22×C4—2121no—
247C22×C4—2121no—
248C22×C4—2121no—
249C22×C4—2122no—
250C22×C4—2122no—
251C22×C4—2122no—
252C22×C4—2122no—
253C22×C4—2122no—
254C22×C4—2122no—
255C22×C4C22⋊#615711yescharacteristic
256C22×C4C22⋊#615711yescharacteristic
257C22×C4C22×#615711yescharacteristic
258C2×C8—2122no—
259C2×C8—2122no—
260C2×C8—2122no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—2122no—
264C4⋊C4—2122no—
265C4⋊C4—2124no—
266C4⋊C4—2122no—
267C22⋊C4—2122no—
268C22⋊C4—2122no—
269C42—22no—
270C42—1062no—
271C42—2121no—
272C42—2121no—
273C23—532no—
274C23—531no—
275C23—531no—
276C23—532no—
277C23—1061no—
278C23—1061no—
279C23—1061no—
280C23—1061no—
281C23C2×C22⋊#615711yescharacteristic
282Q8—84no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—531no—
293C2×C4—531no—
294C2×C4—532no—
295C2×C4—1061no—
296C2×C4—1061no—
297C2×C4—1061no—
298C2×C4—1061no—
299C2×C4—1061no—
300C2×C4—1061no—
301C2×C4—1061no—
302C2×C4—4242no—
303C2×C4—4242no—
304C2×C4—4242no—
305C2×C4—4242no—
306C2×C4—2122no—
307C2×C4—2122no—
308C2×C4—2122no—
309C2×C4—2122no—
310C2×C4—2124no—
311C2×C4—2124no—
312C2×C4—2124no—
313C2×C4—2124no—
314C2×C4—2122no—
315C2×C4—2122no—
316C2×C4—2122no—
317C2×C4#1313327512yes—
318C2×C4#1313327912yes—
319C2×C4Q8×#615712yes—
320C8—4242no—
321C22—532no—
322C22—532no—
323C22—1062no—
324C22—1062no—
325C22—1062no—
326C22—1062no—
327C22—1061no—
328C22—1061no—
329C22—1061no—
330C22—1061no—
331C22(C2×#6157)⋊Q811yescharacteristic
332C22C2×#1313327511yescharacteristic
333C22#1681160411yescharacteristic
334C22C2×#1313327911yescharacteristic
335C22#1681161611yescharacteristic
336C22#1681162111yescharacteristic
337C22#1681162211yescharacteristic
338C4—22no—
339C4—22no—
340C4—22no—
341C4—82no—
342C4—1061no—
343C4—1061no—
344C4—1062no—
345C4—4244no—
346C4—4242no—
347C4—2121no—
348C2—1062no—
349C2—1062no—
350C2#2029012511yescharacteristic
351C2#2029013811yescharacteristic
352C2#2029014611yescharacteristic
353C2#2029015111yescharacteristic
354C2#2029089411yescharacteristic
355C2#2029090311yescharacteristic
356C2#2029090411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4⋊Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C2×D4
cosocleC22×C4×D53C23
hypercenterC2×C4⋊Q8#6157
hypocenterC53#9912
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
15. #16810519C4semidirect
16. #16810493C4semidirect
19. C2×C4⋊Q8×C53C4semidirect
27. (C2×#6157)⋊C4Q8semidirect
183. C2×C4⋊Q8#6157semidirect
198. C53#9912semidirect

Maximal subgroups

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

subgroupindex
#202905722
#202898912
#202895882
#202830622
#991253