Finite Group Databasebeta

← #28306334

Subgroups — #28306334

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—44no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53C2×C412yes—
35C2×#11719919—22no—
36C2×#11719921—22no—
37C2×C4×Dic53C2×C411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C42×D53—42no—
41#13133124—22no—
42#13133124—22no—
43C22×Q8×C53—21no—
44C4⋊Q8×C53—42no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53C2×C412yes—
47C2×C42×C53C2×C411yescharacteristic
48C23×D53C2×D411yescharacteristic
49C22×#6157—24no—
50C22×#6157—42no—
51C22×#6157—42no—
52C22×#6157—42no—
53C22×#6157—42no—
54C22×#6157—42no—
55C22×#6157—42no—
56C2×Q8×C53—42no—
57C2×Q8×C53—42no—
58C2×Q8×C53—42no—
59C22×C4×C53—22no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22×C411yescharacteristic
63C22×Dic53—22no—
64C22×Dic53C22⋊C411yescharacteristic
65C22×Dic53C22⋊C411yescharacteristic
66C22×Dic53C2×D411yescharacteristic
67C2×C4×D53—22no—
68C2×C4×D53—22no—
69C2×C4×D53—41no—
70C2×C4×D53—41no—
71C2×C4×D53—41no—
72C2×C4×D53—41no—
73C2×C4×D53—41no—
74C2×C4×D53SD1612yes—
75C2×C4×D53Q1612yes—
76C2×C4×D53C2×Q812yes—
77C2×C4×D53#4112yes—
78C2×#88324—42no—
79C2×#88324—42no—
80C2×#88324—42no—
81C4⋊C4×C53—42no—
82C4⋊C4×C53—42no—
83C42×C53—22no—
84#11719921—44no—
85C4⋊Dic53—42no—
86C4⋊Dic53—42no—
87C4⋊Dic53—42no—
88#11719919—44no—
89C4×Dic53—22no—
90C4×Dic53—41no—
91C4×Dic53—41no—
92C23×C53C2×C22⋊C411yescharacteristic
93C22×D53—21no—
94C22×D53—21no—
95C22×D53—21no—
96C22×D53—21no—
97C22×D53C22⋊Q812yes—
98C22×D53C2×SD1611yescharacteristic
99C22×D53C2×Q1611yescharacteristic
100C22×D53AGL(1,ℤ/8)12yes—
101C2×#6157—44no—
102C2×#6157—44no—
103C2×#6157—44no—
104C2×#6157—82no—
105C2×#6157—82no—
106C2×#6157—82no—
107C2×#6157—82no—
108Q8×C53—84no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53#9912yes—
119C2×C4×C53#10312yes—
120C2×C4×C53C4×Q812yes—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—42no—
129C2×Dic53—42no—
130C2×Dic53—42no—
131C2×Dic53#10312yes—
132C2×Dic53#12411yescharacteristic
133C2×Dic53C4⋊Q811yescharacteristic
134C4×D53—42no—
135C4×D53—42no—
136C4×D53—42no—
137C4×D53—42no—
138C4×D53—44no—
139#88324—82no—
140#9911—531nomaximal, Sylow
141C22×C53#39111yescharacteristic
142C22×C53C2×#9911yescharacteristic
143C22×C53#41011yescharacteristic
144C22×C53C2×#10311yescharacteristic
145C22×C53#41811yescharacteristic
146C22×C53#42111yescharacteristic
147C22×C53#42111yescharacteristic
148C2×D53—22no—
149C2×D53—22no—
150C2×D53—22no—
151C2×D53—22no—
152C2×D53—21no—
153C2×D53—21no—
154C2×D53—21no—
155C2×D53—21no—
156C2×D53#47612yes—
157C2×D53#47712yes—
158#6157—84no—
159C4×C53—22no—
160C4×C53—22no—
161C4×C53—22no—
162C4×C53—82no—
163Dic53—21no—
164Dic53—21no—
165Dic53—22no—
166Dic53—41no—
167Dic53—82no—
168#2857—531no—
169#1797—531no—
170#1434—531no—
171#1294—534no—
172C2×C53#151011yescharacteristic
173C2×C53#151411yescharacteristic
174C2×C53#151811yescharacteristic
175C2×C53#152011yescharacteristic
176C2×C53#198711yescharacteristic
177C2×C53#199511yescharacteristic
178C2×C53#199611yescharacteristic
179D53—22no—
180D53—22no—
181C2×C4⋊Q8#615711yescharacteristic, complemented
182C2×C22⋊Q8—1061no—
183C22×C42—531no—
184C2×C4⋊C8—1061no—
185C2×C22⋊C8—532no—
186#391—1061no—
187#389—532no—
188#389—1061no—
189#384—531no—
190#380—532no—
191#380—1061no—
192C2×#95—532no—
193C2×#95—1061no—
194C2×#95—1061no—
195#336—1061no—
196C53#991111yescharacteristic, complemented, Sylow
197C22×Q8—21no—
198C23×C4—531no—
199C23×C4—531no—
200C23×C4—531no—
201C23×C4—1062no—
202C22×C8—1062no—
203C4⋊Q8—42no—
204C22⋊Q8—2124no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—532no—
207C2×C4⋊C4—1062no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4C2×#615712yes—
211C2×C22⋊C4—1062no—
212C2×C42—531no—
213C2×C42—531no—
214C2×C42—531no—
215C2×C42—2122no—
216C2×C42C2×#615711yescharacteristic
217C4⋊C8—2122no—
218C22⋊C8—1064no—
219#95—534no—
220#95—1062no—
221#95—1062no—
222#95—1062no—
223#95—2122no—
224#95—2122no—
225C24—531no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C22×C4—22no—
230C22×C4—531no—
231C22×C4—531no—
232C22×C4—532no—
233C22×C4—532no—
234C22×C4—532no—
235C22×C4—532no—
236C22×C4—531no—
237C22×C4—1062no—
238C22×C4—1062no—
239C22×C4—1064no—
240C22×C4—1062no—
241C22×C4—2121no—
242C22×C4—2121no—
243C22×C4—2121no—
244C22×C4—2121no—
245C22×C4—2121no—
246C22×C4—2122no—
247C22×C4—2122no—
248C22×C4—2122no—
249C22×C4—2122no—
250C22×C4—2122no—
251C22×C4—2122no—
252C22×C4C22⋊#615711yescharacteristic
253C22×C4C22⋊#615711yescharacteristic
254C22×C4C22×#615711yescharacteristic
255C2×C8—2122no—
256C2×C8—2122no—
257C2×C8—2122no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—2122no—
261C4⋊C4—2122no—
262C4⋊C4—2122no—
263C4⋊C4—2124no—
264C22⋊C4—2124no—
265C42—22no—
266C42—1062no—
267C42—2121no—
268C42—2121no—
269C23—532no—
270C23—531no—
271C23—531no—
272C23—532no—
273C23—1061no—
274C23—1061no—
275C23—1061no—
276C23—1061no—
277C23C2×C22⋊#615711yescharacteristic
278Q8—84no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—531no—
289C2×C4—531no—
290C2×C4—532no—
291C2×C4—1061no—
292C2×C4—1061no—
293C2×C4—1061no—
294C2×C4—1061no—
295C2×C4—1061no—
296C2×C4—1061no—
297C2×C4—1061no—
298C2×C4—4242no—
299C2×C4—4242no—
300C2×C4—4242no—
301C2×C4—4242no—
302C2×C4—2122no—
303C2×C4—2122no—
304C2×C4—2122no—
305C2×C4—2122no—
306C2×C4—2124no—
307C2×C4—2124no—
308C2×C4—2124no—
309C2×C4—2124no—
310C2×C4—2122no—
311C2×C4—2122no—
312C2×C4—2122no—
313C2×C4#1313327512yes—
314C2×C4#1313328012yes—
315C2×C4Q8×#615712yes—
316C8—4242no—
317C22—532no—
318C22—532no—
319C22—1062no—
320C22—1062no—
321C22—1062no—
322C22—1062no—
323C22—1061no—
324C22—1061no—
325C22—1061no—
326C22—1061no—
327C22(C2×#6157)⋊Q811yescharacteristic
328C22C2×#1313327511yescharacteristic
329C22#1681160411yescharacteristic
330C22C2×#1313328011yescharacteristic
331C22#1681161511yescharacteristic
332C22#1681162111yescharacteristic
333C22#1681162211yescharacteristic
334C4—22no—
335C4—22no—
336C4—22no—
337C4—82no—
338C4—1061no—
339C4—1061no—
340C4—1062no—
341C4—4244no—
342C4—4242no—
343C4—2121no—
344C2—1062no—
345C2—1062no—
346C2#2029012011yescharacteristic
347C2#2029013611yescharacteristic
348C2#2029014711yescharacteristic
349C2#2029015511yescharacteristic
350C2#2029089311yescharacteristic
351C2#2029090311yescharacteristic
352C2#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#9911
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
181. C2×C4⋊Q8#6157semidirect
196. C53#9911semidirect

Maximal subgroups

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

subgroupindex
#202905722
#202898912
#202895882
#202830622
#991153