Finite Group Databasebeta

← #28299466

Subgroups — #28299466

Order 13568.

604 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 126 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
2C4×#13133274C211yesmaximal, characteristic
3#20289534C212yesmaximal, complemented
4#20289533C212yesmaximal, complemented
5#389×D53C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×#13133274—21no—
8C2×#13133274—21no—
9C2×C4×#88324—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×C4⋊C4×D53—21no—
13C2×C4⋊C4×D53C411yescharacteristic
14C2×C4⋊C4×D53C411yescharacteristic
15C2×C42×D53C2211yescharacteristic
16#16810522C2211yescharacteristic
17#16810521C411yescharacteristic
18#16810520—21no—
19#16810519C2211yescharacteristic
20#16810518—21no—
21#95×D53—21no—
22C53×#389C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133284—42no—
26#13133274—24no—
27#13133274—42no—
28C4×#88324—42no—
29C22×C4×D53—21no—
30C22×C4×D53—21no—
31C22×C4×D53C2×C411yescharacteristic
32C22×C4×D53D411yescharacteristic
33C22×C4×D53D411yescharacteristic
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×#11719919—21no—
41C4⋊C4×D53—22no—
42C4⋊C4×D53—44no—
43C4⋊C4×D53—42no—
44C4⋊C4×D53C814yescomplemented
45C2×C4×Dic53D411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53C2311yescharacteristic
48C42×D53—42no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53C2×C411yescharacteristic
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C42×C53C2×C411yescharacteristic
56#95×C53—21no—
57C23×D53C22⋊C411yescharacteristic
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22×C411yescharacteristic
63C22×Dic53—21no—
64C22×Dic53—21no—
65C22×Dic53C22⋊C411yescharacteristic
66C22×Dic53C2×D411yescharacteristic
67C22×Dic53C2×D411yescharacteristic
68C2×C4×D53—22no—
69C2×C4×D53—22no—
70C2×C4×D53—24no—
71C2×C4×D53—22no—
72C2×C4×D53—22no—
73C2×C4×D53—41no—
74C2×C4×D53—41no—
75C2×C4×D53—41no—
76C2×C4×D53—42no—
77C2×C4×D53—42no—
78C2×C4×D53—42no—
79C2×C4×D53—42no—
80C2×C4×D53—42no—
81C2×C4×D53C2×C812yes—
82C2×C4×D53M1612yes—
83C2×C4×D53SD1612yes—
84C2×C4×D53Q1612yes—
85C2×#88324—24no—
86C2×#88324—42no—
87C2×#88324—42no—
88C2×#88324—42no—
89C2×#88324—42no—
90C2×#88324—42no—
91C4⋊C4×C53—21no—
92C4⋊C4×C53—42no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53C2×C812yes—
95C42×C53—41no—
96C4⋊Dic53—21no—
97C4⋊Dic53—42no—
98C4⋊Dic53—41no—
99C4⋊Dic53C2×C812yes—
100#11719919—24no—
101#11719919—42no—
102#11719919—42no—
103#11719919—42no—
104C4×Dic53—22no—
105C4×Dic53—22no—
106C4×Dic53—41no—
107C23×C53C2×C22⋊C411yescharacteristic
108C22×D53—21no—
109C22×D53—21no—
110C22×D53—21no—
111C22×D53—21no—
112C22×D53C22⋊C812yes—
113C22×D53#10312yes—
114C22×D53#10412yes—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—22no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53#10312yes—
129C2×C4×C53C22×C811yescharacteristic
130C2×C4×C53C2×M1611yescharacteristic
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—22no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53C22⋊C812yes—
142C2×Dic53#9912yes—
143C2×Dic53#10112yes—
144C2×Dic53C2×SD1611yescharacteristic
145C2×Dic53C2×Q1611yescharacteristic
146C2×Dic53AGL(1,ℤ/8)12yes—
147C4×D53—24no—
148C4×D53—24no—
149C4×D53—42no—
150C4×D53—42no—
151C4×D53—44no—
152C4×D53—48no—
153C4×D53—84no—
154#88324—44no—
155#88324—82no—
156#7576—531nomaximal, Sylow
157C22×C53C2×C22⋊C811yescharacteristic
158C22×C53C2×#9911yescharacteristic
159C22×C53C2×#10111yescharacteristic
160C22×C53C2×#10311yescharacteristic
161C22×C53#41811yescharacteristic
162C22×C53C2×#10411yescharacteristic
163C22×C53#42111yescharacteristic
164C2×D53—22no—
165C2×D53—22no—
166C2×D53—22no—
167C2×D53—22no—
168C2×D53—22no—
169C2×D53—22no—
170C2×D53Q8⋊C814yes—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—22no—
174C4×C53—42no—
175C4×C53—41no—
176C4×C53—81no—
177Dic53—22no—
178Dic53—21no—
179Dic53—21no—
180Dic53—42no—
181Dic53—81no—
182Dic53#32414yes—
183C2×#389—531no—
184#1797—531no—
185C4×C22⋊C8—531no—
186#1271—532no—
187#1271—532no—
188C2×C53C2×#32411yescharacteristic
189C2×C53C2×Q8⋊C811yescharacteristic
190C2×C53#147811yescharacteristic
191C2×C53#150711yescharacteristic
192C2×C53#152011yescharacteristic
193C2×C53#152811yescharacteristic
194C2×C53#154911yescharacteristic
195D53—24no—
196C22×C4⋊C4—531no—
197C22×C4⋊C4—531no—
198C22×C4⋊C4—1061no—
199C22×C42—531no—
200C2×C4⋊C8—1061no—
201C2×C22⋊C8—1061no—
202C2×C22⋊C8—1061no—
203C2×C4×C8—1061no—
204#389—531no—
205#389—531no—
206#389—531no—
207#389—1061no—
208#389—1061no—
209#389#615711yescharacteristic
210C2×#95—1061no—
211#336—532no—
212#336—532no—
213C53#757611yescharacteristic, complemented, Sylow
214C23×C4—531no—
215C23×C4—531no—
216C23×C4—531no—
217C23×C4—1061no—
218C23×C4—1061no—
219C22×C8—1062no—
220C22×C8—1062no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—531no—
223C2×C4⋊C4—534no—
224C2×C4⋊C4—531no—
225C2×C4⋊C4—1062no—
226C2×C4⋊C4—1061no—
227C2×C4⋊C4—1061no—
228C2×C4⋊C4—1061no—
229C2×C4⋊C4—1061no—
230C2×C4⋊C4—1061no—
231C2×C4⋊C4—2124no—
232C2×C4⋊C4—2122no—
233C2×C4⋊C4C2×#615711yescharacteristic
234C2×C4⋊C4C2×#615711yescharacteristic
235C2×C42—531no—
236C2×C42—531no—
237C2×C42—531no—
238C2×C42—2122no—
239C2×C42C2×#615711yescharacteristic
240C4⋊C8—2122no—
241C22⋊C8—1064no—
242C22⋊C8—2122no—
243C4×C8—2122no—
244#95—21no—
245#95—1061no—
246#95—1061no—
247#95—1061no—
248C24—531no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—531no—
252C22×C4—531no—
253C22×C4—531no—
254C22×C4—532no—
255C22×C4—532no—
256C22×C4—532no—
257C22×C4—532no—
258C22×C4—1062no—
259C22×C4—1062no—
260C22×C4—1062no—
261C22×C4—1062no—
262C22×C4—1064no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—2121no—
266C22×C4—2121no—
267C22×C4—2121no—
268C22×C4—2122no—
269C22×C4—2122no—
270C22×C4—2122no—
271C22×C4—2122no—
272C22×C4—2122no—
273C22×C4C22⋊#615711yescharacteristic
274C22×C4C22⋊#615711yescharacteristic
275C22×C4C22×#615711yescharacteristic
276C2×C8—1064no—
277C2×C8—2122no—
278C2×C8—2122no—
279C2×C8—2122no—
280C2×C8—2122no—
281C2×C8—2122no—
282C4⋊C4—21no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—532no—
286C4⋊C4—1064no—
287C4⋊C4—1061no—
288C4⋊C4—2121no—
289C4⋊C4—2122no—
290C4⋊C4—2122no—
291C4⋊C4—2122no—
292C4⋊C4—2122no—
293C4⋊C4D53⋊C812yes—
294C42—41no—
295C42—1062no—
296C42—1062no—
297C42—2121no—
298C23—532no—
299C23—532no—
300C23—532no—
301C23—1061no—
302C23—1061no—
303C23—1061no—
304C23—1061no—
305C23C2×C22⋊#615711yescharacteristic
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—532no—
320C2×C4—532no—
321C2×C4—532no—
322C2×C4—531no—
323C2×C4—531no—
324C2×C4—532no—
325C2×C4—1064no—
326C2×C4—1064no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—1062no—
332C2×C4—4244no—
333C2×C4—2122no—
334C2×C4—2122no—
335C2×C4—2124no—
336C2×C4—2128no—
337C2×C4—2121no—
338C2×C4—2121no—
339C2×C4—2121no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4#1313328012yes—
343C2×C4C2×D53⋊C811yescharacteristic
344C2×C4D53⋊M1611yescharacteristic
345C8—4242no—
346C8—2124no—
347C22—534no—
348C22—1062no—
349C22—1062no—
350C22—1062no—
351C22—1062no—
352C22—1062no—
353C22—1062no—
354C22D53⋊(C22⋊C8)11yescharacteristic
355C22D53⋊#9911yescharacteristic
356C22D53⋊#10111yescharacteristic
357C22C2×#1313328011yescharacteristic
358C22#1681161511yescharacteristic
359C22D53⋊#10411yescharacteristic
360C22#1681162011yescharacteristic
361C4—22no—
362C4—21no—
363C4—21no—
364C4—41no—
365C4—42no—
366C4—81no—
367C4—534no—
368C4—1062no—
369C4—1061no—
370C4—1061no—
371C4—4241no—
372C4—2122no—
373C2—1064no—
374C2D53⋊#32411yescharacteristic
375C2D53⋊(Q8⋊C8)11yescharacteristic
376C2#2029001511yescharacteristic
377C2#2029011011yescharacteristic
378C2#2029015311yescharacteristic
379C2#2029017811yescharacteristic
380C2#2029023111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#389C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#389#6157
hypocenterC53#7576
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
3. #20289534C2semidirect
4. #20289533C2semidirect
44. C4⋊C4×D53C8semidirect
213. C53#7576semidirect

Maximal subgroups

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

subgroupindex
#202905722
C4×#131332742
#202895342
#202895332
#389×D532
#757653