Finite Group Databasebeta

← #27281501

Subgroups — #27281501

Order 13568.

486 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 109 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#20283798C211yesmaximal, characteristic
2#20279703C211yesmaximal, characteristic
3#20274708C211yesmaximal, characteristic
4#20273388C211yesmaximal, characteristic
5#20273386C211yesmaximal, characteristic
6#20272618C211yesmaximal, characteristic
7C53×#1303C211yesmaximal, characteristic
8C2×#13133217—21no—
9C2×#13133217C2211yescharacteristic
10C2×C4×Dic106C2211yescharacteristic
11#16810753—22no—
12#16810750—22no—
13#16810677—22no—
14#16810676C2211yescharacteristic
15#16810647C2211yescharacteristic
16C2×#13133135—21no—
17#16810555C414yes—
18#16810529—21no—
19#16810522—22no—
20#16810391—21no—
21#16810370—22no—
22#16810368—22no—
23C53×#431C2211yescharacteristic
24C53×#423C2211yescharacteristic
25C4×C4⋊C4×C53C2211yescharacteristic
26C22×Dic106—21no—
27C22×Dic106D411yescharacteristic
28C2×#11719917—21no—
29#13133217—21no—
30#13133217—21no—
31#13133217—21no—
32#13133217—21no—
33#13133217—42no—
34#13133217—42no—
35C4×Dic106—21no—
36C4×Dic106—21no—
37C4×Dic106—21no—
38C4×Dic106—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—24no—
41C2×C4⋊Dic53C2×C412yes—
42C2×C4⋊Dic53D411yescharacteristic
43C2×#11719919C2×C412yes—
44C2×C4×Dic53D412yes—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322D412yes—
48#13133146—24no—
49#13133135—42no—
50C22⋊#88322—21no—
51#13133127—24no—
52C4×#88322—21no—
53#13133124—24no—
54C2×M16×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C42×C53—21no—
57C2×C42×C53C2311yescharacteristic
58C4⋊C8×C53—21no—
59C4⋊C8×C53—21no—
60C4⋊C8×C53D412yes—
61C22⋊C8×C53—21no—
62C4×C8×C53—21no—
63#95×C53C2×C412yes—
64C22×C4×C53—22no—
65C22×C4×C53C22×C411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×Dic53—22no—
69C22×Dic53C22⋊C412yes—
70C22×Dic53C22⋊C412yes—
71C2×Dic106—21no—
72C2×Dic106—21no—
73C2×Dic106—21no—
74C2×Dic106—21no—
75C2×Dic106—21no—
76C2×Dic106—21no—
77C2×Dic106—41no—
78C2×Dic106—41no—
79C2×Dic106—41no—
80C2×Dic106—41no—
81C2×Dic106—41no—
82C2×Dic106—41no—
83M16×C53—41no—
84M16×C53—41no—
85C2×C8×C53—22no—
86C2×C8×C53—41no—
87C4⋊C4×C53—24no—
88C42×C53—41no—
89C42×C53—41no—
90C42×C53C2×D411yescharacteristic
91C42×C53C2×D411yescharacteristic
92C42×C53#4111yescharacteristic
93C42×C53#4111yescharacteristic
94C4⋊Dic53—21no—
95C4⋊Dic53—21no—
96C4⋊Dic53—22no—
97C4⋊Dic53—22no—
98C4⋊Dic53—48no—
99C4⋊Dic53—41no—
100C4⋊Dic53—41no—
101#11719919—22no—
102#11719919—22no—
103C4×Dic53—22no—
104C4×Dic53—22no—
105#11719917—41no—
106#11719917—41no—
107C2×#88322—22no—
108C2×#88322—41no—
109C23×C53C2×C22⋊C411yescharacteristic
110C2×C4×C53—22no—
111C2×C4×C53—24no—
112C2×C4×C53—41no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53C4×D411yescharacteristic
116C2×C4×C53C4×D411yescharacteristic
117C2×C4×C53#12011yescharacteristic
118C2×C4×C53#12111yescharacteristic
119C2×C4×C53#12111yescharacteristic
120C2×C4×C53#12311yescharacteristic
121C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
122C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
123C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C53#13711yescharacteristic
125C2×Dic53—22no—
126C2×Dic53—22no—
127C2×Dic53—22no—
128C2×Dic53—22no—
129C2×Dic53—42no—
130C2×Dic53—42no—
131C2×Dic53—42no—
132C2×Dic53—42no—
133Dic106—41no—
134Dic106—41no—
135Dic106—41no—
136Dic106—41no—
137Dic106—81no—
138Dic106—81no—
139Dic106—81no—
140Dic106—81no—
141C8×C53—42no—
142C8×C53—81no—
143#88322—42no—
144#88322—81no—
145#9402—531nomaximal, Sylow
146C22×C53—21no—
147C22×C53—21no—
148C22×C53#38611yescharacteristic
149C22×C53#41711yescharacteristic
150C22×C53#41811yescharacteristic
151C4×C53—21no—
152C4×C53—21no—
153C4×C53—21no—
154C4×C53—44no—
155C4×C53#45612yes—
156C4×C53#46811yescharacteristic
157C4×C53#46811yescharacteristic
158C4×C53#46911yescharacteristic
159C4×C53#46911yescharacteristic
160Dic53—42no—
161Dic53—42no—
162Dic53—82no—
163#2983—531no—
164#2546—531no—
165#1801—531no—
166#1546—531no—
167#1546—531no—
168#1303—531no—
169#1303D5311yescharacteristic
170C2×C53—21no—
171C2×C53—21no—
172C2×C53#188611yescharacteristic
173C2×C53#189511yescharacteristic
174C2×C53#191511yescharacteristic
175C2×C4⋊Q8—531no—
176C2×C4⋊Q8—1061no—
177C2×C4×Q8—531no—
178#446—1062no—
179#446—1062no—
180#432—1062no—
181#431—531no—
182#431C2×D5311yescharacteristic
183#423—531no—
184#423C2×D5311yescharacteristic
185C2×#101—1061no—
186#398—534no—
187#391—1061no—
188#389—1062no—
189C4×C4⋊C4C2×D5311yescharacteristic
190#343—1061no—
191#332—1062no—
192#332—1062no—
193C53#940211yescharacteristic, complemented, Sylow
194C22×Q8—531no—
195C22×Q8—1061no—
196C2×M16—21no—
197C2×M16—1061no—
198C4⋊Q8—1061no—
199C4⋊Q8—1061no—
200C4⋊Q8—1061no—
201C4⋊Q8—1061no—
202C4⋊Q8—2122no—
203C4⋊Q8—2122no—
204C4×Q8—1061no—
205C4×Q8—1061no—
206C4×Q8—1061no—
207C4×Q8—1061no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—532no—
210C2×C4⋊C4—531no—
211C2×C4⋊C4—532no—
212C2×C4⋊C4—1064no—
213C2×C4⋊C4—1061no—
214C2×C42—21no—
215C2×C42—532no—
216C2×C42C22×D5311yescharacteristic
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—532no—
220C4⋊C8—1064no—
221C4⋊C8—1061no—
222C4⋊C8—1061no—
223C4⋊C8D21212yes—
224#101—2122no—
225C22⋊C8—21no—
226C22⋊C8—1061no—
227#97—1064no—
228C4×C8—21no—
229C4×C8—1061no—
230#95—1064no—
231#95C4×D5312yes—
232C2×Q8—1061no—
233C2×Q8—1061no—
234C2×Q8—1061no—
235C2×Q8—1061no—
236C2×Q8—1061no—
237C2×Q8—1061no—
238C2×Q8—2121no—
239C2×Q8—2121no—
240C2×Q8—2121no—
241C2×Q8—2121no—
242C2×Q8—2121no—
243C2×Q8—2121no—
244C22×C4—22no—
245C22×C4—532no—
246C22×C4—532no—
247C22×C4—1062no—
248C22×C4C2×C4×D5311yescharacteristic
249C22×C4D4×D5311yescharacteristic
250C22×C4D4×D5311yescharacteristic
251M16—41no—
252M16—41no—
253M16—2121no—
254M16—2121no—
255C2×C8—22no—
256C2×C8—41no—
257C2×C8—1062no—
258C2×C8—2121no—
259C4⋊C4—24no—
260C4⋊C4—1062no—
261C4⋊C4—1062no—
262C4⋊C4—1061no—
263C4⋊C4—1061no—
264C4⋊C4—1062no—
265C4⋊C4—1062no—
266C4⋊C4—2128no—
267C4⋊C4—2121no—
268C4⋊C4—2121no—
269C42—41no—
270C42—41no—
271C42—1062no—
272C42—1062no—
273C42C2×D21211yescharacteristic
274C42#1171994511yescharacteristic
275C42D4×D5311yescharacteristic
276C42#1171994911yescharacteristic
277C23C22⋊C4×D5311yescharacteristic
278Q8—4241no—
279Q8—4241no—
280Q8—4241no—
281Q8—4241no—
282Q8—2121no—
283Q8—2121no—
284Q8—2121no—
285Q8—2121no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—1062no—
292C2×C4—1062no—
293C2×C4—1062no—
294C2×C4—1062no—
295C2×C4—2122no—
296C2×C4—2122no—
297C2×C4—2122no—
298C2×C4—2122no—
299C2×C4#1313317911yescharacteristic
300C2×C4C4×D21211yescharacteristic
301C2×C4#1313318711yescharacteristic
302C2×C4#1313319011yescharacteristic
303C2×C4#1313319211yescharacteristic
304C2×C4#1313320211yescharacteristic
305C2×C4#1313324111yescharacteristic
306C2×C4#1313324411yescharacteristic
307C2×C4#1313324411yescharacteristic
308C2×C4#1313324911yescharacteristic
309C8—42no—
310C8—81no—
311C8—4241no—
312C8—2122no—
313C22—21no—
314C22—21no—
315C22#1681050111yescharacteristic
316C22#1681062311yescharacteristic
317C22#1681062611yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—44no—
322C4—4242no—
323C4—2122no—
324C4—2122no—
325C4#1681080712yes—
326C4#1681087011yescharacteristic
327C4#1681087211yescharacteristic
328C4#1681087511yescharacteristic
329C4#1681087711yescharacteristic
330C2—21no—
331C2—21no—
332C2#2027510711yescharacteristic
333C2#2027515211yescharacteristic
334C2#2027526611yescharacteristic
subgroupG/H
centerC22#16810501
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1303C2
socleC22×C53#386
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1303D53
hypocenterC53#9402
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
193. C53#9402semidirect

Maximal subgroups

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

subgroupindex
#202837982
#202797032
#202747082
#202733882
#202733862
#202726182
C53×#13032
#940253