Finite Group Databasebeta

← #27281499

Subgroups — #27281499

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#20279704C211yesmaximal, characteristic
3#20274708C211yesmaximal, characteristic
4#20273387C211yesmaximal, characteristic
5#20272617C211yesmaximal, characteristic
6#20272616C211yesmaximal, characteristic
7C53×#1546C211yesmaximal, characteristic
8C2×#13133215—21no—
9C2×C4×Dic106C2211yescharacteristic
10#16810753—22no—
11#16810750—22no—
12#16810677—22no—
13#16810676C2211yescharacteristic
14#16810647C2211yescharacteristic
15C2×#13133135—21no—
16#16810554C414yes—
17#16810530—21no—
18#16810519—22no—
19C4×C4⋊Dic53C2211yescharacteristic
20#16810391—21no—
21#16810369—22no—
22C2×C4⋊Q8×C53C2211yescharacteristic
23C53×#431C2211yescharacteristic
24C53×#423C2211yescharacteristic
25C53×#332—22no—
26C22×Dic106D411yescharacteristic
27C2×#11719917—21no—
28#13133215—42no—
29#13133215—42no—
30C4×Dic106—21no—
31C4×Dic106—21no—
32C4×Dic106—21no—
33C4×Dic106—21no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53D411yescharacteristic
36C2×#11719919—24no—
37C2×#11719919C2×C412yes—
38C2×C4×Dic53—21no—
39C2×C4×Dic53D412yes—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322D412yes—
43#13133146—24no—
44#13133135—42no—
45C22⋊#88322—21no—
46#13133127—24no—
47C4×#88322—21no—
48#13133124—24no—
49#13133124C2×C412yes—
50C22×Q8×C53—21no—
51C2×M16×C53—21no—
52C4⋊Q8×C53—21no—
53C4⋊Q8×C53—21no—
54C4⋊Q8×C53—21no—
55C4⋊Q8×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53C2×C412yes—
58C2×C42×C53C2311yescharacteristic
59C4⋊C8×C53—21no—
60C4⋊C8×C53—21no—
61C4⋊C8×C53D412yes—
62C22⋊C8×C53—21no—
63C4×C8×C53—21no—
64C2×Q8×C53—41no—
65C2×Q8×C53—41no—
66C2×Q8×C53—41no—
67C2×Q8×C53—41no—
68C2×Q8×C53—41no—
69C2×Q8×C53—41no—
70C22×C4×C53—22no—
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×Dic53—22no—
75C22×Dic53C22⋊C412yes—
76C22×Dic53C22⋊C412yes—
77C2×Dic106—21no—
78C2×Dic106—21no—
79C2×Dic106—21no—
80C2×Dic106—21no—
81C2×Dic106—21no—
82C2×Dic106—21no—
83M16×C53—41no—
84M16×C53—41no—
85C2×C8×C53—22no—
86C2×C8×C53—41no—
87C4⋊C4×C53—22no—
88C4⋊C4×C53—22no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C42×C53C2×D411yescharacteristic
92C42×C53C2×D411yescharacteristic
93C42×C53#4111yescharacteristic
94C42×C53#4111yescharacteristic
95C4⋊Dic53—24no—
96C4⋊Dic53—21no—
97C4⋊Dic53—21no—
98#11719919—22no—
99#11719919—22no—
100#11719919—48no—
101C4×Dic53—22no—
102C4×Dic53—22no—
103C4×Dic53—41no—
104C4×Dic53—41no—
105#11719917—41no—
106#11719917—41no—
107C2×#88322—22no—
108C2×#88322—41no—
109C23×C53C2×C22⋊C411yescharacteristic
110Q8×C53—81no—
111Q8×C53—81no—
112Q8×C53—81no—
113Q8×C53—81no—
114C2×C4×C53—22no—
115C2×C4×C53—41no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53C4×D411yescharacteristic
120C2×C4×C53C4×D411yescharacteristic
121C2×C4×C53#12011yescharacteristic
122C2×C4×C53#12111yescharacteristic
123C2×C4×C53#12111yescharacteristic
124C2×C4×C53#12311yescharacteristic
125C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
128C2×C4×C53#13711yescharacteristic
129C2×Dic53—24no—
130C2×Dic53—22no—
131C2×Dic53—22no—
132C2×Dic53—22no—
133C2×Dic53—22no—
134C2×Dic53—42no—
135C2×Dic53—42no—
136C2×Dic53—42no—
137Dic106—41no—
138Dic106—41no—
139Dic106—41no—
140Dic106—41no—
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—82no—
155C4×C53#45612yes—
156C4×C53#46811yescharacteristic
157C4×C53#46811yescharacteristic
158C4×C53#46911yescharacteristic
159C4×C53#46911yescharacteristic
160Dic53—44no—
161Dic53—42no—
162Dic53—42no—
163#2983—531no—
164#2546—531no—
165#1801—531no—
166#1546—531no—
167#1546D5311yescharacteristic
168#1303—531no—
169#1303—531no—
170C2×C53—21no—
171C2×C53—21no—
172C2×C53#188611yescharacteristic
173C2×C53#189511yescharacteristic
174C2×C53#191511yescharacteristic
175C2×C4⋊Q8—1061no—
176C2×C4⋊Q8C2×D5311yescharacteristic
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⋊C4—531no—
190#343—1061no—
191#332—22no—
192#332—1062no—
193C53#940211yescharacteristic, complemented, Sylow
194C22×Q8—21no—
195C22×Q8—531no—
196C2×M16—21no—
197C2×M16—1061no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
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—1061no—
212C2×C4⋊C4—1064no—
213C2×C4⋊C4C4×D5312yes—
214C2×C42—532no—
215C2×C42—1061no—
216C2×C42C22×D5311yescharacteristic
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—532no—
220C4⋊C8—1064no—
221C4⋊C8—1061no—
222C4⋊C8—1061no—
223C4⋊C8C22⋊D5312yes—
224#101—2122no—
225C22⋊C8—21no—
226C22⋊C8—1061no—
227#97—1064no—
228C4×C8—21no—
229C4×C8—1061no—
230#95—532no—
231#95—1064no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—1061no—
239C2×Q8—1061no—
240C2×Q8—1061no—
241C2×Q8—1061no—
242C2×Q8—1061no—
243C2×Q8—1061no—
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—22no—
260C4⋊C4—22no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—1064no—
264C4⋊C4—1062no—
265C4⋊C4—1062no—
266C4⋊C4—1061no—
267C4⋊C4—1061no—
268C4⋊C4—2128no—
269C42—1062no—
270C42—1062no—
271C42—2121no—
272C42—2121no—
273C42#1171994511yescharacteristic
274C42D4×D5311yescharacteristic
275C42#1171994711yescharacteristic
276C42C2×C22⋊D5311yescharacteristic
277C23C22⋊C4×D5311yescharacteristic
278Q8—81no—
279Q8—81no—
280Q8—81no—
281Q8—81no—
282Q8—2121no—
283Q8—2121no—
284Q8—2121no—
285Q8—2121no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—1064no—
292C2×C4—1062no—
293C2×C4—1062no—
294C2×C4—1062no—
295C2×C4—1062no—
296C2×C4—2122no—
297C2×C4—2122no—
298C2×C4—2122no—
299C2×C4#1313317711yescharacteristic
300C2×C4C4×C22⋊D5311yescharacteristic
301C2×C4#1313318511yescharacteristic
302C2×C4#1313318811yescharacteristic
303C2×C4#1313319011yescharacteristic
304C2×C4#1313319911yescharacteristic
305C2×C4#1313324111yescharacteristic
306C2×C4#1313324111yescharacteristic
307C2×C4#1313324411yescharacteristic
308C2×C4#1313324611yescharacteristic
309C8—42no—
310C8—81no—
311C8—4241no—
312C8—2122no—
313C22—21no—
314C22—21no—
315C22#1681049911yescharacteristic
316C22#1681062111yescharacteristic
317C22#1681062611yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—82no—
322C4—2124no—
323C4—2122no—
324C4—2122no—
325C4#1681080412yes—
326C4#1681086611yescharacteristic
327C4#1681086811yescharacteristic
328C4#1681087311yescharacteristic
329C4#1681087511yescharacteristic
330C2—21no—
331C2—21no—
332C2#2027510511yescharacteristic
333C2#2027515011yescharacteristic
334C2#2027526411yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1546C2
socleC22×C53#386
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1546D53
hypocenterC53#9402
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

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
#202797042
#202747082
#202733872
#202726172
#202726162
C53×#15462
#940253