Finite Group Databasebeta

← #27286363

Subgroups — #27286363

Order 13568.

462 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 101 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#20285866C211yesmaximal, characteristic
2#20285237C211yesmaximal, characteristic
3#20275689C211yesmaximal, characteristic
4#20273280C211yesmaximal, characteristic
5#20273279C211yesmaximal, characteristic
6#20272559C211yesmaximal, characteristic
7C53×#3209C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16811033—21no—
10#16811033—21no—
11#16811029—21no—
12#16811029—21no—
13#16811026—21no—
14#16810975—21no—
15#16810958—22no—
16#16810915—22no—
17#16810676C2211yescharacteristic
18#16810659—22no—
19C2×#13133128C2211yescharacteristic
20#16810489C2211yescharacteristic
21#16810385—22no—
22#16810361—22no—
23#16810360—22no—
24C53×#533C2211yescharacteristic
25C2×C4⋊Q8×C53C2211yescharacteristic
26C53×#494—21no—
27C53×#487—21no—
28C53×#462—22no—
29C53×#451—22no—
30C53×#431C2211yescharacteristic
31C22×#88322D412yes—
32#13133217D411yescharacteristic
33#13133217D411yescharacteristic
34#13133209—21no—
35#13133196—21no—
36#13133196—21no—
37C4×Dic106—21no—
38C4×Dic106—21no—
39#13133176—21no—
40C2×C4⋊Dic53—21no—
41#13133150—24no—
42#13133150—22no—
43#13133150—22no—
44#13133148—22no—
45#13133148—22no—
46C4⋊#88322—21no—
47#13133140—24no—
48C22⋊#88322D412yes—
49#13133128—21no—
50#13133128—21no—
51C4×#88322—21no—
52#13133124—21no—
53#13133124—21no—
54C22×Q8×C53—21no—
55C2×Q16×C53—24no—
56C4⋊Q8×C53—21no—
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53—21no—
59C4⋊Q8×C53D411yescharacteristic
60C4⋊Q8×C53D411yescharacteristic
61C4⋊Q8×C53D412yes—
62#125×C53—21no—
63C22⋊Q8×C53—21no—
64C22⋊Q8×C53—21no—
65C4×Q8×C53—21no—
66C4×Q8×C53—21no—
67#117×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C4⋊C4×C53—21no—
70C2×C42×C53C2311yescharacteristic
71C4⋊C8×C53—21no—
72#103×C53—24no—
73C22⋊C8×C53D412yes—
74C4×C8×C53—21no—
75C2×Q8×C53—22no—
76C2×Q8×C53—21no—
77C2×Q8×C53—21no—
78C2×Q8×C53—41no—
79C2×Q8×C53—82no—
80C22×C4×C53—41no—
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×Dic53—41no—
85C2×Dic106—21no—
86C2×Dic106—21no—
87Q16×C53—84no—
88C2×C8×C53—22no—
89C4⋊C4×C53—21no—
90C4⋊C4×C53—22no—
91C4⋊C4×C53—21no—
92C4⋊C4×C53—21no—
93C4⋊C4×C53—22no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—41no—
99C22⋊C4×C53—41no—
100C42×C53—41no—
101C42×C53C2×D411yescharacteristic
102C42×C53C2×D411yescharacteristic
103C42×C53C2×D411yescharacteristic
104C42×C53#4111yescharacteristic
105C22⋊Dic53—41no—
106C4⋊Dic53—21no—
107C4⋊Dic53—21no—
108C4⋊Dic53—21no—
109C4⋊Dic53—21no—
110C4⋊Dic53—42no—
111#11719919—41no—
112#11719919—41no—
113C4×Dic53—41no—
114C2×#88322—22no—
115C2×#88322—22no—
116C2×#88322—22no—
117C2×#88322Q1614yescomplemented
118C23×C53#12711yescharacteristic
119Q8×C53—44no—
120Q8×C53—42no—
121Q8×C53—82no—
122Q8×C53—81no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—42no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—82no—
129C2×C4×C53#12011yescharacteristic
130C2×C4×C53#12011yescharacteristic
131C2×C4×C53#12011yescharacteristic
132C2×C4×C53#12111yescharacteristic
133C2×C4×C53#12111yescharacteristic
134C2×C4×C53#12111yescharacteristic
135C2×C4×C53C2×Q1612yes—
136C2×C4×C53#13712yes—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—81no—
140C2×Dic53—81no—
141C2×Dic53—81no—
142Dic106—42no—
143Dic106—81no—
144C8×C53—42no—
145#88322—24no—
146#88322—42no—
147#10224—531nomaximal, Sylow
148C22×C53—21no—
149C22×C53—21no—
150C22×C53#39211yescharacteristic
151C22×C53#49411yescharacteristic
152C22×C53#49711yescharacteristic
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—82no—
158C4×C53—81no—
159C4×C53—81no—
160C4×C53#45112yes—
161C4×C53#46212yes—
162Dic53—81no—
163Dic53—81no—
164Dic53—161no—
165#3230—531no—
166#3209D5311yescharacteristic
167#3162—531no—
168#1993—531no—
169#1518—531no—
170#1518—531no—
171#1281—531no—
172C2×C53—21no—
173C2×C53—21no—
174C2×C53#200811yescharacteristic
175C2×C53#200911yescharacteristic
176C2×C53#201611yescharacteristic
177#533—531no—
178#533C2×D5311yescharacteristic
179C2×C4⋊Q8C2×D5311yescharacteristic
180#501—1061no—
181#501—1061no—
182#501—1061no—
183#501—1061no—
184#500—1061no—
185#494—21no—
186#487—21no—
187#487—1061no—
188#484—1062no—
189#477—1062no—
190#462—22no—
191#451—22no—
192#431—531no—
193#431C2×D5311yescharacteristic
194#427—1062no—
195C2×#97—531no—
196#382—531no—
197#340—1062no—
198#328—1062no—
199#328—1062no—
200C53#1022411yescharacteristic, complemented, Sylow
201C22×Q8—21no—
202C2×Q16—24no—
203C22×C8—532no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—531no—
208C4⋊Q8—531no—
209C4⋊Q8C22⋊D5312yes—
210C4⋊Q8C22⋊D5311yescharacteristic
211C4⋊Q8C22⋊D5311yescharacteristic
212#125—21no—
213#125—1061no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C22⋊Q8—1061no—
217C22⋊Q8—1061no—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8—1061no—
221C4×Q8—1061no—
222#117—21no—
223#117—1061no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—1061no—
227C2×C42C22×D5311yescharacteristic
228#107—1062no—
229#107—1062no—
230#107—1064no—
231#106—1062no—
232#106—1062no—
233C4⋊C8—21no—
234C4⋊C8—1061no—
235#103—24no—
236#103—1064no—
237C22⋊C8—532no—
238C22⋊C8C22⋊D5312yes—
239#97—1061no—
240#97—1061no—
241C4×C8—21no—
242C4×C8—1061no—
243#95—1061no—
244#95—1061no—
245C2×Q8—22no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—41no—
249C2×Q8—82no—
250C2×Q8—1061no—
251C2×Q8—1061no—
252C22×C4—41no—
253C22×C4—2121no—
254C22×C4D4×D5311yescharacteristic
255C22×C4D4×D5311yescharacteristic
256C22×C4C2×C22⋊D5311yescharacteristic
257Q16—84no—
258C2×C8—22no—
259C2×C8—534no—
260C2×C8—1062no—
261C2×C8—1062no—
262C2×C8—1062no—
263C4⋊C4—22no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—22no—
268C4⋊C4—21no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4—2122no—
278C4⋊C4—2121no—
279C4⋊C4—2121no—
280C22⋊C4—41no—
281C22⋊C4—2121no—
282C42—41no—
283C42—2121no—
284C42D4×D5311yescharacteristic
285C42#1171994711yescharacteristic
286C42C2×C22⋊D5311yescharacteristic
287C42C2×C22⋊D5311yescharacteristic
288C23#1313321311yescharacteristic
289Q8—42no—
290Q8—44no—
291Q8—81no—
292Q8—82no—
293Q8—4241no—
294Q8—2122no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—82no—
301C2×C4—4241no—
302C2×C4—4241no—
303C2×C4—4241no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4#1313318511yescharacteristic
307C2×C4#1313318511yescharacteristic
308C2×C4#1313318611yescharacteristic
309C2×C4#1313318811yescharacteristic
310C2×C4#1313318911yescharacteristic
311C2×C4#1313318911yescharacteristic
312C2×C4Q16×D5312yes—
313C2×C4#1313324912yes—
314C8—42no—
315C8—1064no—
316C8—2122no—
317C22—21no—
318C22—21no—
319C22#1681053111yescharacteristic
320C22#1681100111yescharacteristic
321C22#1681101611yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—82no—
327C4—81no—
328C4—81no—
329C4—8481no—
330C4—4241no—
331C4—4241no—
332C4#1681077712yes—
333C4#1681083112yes—
334C2—21no—
335C2—21no—
336C2#2027576811yescharacteristic
337C2#2027577311yescharacteristic
338C2#2027581311yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#3209C2
socleC22×C53#392
prime radicalC23×C53#127
hypercenter#3209D53
hypocenterC53#10224
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
117. C2×#88322Q16semidirect
200. C53#10224semidirect

Maximal subgroups

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

subgroupindex
#202858662
#202852372
#202756892
#202732802
#202732792
#202725592
C53×#32092
#1022453