Finite Group Databasebeta

← #27275865

Subgroups — #27275865

Order 13568.

802 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 151 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#20274707C211yesmaximal, characteristic, complemented
2#20273400C212yesmaximal, complemented
3#20273140C212yesmaximal, complemented
4#20273036C211yesmaximal, characteristic, complemented
5C2×C53×#555C211yesmaximal, characteristic
6#16810676C2212yes—
7#16810647C2212yes—
8C2×#13133134—21no—
9#16810392—21no—
10#16810369—24no—
11D4⋊#88322—24no—
12#16810347—22no—
13C53×#555—24no—
14C53×#555—24no—
15C2×C4⋊Q8×C53C2211yescharacteristic
16C2×#124×C53—22no—
17C2×#121×C53—21no—
18C2×#121×C53C412yes—
19C2×C4×D4×C53C2211yescharacteristic
20C2×C4×D4×C53C2211yescharacteristic
21C2×#117×C53—21no—
22C2×#117×C53C412yes—
23C2×#11719917—22no—
24C4⋊#88322—22no—
25C4⋊#88322—22no—
26C4⋊#88322—22no—
27#13133134—42no—
28C22⋊#88322—22no—
29C4×#88322—22no—
30C22×Q8×C53—21no—
31C22×D4×C53—22no—
32C22×D4×C53C2×C411yescharacteristic
33C22×D4×C53D411yescharacteristic
34C23×C4×C53—21no—
35C23×C4×C53C2×C412yes—
36C4⋊Q8×C53—22no—
37C4⋊Q8×C53D414yes—
38#124×C53—216no—
39#121×C53—28no—
40#121×C53—24no—
41#121×C53—24no—
42C4×D4×C53—22no—
43C4×D4×C53—22no—
44C4×D4×C53—22no—
45C4×D4×C53D414yes—
46#117×C53—28no—
47#117×C53—24no—
48#117×C53—24no—
49C2×C4⋊C4×C53—21no—
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53C2×C412yes—
52C2×C4⋊C4×C53D411yescharacteristic
53C2×C22⋊C4×C53—22no—
54C2×C22⋊C4×C53—22no—
55C2×C22⋊C4×C53—21no—
56C2×C22⋊C4×C53D412yes—
57C2×C42×C53—22no—
58C2×C42×C53C2311yescharacteristic
59C24×C53—21no—
60C24×C53C22⋊C412yes—
61C2×Q8×C53—24no—
62C2×Q8×C53—82no—
63C2×D4×C53—28no—
64C2×D4×C53—21no—
65C2×D4×C53—22no—
66C2×D4×C53—22no—
67C2×D4×C53—21no—
68C2×D4×C53—21no—
69C2×D4×C53—42no—
70C2×D4×C53—84no—
71C2×D4×C53C22⋊C412yes—
72C22×C4×C53—21no—
73C22×C4×C53—22no—
74C22×C4×C53—22no—
75C22×C4×C53—22no—
76C22×C4×C53—24no—
77C22×C4×C53—44no—
78C22×C4×C53—42no—
79C22×C4×C53—41no—
80C22×C4×C53—41no—
81C22×C4×C53C22⋊C412yes—
82C22×C4×C53C22×C411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C4⋊C4×C53—21no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—24no—
88C4⋊C4×C53—22no—
89C4⋊C4×C53C22⋊C412yes—
90C4⋊C4×C53C22⋊C414yes—
91C22⋊C4×C53—28no—
92C22⋊C4×C53—28no—
93C22⋊C4×C53—24no—
94C22⋊C4×C53—24no—
95C42×C53—28no—
96C42×C53C2×D412yes—
97C42×C53C2×D412yes—
98#11719917—42no—
99#11719917—42no—
100C2×#88322—42no—
101C2×#88322—42no—
102C23×C53—22no—
103C23×C53—22no—
104C23×C53—22no—
105C23×C53—22no—
106C23×C53—42no—
107C23×C53—42no—
108C23×C53—42no—
109C23×C53—42no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53C2×C22⋊C411yescharacteristic
113Q8×C53—84no—
114D4×C53—24no—
115D4×C53—42no—
116D4×C53—42no—
117D4×C53—42no—
118D4×C53—88no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—24no—
123C2×C4×C53—41no—
124C2×C4×C53—44no—
125C2×C4×C53—44no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—44no—
129C2×C4×C53—44no—
130C2×C4×C53—44no—
131C2×C4×C53—42no—
132C2×C4×C53C2×C22⋊C411yescharacteristic
133C2×C4×C53C2×C22⋊C411yescharacteristic
134C2×C4×C53#12012yes—
135C2×C4×C53#12012yes—
136C2×C4×C53#13512yes—
137C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C53#13711yescharacteristic
139#88322—82no—
140#88322—82no—
141#8397—531nomaximal, Sylow
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—42no—
145C22×C53—42no—
146C22×C53—42no—
147C22×C53—42no—
148C22×C53—42no—
149C22×C53—42no—
150C22×C53—82no—
151C22×C53—82no—
152C22×C53—82no—
153C22×C53—82no—
154C22×C53#37911yescharacteristic
155C22×C53#41611yescharacteristic
156C22×C53#41711yescharacteristic
157C4×C53—22no—
158C4×C53—21no—
159C4×C53—44no—
160C4×C53—42no—
161C4×C53—42no—
162C4×C53#41312yes—
163C4×C53#44912yes—
164C4×C53#45212yes—
165C2×#555D5311yescharacteristic
166#1800—531no—
167#1549—532no—
168#1486—532no—
169#1460—531no—
170C2×C53—21no—
171C2×C53—21no—
172C2×C53—82no—
173C2×C53—82no—
174C2×C53#178511yescharacteristic
175C2×C53#178811yescharacteristic
176C2×C53#179511yescharacteristic
177#555—24no—
178#555—24no—
179C2×C4⋊Q8C2×D5311yescharacteristic
180C2×#124—22no—
181C2×#121—21no—
182C2×#121Dic5312yes—
183C2×C4×D4C2×D5311yescharacteristic
184C2×C4×D4C2×D5311yescharacteristic
185C2×#117—21no—
186C2×#117Dic5312yes—
187#431—532no—
188#423—532no—
189C2×#100—1061no—
190#343—1061no—
191#332—1064no—
192D4⋊C8—1064no—
193#323—1062no—
194C53#839711yescharacteristic, complemented, Sylow
195C22×Q8—21no—
196C22×D4—22no—
197C22×D4C2×Dic5311yescharacteristic
198C22×D4C22⋊D5311yescharacteristic
199C23×C4—21no—
200C23×C4C2×Dic5312yes—
201C2×M16—1062no—
202C4⋊Q8—22no—
203C4⋊Q8C22⋊D5314yes—
204#124—216no—
205#121—28no—
206#121—24no—
207#121—24no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4C22⋊D5314yes—
212#117—28no—
213#117—24no—
214#117—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4C2×Dic5311yescharacteristic
217C2×C4⋊C4C2×Dic5312yes—
218C2×C4⋊C4C22⋊D5311yescharacteristic
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4C22⋊D5312yes—
223C2×C42—22no—
224C2×C42C22×D5311yescharacteristic
225C4⋊C8—1062no—
226C4⋊C8—1062no—
227C4⋊C8—1062no—
228#100—2122no—
229C22⋊C8—1062no—
230C4×C8—1062no—
231C24—21no—
232C24C22⋊Dic5312yes—
233C2×Q8—24no—
234C2×Q8—82no—
235C2×D4—28no—
236C2×D4—21no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—42no—
242C2×D4—84no—
243C2×D4C22⋊Dic5312yes—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4C22⋊Dic5312yes—
254C22×C4C22×Dic5311yescharacteristic
255C22×C4C2×C22⋊D5311yescharacteristic
256C22×C4C2×C22⋊D5311yescharacteristic
257M16—2122no—
258M16—2122no—
259C2×C8—2122no—
260C2×C8—2122no—
261C4⋊C4—21no—
262C4⋊C4—22no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4C22⋊Dic5312yes—
266C4⋊C4C22⋊Dic5314yes—
267C22⋊C4—28no—
268C22⋊C4—28no—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C42—28no—
272C42C2×C22⋊D5312yes—
273C42C2×C22⋊D5312yes—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—41no—
283C23—41no—
284C23C2×C22⋊Dic5311yescharacteristic
285Q8—84no—
286D4—24no—
287D4—42no—
288D4—42no—
289D4—42no—
290D4—88no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—41no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4C2×C22⋊Dic5311yescharacteristic
305C2×C4C2×C22⋊Dic5311yescharacteristic
306C2×C4#1313318612yes—
307C2×C4#1313318612yes—
308C2×C4#1313323712yes—
309C2×C4#1313324211yescharacteristic
310C2×C4#1313324711yescharacteristic
311C8—4242no—
312C8—4242no—
313C22—21no—
314C22—21no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22—82no—
325C22#1681047911yescharacteristic
326C22#1681061911yescharacteristic
327C22#1681062411yescharacteristic
328C4—21no—
329C4—22no—
330C4—42no—
331C4—42no—
332C4—44no—
333C4#1681060712yes—
334C4#1681076812yes—
335C4#1681078512yes—
336C2—21no—
337C2—21no—
338C2—82no—
339C2—82no—
340C2#2027464311yescharacteristic
341C2#2027465511yescharacteristic
342C2#2027468511yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC2×C53×#555C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenterC2×#555D53
hypocenterC53#8397
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. #20274707C2semidirect
2. #20273400C2semidirect
3. #20273140C2semidirect
4. #20273036C2semidirect
194. C53#8397semidirect

Maximal subgroups

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

subgroupindex
#202747072
#202734002
#202731402
#202730362
C2×C53×#5552
#839753