Finite Group Databasebeta

← #27275866

Subgroups — #27275866

Order 13568.

802 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#16811338C211yesmaximal, characteristic
2#20273401C212yesmaximal, complemented
3#20273139C212yesmaximal, complemented
4#20273036C211yesmaximal, characteristic, complemented
5C53×#1800C211yesmaximal, characteristic, complemented
6#16811338—24no—
7#16811338—24no—
8C2×#13133217C2211yescharacteristic
9C2×#13133203—22no—
10C2×#13133191C412yes—
11C2×#13133188—21no—
12C2×C4×D212C2211yescharacteristic
13C2×#13133176C412yes—
14C2×#13133175—21no—
15#16810676C2212yes—
16#16810368—24no—
17#16810350—24no—
18#16810347—22no—
19C2×C4×D4×C53C2211yescharacteristic
20C53×#423C2212yes—
21C2×#100×C53—21no—
22C53×#343—21no—
23C22×Dic106—21no—
24C22×D212C2×C411yescharacteristic
25C22×C22⋊D53—22no—
26C22×C4×D53—21no—
27#13133217—22no—
28#13133217D414yes—
29#13133203—216no—
30#13133191—24no—
31#13133191—24no—
32#13133188—28no—
33C4×D212—22no—
34C4×D212D414yes—
35#13133176—24no—
36#13133176—24no—
37#13133175—28no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×C4⋊Dic53C2×C412yes—
41C2×C22⋊Dic53—22no—
42C2×#11719921—22no—
43C2×#11719921—21no—
44C2×C4×Dic53—22no—
45C4⋊#88322—22no—
46C22⋊#88322—22no—
47C4×#88322—22no—
48C22×D4×C53D411yescharacteristic
49C23×C4×C53C2×C412yes—
50C2×M16×C53—22no—
51C4×D4×C53—22no—
52C4×D4×C53—22no—
53C2×C4⋊C4×C53D411yescharacteristic
54C2×C22⋊C4×C53D412yes—
55C2×C42×C53C2311yescharacteristic
56C4⋊C8×C53—22no—
57C4⋊C8×C53—22no—
58#100×C53—42no—
59C24×C53C22⋊C412yes—
60C23×D53—21no—
61C2×D4×C53—22no—
62C2×D4×C53—22no—
63C2×D4×C53—21no—
64C2×D4×C53—21no—
65C22×C4×C53—22no—
66C22×C4×C53—22no—
67C22×C4×C53—44no—
68C22×C4×C53C22⋊C412yes—
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C2×C22⋊D53—28no—
73C2×C22⋊D53—84no—
74C22×Dic53—21no—
75C22×Dic53—22no—
76C2×D212—21no—
77C2×D212—42no—
78C2×D212C22⋊C412yes—
79C2×C4×D53—24no—
80C2×C4×D53—42no—
81C2×C4×D53—41no—
82C2×C4×D53—41no—
83C2×Dic106—24no—
84C2×Dic106—82no—
85M16×C53—42no—
86M16×C53—42no—
87C2×C8×C53—42no—
88C4⋊C4×C53—22no—
89C22⋊C4×C53—24no—
90C42×C53C2×D412yes—
91C42×C53C2×D412yes—
92C22⋊Dic53—28no—
93#11719921—28no—
94#11719921—24no—
95C4⋊Dic53—21no—
96C4⋊Dic53—22no—
97C4⋊Dic53—24no—
98C4⋊Dic53C22⋊C412yes—
99C4⋊Dic53C22⋊C414yes—
100C4×Dic53—28no—
101C2×#88322—42no—
102C23×C53—22no—
103C23×C53—22no—
104C23×C53—22no—
105C23×C53—42no—
106C23×C53—42no—
107C23×C53C2×C22⋊C411yescharacteristic
108C22×D53—22no—
109C22×D53—42no—
110C22×D53—42no—
111C22×D53—41no—
112C22×D53—41no—
113D4×C53—42no—
114D4×C53—42no—
115C2×C4×C53—22no—
116C2×C4×C53—22no—
117C2×C4×C53—41no—
118C2×C4×C53—44no—
119C2×C4×C53—44no—
120C2×C4×C53—42no—
121C2×C4×C53C2×C22⋊C411yescharacteristic
122C2×C4×C53C2×C22⋊C411yescharacteristic
123C2×C4×C53#12012yes—
124C2×C4×C53#12012yes—
125C2×C4×C53#13512yes—
126C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C53#13711yescharacteristic
128C22⋊D53—88no—
129C2×Dic53—24no—
130C2×Dic53—22no—
131C2×Dic53—44no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134D212—24no—
135D212—42no—
136C4×D53—44no—
137C4×D53—44no—
138Dic106—84no—
139C8×C53—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—82no—
148C22×C53—82no—
149C22×C53#37911yescharacteristic
150C22×C53#41611yescharacteristic
151C22×C53#41711yescharacteristic
152C2×D53—42no—
153C2×D53—42no—
154C2×D53—42no—
155C2×D53—82no—
156C2×D53—82no—
157C4×C53—22no—
158C4×C53—21no—
159C4×C53—42no—
160C4×C53#41312yes—
161C4×C53#44912yes—
162C4×C53#45212yes—
163Dic53—44no—
164Dic53—42no—
165C2×#555—531no—
166#1800D5311yescharacteristic, complemented
167#1549—532no—
168#1486—532no—
169#1460—531no—
170C2×C53—21no—
171C2×C53—21no—
172C2×C53—82no—
173C2×C53#178511yescharacteristic
174C2×C53#178811yescharacteristic
175C2×C53#179511yescharacteristic
176D53—82no—
177#555—1064no—
178#555—1064no—
179C2×C4⋊Q8—531no—
180C2×#124—1062no—
181C2×#121—532no—
182C2×#121—1061no—
183C2×C4×D4—531no—
184C2×C4×D4C2×D5311yescharacteristic
185C2×#117—532no—
186C2×#117—1061no—
187#431—532no—
188#423C2×D5312yes—
189C2×#100—21no—
190#343—21no—
191#332—1064no—
192D4⋊C8—1064no—
193#323—1062no—
194C53#839711yescharacteristic, complemented, Sylow
195C22×Q8—1061no—
196C22×D4—531no—
197C22×D4—1062no—
198C22×D4C22⋊D5311yescharacteristic
199C23×C4—1061no—
200C23×C4C4×D5312yes—
201C2×M16—22no—
202C4⋊Q8—534no—
203C4⋊Q8—1062no—
204#124—10616no—
205#121—1064no—
206#121—1064no—
207#121—1068no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—534no—
211C4×D4—1062no—
212#117—1068no—
213#117—1064no—
214#117—1064no—
215C2×C4⋊C4—531no—
216C2×C4⋊C4—532no—
217C2×C4⋊C4—1061no—
218C2×C4⋊C4C22⋊D5311yescharacteristic
219C2×C22⋊C4—1062no—
220C2×C22⋊C4—1061no—
221C2×C22⋊C4—1062no—
222C2×C22⋊C4D21212yes—
223C2×C42—1062no—
224C2×C42C22×D5311yescharacteristic
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—1062no—
228#100—42no—
229C22⋊C8—1062no—
230C4×C8—1062no—
231C24—1061no—
232C24#1171992112yes—
233C2×Q8—1064no—
234C2×Q8—4242no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—532no—
240C2×D4—1068no—
241C2×D4—1061no—
242C2×D4—4244no—
243C2×D4—2122no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—44no—
247C22×C4—1061no—
248C22×C4—1062no—
249C22×C4—1064no—
250C22×C4—2122no—
251C22×C4—2121no—
252C22×C4—2121no—
253C22×C4#1171992112yes—
254C22×C4C2×C4×D5311yescharacteristic
255C22×C4C2×D21211yescharacteristic
256C22×C4C2×C22⋊D5311yescharacteristic
257M16—42no—
258M16—42no—
259C2×C8—42no—
260C2×C8—2122no—
261C4⋊C4—22no—
262C4⋊C4—532no—
263C4⋊C4—534no—
264C4⋊C4—1061no—
265C4⋊C4—1062no—
266C4⋊C4—1064no—
267C22⋊C4—24no—
268C22⋊C4—1068no—
269C22⋊C4—1068no—
270C22⋊C4—1064no—
271C42—1068no—
272C42D4×D5312yes—
273C42D4×D5312yes—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—42no—
278C23—42no—
279C23—1062no—
280C23—2122no—
281C23—2122no—
282C23—2121no—
283C23—2121no—
284C23C2×#1171992111yescharacteristic
285Q8—4244no—
286D4—42no—
287D4—42no—
288D4—1064no—
289D4—4248no—
290D4—2122no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—1062no—
298C2×C4—1064no—
299C2×C4—2124no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2124no—
303C2×C4—2124no—
304C2×C4C22⋊C4×D5311yescharacteristic
305C2×C4C22⋊C4×D5311yescharacteristic
306C2×C4#1313318512yes—
307C2×C4#1313318712yes—
308C2×C4#1313323712yes—
309C2×C4#1313324311yescharacteristic
310C2×C4#1313324811yescharacteristic
311C8—82no—
312C8—4242no—
313C22—21no—
314C22—21no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—82no—
320C22—4242no—
321C22—4242no—
322C22—2122no—
323C22—2122no—
324C22—2122no—
325C22#1681047811yescharacteristic
326C22#1681061711yescharacteristic
327C22#1681062211yescharacteristic
328C4—21no—
329C4—22no—
330C4—42no—
331C4—2122no—
332C4—2124no—
333C4#1681060412yes—
334C4#1681076612yes—
335C4#1681078312yes—
336C2—21no—
337C2—21no—
338C2—82no—
339C2—4242no—
340C2#2027464011yescharacteristic
341C2#2027465311yescharacteristic
342C2#2027468611yescharacteristic
subgroupG/H
centerC22#16810478
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1800C2
socleC22×C53#379
prime radicalC2×#16811338C2
cosocleC2×C42×C53C23
hypercenter#1800D53
hypocenterC53#8397
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #20273401C2semidirect
3. #20273139C2semidirect
4. #20273036C2semidirect
5. C53×#1800C2semidirect
166. #1800D53semidirect
194. C53#8397semidirect

Maximal subgroups

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

subgroupindex
C2×#168113382
#202734012
#202731392
#202730362
C53×#18002
#839753