Finite Group Databasebeta

← #27368021

Subgroups — #27368021

Order 13568.

702 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811028C211yesmaximal, characteristic, complemented
2#20284202C212yesmaximal
3#20274109C214yesmaximal, complemented
4#20274104C214yesmaximal, complemented
5#20273474C211yesmaximal, characteristic, complemented
6#20273386C212yesmaximal, complemented
7C8×#41×C53C211yesmaximal, characteristic
8C2×#13133247—22no—
9C2×#13133217C2212yes—
10#16811028—21no—
11#16811028—21no—
12#16811028C2214yes—
13#16810833—22no—
14#16810833C2214yes—
15#16810828—22no—
16#16810828C2214yes—
17C2×#13133151—22no—
18#16810647C2212yes—
19#16810631—22no—
20#16810376C2214yes—
21#16810368C2218yes—
22C4×#41×C53C2211yescharacteristic
23C8×Q8×C53C2212yes—
24C8×D4×C53—22no—
25C8×D4×C53C2212yes—
26C53×#431—21no—
27C53×#431C2211yescharacteristic
28C2×C4×C8×C53—21no—
29C2×C4×C8×C53C2211yescharacteristic
30C22×Dic106—22no—
31#13133247—48no—
32C2×#11719925—24no—
33C2×#11719923—24no—
34C2×#11719917—22no—
35#13133217—22no—
36#13133217—22no—
37#13133217—22no—
38#13133217C2314yes—
39C2×C4⋊Dic53—22no—
40C2×C4⋊Dic53—22no—
41#13133151—24no—
42#13133151—24no—
43C4⋊#88322—22no—
44C4⋊#88322C2314yes—
45#13133140—24no—
46#13133140—24no—
47C2×#41×C53D411yescharacteristic
48C22×C8×C53—22no—
49C22×C8×C53D412yes—
50C4×Q8×C53D411yescharacteristic
51C4×Q8×C53C2311yescharacteristic
52C4×D4×C53—21no—
53C4×D4×C53—21no—
54C4×D4×C53D411yescharacteristic
55C4×D4×C53C2311yescharacteristic
56#117×C53—21no—
57#117×C53D411yescharacteristic
58C2×C42×C53—21no—
59C2×C42×C53C2311yescharacteristic
60C4⋊C8×C53—22no—
61C4⋊C8×C53C2312yes—
62C22⋊C8×C53—22no—
63C22⋊C8×C53—21no—
64C4×C8×C53—22no—
65C4×C8×C53—21no—
66C4×C8×C53D411yescharacteristic
67C4×C8×C53D411yescharacteristic
68C4×C8×C53C2312yes—
69#41×C53—22no—
70#41×C53D814yes—
71C2×Q8×C53C2×D411yescharacteristic
72C2×D4×C53—21no—
73C2×D4×C53C2×D411yescharacteristic
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53#4111yescharacteristic
80C22×Dic53—42no—
81C2×Dic106—24no—
82C2×Dic106—44no—
83C2×Dic106—44no—
84C2×Dic106—42no—
85C2×C8×C53—24no—
86C2×C8×C53—22no—
87C2×C8×C53—44no—
88C2×C8×C53C2×D412yes—
89C2×C8×C53C2×D412yes—
90C4⋊C4×C53—21no—
91C4⋊C4×C53—21no—
92C4⋊C4×C53C2×D411yescharacteristic
93C4⋊C4×C53C2×D411yescharacteristic
94C22⋊C4×C53—21no—
95C22⋊C4×C53—21no—
96C22⋊C4×C53—21no—
97C42×C53—21no—
98C42×C53—21no—
99C42×C53—21no—
100C42×C53C2×D411yescharacteristic
101C42×C53C2×D411yescharacteristic
102C42×C53#4111yescharacteristic
103C42×C53C2411yescharacteristic
104#11719925—48no—
105#11719923—48no—
106C4⋊Dic53—24no—
107C4⋊Dic53—24no—
108C4⋊Dic53—42no—
109C4⋊Dic53—42no—
110#11719917—44no—
111C2×#88322—24no—
112C23×C53—21no—
113C23×C53#12111yescharacteristic
114Q8×C53—21no—
115Q8×C53C2×D812yes—
116D4×C53—24no—
117D4×C53—21no—
118D4×C53C2×D812yes—
119C2×C4×C53—21no—
120C2×C4×C53—22no—
121C2×C4×C53—24no—
122C2×C4×C53—22no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53#12111yescharacteristic
128C2×C4×C53#12111yescharacteristic
129C2×C4×C53#12111yescharacteristic
130C2×C4×C53C2×D812yes—
131C2×C4×C53#13512yes—
132C2×C4×C53C22×D411yescharacteristic
133C2×C4×C53C22×D411yescharacteristic
134C2×C4×C53C2×#4111yescharacteristic
135C2×Dic53—44no—
136C2×Dic53—84no—
137Dic106—44no—
138Dic106—44no—
139Dic106—84no—
140C8×C53—22no—
141C8×C53—42no—
142C8×C53#13714yescomplemented
143#88322—44no—
144#19271—531nomaximal, Sylow
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—22no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53#46612yes—
151C22×C53C2×#12111yescharacteristic
152C4×C53—21no—
153C4×C53—22no—
154C4×C53—21no—
155C4×C53—41no—
156C4×C53—41no—
157C4×C53#46612yes—
158C4×C53#47312yes—
159C4×C53C22×D811yescharacteristic
160C4×C53C2×#13511yescharacteristic
161C4×C53C2×#13712yes—
162Dic53—84no—
163C2×#500—531no—
164#3039—532no—
165C8×#41D5311yescharacteristic
166#1674—534no—
167#1673—534no—
168#1567—531no—
169#1546—532no—
170C2×C53—22no—
171C2×C53—42no—
172C2×C53#303911yescharacteristic
173C2×C53C2×#46611yescharacteristic
174C2×C53C2×#47311yescharacteristic
175C2×#137—1062no—
176C2×C4⋊Q8—532no—
177C4×#41C2×D5311yescharacteristic
178#500—534no—
179#500—1061no—
180#500—1061no—
181#462—534no—
182#462—1062no—
183#461—534no—
184#461—1062no—
185C8×Q8C2×D5312yes—
186C8×D4—22no—
187C8×D4C2×D5312yes—
188#431—21no—
189#431C2×D5311yescharacteristic
190C2×#107—1062no—
191#423—532no—
192#419—1062no—
193C2×C4×C8—21no—
194C2×C4×C8C2×D5311yescharacteristic
195#335—534no—
196#332—538no—
197C53#1927111yescharacteristic, complemented, Sylow
198C2×#41D21211yescharacteristic
199C22×Q8—1062no—
200#137—2128no—
201C2×Q16—1064no—
202C2×SD16—1064no—
203C2×M16—1062no—
204C22×C8—22no—
205C22×C8C22⋊D5312yes—
206C4⋊Q8—534no—
207C4⋊Q8—1062no—
208C4⋊Q8—1062no—
209C4⋊Q8—1062no—
210C4×Q8D21211yescharacteristic
211C4×Q8C22×D5311yescharacteristic
212C4×D4—21no—
213C4×D4—21no—
214C4×D4D21211yescharacteristic
215C4×D4C22×D5311yescharacteristic
216#117—21no—
217#117D21211yescharacteristic
218C2×C4⋊C4—1062no—
219C2×C4⋊C4—1062no—
220C2×C42—21no—
221C2×C42C22×D5311yescharacteristic
222#107—1064no—
223#107—1064no—
224C4⋊C8—22no—
225C4⋊C8—534no—
226C4⋊C8—1062no—
227C4⋊C8C22×D5312yes—
228#103—1064no—
229#103—1064no—
230C22⋊C8—21no—
231C22⋊C8—22no—
232C4×C8—22no—
233C4×C8—21no—
234C4×C8C22⋊D5311yescharacteristic
235C4×C8C22⋊D5311yescharacteristic
236C4×C8C22×D5312yes—
237#41—22no—
238#41D42414yes—
239C2×Q8—1064no—
240C2×Q8—2124no—
241C2×Q8—2124no—
242C2×Q8—2122no—
243C2×Q8C2×D21211yescharacteristic
244C2×D4—21no—
245C2×D4C2×D21211yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—2122no—
250C22×C4C2×D21211yescharacteristic
251C22×C4#1171994511yescharacteristic
252C22×C4C2×C22⋊D5311yescharacteristic
253Q16—2128no—
254SD16—2128no—
255M16—2124no—
256C2×C8—24no—
257C2×C8—22no—
258C2×C8—44no—
259C2×C8—1064no—
260C2×C8C2×C22⋊D5312yes—
261C2×C8C2×C22⋊D5312yes—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—1064no—
265C4⋊C4—1064no—
266C4⋊C4—2122no—
267C4⋊C4—2122no—
268C4⋊C4C2×D21211yescharacteristic
269C4⋊C4C2×D21211yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—21no—
273C42—21no—
274C42—21no—
275C42—21no—
276C42C2×D21211yescharacteristic
277C42#1171994511yescharacteristic
278C42C2×C22⋊D5311yescharacteristic
279C42C23×D5311yescharacteristic
280C23—21no—
281C23#1313319111yescharacteristic
282Q8—21no—
283Q8—4244no—
284Q8—2124no—
285Q8—2124no—
286Q8C2×D42412yes—
287D4—24no—
288D4—21no—
289D4C2×D42412yes—
290C2×C4—21no—
291C2×C4—24no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—4244no—
299C2×C4—2124no—
300C2×C4#1313319111yescharacteristic
301C2×C4#1313319111yescharacteristic
302C2×C4#1313319111yescharacteristic
303C2×C4C2×D42412yes—
304C2×C4#1313323912yes—
305C2×C4C22×C22⋊D5311yescharacteristic
306C2×C4C22×D21211yescharacteristic
307C2×C4C2×#1171994511yescharacteristic
308C8—22no—
309C8—42no—
310C8—2124no—
311C8#1313324714yescomplemented
312C22—21no—
313C22—21no—
314C22—22no—
315C22—41no—
316C22—41no—
317C22#1681085912yes—
318C22C2×#1313319111yescharacteristic
319C4—21no—
320C4—21no—
321C4—22no—
322C4—41no—
323C4—41no—
324C4—4244no—
325C4#1681085912yes—
326C4#1681089812yes—
327C4C22×D42411yescharacteristic
328C4C2×#1313323911yescharacteristic
329C4C2×#1313324712yes—
330C2—22no—
331C2—42no—
332C2#2028420211yescharacteristic
333C2C2×#1681085911yescharacteristic
334C2C2×#1681089811yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC8×#41×C53C2
socleC22×C53C2×#121
prime radicalC2×#41×C53D4
hypercenterC8×#41D53
hypocenterC53#19271
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16811028C2semidirect
3. #20274109C2semidirect
4. #20274104C2semidirect
5. #20273474C2semidirect
6. #20273386C2semidirect
142. C8×C53#137semidirect
197. C53#19271semidirect
311. C8#13133247semidirect

Maximal subgroups

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

subgroupindex
C2×#168110282
#202842022
#202741092
#202741042
#202734742
#202733862
C8×#41×C532
#1927153