Finite Group Databasebeta

← #27275652

Subgroups — #27275652

Order 13568.

774 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 774 conjugacy classes of subgroups — too many to draw. Only its 143 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#20288518C211yesmaximal, characteristic
2#20275510C211yesmaximal, characteristic
3#20273406C212yesmaximal, complemented
4#20273169C212yesmaximal, complemented
5C53×#1800C211yesmaximal, characteristic
6#16811332—24no—
7#16811330—22no—
8D4×Dic106—22no—
9#16811254—21no—
10#16811249C414yes—
11#16811180C2212yes—
12C2×#13133196—22no—
13#16811098—21no—
14C2×C4⋊#88322C2211yescharacteristic
15C2×C22⋊#88322—22no—
16C2×C4×#88322C2211yescharacteristic
17#16810371—22no—
18#16810368—22no—
19#16810353—24no—
20C2×C4×D4×C53C2211yescharacteristic
21C53×#423C2212yes—
22C2×#100×C53—21no—
23C53×#343—21no—
24C22×Dic106—22no—
25C22×#88322—21no—
26C22×#88322—21no—
27#13133217D412yes—
28#13133217D412yes—
29#13133216—24no—
30#13133209—22no—
31#13133208—24no—
32#13133196—24no—
33#13133196—24no—
34#13133196—24no—
35#13133196—22no—
36#13133196C2×C414yes—
37#13133194—28no—
38#13133194—28no—
39C4×Dic106—22no—
40C4×Dic106D414yes—
41D4×Dic53—24no—
42D4×Dic53—24no—
43#13133176—22no—
44C2×C4⋊Dic53—22no—
45C2×#11719919—24no—
46C2×C22⋊Dic53—24no—
47C4⋊#88322—22no—
48C22⋊#88322—44no—
49C4×#88322—22no—
50C22×D4×C53D411yescharacteristic
51C23×C4×C53D412yes—
52C2×M16×C53—22no—
53C4×D4×C53—22no—
54C4×D4×C53—22no—
55C2×C4⋊C4×C53D411yescharacteristic
56C2×C22⋊C4×C53C2×C412yes—
57C2×C42×C53C2311yescharacteristic
58C4⋊C8×C53—22no—
59C4⋊C8×C53—22no—
60#100×C53—42no—
61C24×C53C22⋊C412yes—
62C2×D4×C53—22no—
63C2×D4×C53—22no—
64C2×D4×C53SD1612yes—
65C2×D4×C53Q1612yes—
66C22×C4×C53—22no—
67C22×C4×C53—22no—
68C22×C4×C53—42no—
69C22×C4×C53—42no—
70C22×C4×C53C22⋊C412yes—
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×Dic53—24no—
75C22×Dic53—24no—
76C2×Dic106—22no—
77C2×Dic106—24no—
78C2×Dic106—82no—
79C2×Dic106—82no—
80C2×Dic106C22⋊C414yes—
81M16×C53—42no—
82M16×C53—42no—
83C2×C8×C53—42no—
84C4⋊C4×C53—22no—
85C22⋊C4×C53—24no—
86C42×C53C2×D412yes—
87C42×C53C2×D412yes—
88C22⋊Dic53—28no—
89C22⋊Dic53—24no—
90C4⋊Dic53—24no—
91C4⋊Dic53—22no—
92C4⋊Dic53—22no—
93C4⋊Dic53—22no—
94C4⋊Dic53C22⋊C414yes—
95#11719919—28no—
96#11719919—24no—
97#11719919—24no—
98C4×Dic53—24no—
99C2×#88322—22no—
100C2×#88322—41no—
101C2×#88322—41no—
102C2×#88322—42no—
103C2×#88322—41no—
104C23×C53—22no—
105C23×C53—22no—
106C23×C53—42no—
107C23×C53—42no—
108C23×C53#10314yes—
109C23×C53C2×C22⋊C411yescharacteristic
110D4×C53—24no—
111D4×C53—24no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—22no—
115C2×C4×C53—41no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—82no—
120C2×C4×C53C2×C22⋊C412yes—
121C2×C4×C53#12012yes—
122C2×C4×C53#12011yescharacteristic
123C2×C4×C53#12011yescharacteristic
124C2×C4×C53C2×SD1611yescharacteristic
125C2×C4×C53C2×Q1611yescharacteristic
126C2×C4×C53#13712yes—
127C2×Dic53—24no—
128C2×Dic53—24no—
129C2×Dic53—84no—
130C2×Dic53—84no—
131Dic106—42no—
132Dic106—42no—
133Dic106—44no—
134Dic106—84no—
135C8×C53—82no—
136#88322—42no—
137#88322—81no—
138#8355—531nomaximal, Sylow
139C22×C53—21no—
140C22×C53—21no—
141C22×C53—24no—
142C22×C53—44no—
143C22×C53—42no—
144C22×C53—42no—
145C22×C53—82no—
146C22×C53#37911yescharacteristic
147C22×C53C2×#10311yescharacteristic
148C22×C53#41911yescharacteristic
149C4×C53—21no—
150C4×C53—22no—
151C4×C53—42no—
152C4×C53#44812yes—
153C4×C53#45112yes—
154C4×C53#45311yescharacteristic
155C4×C53#45511yescharacteristic
156Dic53—44no—
157Dic53—44no—
158#3501—531no—
159#1954—531no—
160#1800D5311yescharacteristic
161#1550—532no—
162#1493—532no—
163C2×C53—21no—
164C2×C53—21no—
165C2×C53—44no—
166C2×C53#178711yescharacteristic
167C2×C53#179111yescharacteristic
168C2×C53#179211yescharacteristic
169#554—1064no—
170#554—1062no—
171D4×Q8—1062no—
172#544—1061no—
173#543—534no—
174#533—532no—
175C2×C22⋊Q8—1062no—
176#518—1061no—
177C2×C4×D4C2×D5311yescharacteristic
178#423C2×D5312yes—
179C2×C4⋊C8—531no—
180C2×#100—21no—
181C2×C22⋊C8—1062no—
182C2×C4×C8—531no—
183#343—21no—
184#333—1062no—
185#332—1062no—
186Q8⋊C8—1064no—
187C53#835511yescharacteristic, complemented, Sylow
188C22×Q8—1062no—
189C22×D4C22⋊D5311yescharacteristic
190C23×C4D21212yes—
191C2×M16—22no—
192C22×C8—1061no—
193C22×C8—1061no—
194C4⋊Q8—532no—
195C4⋊Q8—532no—
196C4⋊Q8—1064no—
197#125—1064no—
198#125—1062no—
199C22⋊Q8—534no—
200C22⋊Q8—1068no—
201C22⋊Q8—1068no—
202C22⋊Q8—1064no—
203C22⋊Q8—1064no—
204C22⋊Q8—1062no—
205C22⋊Q8—1064no—
206C4×Q8—534no—
207C4×Q8—1062no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—1064no—
211C4×D4—1064no—
212#117—1062no—
213C2×C4⋊C4—1064no—
214C2×C4⋊C4—1062no—
215C2×C4⋊C4C22⋊D5311yescharacteristic
216C2×C22⋊C4—1064no—
217C2×C22⋊C4C4×D5312yes—
218C2×C42C22×D5311yescharacteristic
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—1062no—
222#100—42no—
223C22⋊C8—2124no—
224C4×C8—1062no—
225C24#1171992112yes—
226C2×Q8—534no—
227C2×Q8—1064no—
228C2×Q8—1062no—
229C2×Q8—4242no—
230C2×Q8—4242no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4#1171992312yes—
234C2×D4#1171992512yes—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—1064no—
240C22×C4—1064no—
241C22×C4#1171992112yes—
242C22×C4C2×C4×D5311yescharacteristic
243C22×C4C2×D21211yescharacteristic
244C22×C4C2×C22⋊D5311yescharacteristic
245M16—42no—
246M16—42no—
247C2×C8—42no—
248C2×C8—1062no—
249C2×C8—2122no—
250C2×C8—2121no—
251C2×C8—2121no—
252C2×C8—2121no—
253C4⋊C4—22no—
254C4⋊C4—534no—
255C4⋊C4—1064no—
256C4⋊C4—1068no—
257C4⋊C4—1064no—
258C4⋊C4—1062no—
259C4⋊C4—1062no—
260C4⋊C4—1064no—
261C4⋊C4—1062no—
262C22⋊C4—24no—
263C22⋊C4—1068no—
264C22⋊C4—1064no—
265C42—1064no—
266C42D4×D5312yes—
267C42D4×D5312yes—
268C23—22no—
269C23—22no—
270C23—42no—
271C23—42no—
272C23#1313314014yes—
273C23C2×#1171992111yescharacteristic
274Q8—4244no—
275Q8—2124no—
276Q8—2122no—
277Q8—2122no—
278D4—24no—
279D4—24no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—82no—
288C2×C4—1064no—
289C2×C4—1064no—
290C2×C4—4244no—
291C2×C4—4244no—
292C2×C4C22⋊C4×D5312yes—
293C2×C4#1313318512yes—
294C2×C4#1313318711yescharacteristic
295C2×C4#1313318711yescharacteristic
296C2×C4C2×#1171992311yescharacteristic
297C2×C4C2×#1171992511yescharacteristic
298C2×C4#1313324712yes—
299C8—82no—
300C8—4241no—
301C8—2122no—
302C22—21no—
303C22—21no—
304C22—24no—
305C22—44no—
306C22—42no—
307C22—42no—
308C22—82no—
309C22#1681047811yescharacteristic
310C22C2×#1313314011yescharacteristic
311C22#1681063111yescharacteristic
312C4—22no—
313C4—21no—
314C4—42no—
315C4—2124no—
316C4—2124no—
317C4#1681076112yes—
318C4#1681077812yes—
319C4#1681079311yescharacteristic
320C4#1681080311yescharacteristic
321C2—21no—
322C2—21no—
323C2—44no—
324C2#2027464911yescharacteristic
325C2#2027466411yescharacteristic
326C2#2027467111yescharacteristic
subgroupG/H
centerC22#16810478
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1800C2
socleC22×C53#379
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#1800D53
hypocenterC53#8355
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
3. #20273406C2semidirect
4. #20273169C2semidirect
187. C53#8355semidirect

Maximal subgroups

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

subgroupindex
#202885182
#202755102
#202734062
#202731692
C53×#18002
#835553