Finite Group Databasebeta

← #27297628

Subgroups — #27297628

Order 13568.

362 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 75 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#20276970C211yesmaximal, characteristic, complemented
2#20276760C211yesmaximal, characteristic, complemented
3#20274085C211yesmaximal, characteristic, complemented
4#20272761C211yesmaximal, characteristic, complemented
5Q8⋊#11719908C211yesmaximal, characteristic, complemented
6#20272676C211yesmaximal, characteristic, complemented
7C53×#3391C211yesmaximal, characteristic
8C2×#13133161—21no—
9C2×#13133161—22no—
10#16811003C2211yescharacteristic
11#16810908—21no—
12C4⋊#11719908C2211yescharacteristic
13#16810426—21no—
14C4×#11719908C2211yescharacteristic
15#16810376C2211yescharacteristic
16#16810370—21no—
17C53×#550—21no—
18C53×#493—21no—
19C53×#493C2211yescharacteristic
20C53×#459—22no—
21C53×#459—21no—
22C8×Q8×C53C2211yescharacteristic
23C4×D8×C53—21no—
24C4×D8×C53C2211yescharacteristic
25C2×Dic212—21no—
26C2×Dic212—21no—
27#13133217—21no—
28#13133161—44no—
29#13133161—42no—
30C2×#11719908—21no—
31C2×#11719908—21no—
32#13133148—41no—
33C4⋊#88322—21no—
34#13133141—41no—
35C2×#41×C53—41no—
36C2×D8×C53—22no—
37C2×D8×C53—21no—
38C2×D8×C53D411yescharacteristic
39C2×D8×C53D412yes—
40#127×C53—22no—
41#127×C53—21no—
42#121×C53—42no—
43#121×C53—41no—
44C4×Q8×C53D411yescharacteristic
45C4×D4×C53—22no—
46C4×D4×C53—21no—
47#107×C53D411yescharacteristic
48C4⋊C8×C53—21no—
49C4⋊C8×C53D411yescharacteristic
50#102×C53—22no—
51#102×C53—21no—
52C4×C8×C53—21no—
53C4×C8×C53C2311yescharacteristic
54#41×C53—82no—
55#41×C53—82no—
56C2×Q8×C53D811yescharacteristic
57C2×D4×C53—21no—
58C2×D4×C53—22no—
59C2×D4×C53—42no—
60C2×D4×C53—42no—
61C2×D4×C53—41no—
62C2×D4×C53—41no—
63C22×C4×C53—42no—
64C22×C4×C53—41no—
65C2×Dic106—41no—
66D8×C53—21no—
67D8×C53—21no—
68D8×C53—42no—
69D8×C53—41no—
70D8×C53—41no—
71D8×C53—82no—
72C2×C8×C53—21no—
73C2×C8×C53C2×D411yescharacteristic
74C2×C8×C53C2×D411yescharacteristic
75C4⋊C4×C53—21no—
76C4⋊C4×C53—21no—
77C4⋊C4×C53D811yescharacteristic
78C22⋊C4×C53—42no—
79C22⋊C4×C53—41no—
80C42×C53—21no—
81C42×C53C2×D411yescharacteristic
82C4⋊Dic53—41no—
83C4⋊Dic53—41no—
84C2×#88322—41no—
85Dic212—42no—
86Dic212—41no—
87Dic212—41no—
88#11719908—22no—
89#11719908—41no—
90C23×C53—42no—
91C23×C53—41no—
92Q8×C53—21no—
93Q8×C53SD3212yescomplemented
94D4×C53—41no—
95D4×C53—42no—
96D4×C53—42no—
97D4×C53—41no—
98D4×C53—84no—
99D4×C53—81no—
100D4×C53—81no—
101D4×C53—161no—
102D4×C53—162no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—41no—
106C2×C4×C53—84no—
107C2×C4×C53—81no—
108C2×C4×C53—81no—
109C2×C4×C53#12011yescharacteristic
110C2×C4×C53C2×D811yescharacteristic
111C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
112C2×Dic53—81no—
113Dic106—81no—
114Dic106—81no—
115C8×C53—21no—
116C8×C53—41no—
117C8×C53#13511yescharacteristic
118C8×C53AGL(1,ℤ/8)11yescharacteristic
119#88322—81no—
120#12332—531nomaximal, Sylow
121C22×C53—81no—
122C22×C53—82no—
123C22×C53—82no—
124C22×C53—81no—
125C22×C53—82no—
126C22×C53—81no—
127C22×C53#44711yescharacteristic
128C4×C53—22no—
129C4×C53—21no—
130C4×C53—41no—
131C4×C53—81no—
132C4×C53#44911yescharacteristic
133C4×C53#45311yescharacteristic
134C4×C53C2×SD3211yescharacteristic
135C4×C53#50911yescharacteristic
136Dic53—161no—
137#3391D5311yescharacteristic
138#2248—531no—
139#2208—531no—
140#1670—531no—
141#1361—531no—
142Q8⋊C16—531no—
143#1331—531no—
144C2×C53—161no—
145C2×C53—162no—
146C2×C53#162111yescharacteristic
147C2×C53#218711yescharacteristic
148C2×C53#219411yescharacteristic
149#550—21no—
150C2×SD32—1061no—
151C2×SD32—1062no—
152#494—531no—
153#493—21no—
154#493C2×D5311yescharacteristic
155#475—1061no—
156#459—22no—
157#459—21no—
158C8×Q8C2×D5311yescharacteristic
159C4×D8—21no—
160C4×D8C2×D5311yescharacteristic
161C4⋊C16—531no—
162#358—1061no—
163C4×C16—531no—
164#335—531no—
165#332—1061no—
166C53#1233211yescharacteristic, complemented, Sylow
167C2×#41—41no—
168C2×Q16—1061no—
169C2×Q16—1061no—
170C2×D8—22no—
171C2×D8—21no—
172C2×D8C22⋊D5311yescharacteristic
173C2×D8C22⋊D5312yes—
174C4⋊Q8—1061no—
175#127—22no—
176#127—21no—
177#121—42no—
178#121—41no—
179C4×Q8C22⋊D5311yescharacteristic
180C4×D4—22no—
181C4×D4—21no—
182SD32—2124no—
183SD32—2122no—
184C2×C16—1061no—
185C2×C16—1061no—
186#107C22⋊D5311yescharacteristic
187#106—2121no—
188C4⋊C8—21no—
189C4⋊C8—1061no—
190C4⋊C8C22⋊D5311yescharacteristic
191#103—2121no—
192#102—22no—
193#102—21no—
194C4×C8—21no—
195C4×C8C22×D5311yescharacteristic
196#41—82no—
197#41—82no—
198C2×Q8—2121no—
199C2×Q8#1171992211yescharacteristic
200C2×D4—21no—
201C2×D4—22no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—41no—
205C2×D4—41no—
206C22×C4—42no—
207C22×C4—41no—
208Q16—2122no—
209Q16—2121no—
210Q16—2121no—
211D8—21no—
212D8—21no—
213D8—42no—
214D8—41no—
215D8—41no—
216D8—82no—
217C2×C8—21no—
218C2×C8—2121no—
219C2×C8C2×C22⋊D5311yescharacteristic
220C2×C8C2×C22⋊D5311yescharacteristic
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—2121no—
224C4⋊C4—2121no—
225C4⋊C4#1171992211yescharacteristic
226C22⋊C4—42no—
227C22⋊C4—41no—
228C42—21no—
229C42C2×C22⋊D5311yescharacteristic
230C16—1062no—
231C16—2121no—
232C23—42no—
233C23—41no—
234Q8—21no—
235Q8—4241no—
236Q8—4241no—
237Q8#1313316112yescomplemented
238D4—41no—
239D4—42no—
240D4—42no—
241D4—41no—
242D4—84no—
243D4—81no—
244D4—81no—
245D4—161no—
246D4—162no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—41no—
250C2×C4—84no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—4241no—
254C2×C4#1313318611yescharacteristic
255C2×C4C2×#1171992211yescharacteristic
256C2×C4#1313324311yescharacteristic
257C8—21no—
258C8—41no—
259C8—4241no—
260C8#1313323711yescharacteristic
261C8#1313324211yescharacteristic
262C22—81no—
263C22—82no—
264C22—82no—
265C22—81no—
266C22—82no—
267C22—81no—
268C22#1681075711yescharacteristic
269C4—22no—
270C4—21no—
271C4—41no—
272C4—81no—
273C4—8481no—
274C4#1681076811yescharacteristic
275C4#1681079111yescharacteristic
276C4C2×#1313316111yescharacteristic
277C4#1681105911yescharacteristic
278C2—161no—
279C2—162no—
280C2#2027378011yescharacteristic
281C2#2027664411yescharacteristic
282C2#2027668611yescharacteristic
subgroupG/H
centerC22#16810757
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
Fitting, prime radicalC53×#3391C2
socleC22×C53#447
hypercenter#3391D53
hypocenterC53#12332
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3391

Complements

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

NH ≅ G/Ntype
1. #20276970C2semidirect
2. #20276760C2semidirect
3. #20274085C2semidirect
4. #20272761C2semidirect
5. Q8⋊#11719908C2semidirect
6. #20272676C2semidirect
93. Q8×C53SD32semidirect
166. C53#12332semidirect
237. Q8#13133161semidirect

Maximal subgroups

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

subgroupindex
#202769702
#202767602
#202740852
#202727612
Q8⋊#117199082
#202726762
C53×#33912
#1233253