Finite Group Databasebeta

← #27297600

Subgroups — #27297600

Order 13568.

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

normal subgroups only

This group has 378 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#20276964C211yesmaximal, characteristic, complemented
2#20276750C211yesmaximal, characteristic, complemented
3#20274075C211yesmaximal, characteristic, complemented
4#20272761C211yesmaximal, characteristic, complemented
5Q8⋊#11719908C211yesmaximal, characteristic, complemented
6#20272676C211yesmaximal, characteristic, complemented
7C53×#3391C211yesmaximal, characteristic, complemented
8C2×#13133158—21no—
9C2×#13133158—22no—
10#16811000C2211yescharacteristic
11#16810823—21no—
12C4⋊#11719908C2211yescharacteristic
13#16810423—21no—
14C4×#11719908C2211yescharacteristic
15#16810376C2211yescharacteristic
16#16810366—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×#11719924—41no—
26C2×D424—21no—
27C2×D424—21no—
28#13133214—21no—
29#13133158—44no—
30#13133158—42no—
31C2×#11719908—21no—
32C2×#11719908—21no—
33C4⋊#88322—21no—
34#13133137—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×D212—41no—
66C2×D212—41no—
67C2×D212—41no—
68D8×C53—21no—
69D8×C53—21no—
70D8×C53—42no—
71D8×C53—41no—
72D8×C53—41no—
73D8×C53—82no—
74C2×C8×C53—21no—
75C2×C8×C53C2×D411yescharacteristic
76C2×C8×C53C2×D411yescharacteristic
77C4⋊C4×C53—21no—
78C4⋊C4×C53—21no—
79C4⋊C4×C53D811yescharacteristic
80C22⋊C4×C53—42no—
81C22⋊C4×C53—41no—
82C42×C53—21no—
83C42×C53C2×D411yescharacteristic
84#11719924—82no—
85C2×#88322—41no—
86D424—42no—
87D424—41no—
88D424—41no—
89#11719908—22no—
90#11719908—41no—
91C23×C53—42no—
92C23×C53—41no—
93C22×D53—81no—
94Q8×C53—21no—
95Q8×C53D1612yescomplemented
96D4×C53—41no—
97D4×C53—42no—
98D4×C53—42no—
99D4×C53—41no—
100D4×C53—84no—
101D4×C53—81no—
102D4×C53—81no—
103D4×C53—161no—
104D4×C53—162no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—41no—
108C2×C4×C53—84no—
109C2×C4×C53—81no—
110C2×C4×C53—81no—
111C2×C4×C53#12011yescharacteristic
112C2×C4×C53C2×D811yescharacteristic
113C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
114D212—81no—
115D212—81no—
116D212—81no—
117D212—81no—
118D212—161no—
119C8×C53—21no—
120C8×C53—41no—
121C8×C53#13511yescharacteristic
122C8×C53AGL(1,ℤ/8)11yescharacteristic
123#88322—81no—
124#12328—531nomaximal, Sylow
125C22×C53—81no—
126C22×C53—82no—
127C22×C53—82no—
128C22×C53—81no—
129C22×C53—82no—
130C22×C53—81no—
131C22×C53#44711yescharacteristic
132C2×D53—161no—
133C2×D53—161no—
134C2×D53—161no—
135C4×C53—22no—
136C4×C53—21no—
137C4×C53—41no—
138C4×C53—81no—
139C4×C53#44911yescharacteristic
140C4×C53#45311yescharacteristic
141C4×C53C2×D1611yescharacteristic
142C4×C53#50911yescharacteristic
143#3391D5311yescharacteristic, complemented
144#2246—531no—
145#2206—531no—
146#1668—531no—
147#1361—531no—
148Q8⋊C16—531no—
149#1331—531no—
150C2×C53—161no—
151C2×C53—162no—
152C2×C53#162111yescharacteristic
153C2×C53#218411yescharacteristic
154C2×C53#219211yescharacteristic
155D53—321no—
156#550—21no—
157C2×D16—1061no—
158C2×D16—1062no—
159#493—21no—
160#493—531no—
161#493C2×D5311yescharacteristic
162#460—1061no—
163#459—22no—
164#459—21no—
165C8×Q8C2×D5311yescharacteristic
166C4×D8—21no—
167C4×D8C2×D5311yescharacteristic
168C4⋊C16—531no—
169#357—1061no—
170C4×C16—531no—
171#335—531no—
172#331—1061no—
173C53#1232811yescharacteristic, complemented, Sylow
174C2×#41—41no—
175C2×SD16—2121no—
176C2×D8—22no—
177C2×D8—21no—
178C2×D8—1061no—
179C2×D8—1061no—
180C2×D8C22⋊D5311yescharacteristic
181C2×D8C22⋊D5312yes—
182#127—22no—
183#127—21no—
184#127—1061no—
185#121—42no—
186#121—41no—
187C4×Q8C22⋊D5311yescharacteristic
188C4×D4—22no—
189C4×D4—21no—
190D16—2124no—
191D16—2122no—
192C2×C16—1061no—
193C2×C16—1061no—
194#107C22⋊D5311yescharacteristic
195C4⋊C8—21no—
196C4⋊C8—1061no—
197C4⋊C8C22⋊D5311yescharacteristic
198#102—22no—
199#102—21no—
200#102—2121no—
201C4×C8—21no—
202C4×C8C22×D5311yescharacteristic
203#41—82no—
204#41—82no—
205C2×Q8#1171992211yescharacteristic
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—2121no—
213C2×D4—2121no—
214C2×D4—2121no—
215C22×C4—42no—
216C22×C4—41no—
217SD16—4242no—
218D8—21no—
219D8—21no—
220D8—42no—
221D8—41no—
222D8—41no—
223D8—82no—
224D8—2122no—
225D8—2121no—
226D8—2121no—
227C2×C8—21no—
228C2×C8—2121no—
229C2×C8C2×C22⋊D5311yescharacteristic
230C2×C8C2×C22⋊D5311yescharacteristic
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4#1171992211yescharacteristic
234C22⋊C4—42no—
235C22⋊C4—41no—
236C42—21no—
237C42C2×C22⋊D5311yescharacteristic
238C16—1062no—
239C16—2121no—
240C23—42no—
241C23—41no—
242C23—4241no—
243Q8—21no—
244Q8#1313315812yescomplemented
245D4—41no—
246D4—42no—
247D4—42no—
248D4—41no—
249D4—84no—
250D4—81no—
251D4—81no—
252D4—161no—
253D4—162no—
254D4—8481no—
255D4—4241no—
256D4—4241no—
257D4—4241no—
258D4—4241no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—41no—
262C2×C4—84no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4#1313318611yescharacteristic
266C2×C4C2×#1171992211yescharacteristic
267C2×C4#1313324311yescharacteristic
268C8—21no—
269C8—41no—
270C8—4241no—
271C8#1313323711yescharacteristic
272C8#1313324211yescharacteristic
273C22—81no—
274C22—82no—
275C22—82no—
276C22—81no—
277C22—82no—
278C22—81no—
279C22—8481no—
280C22—8481no—
281C22—8481no—
282C22#1681075711yescharacteristic
283C4—22no—
284C4—21no—
285C4—41no—
286C4—81no—
287C4#1681076811yescharacteristic
288C4#1681079111yescharacteristic
289C4C2×#1313315811yescharacteristic
290C4#1681105911yescharacteristic
291C2—161no—
292C2—162no—
293C2—16961no—
294C2#2027378011yescharacteristic
295C2#2027662711yescharacteristic
296C2#2027667211yescharacteristic
subgroupG/H
centerC22#16810757
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#3391C2
socleC22×C53#447
hypercenter#3391D53
hypocenterC53#12328
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. #20276964C2semidirect
2. #20276750C2semidirect
3. #20274075C2semidirect
4. #20272761C2semidirect
5. Q8⋊#11719908C2semidirect
6. #20272676C2semidirect
7. C53×#3391C2semidirect
95. Q8×C53D16semidirect
143. #3391D53semidirect
173. C53#12328semidirect
244. Q8#13133158semidirect

Maximal subgroups

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

subgroupindex
#202769642
#202767502
#202740752
#202727612
Q8⋊#117199082
#202726762
C53×#33912
#1232853