Finite Group Databasebeta

← #27297565

Subgroups — #27297565

Order 13568.

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

normal subgroups only

This group has 374 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#20276967C211yesmaximal, characteristic, complemented
2#20276760C211yesmaximal, characteristic, complemented
3#20274080C211yesmaximal, characteristic, complemented
4#20272764C211yesmaximal, characteristic, complemented
5#20272676C211yesmaximal, characteristic, complemented
6D4⋊#11719908C211yesmaximal, characteristic, complemented
7C53×#3283C211yesmaximal, characteristic
8C2×#13133163—22no—
9C2×#13133161—21no—
10#16811003C2211yescharacteristic
11#16810903—21no—
12C4⋊#11719908C2211yescharacteristic
13#16810426—21no—
14C4×#11719908C2211yescharacteristic
15#16810376C2211yescharacteristic
16#16810370—21no—
17C2×#135×C53—21no—
18C53×#547—21no—
19C53×#494C2211yescharacteristic
20C53×#467—21no—
21C53×#461—21no—
22C53×#452—22no—
23C4×D8×C53C2211yescharacteristic
24C8×D4×C53C2211yescharacteristic
25C2×Dic212—21no—
26C2×Dic212—21no—
27#13133217—21no—
28#13133163—44no—
29#13133161—42no—
30C2×#11719908—21no—
31C2×#11719908—21no—
32#13133150—41no—
33C4⋊#88322—21no—
34#13133138—41no—
35C2×#41×C53—22no—
36#135×C53—42no—
37#135×C53—42no—
38C2×Q16×C53—21no—
39C2×Q16×C53D412yes—
40C2×SD16×C53—22no—
41C2×D8×C53D411yescharacteristic
42C22×C8×C53—21no—
43C4⋊Q8×C53—21no—
44#123×C53—42no—
45C22⋊Q8×C53—22no—
46#121×C53—41no—
47C4×D4×C53—21no—
48C4×D4×C53D411yescharacteristic
49C2×C4⋊C4×C53—41no—
50#107×C53D411yescharacteristic
51C4⋊C8×C53D411yescharacteristic
52#103×C53—22no—
53#102×C53—21no—
54C22⋊C8×C53—21no—
55C4×C8×C53C2311yescharacteristic
56#41×C53—44no—
57#41×C53—82no—
58C2×Q8×C53—22no—
59C2×D4×C53—21no—
60C2×D4×C53—42no—
61C2×D4×C53D811yescharacteristic
62C22×C4×C53—21no—
63C22×C4×C53—42no—
64C22×C4×C53—41no—
65C2×Dic106—41no—
66Q16×C53—42no—
67Q16×C53—41no—
68Q16×C53—41no—
69SD16×C53—44no—
70D8×C53—21no—
71D8×C53D812yes—
72C2×C8×C53—21no—
73C2×C8×C53—41no—
74C2×C8×C53—41no—
75C2×C8×C53C2×D411yescharacteristic
76C2×C8×C53C2×D411yescharacteristic
77C4⋊C4×C53—21no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53D811yescharacteristic
82C22⋊C4×C53—21no—
83C22⋊C4×C53—42no—
84C22⋊C4×C53—41no—
85C42×C53C2×D411yescharacteristic
86C4⋊Dic53—41no—
87C4⋊Dic53—41no—
88C2×#88322—41no—
89Dic212—42no—
90Dic212—41no—
91Dic212—41no—
92#11719908—22no—
93#11719908—41no—
94C23×C53—21no—
95C23×C53—41no—
96Q8×C53—42no—
97Q8×C53—42no—
98D4×C53—22no—
99D4×C53—21no—
100D4×C53—21no—
101D4×C53—41no—
102D4×C53—84no—
103C2×C4×C53—21no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—42no—
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
114C2×Dic53—81no—
115Dic106—81no—
116Dic106—81no—
117C8×C53—21no—
118C8×C53—41no—
119C8×C53C2×D811yescharacteristic
120C8×C53AGL(1,ℤ/8)11yescharacteristic
121#88322—81no—
122#12323—531nomaximal, Sylow
123C22×C53—41no—
124C22×C53—42no—
125C22×C53—41no—
126C22×C53—41no—
127C22×C53—81no—
128C22×C53—81no—
129C22×C53#44711yescharacteristic
130C4×C53—21no—
131C4×C53—41no—
132C4×C53—82no—
133C4×C53—81no—
134C4×C53#44711yescharacteristic
135C4×C53#45311yescharacteristic
136C4×C53#50811yescharacteristic
137C4×C53#51011yescharacteristic
138Dic53—161no—
139#3283D5311yescharacteristic
140#2247—531no—
141#2208—531no—
142#1669—531no—
143#1362—531no—
144#1331—531no—
145D4⋊C16—531no—
146C2×C53—81no—
147C2×C53—82no—
148C2×C53#161911yescharacteristic
149C2×C53#218911yescharacteristic
150C2×C53#219511yescharacteristic
151C2×#135—21no—
152#547—21no—
153C2×Q32—1062no—
154C2×SD32—1061no—
155#494—531no—
156#494C2×D5311yescharacteristic
157#474—1061no—
158#467—21no—
159#461—21no—
160#452—22no—
161C4×D8C2×D5311yescharacteristic
162C8×D4C2×D5311yescharacteristic
163C4⋊C16—531no—
164#358—1061no—
165C4×C16—531no—
166#335—531no—
167#332—1061no—
168C53#1232311yescharacteristic, complemented, Sylow
169C2×#41—22no—
170#135—42no—
171#135—42no—
172C2×Q16—21no—
173C2×Q16—1061no—
174C2×Q16—1061no—
175C2×Q16C22⋊D5312yes—
176C2×SD16—22no—
177C2×D8C22⋊D5311yescharacteristic
178C22×C8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—1061no—
181#123—42no—
182C22⋊Q8—22no—
183#121—41no—
184C4×D4—21no—
185C4×D4C22⋊D5311yescharacteristic
186C2×C4⋊C4—41no—
187Q32—2124no—
188SD32—2122no—
189C2×C16—1061no—
190C2×C16—1061no—
191#107—2121no—
192#107C22⋊D5311yescharacteristic
193C4⋊C8—1061no—
194C4⋊C8C22⋊D5311yescharacteristic
195#103—22no—
196#102—21no—
197#102—2121no—
198C22⋊C8—21no—
199C4×C8C22×D5311yescharacteristic
200#41—44no—
201#41—82no—
202C2×Q8—22no—
203C2×Q8—2121no—
204C2×D4—21no—
205C2×D4—42no—
206C2×D4#1171992211yescharacteristic
207C22×C4—21no—
208C22×C4—42no—
209C22×C4—41no—
210Q16—42no—
211Q16—41no—
212Q16—41no—
213Q16—2122no—
214Q16—2121no—
215Q16—2121no—
216SD16—44no—
217D8—21no—
218D8#1171992212yes—
219C2×C8—21no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—2121no—
223C2×C8C2×C22⋊D5311yescharacteristic
224C2×C8C2×C22⋊D5311yescharacteristic
225C4⋊C4—21no—
226C4⋊C4—42no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—2121no—
230C4⋊C4—2121no—
231C4⋊C4#1171992211yescharacteristic
232C22⋊C4—21no—
233C22⋊C4—42no—
234C22⋊C4—41no—
235C42C2×C22⋊D5311yescharacteristic
236C16—1062no—
237C16—2121no—
238C23—21no—
239C23—41no—
240Q8—42no—
241Q8—42no—
242Q8—4241no—
243Q8—4241no—
244D4—22no—
245D4—21no—
246D4—21no—
247D4—41no—
248D4—84no—
249C2×C4—21no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—84no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—4241no—
258C2×C4#1313318611yescharacteristic
259C2×C4C2×#1171992211yescharacteristic
260C2×C4#1313324311yescharacteristic
261C8—21no—
262C8—41no—
263C8—4241no—
264C8C2×#1171992211yescharacteristic
265C8#1313324311yescharacteristic
266C22—41no—
267C22—42no—
268C22—41no—
269C22—41no—
270C22—81no—
271C22—81no—
272C22#1681075711yescharacteristic
273C4—21no—
274C4—41no—
275C4—82no—
276C4—81no—
277C4—8481no—
278C4#1681075711yescharacteristic
279C4#1681079111yescharacteristic
280C4#1681105311yescharacteristic
281C4#1681106311yescharacteristic
282C2—81no—
283C2—82no—
284C2#2027376911yescharacteristic
285C2#2027665511yescharacteristic
286C2#2027669311yescharacteristic
subgroupG/H
centerC22#16810757
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
Fitting, prime radicalC53×#3283C2
socleC22×C53#447
hypercenter#3283D53
hypocenterC53#12323
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ #3283

Complements

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

NH ≅ G/Ntype
1. #20276967C2semidirect
2. #20276760C2semidirect
3. #20274080C2semidirect
4. #20272764C2semidirect
5. #20272676C2semidirect
6. D4⋊#11719908C2semidirect
168. C53#12323semidirect

Maximal subgroups

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

subgroupindex
#202769672
#202767602
#202740802
#202727642
#202726762
D4⋊#117199082
C53×#32832
#1232353