Finite Group Databasebeta

← #27281948

Subgroups — #27281948

Order 13568.

518 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 105 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#20283755C211yesmaximal, characteristic
2#20280073C211yesmaximal, characteristic
3#20274704C211yesmaximal, characteristic
4#20273144C211yesmaximal, characteristic, complemented
5#20272575C211yesmaximal, characteristic, complemented
6#20272574C211yesmaximal, characteristic, complemented
7C53×#1487C211yesmaximal, characteristic, complemented
8C2×C4×D212C2211yescharacteristic
9#16810702—22no—
10#16810700—22no—
11#16810676C2211yescharacteristic
12#16810647C2211yescharacteristic
13C2×#13133133—21no—
14#16810585—22no—
15#16810558C412yes—
16#16810534C412yes—
17#16810525—21no—
18#16810521—21no—
19#16810494—22no—
20#16810489C2211yescharacteristic
21#16810391—21no—
22Q8⋊#88322—22no—
23C2×C4×Q8×C53C2211yescharacteristic
24C53×#431C2211yescharacteristic
25C53×#423C2211yescharacteristic
26C53×Q8⋊C8—22no—
27C22×D212D411yescharacteristic
28C22×C4×D53D412yes—
29C2×#11719917—21no—
30C2×#11719912—24no—
31C4×D212—21no—
32C4×D212—21no—
33C4×D212—21no—
34C4×D212—21no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53D411yescharacteristic
37C2×#11719919—22no—
38C2×#11719921—22no—
39C2×#11719921—22no—
40C2×#11719921C2×C412yes—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44#13133133—42no—
45C22⋊#88322—21no—
46#13133129—24no—
47C4×#88322D412yes—
48#13133124—21no—
49#13133124—22no—
50#13133124C2×C412yes—
51C22×Q8×C53C2×C411yescharacteristic
52C2×M16×C53—21no—
53C4×Q8×C53—21no—
54C4×Q8×C53—21no—
55C4×Q8×C53—21no—
56C4×Q8×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53C2×C411yescharacteristic
59C2×C42×C53—21no—
60C2×C42×C53C2311yescharacteristic
61C4⋊C8×C53—21no—
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64C22⋊C8×C53—21no—
65C4×C8×C53D412yes—
66C23×D53C22⋊C412yes—
67C2×Q8×C53—21no—
68C2×Q8×C53—21no—
69C2×Q8×C53—42no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×Dic53—22no—
76C22×Dic53C22⋊C412yes—
77C2×D212—21no—
78C2×D212—21no—
79C2×D212—21no—
80C2×D212—21no—
81C2×D212—21no—
82C2×D212—21no—
83C2×C4×D53—22no—
84C2×C4×D53—22no—
85C2×C4×D53—82no—
86M16×C53—41no—
87M16×C53—41no—
88C2×C8×C53—22no—
89C2×C8×C53—41no—
90C4⋊C4×C53—21no—
91C4⋊C4×C53—21no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—41no—
94C42×C53—41no—
95C42×C53—41no—
96C42×C53C2×D411yescharacteristic
97C42×C53C2×D411yescharacteristic
98C42×C53#4111yescharacteristic
99C42×C53#4111yescharacteristic
100#11719921—22no—
101#11719921—22no—
102#11719921—44no—
103#11719921—82no—
104C4⋊Dic53—21no—
105C4⋊Dic53—21no—
106C4⋊Dic53—41no—
107C4⋊Dic53—41no—
108#11719919—82no—
109#11719917—41no—
110#11719917—41no—
111C2×#88322—22no—
112C2×#88322—41no—
113#11719912—84no—
114C23×C53C2×C22⋊C411yescharacteristic
115C22×D53—22no—
116C22×D53—22no—
117C22×D53—22no—
118C22×D53—42no—
119C22×D53—42no—
120Q8×C53—22no—
121Q8×C53—22no—
122Q8×C53—41no—
123Q8×C53—41no—
124C2×C4×C53—22no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53C4×D411yescharacteristic
133C2×C4×C53C4×D411yescharacteristic
134C2×C4×C53#12011yescharacteristic
135C2×C4×C53#12111yescharacteristic
136C2×C4×C53#12111yescharacteristic
137C2×C4×C53#12311yescharacteristic
138C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C53#13711yescharacteristic
140C2×C4×C53#13711yescharacteristic
141C2×C4×C53#13711yescharacteristic
142C2×Dic53—22no—
143C2×Dic53—42no—
144C2×Dic53—42no—
145C2×Dic53—42no—
146C2×Dic53—42no—
147D212—41no—
148D212—41no—
149D212—41no—
150D212—41no—
151C4×D53—44no—
152C4×D53—82no—
153C8×C53—42no—
154C8×C53—81no—
155#88322—42no—
156#88322—81no—
157#9487—531nomaximal, Sylow
158C22×C53—21no—
159C22×C53—21no—
160C22×C53#38611yescharacteristic
161C22×C53#41711yescharacteristic
162C22×C53#41911yescharacteristic
163C2×D53—44no—
164C2×D53—42no—
165C2×D53—42no—
166C2×D53—42no—
167C2×D53—82no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—42no—
173C4×C53—81no—
174C4×C53#46811yescharacteristic
175C4×C53#46811yescharacteristic
176C4×C53#47011yescharacteristic
177C4×C53#47011yescharacteristic
178Dic53—42no—
179Dic53—82no—
180#2975—531no—
181#2580—531no—
182#1800—531no—
183#1487—531no—
184#1487D5311yescharacteristic, complemented
185#1286—531no—
186#1286—531no—
187C2×C53—21no—
188C2×C53—21no—
189C2×C53#189111yescharacteristic
190C2×C53#189411yescharacteristic
191C2×C53#191111yescharacteristic
192D53—82no—
193C2×C4×Q8C2×D5311yescharacteristic
194C2×C4×D4—531no—
195#436—1062no—
196#436—1062no—
197#431—531no—
198#431C2×D5311yescharacteristic
199#423—531no—
200#423C2×D5311yescharacteristic
201C2×#100—1061no—
202#407—1062no—
203#399—532no—
204#393—532no—
205#390—1061no—
206#389—1061no—
207#385—1062no—
208#382—531no—
209#343—1061no—
210Q8⋊C8—22no—
211Q8⋊C8—1062no—
212C53#948711yescharacteristic, complemented, Sylow
213C22×Q8C4×D5311yescharacteristic
214C22×D4—531no—
215C23×C4—532no—
216C2×M16—21no—
217C2×M16—1064no—
218C2×M16—1061no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×D4—1061no—
224C4×D4—1061no—
225C4×D4—1061no—
226C4×D4—1061no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—531no—
229C2×C4⋊C4—1062no—
230C2×C4⋊C4—1061no—
231C2×C4⋊C4C4×D5311yescharacteristic
232C2×C22⋊C4—532no—
233C2×C22⋊C4—1062no—
234C2×C22⋊C4—1062no—
235C2×C42—21no—
236C2×C42C22×D5311yescharacteristic
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—1061no—
241C4⋊C8—1061no—
242C4⋊C8—1061no—
243#100—2122no—
244C22⋊C8—21no—
245C22⋊C8—1064no—
246C22⋊C8—1061no—
247C4×C8—532no—
248C4×C8C22⋊D5312yes—
249#95—532no—
250#95—1062no—
251#95—1061no—
252C24—532no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—42no—
256C2×D4—1061no—
257C2×D4—1061no—
258C2×D4—1061no—
259C2×D4—1061no—
260C2×D4—1061no—
261C2×D4—1061no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—532no—
265C22×C4—1062no—
266C22×C4—1062no—
267C22×C4—1062no—
268C22×C4—4242no—
269C22×C4C2×C4×D5311yescharacteristic
270C22×C4D4×D5311yescharacteristic
271C22×C4D4×D5311yescharacteristic
272M16—41no—
273M16—41no—
274M16—4244no—
275M16—2121no—
276M16—2121no—
277C2×C8—22no—
278C2×C8—41no—
279C2×C8—1062no—
280C2×C8—2121no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—1061no—
286C4⋊C4—1061no—
287C4⋊C4—4242no—
288C4⋊C4—2121no—
289C4⋊C4—2121no—
290C22⋊C4—1062no—
291C22⋊C4—1062no—
292C22⋊C4—4242no—
293C22⋊C4—2124no—
294C42—41no—
295C42—41no—
296C42#1171994511yescharacteristic
297C42D4×D5311yescharacteristic
298C42#1171994711yescharacteristic
299C42C2×C22⋊D5311yescharacteristic
300C23—1062no—
301C23—1062no—
302C23—1062no—
303C23—2122no—
304C23—2122no—
305C23C22⋊C4×D5311yescharacteristic
306Q8—22no—
307Q8—22no—
308Q8—41no—
309Q8—41no—
310D4—2121no—
311D4—2121no—
312D4—2121no—
313D4—2121no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—1062no—
323C2×C4—4242no—
324C2×C4—2124no—
325C2×C4—2122no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4—2122no—
329C2×C4#1313317711yescharacteristic
330C2×C4C4×C22⋊D5311yescharacteristic
331C2×C4#1313318511yescharacteristic
332C2×C4#1313318811yescharacteristic
333C2×C4#1313319011yescharacteristic
334C2×C4#1313319911yescharacteristic
335C2×C4#1313324411yescharacteristic
336C2×C4#1313324611yescharacteristic
337C2×C4#1313324911yescharacteristic
338C2×C4#1313324911yescharacteristic
339C8—42no—
340C8—81no—
341C8—4241no—
342C8—2122no—
343C22—21no—
344C22—21no—
345C22—4242no—
346C22—2124no—
347C22—2122no—
348C22—2122no—
349C22—2122no—
350C22#1681049911yescharacteristic
351C22#1681062311yescharacteristic
352C22#1681063011yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—42no—
358C4—81no—
359C4—4242no—
360C4—2122no—
361C4#1681086611yescharacteristic
362C4#1681087211yescharacteristic
363C4#1681087811yescharacteristic
364C4#1681088011yescharacteristic
365C2—21no—
366C2—21no—
367C2—4242no—
368C2#2027513011yescharacteristic
369C2#2027514511yescharacteristic
370C2#2027524411yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1487C2
socleC22×C53#386
prime radicalC22×D212D4
cosocleC2×C42×C53C23
hypercenter#1487D53
hypocenterC53#9487
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
4. #20273144C2semidirect
5. #20272575C2semidirect
6. #20272574C2semidirect
7. C53×#1487C2semidirect
184. #1487D53semidirect
212. C53#9487semidirect

Maximal subgroups

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

subgroupindex
#202837552
#202800732
#202747042
#202731442
#202725752
#202725742
C53×#14872
#948753