Finite Group Databasebeta

← #27400871

Subgroups — #27400871

Order 13568.

1066 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811358C211yesmaximal, characteristic
2M16×Dic106C212yesmaximal, complemented
3#20275563C214yesmaximal, complemented
4#20275555C212yesmaximal
5#20274711C211yesmaximal, characteristic, complemented
6#20274706C212yesmaximal, complemented
7#20274506C212yesmaximal, complemented
8C4×#101×C53C211yesmaximal, characteristic, complemented
9#16811358—22no—
10#16811358—22no—
11#16811358—24no—
12C2×#13133214C2211yescharacteristic
13C2×#13133206C2212yes—
14C2×#13133205C412yes—
15C2×#13133202C414yes—
16C2×C4×Dic106C2211yescharacteristic
17C2×C4×D212C412yes—
18C2×#13133179C2212yes—
19#16810753—22no—
20#16810751—24no—
21C8×Dic106—22no—
22#16810647C2212yes—
23#16810646C2214yes—
24C2×#13133136C2212yes—
25C2×#13133133C2214yes—
26C2×#13133133C2214yes—
27M16×Dic53C2214yes—
28#16810392C2212yes—
29C2×C4×Q8×C53C2211yescharacteristic
30C2×#101×C53C2212yes—
31C4×M16×C53C2212yes—
32C53×#343C2211yescharacteristic
33C53×#343C2211yescharacteristic
34C22×Dic106C2×C411yescharacteristic
35C22×D212C2×C412yes—
36C22×D212C2×C412yes—
37C22×D212C2312yes—
38C22×C4×D53C2×C414yes—
39C2×#11719917C2314yes—
40#13133214—22no—
41#13133214—22no—
42#13133206—24no—
43#13133206—24no—
44#13133205—24no—
45#13133205—24no—
46#13133202—28no—
47#13133202—28no—
48C4×Dic106D412yes—
49C4×Dic106D412yes—
50C4×Dic106D414yes—
51C4×D212—24no—
52C4×D212—22no—
53C4×D212—22no—
54#13133179—24no—
55#13133179—24no—
56C2×C4⋊Dic53C2×C411yescharacteristic
57C2×#11719919C2×C412yes—
58C2×#11719921C2×C414yes—
59C2×#11719921C2×C414yes—
60C2×#11719921C2×C414yes—
61C2×C4×Dic53C2312yes—
62C4⋊#88322—22no—
63C4⋊#88322—22no—
64#13133146—24no—
65#13133146—24no—
66#13133136—42no—
67#13133133—28no—
68#13133133—44no—
69#13133127—24no—
70C8×Dic53—24no—
71C22×Q8×C53C2311yescharacteristic
72C2×M16×C53C2312yes—
73C2×M16×C53C2312yes—
74C4×Q8×C53—22no—
75C4×Q8×C53—22no—
76C2×C4⋊C4×C53C2×C412yes—
77C2×C4⋊C4×C53C2311yescharacteristic
78C2×C42×C53C2×C412yes—
79C2×C42×C53C2311yescharacteristic
80#101×C53—24no—
81#97×C53—22no—
82C4×C8×C53—22no—
83C23×D53C22×C414yes—
84C2×Q8×C53—22no—
85C2×Q8×C53—22no—
86C2×Q8×C53—21no—
87C2×Q8×C53—21no—
88C22×C4×C53C22×C412yes—
89C22×C4×C53C22×C411yescharacteristic
90C22×C4×C53C22×C412yes—
91C22×C4×C53C22×C411yescharacteristic
92C22×C4×C53C2411yescharacteristic
93C22×Dic53C22×C412yes—
94C22×Dic53C22×C412yes—
95C2×D212—22no—
96C2×D212—22no—
97C2×D212—24no—
98C2×D212—24no—
99C2×D212—22no—
100C2×D212—22no—
101C2×D212—24no—
102C2×D212—44no—
103C2×D212—82no—
104C2×C4×D53—24no—
105C2×C4×D53—24no—
106C2×C4×D53—84no—
107C2×Dic106—42no—
108C2×Dic106C22⋊C412yes—
109C2×Dic106C22⋊C412yes—
110M16×C53—22no—
111M16×C53—22no—
112M16×C53—22no—
113M16×C53#4114yes—
114C2×C8×C53—22no—
115C2×C8×C53—22no—
116C4⋊C4×C53—24no—
117C4⋊C4×C53—22no—
118C42×C53—22no—
119C42×C53—22no—
120C42×C53C2×D412yes—
121C42×C53C2×D412yes—
122#11719921—28no—
123#11719921—24no—
124#11719921—24no—
125#11719921—28no—
126C4⋊Dic53C22⋊C412yes—
127C4⋊Dic53C22⋊C411yescharacteristic
128C4⋊Dic53C22⋊C411yescharacteristic
129#11719919C22⋊C414yes—
130#11719919C22⋊C414yes—
131C4×Dic53C2×D414yes—
132C4×Dic53C2×D414yes—
133#11719917—24no—
134#11719917—24no—
135C2×#88322—24no—
136C23×C53C23×C411yescharacteristic
137C22×D53—24no—
138C22×D53—24no—
139C22×D53—24no—
140C22×D53—44no—
141C22×D53—44no—
142Q8×C53—42no—
143Q8×C53—42no—
144C2×C4×C53—22no—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53—22no—
148C2×C4×C53—22no—
149C2×C4×C53—41no—
150C2×C4×C53—42no—
151C2×C4×C53C2×C22⋊C411yescharacteristic
152C2×C4×C53C2×C22⋊C412yes—
153C2×C4×C53C2×C22⋊C411yescharacteristic
154C2×C4×C53C22×D412yes—
155C2×C4×C53C2×#4112yes—
156C2×C4×C53H(5,𝔽2)11yescharacteristic
157C2×C4×C532−1+411yescharacteristic
158C2×Dic53—42no—
159C2×Dic53—42no—
160C2×Dic53C2×C22⋊C414yes—
161C2×Dic53C2×C22⋊C412yes—
162C2×Dic53C2×C22⋊C412yes—
163D212—44no—
164D212—44no—
165D212—44no—
166D212—44no—
167D212—84no—
168C4×D53—84no—
169C4×D53—84no—
170Dic106—24no—
171Dic106—42no—
172C8×C53—24no—
173C8×C53—42no—
174#88322—44no—
175#22539—531nomaximal, Sylow
176C22×C53—21no—
177C22×C53—21no—
178C22×C53C22×C22⋊C411yescharacteristic
179C22×C53C4×#4111yescharacteristic
180C22×C53#52011yescharacteristic
181C2×D53—44no—
182C2×D53—44no—
183C2×D53—44no—
184C2×D53—84no—
185C2×D53—84no—
186C4×C53—21no—
187C4×C53—22no—
188C4×C53—42no—
189C4×C53—42no—
190C4×C53#41312yes—
191C4×C53#54812yes—
192C4×C53#55012yes—
193Dic53—24no—
194Dic53—22no—
195Dic53—22no—
196C2×#559—531no—
197Q8×M16—532no—
198#1967—534no—
199#1965—532no—
200#1801—531no—
201#1800—532no—
202C4×#101D5311yescharacteristic, complemented
203C4×#100—532no—
204C2×C53—21no—
205C2×C53—21no—
206C2×C53#234111yescharacteristic
207C2×C53C2×#41311yescharacteristic
208C2×C53#288811yescharacteristic
209D53—84no—
210#559—1062no—
211#559—1064no—
212#559—1062no—
213C2×#127—531no—
214C2×#124—532no—
215C2×#124—532no—
216C2×#123—534no—
217C2×C4×Q8—531no—
218C2×C4×Q8C2×D5311yescharacteristic
219C2×C4×D4—532no—
220C2×C4×D4—532no—
221#446—1062no—
222#446—1064no—
223C8×Q8—1062no—
224#423—534no—
225#423—532no—
226C2×#101—532no—
227C2×#101C2×D5312yes—
228C2×#100—534no—
229C2×#100—534no—
230C4×M16—534no—
231C4×M16C2×D5312yes—
232#343—532no—
233#343C2×D5311yescharacteristic
234#343C2×D5311yescharacteristic
235C53#2253911yescharacteristic, complemented, Sylow
236C22×Q8—531no—
237C22×Q8C22×D5311yescharacteristic
238C22×D4—532no—
239C22×D4—532no—
240C22×D4—532no—
241C23×C4—534no—
242C2×M16—534no—
243C2×M16C22×D5312yes—
244C2×M16C22×D5312yes—
245#127—1062no—
246#127—1062no—
247#124—1064no—
248#124—1064no—
249#124—1064no—
250#124—1064no—
251#123—1068no—
252#123—1068no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—532no—
256C4×Q8—534no—
257C4×Q8—532no—
258C4×D4—1064no—
259C4×D4—1062no—
260C4×D4—1062no—
261C4×D4—1064no—
262C4×D4—1064no—
263C2×C4⋊C4—531no—
264C2×C4⋊C4—532no—
265C2×C4⋊C4C4×D5312yes—
266C2×C4⋊C4C22×D5311yescharacteristic
267C2×C22⋊C4—534no—
268C2×C22⋊C4—534no—
269C2×C22⋊C4—534no—
270C2×C42—532no—
271C2×C42C4×D5312yes—
272C2×C42C22×D5311yescharacteristic
273C4⋊C8—1064no—
274C4⋊C8—1064no—
275C4⋊C8—1062no—
276C4⋊C8—1062no—
277#101—24no—
278#101—2122no—
279#100—1068no—
280#100—2124no—
281#97—22no—
282#97—1064no—
283C4×C8—22no—
284C4×C8—1064no—
285C24—534no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—532no—
291C2×Q8—532no—
292C2×Q8—2122no—
293C2×D4—1064no—
294C2×D4—1062no—
295C2×D4—1062no—
296C2×D4—1064no—
297C2×D4—1064no—
298C2×D4—1062no—
299C2×D4—1062no—
300C2×D4—4242no—
301C2×D4—2124no—
302C22×C4—532no—
303C22×C4—532no—
304C22×C4—1064no—
305C22×C4—1064no—
306C22×C4—4244no—
307C22×C4C2×C4×D5312yes—
308C22×C4C2×C4×D5311yescharacteristic
309C22×C4C2×C4×D5312yes—
310C22×C4C2×C4×D5311yescharacteristic
311C22×C4C23×D5311yescharacteristic
312M16—22no—
313M16—22no—
314M16—22no—
315M16—1064no—
316M16—1064no—
317M16#1171994514yes—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—1064no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—532no—
324C4⋊C4—531no—
325C4⋊C4—531no—
326C4⋊C4—534no—
327C4⋊C4—534no—
328C22⋊C4—1064no—
329C22⋊C4—1064no—
330C22⋊C4—1068no—
331C22⋊C4—1068no—
332C42—22no—
333C42—22no—
334C42—534no—
335C42—534no—
336C42D4×D5312yes—
337C42D4×D5312yes—
338C23—1064no—
339C23—1064no—
340C23—1064no—
341C23—2124no—
342C23—2124no—
343C23C22×C4×D5311yescharacteristic
344Q8—42no—
345Q8—42no—
346Q8—1064no—
347Q8—2122no—
348D4—4244no—
349D4—2124no—
350D4—2124no—
351D4—2124no—
352D4—2124no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—534no—
361C2×C4—532no—
362C2×C4—532no—
363C2×C4—4244no—
364C2×C4—4244no—
365C2×C4—2122no—
366C2×C4—2122no—
367C2×C4C22⋊C4×D5311yescharacteristic
368C2×C4C22⋊C4×D5312yes—
369C2×C4C22⋊C4×D5311yescharacteristic
370C2×C4C2×D4×D5312yes—
371C2×C4C2×#1171994512yes—
372C2×C4#1313326311yescharacteristic
373C2×C4#1313326411yescharacteristic
374C8—24no—
375C8—42no—
376C8—2124no—
377C22—21no—
378C22—21no—
379C22—4244no—
380C22—4244no—
381C22—2124no—
382C22—2124no—
383C22—2124no—
384C22C2×C22⋊C4×D5311yescharacteristic
385C22C4×#1171994511yescharacteristic
386C22#1681110311yescharacteristic
387C4—21no—
388C4—22no—
389C4—42no—
390C4—42no—
391C4—1064no—
392C4—1062no—
393C4—1062no—
394C4#1681060612yes—
395C4#1681128212yes—
396C4#1681129412yes—
397C2—21no—
398C2—21no—
399C2—4244no—
400C2#2027755511yescharacteristic
401C2C2×#1681060611yescharacteristic
402C2#2028319211yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC22×C4C23×D53
FittingC4×#101×C53C2
socleC22×C53C22×C22⋊C4
prime radicalC2×#13133214C22
cosocleC22×C4×C53C24
hypercenterC4×#101D53
hypocenterC53#22539
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
2. M16×Dic106C2semidirect
3. #20275563C2semidirect
5. #20274711C2semidirect
6. #20274706C2semidirect
7. #20274506C2semidirect
8. C4×#101×C53C2semidirect
202. C4×#101D53semidirect
235. C53#22539semidirect

Maximal subgroups

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

subgroupindex
C2×#168113582
M16×Dic1062
#202755632
#202755552
#202747112
#202747062
#202745062
C4×#101×C532
#2253953