Finite Group Databasebeta

← #27276625

Subgroups — #27276625

Order 13568.

694 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20273432C212yesmaximal, complemented
2#20273342C212yesmaximal, complemented
3#20273049C211yesmaximal, characteristic
4#20273049C211yesmaximal, characteristic
5C2×C53×#544C211yesmaximal, characteristic
6#16810680C2212yes—
7#16810680C2212yes—
8#16810369—24no—
9#16810365—24no—
10#16810349—21no—
11#16810349—21no—
12#16810347—21no—
13#16810347—21no—
14C53×#544—24no—
15C53×#544—24no—
16C2×C4⋊Q8×C53—21no—
17C2×C4⋊Q8×C53C411yescharacteristic
18C2×C4⋊Q8×C53C2211yescharacteristic
19C2×#125×C53—21no—
20C2×#125×C53C411yescharacteristic
21C2×#125×C53C2211yescharacteristic
22C2×C22⋊Q8×C53—21no—
23C2×C22⋊Q8×C53—21no—
24C2×#117×C53C411yescharacteristic
25C2×#117×C53C2211yescharacteristic
26C22×C4⋊C4×C53C411yescharacteristic
27C4⋊#88322—22no—
28C4⋊#88322—22no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31#13133128—22no—
32#13133128—22no—
33C22×Q8×C53—21no—
34C22×Q8×C53—21no—
35C23×C4×C53—21no—
36C23×C4×C53C2×C411yescharacteristic
37C4⋊Q8×C53—28no—
38C4⋊Q8×C53—22no—
39C4⋊Q8×C53—22no—
40C4⋊Q8×C53—22no—
41C4⋊Q8×C53D414yes—
42#125×C53—28no—
43#125×C53—22no—
44#125×C53—22no—
45#125×C53—22no—
46#125×C53D414yes—
47C22⋊Q8×C53—28no—
48C22⋊Q8×C53—28no—
49#117×C53—22no—
50#117×C53—22no—
51#117×C53—22no—
52#117×C53—22no—
53C2×C4⋊C4×C53—22no—
54C2×C4⋊C4×C53—22no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53C2×C411yescharacteristic
63C2×C4⋊C4×C53C2×C411yescharacteristic
64C2×C4⋊C4×C53C2×C411yescharacteristic
65C2×C4⋊C4×C53C2×C411yescharacteristic
66C2×C4⋊C4×C53D411yescharacteristic
67C2×C4⋊C4×C53D411yescharacteristic
68C2×C22⋊C4×C53—21no—
69C2×C22⋊C4×C53D411yescharacteristic
70C2×C22⋊C4×C53D411yescharacteristic
71C2×C42×C53—21no—
72C2×C42×C53C2×C411yescharacteristic
73C2×C42×C53C2311yescharacteristic
74C24×C53C22⋊C411yescharacteristic
75C2×Q8×C53—24no—
76C2×Q8×C53—24no—
77C2×Q8×C53—82no—
78C2×Q8×C53—82no—
79C22×C4×C53—22no—
80C22×C4×C53—22no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—24no—
88C22×C4×C53—82no—
89C22×C4×C53C22⋊C411yescharacteristic
90C22×C4×C53C22⋊C411yescharacteristic
91C22×C4×C53C22⋊C411yescharacteristic
92C22×C4×C53C22×C411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C4⋊C4×C53—24no—
96C4⋊C4×C53—24no—
97C4⋊C4×C53—24no—
98C4⋊C4×C53—24no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—24no—
104C4⋊C4×C53—24no—
105C4⋊C4×C53—24no—
106C4⋊C4×C53—22no—
107C4⋊C4×C53—22no—
108C4⋊C4×C53C22⋊C412yes—
109C4⋊C4×C53C22⋊C412yes—
110C4⋊C4×C53C22⋊C412yes—
111C4⋊C4×C53C22⋊C412yes—
112C22⋊C4×C53—24no—
113C22⋊C4×C53—22no—
114C22⋊C4×C53—22no—
115C42×C53—24no—
116C42×C53—21no—
117C42×C53—21no—
118C42×C53C2×D412yes—
119C42×C53C2×D412yes—
120C2×#88322—42no—
121C2×#88322—42no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53#10112yes—
127C23×C53C2×C22⋊C411yescharacteristic
128Q8×C53—84no—
129Q8×C53—84no—
130C2×C4×C53—24no—
131C2×C4×C53—24no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—22no—
139C2×C4×C53—22no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—42no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—84no—
151C2×C4×C53#10112yes—
152C2×C4×C53C2×C22⋊C411yescharacteristic
153C2×C4×C53C2×C22⋊C411yescharacteristic
154C2×C4×C53#12012yes—
155C2×C4×C53#12012yes—
156#88322—82no—
157#88322—82no—
158#8525—531nomaximal, Sylow
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—22no—
162C22×C53—42no—
163C22×C53—42no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53#37911yescharacteristic
167C22×C53C2×#10111yescharacteristic
168C22×C53C2×#10111yescharacteristic
169C4×C53—22no—
170C4×C53—22no—
171C4×C53—22no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—42no—
175C4×C53—42no—
176C4×C53—42no—
177C4×C53—41no—
178C4×C53—41no—
179C4×C53—42no—
180C2×#544D5311yescharacteristic
181#1556—532no—
182#1534—532no—
183#1464—531no—
184#1464—531no—
185C2×C53—21no—
186C2×C53—21no—
187C2×C53—42no—
188C2×C53#178411yescharacteristic
189C2×C53#179511yescharacteristic
190C2×C53#179611yescharacteristic
191#544—24no—
192#544—24no—
193C2×C4⋊Q8—21no—
194C2×C4⋊Q8Dic5311yescharacteristic
195C2×C4⋊Q8C2×D5311yescharacteristic
196C2×#125—21no—
197C2×#125Dic5311yescharacteristic
198C2×#125C2×D5311yescharacteristic
199C2×C22⋊Q8—21no—
200C2×C22⋊Q8—21no—
201C2×#117Dic5311yescharacteristic
202C2×#117C2×D5311yescharacteristic
203C22×C4⋊C4Dic5311yescharacteristic
204#432—532no—
205#432—532no—
206#332—1064no—
207#330—1064no—
208#324—1061no—
209#324—1061no—
210#323—1061no—
211#323—1061no—
212C53#852511yescharacteristic, complemented, Sylow
213C22×Q8—21no—
214C22×Q8—21no—
215C23×C4—21no—
216C23×C4C2×Dic5311yescharacteristic
217C4⋊Q8—28no—
218C4⋊Q8—22no—
219C4⋊Q8—22no—
220C4⋊Q8—22no—
221C4⋊Q8C22⋊D5314yes—
222#125—28no—
223#125—22no—
224#125—22no—
225#125—22no—
226#125C22⋊D5314yes—
227C22⋊Q8—28no—
228C22⋊Q8—28no—
229#117—22no—
230#117—22no—
231#117—22no—
232#117—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4C2×Dic5311yescharacteristic
243C2×C4⋊C4C2×Dic5311yescharacteristic
244C2×C4⋊C4C2×Dic5311yescharacteristic
245C2×C4⋊C4C2×Dic5311yescharacteristic
246C2×C4⋊C4C22⋊D5311yescharacteristic
247C2×C4⋊C4C22⋊D5311yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4C22⋊D5311yescharacteristic
250C2×C22⋊C4C22⋊D5311yescharacteristic
251C2×C42—21no—
252C2×C42C2×Dic5311yescharacteristic
253C2×C42C22×D5311yescharacteristic
254C4⋊C8—1062no—
255C4⋊C8—1062no—
256C22⋊C8—1062no—
257C22⋊C8—1062no—
258#97—1062no—
259#97—1062no—
260C24C22⋊Dic5311yescharacteristic
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—82no—
264C2×Q8—82no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—24no—
274C22×C4—82no—
275C22×C4C22⋊Dic5311yescharacteristic
276C22×C4C22⋊Dic5311yescharacteristic
277C22×C4C22⋊Dic5311yescharacteristic
278C22×C4C22×Dic5311yescharacteristic
279C22×C4C2×C22⋊D5311yescharacteristic
280C22×C4C2×C22⋊D5311yescharacteristic
281C2×C8—2122no—
282C2×C8—2122no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4C22⋊Dic5312yes—
297C4⋊C4C22⋊Dic5312yes—
298C4⋊C4C22⋊Dic5312yes—
299C4⋊C4C22⋊Dic5312yes—
300C22⋊C4—24no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C42—24no—
304C42—21no—
305C42—21no—
306C42C2×C22⋊D5312yes—
307C42C2×C22⋊D5312yes—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23#1313313612yes—
313C23C2×C22⋊Dic5311yescharacteristic
314Q8—84no—
315Q8—84no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—84no—
337C2×C4#1313313612yes—
338C2×C4C2×C22⋊Dic5311yescharacteristic
339C2×C4C2×C22⋊Dic5311yescharacteristic
340C2×C4#1313318612yes—
341C2×C4#1313318612yes—
342C8—4242no—
343C8—4242no—
344C22—21no—
345C22—21no—
346C22—22no—
347C22—42no—
348C22—42no—
349C22—41no—
350C22—41no—
351C22#1681047911yescharacteristic
352C22C2×#1313313611yescharacteristic
353C22C2×#1313313611yescharacteristic
354C4—22no—
355C4—22no—
356C4—21no—
357C4—21no—
358C4—22no—
359C4—42no—
360C4—41no—
361C4—42no—
362C4—42no—
363C4—42no—
364C4—41no—
365C2—21no—
366C2—21no—
367C2—42no—
368C2#2027463911yescharacteristic
369C2#2027468511yescharacteristic
370C2#2027469011yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#544C2
socleC22×C53#379
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenterC2×#544D53
hypocenterC53#8525
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
1. #20273432C2semidirect
2. #20273342C2semidirect
212. C53#8525semidirect

Maximal subgroups

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

subgroupindex
#202734322
#202733422
#202730492
C2×C53×#5442
#852553