Finite Group Databasebeta

← C2×#20272739 #27294367

Subgroups — C2×#20272739 #27294367

Order 13568.

712 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#16810390C211yesmaximal, characteristic
2C2×#16810347C211yesmaximal, characteristic
3#20272739C214yesmaximal, direct product factor
4C2×C53×#399C211yesmaximal, characteristic
5C22×C22⋊Dic53—21no—
6C2×C22⋊#88322—21no—
7#16810390—21no—
8#16810390C2212yes—
9#16810347—21no—
10#16810347C2212yes—
11C22×C4⋊C4×C53—21no—
12C22×C4⋊C4×C53C411yescharacteristic, complemented
13C22×C22⋊C4×C53—21no—
14C22×C22⋊C4×C53C2211yescharacteristic
15C53×#399—21no—
16C53×#399—21no—
17C53×#399C412yescomplemented
18C53×#399C2212yes—
19C2×#95×C53C411yescharacteristic, complemented
20C23×Dic53—41no—
21C22×#88322—41no—
22C2×C22⋊Dic53—24no—
23C2×C22⋊Dic53—22no—
24C2×C22⋊Dic53—41no—
25C2×C22⋊Dic53—42no—
26C22⋊#88322—22no—
27C22⋊#88322—41no—
28C25×C53D411yescharacteristic
29C23×C4×C53—21no—
30C23×C4×C53—21no—
31C23×C4×C53C2×C411yescharacteristic
32C23×C4×C53D411yescharacteristic
33C2×C4⋊C4×C53—22no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53—44no—
37C2×C4⋊C4×C53—41no—
38C2×C4⋊C4×C53—41no—
39C2×C4⋊C4×C53C2×C412yescomplemented
40C2×C22⋊C4×C53—22no—
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53—41no—
43C2×C22⋊C4×C53—41no—
44C2×C22⋊C4×C53—44no—
45C2×C22⋊C4×C53C2×C411yescharacteristic
46C2×C22⋊C4×C53D411yescharacteristic
47C2×C22⋊C4×C53D411yescharacteristic
48C2×C22⋊C4×C53C2311yescharacteristic
49#95×C53—21no—
50#95×C53C2×C412yescomplemented
51C24×C53—21no—
52C24×C53—21no—
53C24×C53—22no—
54C24×C53—42no—
55C24×C53—42no—
56C24×C53C22⋊C411yescharacteristic
57C24×C53C22⋊C411yescharacteristic
58C24×C53SD1612yes—
59C24×C53Q1612yes—
60C24×C53C2×D411yescharacteristic
61C22×C4×C53—24no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—22no—
66C22×C4×C53—21no—
67C22×C4×C53—22no—
68C22×C4×C53—42no—
69C22×C4×C53—42no—
70C22×C4×C53—42no—
71C22×C4×C53—42no—
72C22×C4×C53—42no—
73C22×C4×C53—42no—
74C22×C4×C53—41no—
75C22×C4×C53—41no—
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22×C411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×Dic53—44no—
81C22×Dic53—42no—
82C22×Dic53—82no—
83C22×Dic53—82no—
84C4⋊C4×C53—44no—
85C4⋊C4×C53—44no—
86C4⋊C4×C53—42no—
87C4⋊C4×C53—42no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—41no—
92C22⋊C4×C53—44no—
93C22⋊C4×C53—44no—
94C22⋊Dic53—24no—
95C22⋊Dic53—44no—
96C22⋊Dic53—42no—
97C2×#88322—42no—
98C2×#88322—82no—
99C23×C53—22no—
100C23×C53—22no—
101C23×C53—22no—
102C23×C53—22no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—22no—
106C23×C53—22no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—44no—
112C23×C53—44no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53—42no—
116C23×C53—42no—
117C23×C53—81no—
118C23×C53—84no—
119C23×C53—81no—
120C23×C53#9911yescharacteristic
121C23×C53#9911yescharacteristic
122C23×C53#10312yes—
123C23×C53#10312yes—
124C23×C53#10412yes—
125C23×C53C2×C22⋊C411yescharacteristic
126C23×C53C2×SD1611yescharacteristic
127C23×C53C2×Q1611yescharacteristic
128C2×C4×C53—24no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—42no—
134C2×C4×C53—48no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—42no—
144C2×C4×C53—42no—
145C2×C4×C53—42no—
146C2×C4×C53—81no—
147C2×C4×C53—82no—
148C2×C4×C53—82no—
149C2×Dic53—44no—
150C2×Dic53—82no—
151C2×Dic53—88no—
152C2×Dic53—82no—
153#88322—82no—
154C2×#1352—531nomaximal, Sylow
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—24no—
158C22×C53—24no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—22no—
163C22×C53—21no—
164C22×C53—42no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—44no—
170C22×C53—44no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—81no—
174C22×C53—81no—
175C22×C53—82no—
176C22×C53—82no—
177C22×C53#32812yes—
178C22×C53#32812yes—
179C22×C53#35212yes—
180C22×C53#35312yes—
181C22×C53C2×#9911yescharacteristic
182C22×C53C2×#10311yescharacteristic
183C22×C53C2×#10411yescharacteristic
184C4×C53—44no—
185C4×C53—41no—
186C4×C53—41no—
187C4×C53—41no—
188C4×C53—41no—
189C4×C53—82no—
190Dic53—84no—
191C2×#399D5311yescharacteristic
192C2×#342—531no—
193C2×#323—531no—
194#1352—534no—
195C2×C53—21no—
196C2×C53—21no—
197C2×C53—22no—
198C2×C53—41no—
199C2×C53—42no—
200C2×C53—41no—
201C2×C53#135214yescomplemented
202C2×C53C2×#32811yescharacteristic
203C2×C53C2×#35211yescharacteristic
204C2×C53C2×#35311yescharacteristic
205C22×C4⋊C4—21no—
206C22×C4⋊C4Dic5311yescharacteristic, complemented
207C22×C22⋊C4—21no—
208C22×C22⋊C4—1061no—
209C22×C22⋊C4C2×D5311yescharacteristic
210C2×C22⋊C8—1061no—
211#399—21no—
212#399—21no—
213#399Dic5312yescomplemented
214#399C2×D5312yes—
215C2×#95Dic5311yescharacteristic, complemented
216#342—532no—
217#342—1061no—
218#323—532no—
219#323—1061no—
220C53C2×#135211yescharacteristic, complemented, Sylow
221C25C22⋊D5311yescharacteristic
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—2121no—
225C23×C4C2×Dic5311yescharacteristic
226C23×C4C22⋊D5311yescharacteristic
227C22×C8—2121no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4C2×Dic5312yescomplemented
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—1064no—
241C2×C22⋊C4—1062no—
242C2×C22⋊C4—2121no—
243C2×C22⋊C4—2122no—
244C2×C22⋊C4C2×Dic5311yescharacteristic
245C2×C22⋊C4C22⋊D5311yescharacteristic
246C2×C22⋊C4C22⋊D5311yescharacteristic
247C2×C22⋊C4C22×D5311yescharacteristic
248C22⋊C8—1062no—
249C22⋊C8—2121no—
250#95—21no—
251#95C2×Dic5312yescomplemented
252C24—21no—
253C24—21no—
254C24—22no—
255C24—42no—
256C24—42no—
257C24#1171992412yes—
258C24#1171992512yes—
259C24C22⋊Dic5311yescharacteristic
260C24C22⋊Dic5311yescharacteristic
261C24C2×C22⋊D5311yescharacteristic
262C22×C4—24no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—4242no—
278C22×C4—4242no—
279C22×C4—2122no—
280C22×C4—2124no—
281C22×C4C22⋊Dic5311yescharacteristic
282C22×C4C22⋊Dic5311yescharacteristic
283C22×C4C22×Dic5311yescharacteristic
284C22×C4C2×C22⋊D5311yescharacteristic
285C2×C8—4242no—
286C2×C8—2122no—
287C4⋊C4—42no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—44no—
296C22⋊C4—44no—
297C22⋊C4—1064no—
298C22⋊C4—2124no—
299C22⋊C4—2122no—
300C23—22no—
301C23—22no—
302C23—21no—
303C23—21no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—44no—
314C23—44no—
315C23—41no—
316C23—42no—
317C23—42no—
318C23—84no—
319C23—81no—
320C23—81no—
321C23#1313313211yescharacteristic
322C23#1313313211yescharacteristic
323C23#1313314112yes—
324C23#1313314112yes—
325C23#1313314412yes—
326C23C2×C22⋊Dic5311yescharacteristic
327C23C2×#1171992411yescharacteristic
328C23C2×#1171992511yescharacteristic
329C2×C4—24no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—48no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—81no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—4248no—
351C2×C4—4242no—
352C2×C4—4242no—
353C2×C4—2124no—
354C8—4242no—
355C22—24no—
356C22—24no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—22no—
363C22—21no—
364C22—42no—
365C22—41no—
366C22—41no—
367C22—44no—
368C22—44no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—81no—
374C22—81no—
375C22—82no—
376C22—82no—
377C22#1681036012yes—
378C22#1681036012yes—
379C22#1681041312yes—
380C22#1681041612yes—
381C22C2×#1313313211yescharacteristic
382C22C2×#1313314111yescharacteristic
383C22C2×#1313314411yescharacteristic
384C4—44no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—82no—
390C4—4244no—
391C2—21no—
392C2—21no—
393C2—22no—
394C2—41no—
395C2—42no—
396C2—41no—
397C2#2027273914yesdirect product factor
398C2C2×#1681036011yescharacteristic
399C2C2×#1681041311yescharacteristic
400C2C2×#1681041611yescharacteristic
subgroupG/H
centerC23#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC2×C53×#399C2
socleC23×C53#99
prime radicalC25×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenterC2×#399D53
hypocenterC53C2×#1352
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
3. #20272739C2direct (N × H)
12. C22×C4⋊C4×C53C4semidirect
17. C53×#399C4semidirect
19. C2×#95×C53C4semidirect
39. C2×C4⋊C4×C53C2×C4semidirect
50. #95×C53C2×C4semidirect
201. C2×C53#1352semidirect
206. C22×C4⋊C4Dic53semidirect
213. #399Dic53semidirect
215. C2×#95Dic53semidirect
220. C53C2×#1352semidirect
234. C2×C4⋊C4C2×Dic53semidirect
251. #95C2×Dic53semidirect
397. C2#20272739direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168103902
C2×#168103472
#202727392
C2×C53×#3992
C2×#135253