Finite Group Databasebeta

← C2×#20272646 #27293190

Subgroups — C2×#20272646 #27293190

Order 13568.

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

normal subgroups only

This group has 834 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×#16810524C211yesmaximal, characteristic
2C2×#16810349C211yesmaximal, characteristic, complemented
3#20272646C214yesmaximal, direct product factor
4C2×C53×#324C211yesmaximal, characteristic, complemented
5C23×D212C411yescharacteristic
6C22×#11719921—21no—
7C22×#11719921—21no—
8C22×C4×Dic53C411yescharacteristic
9C2×C22⋊#88322—21no—
10#16810524—21no—
11#16810524—21no—
12#16810524C412yes—
13#16810524C2212yes—
14#16810349—21no—
15#16810349C2212yescomplemented
16C22×C4⋊C4×C53C2211yescharacteristic
17C2×C22⋊C8×C53—21no—
18C53×#324—21no—
19C53×#324C2212yescomplemented
20C24×D53—21no—
21C22×D212—21no—
22C22×D212—22no—
23C22×D212—41no—
24C22×D212—41no—
25C22×D212—42no—
26C22×D212C814yescomplemented
27C22×D212C2×C412yes—
28C23×Dic53—21no—
29C22×#88322—41no—
30C2×#11719921—22no—
31C2×#11719921—22no—
32C2×#11719921—22no—
33C2×#11719921—22no—
34C2×#11719921—44no—
35C2×#11719921—44no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—42no—
38C2×C4×Dic53C2×C412yes—
39C22⋊#88322—22no—
40C22⋊#88322—41no—
41C23×C4×C53C2×C411yescharacteristic
42C23×C4×C53D411yescharacteristic
43C23×C4×C53D411yescharacteristic
44C22×C8×C53—41no—
45C2×C4⋊C4×C53—41no—
46C2×C4⋊C4×C53—41no—
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C4⋊C4×C53C2311yescharacteristic
51C22⋊C8×C53—22no—
52C22⋊C8×C53—41no—
53C24×C53C22⋊C411yescharacteristic
54C23×D53—24no—
55C23×D53—22no—
56C23×D53—42no—
57C23×D53—42no—
58C23×D53—42no—
59C23×D53—42no—
60C23×D53—41no—
61C23×D53—41no—
62C23×D53—42no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—22no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—42no—
71C22×C4×C53—42no—
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C2×C812yes—
76C22×C4×C53M1612yes—
77C22×C4×C53C22×C411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×Dic53—24no—
81C22×Dic53—22no—
82C22×Dic53—42no—
83C22×Dic53—42no—
84C2×D212—24no—
85C2×D212—22no—
86C2×D212—44no—
87C2×D212—44no—
88C2×D212—42no—
89C2×D212—42no—
90C2×D212—44no—
91C2×D212—44no—
92C2×D212—42no—
93C2×D212—42no—
94C2×D212C2×C814yescomplemented
95C2×C8×C53—42no—
96C2×C8×C53—82no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—41no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101#11719921—44no—
102#11719921—44no—
103#11719921—44no—
104#11719921—44no—
105C4×Dic53—42no—
106C4×Dic53—42no—
107C2×#88322—42no—
108C2×#88322—82no—
109C23×C53—22no—
110C23×C53—22no—
111C23×C53C22⋊C812yes—
112C23×C53#9911yescharacteristic
113C23×C53#9911yescharacteristic
114C23×C53#10012yes—
115C23×C53C2×C22⋊C411yescharacteristic
116C22×D53—24no—
117C22×D53—48no—
118C22×D53—48no—
119C22×D53—42no—
120C22×D53—42no—
121C22×D53—42no—
122C22×D53—42no—
123C22×D53—44no—
124C22×D53—44no—
125C22×D53—42no—
126C22×D53—42no—
127C22×D53—84no—
128C22×D53—84no—
129C22×D53—82no—
130C22×D53—82no—
131C22×D53—82no—
132C22×D53—82no—
133C2×C4×C53—22no—
134C2×C4×C53—22no—
135C2×C4×C53—22no—
136C2×C4×C53—22no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—81no—
152C2×C4×C53—81no—
153C2×C4×C53C22⋊C812yes—
154C2×C4×C53C22×C811yescharacteristic
155C2×C4×C53C2×M1611yescharacteristic
156C2×Dic53—24no—
157C2×Dic53—48no—
158C2×Dic53—42no—
159C2×Dic53—42no—
160D212—44no—
161D212—48no—
162D212—44no—
163C8×C53—82no—
164#88322—82no—
165C2×#1318—531nomaximal, Sylow
166C22×C53—24no—
167C22×C53—24no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53#32312yes—
173C22×C53#32312yes—
174C22×C53#35312yes—
175C22×C53#35512yes—
176C22×C53C2×C22⋊C811yescharacteristic
177C22×C53C2×#9911yescharacteristic
178C22×C53C2×#10011yescharacteristic
179C2×D53—44no—
180C2×D53—44no—
181C2×D53—44no—
182C2×D53—82no—
183C2×D53—82no—
184C2×D53—88no—
185C2×D53—88no—
186C2×D53—82no—
187C2×D53—82no—
188C4×C53—22no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—41no—
192C4×C53—41no—
193C4×C53—41no—
194C4×C53—41no—
195Dic53—44no—
196C2×#390—531no—
197C2×#324—531no—
198C2×#324D5311yescharacteristic, complemented
199#1318—534no—
200C2×C53—21no—
201C2×C53—21no—
202C2×C53—22no—
203C2×C53#131814yescomplemented
204C2×C53C2×#32311yescharacteristic
205C2×C53C2×#35311yescharacteristic
206C2×C53C2×#35511yescharacteristic
207D53—84no—
208C23×D4—531no—
209C22×C4⋊C4C2×D5311yescharacteristic
210C22×C22⋊C4—1061no—
211C22×C22⋊C4—1061no—
212C22×C42—531no—
213C2×C22⋊C8—21no—
214C2×C22⋊C8—1061no—
215#390—532no—
216#390—532no—
217#390—1061no—
218#390—1061no—
219#324—21no—
220#324—532no—
221#324—1061no—
222#324C2×D5312yescomplemented
223C53C2×#131811yescharacteristic, complemented, Sylow
224C25—1061no—
225C22×D4—532no—
226C22×D4—534no—
227C22×D4—1062no—
228C22×D4—1061no—
229C22×D4—2121no—
230C22×D4—2121no—
231C22×D4—2122no—
232C23×C4—1061no—
233C23×C4C4×D5311yescharacteristic
234C23×C4D21211yescharacteristic
235C23×C4C22⋊D5311yescharacteristic
236C22×C8—41no—
237C22×C8—2121no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4C4×D5311yescharacteristic
241C2×C4⋊C4D21211yescharacteristic
242C2×C4⋊C4C22⋊D5311yescharacteristic
243C2×C4⋊C4C22×D5311yescharacteristic
244C2×C22⋊C4—1062no—
245C2×C22⋊C4—1062no—
246C2×C22⋊C4—1062no—
247C2×C22⋊C4—1062no—
248C2×C22⋊C4—2124no—
249C2×C22⋊C4—2124no—
250C2×C42—532no—
251C2×C42—1061no—
252C2×C42—2122no—
253C22⋊C8—22no—
254C22⋊C8—41no—
255C22⋊C8—1062no—
256C22⋊C8—2121no—
257C24—1064no—
258C24—1062no—
259C24—2122no—
260C24—2122no—
261C24—2122no—
262C24—2122no—
263C24—2121no—
264C24—2121no—
265C24—2122no—
266C24#1171992111yescharacteristic
267C2×D4—534no—
268C2×D4—1064no—
269C2×D4—1062no—
270C2×D4—2122no—
271C2×D4—2122no—
272C2×D4—2124no—
273C2×D4—2124no—
274C2×D4—2122no—
275C2×D4—2122no—
276C2×D4—2124no—
277C2×D4—2124no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—1064no—
288C22×C4—1062no—
289C22×C4—2122no—
290C22×C4—2122no—
291C22×C4C8×D5312yes—
292C22×C4#1171991212yes—
293C22×C4#1171992111yescharacteristic
294C22×C4#1171992111yescharacteristic
295C22×C4#1171992111yescharacteristic
296C22×C4C2×C4×D5311yescharacteristic
297C22×C4C2×D21211yescharacteristic
298C22×C4C2×C22⋊D5311yescharacteristic
299C2×C8—42no—
300C2×C8—82no—
301C2×C8—4242no—
302C2×C8—2122no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C22⋊C4—2124no—
308C22⋊C4—2124no—
309C22⋊C4—2124no—
310C22⋊C4—2124no—
311C42—2122no—
312C42—2122no—
313C23—22no—
314C23—22no—
315C23—1064no—
316C23—4244no—
317C23—4244no—
318C23—4242no—
319C23—4242no—
320C23—4242no—
321C23—4242no—
322C23—2122no—
323C23—2122no—
324C23—2128no—
325C23—2128no—
326C23—2122no—
327C23—2122no—
328C23—2122no—
329C23—2122no—
330C23—2124no—
331C23—2124no—
332C23#1313312912yes—
333C23#1313313111yescharacteristic
334C23#1313313111yescharacteristic
335C23#1313313312yes—
336C23C2×#1171992111yescharacteristic
337D4—2124no—
338D4—2128no—
339D4—2124no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—1064no—
361C2×C4—2128no—
362C2×C4—2122no—
363C2×C4—2122no—
364C2×C4#1313312912yes—
365C2×C4C2×C8×D5311yescharacteristic
366C2×C4C2×#1171991211yescharacteristic
367C8—82no—
368C8—4242no—
369C22—24no—
370C22—24no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—4242no—
376C22—4242no—
377C22—4248no—
378C22—4248no—
379C22—4242no—
380C22—4242no—
381C22—2124no—
382C22—2124no—
383C22—2124no—
384C22#1681034612yes—
385C22#1681034612yes—
386C22#1681041512yes—
387C22#1681041912yes—
388C22C2×#1313312911yescharacteristic
389C22C2×#1313313111yescharacteristic
390C22C2×#1313313311yescharacteristic
391C4—22no—
392C4—21no—
393C4—21no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—2124no—
399C2—21no—
400C2—22no—
401C2—21no—
402C2—4244no—
403C2#2027264614yesdirect product factor
404C2C2×#1681034611yescharacteristic
405C2C2×#1681041511yescharacteristic
406C2C2×#1681041911yescharacteristic
subgroupG/H
centerC23#13133131
commutatorC2×C4×C53C22×C8
FrattiniC2×C4⋊C4C22×D53
FittingC2×C53×#324C2
socleC23×C53#99
prime radicalC23×D212C4
cosocleC2×C4⋊C4×C53C23
hypercenterC2×#324D53
hypocenterC53C2×#1318
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. C2×#16810349C2semidirect
3. #20272646C2direct (N × H)
4. C2×C53×#324C2semidirect
15. #16810349C22semidirect
19. C53×#324C22semidirect
26. C22×D212C8semidirect
94. C2×D212C2×C8semidirect
198. C2×#324D53semidirect
203. C2×C53#1318semidirect
222. #324C2×D53semidirect
223. C53C2×#1318semidirect
403. C2#20272646direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168105242
C2×#168103492
#202726462
C2×C53×#3242
C2×#131853