Finite Group Databasebeta

← C2×#20273757 #27356327

Subgroups — C2×#20273757 #27356327

Order 13568.

984 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810956C212yesmaximal
2C2×#16810562C211yesmaximal, characteristic
3#20273757C218yesmaximal, direct product factor
4C2×#16810360C212yesmaximal
5C2×#16810349C211yesmaximal, characteristic
6C2×C53×#544C211yesmaximal, characteristic
7C22×C4⋊Dic53C2211yescharacteristic
8#16810956—24no—
9#16810956C2218yes—
10C2×#13133150—22no—
11C2×#13133141—22no—
12C2×C22⋊#88322C2212yes—
13#16810562—22no—
14#16810562C2214yes—
15C2×#13133124—21no—
16C2×#13133124—21no—
17C2×#13133124C2211yescharacteristic
18#16810360C2218yes—
19#16810349C2214yes—
20C53×#544—24no—
21C53×#544—22no—
22C53×#544C2214yes—
23C2×C4⋊Q8×C53—22no—
24C2×#125×C53—22no—
25C2×C22⋊Q8×C53—21no—
26C2×C22⋊Q8×C53C2212yes—
27C2×#117×C53—21no—
28C22×C4⋊C4×C53C2211yescharacteristic
29C23×Dic53—21no—
30C23×Dic53—21no—
31C22×#88322—22no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—22no—
34C2×C4⋊Dic53—22no—
35C2×C4⋊Dic53C2312yes—
36#13133150—28no—
37#13133141—28no—
38C22⋊#88322—22no—
39C22⋊#88322C2314yes—
40#13133124—21no—
41#13133124—24no—
42#13133124—24no—
43#13133124C2312yes—
44C22×Q8×C53—21no—
45C22×Q8×C53D412yes—
46C23×C4×C53D411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C23×C4×C53C2311yescharacteristic
49C4⋊Q8×C53—28no—
50C4⋊Q8×C53—28no—
51#125×C53—28no—
52#125×C53—28no—
53C22⋊Q8×C53—24no—
54C22⋊Q8×C53—24no—
55C22⋊Q8×C53—22no—
56C22⋊Q8×C53—22no—
57C22⋊Q8×C53D414yes—
58C22⋊Q8×C53C2314yes—
59#117×C53—24no—
60#117×C53—24no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—22no—
65C2×C4⋊C4×C53—22no—
66C2×C4⋊C4×C53—22no—
67C2×C4⋊C4×C53—22no—
68C2×C4⋊C4×C53D412yes—
69C2×C4⋊C4×C53D412yes—
70C2×C4⋊C4×C53C2312yes—
71C2×C22⋊C4×C53—22no—
72C2×C42×C53—22no—
73C24×C53C2×D411yescharacteristic
74C2×Q8×C53—24no—
75C2×Q8×C53—22no—
76C2×Q8×C53—44no—
77C2×Q8×C53—82no—
78C2×Q8×C53C2×D414yes—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—22no—
82C22×C4×C53—22no—
83C22×C4×C53—22no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—42no—
87C22×C4×C53—42no—
88C22×C4×C53Q1614yes—
89C22×C4×C53C2×D412yes—
90C22×C4×C53C2×D412yes—
91C22×C4×C53C2×D412yes—
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D412yes—
94C22×C4×C53C2411yescharacteristic
95C22×Dic53—22no—
96C22×Dic53—22no—
97C22×Dic53—42no—
98C22×Dic53—42no—
99C22×Dic53—44no—
100C22×Dic53—41no—
101C22×Dic53—41no—
102C22×Dic53—42no—
103C4⋊C4×C53—24no—
104C4⋊C4×C53—24no—
105C4⋊C4×C53—24no—
106C4⋊C4×C53—28no—
107C4⋊C4×C53—24no—
108C4⋊C4×C53—24no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—24no—
111C4⋊C4×C53—24no—
112C4⋊C4×C53C2×D414yes—
113C22⋊C4×C53—24no—
114C22⋊C4×C53—24no—
115C42×C53—28no—
116C4⋊Dic53—24no—
117C4⋊Dic53—42no—
118C2×#88322—24no—
119C2×#88322—44no—
120C23×C53—22no—
121C23×C53—22no—
122C23×C53#12011yescharacteristic
123C23×C53#12011yescharacteristic
124C23×C53C2×Q1612yes—
125C23×C53#13712yes—
126C23×C53C22×D411yescharacteristic
127Q8×C53—44no—
128Q8×C53—82no—
129Q8×C53—84no—
130C2×C4×C53—22no—
131C2×C4×C53—22no—
132C2×C4×C53—24no—
133C2×C4×C53—24no—
134C2×C4×C53—24no—
135C2×C4×C53—22no—
136C2×C4×C53—24no—
137C2×C4×C53—24no—
138C2×C4×C53—42no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—44no—
142C2×C4×C53—42no—
143C2×C4×C53—44no—
144C2×C4×C53—42no—
145C2×C4×C53—81no—
146C2×C4×C53#12012yes—
147C2×C4×C53C2×Q1614yes—
148C2×C4×C53C22×D412yes—
149C2×Dic53—42no—
150C2×Dic53—42no—
151C2×Dic53—44no—
152C2×Dic53—42no—
153C2×Dic53—42no—
154C2×Dic53—84no—
155C2×Dic53—82no—
156C2×Dic53—82no—
157#88322—44no—
158C2×#1617—531nomaximal, Sylow
159C22×C53—24no—
160C22×C53—24no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53#45112yes—
166C22×C53#45112yes—
167C22×C53#45412yes—
168C22×C53#45812yes—
169C22×C53C2×#12011yescharacteristic
170C22×C53C22×Q1611yescharacteristic
171C22×C53C2×#13711yescharacteristic
172C4×C53—24no—
173C4×C53—22no—
174C4×C53—22no—
175C4×C53—44no—
176C4×C53—42no—
177C4×C53—44no—
178Dic53—82no—
179Dic53—82no—
180C2×#544D5311yescharacteristic
181C2×#484—532no—
182C2×#400—531no—
183#1617—538no—
184C2×#328—532no—
185C2×#324—531no—
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—22no—
189C2×C53#161714yescomplemented
190C2×C53C2×#45111yescharacteristic
191C2×C53C2×#45411yescharacteristic
192C2×C53C2×#45811yescharacteristic
193#544—24no—
194#544—22no—
195#544C2×D5314yes—
196C2×C4⋊Q8—22no—
197C2×#125—22no—
198C2×C22⋊Q8—21no—
199C2×C22⋊Q8C2×D5312yes—
200C2×#117—21no—
201C22×C4⋊C4—531no—
202C22×C4⋊C4C2×D5311yescharacteristic
203#484—538no—
204#484—1064no—
205C2×#107—1062no—
206C2×#103—1062no—
207C2×C22⋊C8—532no—
208#400—534no—
209#400—1062no—
210C2×#95—531no—
211C2×#95—1061no—
212C2×#95—1061no—
213#328—538no—
214#324—534no—
215C53C2×#161711yescharacteristic, complemented, Sylow
216C22×Q8—21no—
217C22×Q8C22⋊D5312yes—
218C23×C4—1061no—
219C23×C4—1061no—
220C23×C4C22⋊D5311yescharacteristic
221C23×C4C22⋊D5311yescharacteristic
222C23×C4C22×D5311yescharacteristic
223C22×C8—1062no—
224C4⋊Q8—28no—
225C4⋊Q8—28no—
226#125—28no—
227#125—28no—
228C22⋊Q8—24no—
229C22⋊Q8—24no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8C22⋊D5314yes—
233C22⋊Q8C22×D5314yes—
234#117—24no—
235#117—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—532no—
244C2×C4⋊C4—1062no—
245C2×C4⋊C4—1062no—
246C2×C4⋊C4—1061no—
247C2×C4⋊C4C22⋊D5312yes—
248C2×C4⋊C4C22⋊D5312yes—
249C2×C4⋊C4C22×D5312yes—
250C2×C22⋊C4—22no—
251C2×C42—22no—
252#107—1068no—
253#103—1068no—
254C22⋊C8—534no—
255C22⋊C8—1062no—
256#95—532no—
257#95—1061no—
258#95—1064no—
259#95—1064no—
260C24C2×C22⋊D5311yescharacteristic
261C2×Q8—24no—
262C2×Q8—22no—
263C2×Q8—44no—
264C2×Q8—82no—
265C2×Q8C2×C22⋊D5314yes—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—1062no—
276C22×C4—1062no—
277C22×C4—2122no—
278C22×C4—2122no—
279C22×C4—2124no—
280C22×C4—2121no—
281C22×C4—2121no—
282C22×C4—2122no—
283C22×C4#1171992514yes—
284C22×C4C2×C22⋊D5312yes—
285C22×C4C2×C22⋊D5312yes—
286C22×C4C2×C22⋊D5312yes—
287C22×C4C2×C22⋊D5311yescharacteristic
288C22×C4C2×C22⋊D5312yes—
289C22×C4C23×D5311yescharacteristic
290C2×C8—1064no—
291C2×C8—2124no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—28no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—1064no—
302C4⋊C4—2122no—
303C4⋊C4C2×C22⋊D5314yes—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C42—28no—
307C23—22no—
308C23—22no—
309C23#1313318611yescharacteristic
310C23#1313318611yescharacteristic
311C23C2×#1171992512yes—
312C23#1313324812yes—
313C23C22×C22⋊D5311yescharacteristic
314Q8—44no—
315Q8—82no—
316Q8—84no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—81no—
333C2×C4—4244no—
334C2×C4—4242no—
335C2×C4—4242no—
336C2×C4—2122no—
337C2×C4—2122no—
338C2×C4—2124no—
339C2×C4—2122no—
340C2×C4—2122no—
341C2×C4#1313318612yes—
342C2×C4C2×#1171992514yes—
343C2×C4C22×C22⋊D5312yes—
344C8—2124no—
345C22—24no—
346C22—24no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22#1681077912yes—
352C22#1681077912yes—
353C22#1681079612yes—
354C22#1681081312yes—
355C22C2×#1313318611yescharacteristic
356C22C22×#1171992511yescharacteristic
357C22C2×#1313324811yescharacteristic
358C4—24no—
359C4—22no—
360C4—22no—
361C4—44no—
362C4—42no—
363C4—44no—
364C4—4242no—
365C4—4242no—
366C2—21no—
367C2—22no—
368C2—21no—
369C2#2027375714yesdirect product factor
370C2C2×#1681077911yescharacteristic
371C2C2×#1681079611yescharacteristic
372C2C2×#1681081311yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#544C2
socleC23×C53#120
prime radicalC24×C53C2×D4
hypercenterC2×#544D53
hypocenterC53C2×#1617
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #20273757C2direct (N × H)
189. C2×C53#1617semidirect
215. C53C2×#1617semidirect
369. C2#20273757direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168109562
C2×#168105622
#202737572
C2×#168103602
C2×#168103492
C2×C53×#5442
C2×#161753