Finite Group Databasebeta

← C2×#20276161 #27412989

Subgroups — C2×#20276161 #27412989

Order 13568.

950 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810622C212yesmaximal
2C2×#16810589C211yesmaximal, characteristic
3#20276161C218yesmaximal, direct product factor
4C2×#16810392C212yesmaximal, complemented
5C2×#16810388C211yesmaximal, characteristic, complemented
6C22×#106×C53C211yesmaximal, characteristic, complemented
7C22×#11719945C2211yescharacteristic
8C22×#11719917C2212yes—
9C22×#11719917C2211yescharacteristic
10#16810622—24no—
11#16810622C2218yes—
12C2×#13133140—22no—
13C2×#13133137—22no—
14#16810589—22no—
15#16810589C2214yes—
16C2×#13133129—22no—
17#16810392C2218yescomplemented
18#16810388C2214yescomplemented
19C23×C8×C53C2211yescharacteristic
20C22×C4⋊C4×C53C2212yes—
21C2×#106×C53—22no—
22C2×#106×C53—22no—
23C2×#106×C53C2214yescomplemented
24C2×#11719945—21no—
25C2×#11719945—21no—
26C2×#11719945—24no—
27C2×#11719945—24no—
28C2×#11719945D412yes—
29C2×#11719945C2312yes—
30C22×Dic106D411yescharacteristic
31C22×D212D411yescharacteristic
32C22×C22⋊D53—21no—
33C22×C4×D53—21no—
34C2×#11719917—21no—
35C2×#11719917—22no—
36C2×#11719917—24no—
37C2×#11719917—24no—
38C2×#11719917—24no—
39C2×#11719917C2312yes—
40C2×#11719917C2314yes—
41C22×#88322—21no—
42C22×#88322—22no—
43#13133140—28no—
44#13133137—28no—
45#13133129—28no—
46C23×C4×C53—22no—
47C23×C4×C53C2311yescharacteristic
48C22×C8×C53—24no—
49C22×C8×C53D412yes—
50C22×C8×C53D412yes—
51C22×C8×C53C2312yes—
52C2×C4⋊C4×C53—22no—
53C2×C4⋊C4×C53—22no—
54C2×C4⋊C4×C53—22no—
55C2×C4⋊C4×C53—22no—
56C2×C4⋊C4×C53—22no—
57C2×C4⋊C4×C53C2314yes—
58#106×C53—28no—
59C24×C53#4111yescharacteristic
60C23×D53—21no—
61C22×C4×C53—24no—
62C22×C4×C53—42no—
63C22×C4×C53—42no—
64C22×C4×C53—84no—
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53#4111yescharacteristic
70C22×C4×C53#4114yes—
71C22×C4×C53#4114yes—
72C22×C4×C53C2411yescharacteristic
73C2×C22⋊D53—22no—
74C2×C22⋊D53—22no—
75C2×C22⋊D53—44no—
76C22×Dic53—21no—
77#11719945—28no—
78#11719945—24no—
79#11719945—24no—
80#11719945—42no—
81#11719945—42no—
82#11719945—44no—
83C2×D212—24no—
84C2×D212—21no—
85C2×D212C2×D412yes—
86C2×C4×D53—22no—
87C2×C4×D53—22no—
88C2×C4×D53—42no—
89C2×C4×D53—42no—
90C2×Dic106—24no—
91C2×Dic106—21no—
92C2×Dic106C2×D412yes—
93C2×C8×C53—24no—
94C2×C8×C53—24no—
95C2×C8×C53—24no—
96C2×C8×C53C2×D414yes—
97C4⋊C4×C53—24no—
98C4⋊C4×C53—24no—
99C4⋊C4×C53—42no—
100C4⋊C4×C53—42no—
101#11719917—24no—
102#11719917—24no—
103#11719917—24no—
104#11719917—44no—
105#11719917—42no—
106C2×#88322—22no—
107C2×#88322—24no—
108C2×#88322—44no—
109C2×#88322—42no—
110C23×C53—24no—
111C23×C53#12111yescharacteristic
112C23×C53#12111yescharacteristic
113C23×C53#12614yes—
114C23×C53C2×#4111yescharacteristic
115C22×D53—22no—
116C22×D53—42no—
117C22×D53—44no—
118C2×C4×C53—44no—
119C2×C4×C53—84no—
120C2×C4×C53—84no—
121C2×C4×C53—84no—
122C2×C4×C53#12111yescharacteristic
123C2×C4×C53#12111yescharacteristic
124C2×C4×C53#12312yes—
125C2×C4×C53#12312yes—
126C2×C4×C53#12314yes—
127C2×C4×C53#12314yes—
128C2×C4×C53#12412yes—
129C2×C4×C53#12412yes—
130C2×C4×C53#12614yes—
131C2×C4×C53C22×D411yescharacteristic
132C2×C4×C53C22×D411yescharacteristic
133C2×C4×C53C2×#4112yes—
134C2×C4×C53C2×#4112yes—
135C22⋊D53—44no—
136C22⋊D53—44no—
137C2×Dic53—22no—
138C2×Dic53—42no—
139D212—24no—
140D212—42no—
141C4×D53—42no—
142C4×D53—42no—
143C4×D53—44no—
144Dic106—24no—
145Dic106—42no—
146C8×C53—24no—
147#88322—44no—
148#88322—42no—
149C2×#2079—531nomaximal, Sylow
150C22×C53—28no—
151C22×C53—22no—
152C22×C53—22no—
153C22×C53#39712yes—
154C22×C53#39712yes—
155C22×C53#47114yes—
156C22×C53C2×#12111yescharacteristic
157C22×C53C2×#12612yes—
158C2×D53—42no—
159C2×D53—44no—
160C2×D53—84no—
161C4×C53—84no—
162C4×C53#39712yes—
163C4×C53#39712yes—
164C4×C53C2×#12311yescharacteristic
165C4×C53C2×#12312yes—
166C4×C53C2×#12411yescharacteristic
167Dic53—42no—
168C22×#106D5311yescharacteristic, complemented
169C2×#417—532no—
170C2×#408—531no—
171#2079—538no—
172C2×#343—532no—
173C2×#341—531no—
174C2×C53—22no—
175C2×C53—22no—
176C2×C53#207914yescomplemented
177C2×C53C2×#39711yescharacteristic
178C2×C53C2×#47112yes—
179D53—82no—
180C22×#41—531no—
181C22×M16—531no—
182C22×M16—532no—
183C23×C8C2×D5311yescharacteristic
184C22×C4⋊C4C2×D5312yes—
185C2×#106—22no—
186C2×#106—22no—
187C2×#106C2×D5314yescomplemented
188#417—538no—
189#417—1064no—
190C2×#103—1062no—
191C2×#102—1062no—
192#408—534no—
193#408—1062no—
194C2×C22⋊C8—1062no—
195#343—538no—
196#341—534no—
197C53C2×#207911yescharacteristic, complemented, Sylow
198C2×#41—532no—
199C2×#41—532no—
200C2×#41—1061no—
201C2×#41—1061no—
202C2×#41—1064no—
203C2×#41—1064no—
204C22×Q8—531no—
205C22×D4—531no—
206C22×D4—1061no—
207C23×C4—22no—
208C23×C4—1061no—
209C23×C4C22×D5311yescharacteristic
210C2×M16—534no—
211C2×M16—532no—
212C2×M16—1062no—
213C2×M16—1061no—
214C2×M16—1064no—
215C2×M16—1064no—
216C2×M16—1064no—
217C22×C8—24no—
218C22×C8—1062no—
219C22×C8—1061no—
220C22×C8C22⋊D5312yes—
221C22×C8C22⋊D5312yes—
222C22×C8C22×D5312yes—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4C22×D5314yes—
229#106—28no—
230#103—1068no—
231#102—1068no—
232C22⋊C8—1068no—
233C24—1061no—
234C24#1171994711yescharacteristic
235#41—1068no—
236#41—1064no—
237#41—1064no—
238#41—2122no—
239#41—2122no—
240#41—2124no—
241C2×Q8—532no—
242C2×Q8—1061no—
243C2×Q8—1064no—
244C2×D4—532no—
245C2×D4—1062no—
246C2×D4—1062no—
247C2×D4—1061no—
248C2×D4—1064no—
249C2×D4—2124no—
250C22×C4—24no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—84no—
254C22×C4—1061no—
255C22×C4—1062no—
256C22×C4—1062no—
257C22×C4—2122no—
258C22×C4—2122no—
259C22×C4#1171994514yes—
260C22×C4D4×D5311yescharacteristic
261C22×C4D4×D5311yescharacteristic
262C22×C4#1171994711yescharacteristic
263C22×C4#1171994914yes—
264C22×C4C2×C22⋊D5311yescharacteristic
265C22×C4C2×C22⋊D5311yescharacteristic
266C22×C4C23×D5311yescharacteristic
267M16—1064no—
268M16—1064no—
269M16—1064no—
270M16—2124no—
271M16—2122no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—1064no—
276C2×C8—1062no—
277C2×C8—2124no—
278C2×C8—2122no—
279C2×C8C2×C22⋊D5314yes—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C23—24no—
285C23—1062no—
286C23—2122no—
287C23—2124no—
288C23#1313318811yescharacteristic
289C23#1313318811yescharacteristic
290C23#1313321214yes—
291C23C2×#1171994711yescharacteristic
292Q8—1064no—
293Q8—2122no—
294D4—1064no—
295D4—2124no—
296D4—2124no—
297D4—2122no—
298C2×C4—44no—
299C2×C4—84no—
300C2×C4—84no—
301C2×C4—84no—
302C2×C4—1062no—
303C2×C4—2122no—
304C2×C4—2124no—
305C2×C4—2122no—
306C2×C4—2122no—
307C2×C4#1313318811yescharacteristic
308C2×C4#1313318811yescharacteristic
309C2×C4#1313320012yes—
310C2×C4#1313320012yes—
311C2×C4#1313320214yes—
312C2×C4#1313320214yes—
313C2×C4#1313320612yes—
314C2×C4#1313320612yes—
315C2×C4#1313321214yes—
316C2×C4C2×D4×D5311yescharacteristic
317C2×C4C22×C22⋊D5311yescharacteristic
318C2×C4C2×#1171994512yes—
319C2×C4C2×#1171994912yes—
320C8—24no—
321C8—2124no—
322C8—2122no—
323C22—28no—
324C22—22no—
325C22—22no—
326C22—4244no—
327C22—2122no—
328C22—2124no—
329C22#1681054912yes—
330C22#1681054912yes—
331C22#1681088314yes—
332C22C2×#1313318811yescharacteristic
333C22C2×#1313321212yes—
334C4—84no—
335C4—2122no—
336C4#1681054912yes—
337C4#1681054912yes—
338C4C2×#1313320011yescharacteristic
339C4C2×#1313320212yes—
340C4C2×#1313320611yescharacteristic
341C2—22no—
342C2—22no—
343C2—4242no—
344C2#2027616114yesdirect product factor
345C2C2×#1681054911yescharacteristic
346C2C2×#1681088312yes—
subgroupG/H
centerC23#13133188
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC22×#106×C53C2
socleC23×C53#121
prime radicalC22×#11719945C22
hypercenterC22×#106D53
hypocenterC53C2×#2079
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#106

Complements

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

NH ≅ G/Ntype
3. #20276161C2direct (N × H)
4. C2×#16810392C2semidirect
5. C2×#16810388C2semidirect
6. C22×#106×C53C2semidirect
17. #16810392C22semidirect
18. #16810388C22semidirect
23. C2×#106×C53C22semidirect
168. C22×#106D53semidirect
176. C2×C53#2079semidirect
187. C2×#106C2×D53semidirect
197. C53C2×#2079semidirect
344. C2#20276161direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168106222
C2×#168105892
#202761612
C2×#168103922
C2×#168103882
C22×#106×C532
C2×#207953