Finite Group Databasebeta

← C2×#20273688 #27356275

Subgroups — C2×#20273688 #27356275

Order 13568.

1200 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1200 conjugacy classes of subgroups — too many to draw. Only its 203 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×#16811254C211yesmaximal, characteristic
2C2×#16810779C212yesmaximal
3#20273688C218yesmaximal, direct product factor
4C2×#16810359C212yesmaximal
5C2×#16810349C211yesmaximal, characteristic
6C2×C53×#395C211yesmaximal, characteristic
7C22×#11719925—22no—
8#16811254—24no—
9#16811254—22no—
10#16811254C2214yes—
11C2×#13133216—22no—
12C2×#13133208—22no—
13C2×#13133196—21no—
14C2×#13133196C2212yes—
15C2×#13133176—21no—
16#16810779—24no—
17#16810779C2218yes—
18C2×#13133141—22no—
19C2×C22⋊#88322C2212yes—
20#16810359C2218yes—
21#16810349C2214yes—
22C23×Q8×C53C2211yescharacteristic
23C22×C4⋊C4×C53—21no—
24C22×C4⋊C4×C53C2211yescharacteristic
25C53×#395—22no—
26C53×#395C2214yes—
27C2×#95×C53—21no—
28C2×#95×C53C2211yescharacteristic
29C22×Dic106—21no—
30C22×Dic106D412yes—
31C2×#11719925—28no—
32C2×#11719925—48no—
33C22×#88322—22no—
34#13133216—28no—
35#13133216—28no—
36#13133208—28no—
37#13133208—28no—
38#13133196—24no—
39#13133196—24no—
40#13133196—22no—
41#13133196—22no—
42#13133196D414yes—
43#13133196C2314yes—
44#13133176—24no—
45#13133176—24no—
46C2×C4⋊Dic53—21no—
47C2×C4⋊Dic53—21no—
48C2×C4⋊Dic53—21no—
49C2×C4⋊Dic53D412yes—
50C2×#11719919—22no—
51C2×#11719919—22no—
52C2×C22⋊Dic53—22no—
53C2×C4×Dic53—22no—
54#13133141—28no—
55C22⋊#88322—22no—
56C22⋊#88322C2314yes—
57C22×Q8×C53—21no—
58C22×Q8×C53—22no—
59C22×Q8×C53—22no—
60C22×Q8×C53—44no—
61C22×Q8×C53C2312yes—
62C23×C4×C53—21no—
63C23×C4×C53—21no—
64C23×C4×C53D411yescharacteristic
65C23×C4×C53D411yescharacteristic
66C23×C4×C53C2311yescharacteristic
67C2×C4⋊C4×C53—22no—
68C2×C4⋊C4×C53—22no—
69C2×C4⋊C4×C53—24no—
70C2×C4⋊C4×C53—44no—
71C2×C4⋊C4×C53D412yes—
72C2×C4⋊C4×C53C2312yes—
73#95×C53—24no—
74#95×C53—21no—
75#95×C53C2312yes—
76C24×C53C2×D411yescharacteristic
77C2×Q8×C53—28no—
78C2×Q8×C53—24no—
79C2×Q8×C53—44no—
80C2×Q8×C53—44no—
81C2×Q8×C53—44no—
82C2×Q8×C53—48no—
83C2×Q8×C53—42no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—22no—
87C22×C4×C53—24no—
88C22×C4×C53—24no—
89C22×C4×C53—21no—
90C22×C4×C53—22no—
91C22×C4×C53—22no—
92C22×C4×C53—21no—
93C22×C4×C53—42no—
94C22×C4×C53—42no—
95C22×C4×C53—42no—
96C22×C4×C53—41no—
97C22×C4×C53—41no—
98C22×C4×C53—42no—
99C22×C4×C53—42no—
100C22×C4×C53Q1614yescomplemented
101C22×C4×C53C2×D412yes—
102C22×C4×C53C2×D412yes—
103C22×C4×C53C2×D412yes—
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D412yes—
106C22×C4×C53C2411yescharacteristic
107C22×Dic53—22no—
108C22×Dic53—22no—
109C2×Dic106—24no—
110C2×Dic106—22no—
111C2×Dic106—44no—
112C2×Dic106—82no—
113C2×Dic106C2×D414yes—
114C4⋊C4×C53—24no—
115C4⋊C4×C53—24no—
116C4⋊C4×C53—48no—
117C22⋊Dic53—24no—
118C22⋊Dic53—24no—
119#11719925—416no—
120C4⋊Dic53—24no—
121C4⋊Dic53—24no—
122C4⋊Dic53—24no—
123C4⋊Dic53—22no—
124C4⋊Dic53C2×D414yes—
125#11719919—28no—
126#11719919—24no—
127#11719919—24no—
128C4×Dic53—28no—
129C2×#88322—24no—
130C2×#88322—44no—
131C23×C53—22no—
132C23×C53—22no—
133C23×C53#12011yescharacteristic
134C23×C53#12011yescharacteristic
135C23×C53C2×Q1612yes—
136C23×C53#13712yes—
137C23×C53C22×D411yescharacteristic
138Q8×C53—28no—
139Q8×C53—44no—
140Q8×C53—48no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—24no—
144C2×C4×C53—24no—
145C2×C4×C53—24no—
146C2×C4×C53—24no—
147C2×C4×C53—24no—
148C2×C4×C53—22no—
149C2×C4×C53—42no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—48no—
153C2×C4×C53—42no—
154C2×C4×C53—44no—
155C2×C4×C53—42no—
156C2×C4×C53—42no—
157C2×C4×C53—42no—
158C2×C4×C53—42no—
159C2×C4×C53—44no—
160C2×C4×C53—42no—
161C2×C4×C53—81no—
162C2×C4×C53#12012yes—
163C2×C4×C53C2×Q1614yescomplemented
164C2×C4×C53C22×D412yes—
165C2×Dic53—24no—
166C2×Dic53—24no—
167C2×Dic53—44no—
168C2×Dic53—44no—
169Dic106—44no—
170Dic106—82no—
171Dic106—84no—
172#88322—44no—
173C2×#1605—531nomaximal, Sylow
174C22×C53—24no—
175C22×C53—24no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53#45112yes—
181C22×C53#45112yes—
182C22×C53#45612yes—
183C22×C53B(4,𝔽2)12yes—
184C22×C53C2×#12011yescharacteristic
185C22×C53C22×Q1611yescharacteristic
186C22×C53C2×#13711yescharacteristic
187C4×C53—24no—
188C4×C53—22no—
189C4×C53—22no—
190C4×C53—44no—
191C4×C53—42no—
192C4×C53—44no—
193Dic53—44no—
194Dic53—44no—
195C2×#544—531no—
196C2×#451—532no—
197C2×#395D5311yescharacteristic
198#1605—538no—
199C2×#328—532no—
200C2×#324—531no—
201C2×C53—21no—
202C2×C53—21no—
203C2×C53—22no—
204C2×C53#160514yescomplemented
205C2×C53C2×#45111yescharacteristic
206C2×C53C2×#45611yescharacteristic
207C2×C53C2×B(4,𝔽2)11yescharacteristic
208C23×Q8C2×D5311yescharacteristic
209C22×Q16—1062no—
210#544—534no—
211#544—1064no—
212#544—1062no—
213C2×C4⋊Q8—1062no—
214C2×#125—1062no—
215C2×C22⋊Q8—532no—
216C2×C22⋊Q8—1061no—
217C2×#117—1061no—
218C22×C4⋊C4—21no—
219C22×C4⋊C4C2×D5311yescharacteristic
220#451—538no—
221#451—1064no—
222C2×#103—1062no—
223C2×C22⋊C8—532no—
224#395—22no—
225#395C2×D5314yes—
226C2×#95—21no—
227C2×#95C2×D5311yescharacteristic
228#328—538no—
229#324—534no—
230C53C2×#160511yescharacteristic, complemented, Sylow
231C22×Q8—21no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—44no—
235C22×Q8—532no—
236C22×Q8—1061no—
237C22×Q8C22×D5312yes—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4C22⋊D5311yescharacteristic
241C23×C4C22⋊D5311yescharacteristic
242C23×C4C22×D5311yescharacteristic
243C2×Q16—1068no—
244C2×Q16—2128no—
245C22×C8—1062no—
246C4⋊Q8—1068no—
247C4⋊Q8—1068no—
248#125—1068no—
249#125—1068no—
250C22⋊Q8—534no—
251C22⋊Q8—534no—
252C22⋊Q8—1064no—
253C22⋊Q8—1062no—
254C22⋊Q8—1062no—
255C22⋊Q8—1064no—
256#117—1064no—
257#117—1064no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—532no—
263C2×C4⋊C4—1061no—
264C2×C4⋊C4—1061no—
265C2×C4⋊C4—1061no—
266C2×C4⋊C4—1062no—
267C2×C4⋊C4—1062no—
268C2×C4⋊C4C22⋊D5312yes—
269C2×C4⋊C4C22×D5312yes—
270C2×C22⋊C4—1062no—
271C2×C42—1062no—
272#103—1068no—
273C22⋊C8—534no—
274C22⋊C8—1062no—
275#95—24no—
276#95—21no—
277#95C22×D5312yes—
278C24C2×C22⋊D5311yescharacteristic
279C2×Q8—28no—
280C2×Q8—24no—
281C2×Q8—44no—
282C2×Q8—44no—
283C2×Q8—44no—
284C2×Q8—48no—
285C2×Q8—42no—
286C2×Q8—534no—
287C2×Q8—1064no—
288C2×Q8—1062no—
289C2×Q8—4242no—
290C2×Q8—2124no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—1062no—
308C22×C4—1062no—
309C22×C4#1171992514yescomplemented
310C22×C4D4×D5312yes—
311C22×C4D4×D5312yes—
312C22×C4C2×C22⋊D5312yes—
313C22×C4C2×C22⋊D5312yes—
314C22×C4C2×C22⋊D5311yescharacteristic
315C22×C4C23×D5311yescharacteristic
316Q16—21216no—
317C2×C8—1064no—
318C2×C8—2124no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—48no—
322C4⋊C4—534no—
323C4⋊C4—1064no—
324C4⋊C4—1064no—
325C4⋊C4—1064no—
326C4⋊C4—1068no—
327C4⋊C4—1062no—
328C4⋊C4—1064no—
329C4⋊C4—1064no—
330C22⋊C4—1064no—
331C22⋊C4—1064no—
332C42—1068no—
333C23—22no—
334C23—22no—
335C23#1313318511yescharacteristic
336C23#1313318511yescharacteristic
337C23C2×#1171992512yes—
338C23#1313324712yes—
339C23C22×C22⋊D5311yescharacteristic
340Q8—28no—
341Q8—44no—
342Q8—48no—
343Q8—4242no—
344Q8—4244no—
345Q8—2124no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—48no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—81no—
367C2×C4—1064no—
368C2×C4—1064no—
369C2×C4—2124no—
370C2×C4—2124no—
371C2×C4#1313318512yes—
372C2×C4C2×#1171992514yescomplemented
373C2×C4C2×D4×D5312yes—
374C8—2124no—
375C22—24no—
376C22—24no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22#1681077812yes—
382C22#1681077812yes—
383C22#1681080512yes—
384C22#1681080912yes—
385C22C2×#1313318511yescharacteristic
386C22C22×#1171992511yescharacteristic
387C22C2×#1313324711yescharacteristic
388C4—24no—
389C4—22no—
390C4—22no—
391C4—44no—
392C4—44no—
393C4—42no—
394C4—2124no—
395C4—2124no—
396C2—21no—
397C2—22no—
398C2—21no—
399C2#2027368814yesdirect product factor
400C2C2×#1681077811yescharacteristic
401C2C2×#1681080511yescharacteristic
402C2C2×#1681080911yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#395C2
socleC23×C53#120
prime radicalC24×C53C2×D4
hypercenterC2×#395D53
hypocenterC53C2×#1605
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
3. #20273688C2direct (N × H)
100. C22×C4×C53Q16semidirect
163. C2×C4×C53C2×Q16semidirect
204. C2×C53#1605semidirect
230. C53C2×#1605semidirect
309. C22×C4#11719925semidirect
372. C2×C4C2×#11719925semidirect
399. C2#20273688direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168112542
C2×#168107792
#202736882
C2×#168103592
C2×#168103492
C2×C53×#3952
C2×#160553