Finite Group Databasebeta

← C2×#20276794 #27538476

Subgroups — C2×#20276794 #27538476

Order 13568.

686 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811063C211yesmaximal, characteristic
2C2×#16811001C211yesmaximal, characteristic, complemented
3C2×#16810740C211yesmaximal, characteristic
4#20276794C218yesmaximal, direct product factor
5C2×#16810443C211yesmaximal, characteristic, complemented
6C2×#16810440C211yesmaximal, characteristic, complemented
7C2×#16810439C211yesmaximal, characteristic, complemented
8C2×C53×#510C211yesmaximal, characteristic
9C22×Dic212C2211yescharacteristic
10C22×#11719925—21no—
11C2×#13133224—21no—
12C2×#13133215—21no—
13#16811063—22no—
14#16811063C2214yes—
15C2×#13133163—21no—
16C2×#13133161—21no—
17C2×#13133157C2211yescharacteristic
18#16811001—22no—
19#16811001C2214yescomplemented
20#16810740—22no—
21#16810740C2214yes—
22C2×#13133153C2211yescharacteristic
23C2×#13133144—21no—
24C2×C8×Dic53C2211yescharacteristic
25#16810443C2214yescomplemented
26#16810440C2214yescomplemented
27#16810439C2214yescomplemented
28C2×#135×C53C2211yescharacteristic
29C22×Q16×C53C2211yescharacteristic
30C53×#510—22no—
31C53×#510C2214yes—
32C2×Q32×C53—21no—
33C2×SD32×C53—21no—
34C2×M32×C53C2211yescharacteristic
35C22×Dic106—21no—
36C2×Dic212—21no—
37C2×Dic212—21no—
38C2×Dic212—21no—
39C2×Dic212—21no—
40C2×Dic212—21no—
41C2×Dic212C2312yes—
42C2×#11719925—24no—
43C2×#11719925—41no—
44C2×#11719925—41no—
45C2×#11719925—42no—
46#13133224—24no—
47#13133224—42no—
48C2×#11719917—21no—
49C2×#11719917—41no—
50C22×#88322—41no—
51C22×#88322D411yescharacteristic
52#13133215—24no—
53#13133215—42no—
54C2×#11719919—42no—
55C2×C4×Dic53D411yescharacteristic
56#13133163—24no—
57#13133161—24no—
58#13133157—21no—
59#13133157C2312yes—
60C2×#11719908—21no—
61#13133153—21no—
62#13133153C2312yes—
63#13133144—24no—
64C8×Dic53D412yes—
65C8×Dic53C2312yes—
66C2×#41×C53—21no—
67C22×Q8×C53—21no—
68#135×C53—21no—
69#135×C53—21no—
70#135×C53D412yes—
71#135×C53C2312yes—
72C2×Q16×C53—21no—
73C2×Q16×C53—21no—
74C2×Q16×C53—21no—
75C2×Q16×C53—21no—
76C2×Q16×C53—21no—
77C2×Q16×C53D411yescharacteristic
78C2×Q16×C53C2312yes—
79C2×SD16×C53—21no—
80C2×D8×C53D411yescharacteristic
81C22×C8×C53C2311yescharacteristic
82Q32×C53—24no—
83SD32×C53—24no—
84M32×C53—21no—
85M32×C53C2312yes—
86C2×C16×C53—21no—
87#41×C53—22no—
88#41×C53—22no—
89#41×C53—41no—
90#41×C53—41no—
91C2×Q8×C53—21no—
92C2×Q8×C53—22no—
93C2×Q8×C53—41no—
94C2×Q8×C53—41no—
95C2×Q8×C53—41no—
96C2×Q8×C53—41no—
97C2×Q8×C53—41no—
98C2×D4×C53—21no—
99C2×D4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53C2×D411yescharacteristic
103C22×Dic53—21no—
104C22×Dic53—41no—
105C2×Dic106—22no—
106C2×Dic106—41no—
107C2×Dic106—41no—
108C2×Dic106—41no—
109C2×Dic106—41no—
110C2×Dic106—41no—
111Q16×C53—21no—
112Q16×C53—22no—
113Q16×C53—22no—
114Q16×C53—41no—
115Q16×C53—41no—
116Q16×C53C2×D412yes—
117SD16×C53—22no—
118SD16×C53—22no—
119D8×C53—21no—
120D8×C53C2×D412yes—
121C2×C8×C53C2×D411yescharacteristic
122C2×C8×C53C2×D411yescharacteristic
123C2×C8×C53C2×D411yescharacteristic
124C2×C8×C53#4111yescharacteristic
125C2×C8×C53#4111yescharacteristic
126C2×C8×C53C2411yescharacteristic
127#11719925—44no—
128#11719925—44no—
129#11719919—48no—
130C4×Dic53—21no—
131C4×Dic53C2×D412yes—
132#11719917—22no—
133#11719917—41no—
134#11719917—44no—
135C2×#88322—41no—
136C2×#88322—44no—
137C2×#88322D811yescharacteristic
138C2×#88322D811yescharacteristic
139C2×#88322D812yes—
140C2×#88322C2×D412yes—
141Dic212—22no—
142Dic212—22no—
143Dic212—41no—
144Dic212—41no—
145C16×C53—22no—
146#11719908—22no—
147C23×C53—41no—
148C23×C53AGL(1,ℤ/8)11yescharacteristic
149Q8×C53—22no—
150Q8×C53—41no—
151Q8×C53—42no—
152Q8×C53—42no—
153Q8×C53—81no—
154Q8×C53—81no—
155D4×C53—22no—
156D4×C53—42no—
157D4×C53—42no—
158D4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—42no—
161C2×C4×C53—42no—
162C2×C4×C53—42no—
163C2×C4×C53—81no—
164C2×C4×C53—81no—
165C2×C4×C53#12111yescharacteristic
166C2×C4×C53#12111yescharacteristic
167C2×C4×C53C2×D811yescharacteristic
168C2×C4×C53C2×D811yescharacteristic
169C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C53C22×D411yescharacteristic
171C2×Dic53—24no—
172C2×Dic53—22no—
173C2×Dic53—42no—
174C2×Dic53—81no—
175C2×Dic53—81no—
176Dic106—42no—
177Dic106—42no—
178Dic106—81no—
179Dic106—81no—
180C8×C53#12111yescharacteristic
181C8×C53#12111yescharacteristic
182C8×C53C22×D411yescharacteristic
183C8×C53C2×#4111yescharacteristic
184#88322—42no—
185#88322C2×D812yes—
186#88322C2×D812yes—
187C2×#2214—531nomaximal, Sylow
188C22×C53—22no—
189C22×C53—42no—
190C22×C53—82no—
191C22×C53#45911yescharacteristic
192C22×C53#45911yescharacteristic
193C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
194C4×C53—42no—
195C4×C53—82no—
196C4×C53#45911yescharacteristic
197C4×C53#45911yescharacteristic
198C4×C53C2×#12111yescharacteristic
199C4×C53C22×D811yescharacteristic
200Dic53—24no—
201Dic53—82no—
202C2×#510—531no—
203C2×#510D5311yescharacteristic
204C2×#494—531no—
205C2×#443—531no—
206#2214—538no—
207C2×#364—531no—
208C2×#362—531no—
209C2×#362—531no—
210C2×C53—21no—
211C2×C53—21no—
212C2×C53—82no—
213C2×C53#221412yescomplemented
214C2×C53C2×#45911yescharacteristic
215C2×#135C2×D5311yescharacteristic
216C22×Q16—531no—
217C22×Q16—1061no—
218C22×Q16C2×D5311yescharacteristic
219C2×#131—1061no—
220C2×C4⋊Q8—1061no—
221#510—22no—
222#510—534no—
223#510—1062no—
224#510C2×D5314yes—
225C2×Q32—21no—
226C2×Q32—1061no—
227C2×SD32—21no—
228C2×SD32—1061no—
229C2×M32—531no—
230C2×M32C2×D5311yescharacteristic
231#494—534no—
232#494—1062no—
233#443—534no—
234#443—1062no—
235C2×#108—531no—
236C2×#104—1061no—
237C2×C4×C8—531no—
238#364—534no—
239#362—534no—
240#362—534no—
241C53C2×#221411yescharacteristic, complemented, Sylow
242C2×#41—21no—
243C22×Q8—21no—
244C22×Q8—1061no—
245#135—21no—
246#135—21no—
247#135C22⋊D5312yes—
248#135C22×D5312yes—
249C2×Q16—21no—
250C2×Q16—21no—
251C2×Q16—21no—
252C2×Q16—21no—
253C2×Q16—21no—
254C2×Q16—532no—
255C2×Q16—1064no—
256C2×Q16—1061no—
257C2×Q16—1061no—
258C2×Q16—1061no—
259C2×Q16—1061no—
260C2×Q16—1061no—
261C2×Q16—2121no—
262C2×Q16—2121no—
263C2×Q16—2122no—
264C2×Q16C22⋊D5311yescharacteristic
265C2×Q16C22×D5312yes—
266C2×SD16—21no—
267C2×D8C22⋊D5311yescharacteristic
268#131—1064no—
269#131—2122no—
270C2×M16—1061no—
271C2×M16—2121no—
272C22×C8—531no—
273C22×C8—2121no—
274C22×C8C22×D5311yescharacteristic
275C4⋊Q8—1064no—
276C4⋊Q8—2122no—
277C2×C4⋊C4—2122no—
278C2×C42—531no—
279Q32—24no—
280Q32—1064no—
281SD32—24no—
282SD32—1064no—
283M32—21no—
284M32—532no—
285M32—1061no—
286M32C22×D5312yes—
287C2×C16—21no—
288C2×C16—1061no—
289#108—532no—
290#108—1061no—
291#104—1064no—
292C4×C8—532no—
293C4×C8—532no—
294#41—22no—
295#41—22no—
296#41—41no—
297#41—41no—
298C2×Q8—21no—
299C2×Q8—22no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—1062no—
306C2×Q8—2121no—
307C2×Q8—2121no—
308C2×Q8—2121no—
309C2×Q8—2121no—
310C2×Q8—2121no—
311C2×D4—21no—
312C2×D4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—1061no—
316C22×C4—2121no—
317C22×C4D4×D5311yescharacteristic
318Q16—21no—
319Q16—22no—
320Q16—22no—
321Q16—41no—
322Q16—41no—
323Q16—1062no—
324Q16—1062no—
325Q16—2124no—
326Q16—2124no—
327Q16—2121no—
328Q16—2121no—
329Q16C2×C22⋊D5312yes—
330SD16—22no—
331SD16—22no—
332D8—21no—
333D8C2×C22⋊D5312yes—
334M16—1062no—
335M16—2124no—
336M16—2121no—
337C2×C8—531no—
338C2×C8—531no—
339C2×C8—532no—
340C2×C8—532no—
341C2×C8—2124no—
342C2×C8—2121no—
343C2×C8D4×D5311yescharacteristic
344C2×C8#1171994711yescharacteristic
345C2×C8#1171994711yescharacteristic
346C2×C8C2×C22⋊D5311yescharacteristic
347C2×C8C2×C22⋊D5311yescharacteristic
348C2×C8C23×D5311yescharacteristic
349C4⋊C4—2128no—
350C42—532no—
351C42—1061no—
352C16—22no—
353C16—1062no—
354C23—41no—
355C23#1313324111yescharacteristic
356Q8—22no—
357Q8—41no—
358Q8—42no—
359Q8—42no—
360Q8—81no—
361Q8—81no—
362Q8—4241no—
363Q8—4241no—
364Q8—2122no—
365Q8—2122no—
366D4—22no—
367D4—42no—
368D4—42no—
369D4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—1064no—
377C2×C4—1062no—
378C2×C4—4241no—
379C2×C4—4241no—
380C2×C4—2122no—
381C2×C4#1313318911yescharacteristic
382C2×C4#1313318911yescharacteristic
383C2×C4D8×D5311yescharacteristic
384C2×C4D8×D5311yescharacteristic
385C2×C4#1313324111yescharacteristic
386C2×C4C2×D4×D5311yescharacteristic
387C8—532no—
388C8—532no—
389C8—2122no—
390C8#1313318911yescharacteristic
391C8#1313318911yescharacteristic
392C8C22×C22⋊D5311yescharacteristic
393C8C2×#1171994711yescharacteristic
394C22—22no—
395C22—42no—
396C22—82no—
397C22#1681081611yescharacteristic
398C22#1681081611yescharacteristic
399C22C2×#1313324111yescharacteristic
400C4—42no—
401C4—82no—
402C4—1064no—
403C4—4242no—
404C4#1681081611yescharacteristic
405C4#1681081611yescharacteristic
406C4C2×#1313318911yescharacteristic
407C4C2×D8×D5311yescharacteristic
408C2—21no—
409C2—21no—
410C2—82no—
411C2#2027679412yesdirect product factor
412C2C2×#1681081611yescharacteristic
subgroupG/H
centerC22#16810816
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC2×C53×#510C2
socleC22×C53#459
prime radicalC2×#135×C53C22
hypercenterC2×#510D53
hypocenterC53C2×#2214
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#510

Complements

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

NH ≅ G/Ntype
2. C2×#16811001C2semidirect
4. #20276794C2direct (N × H)
5. C2×#16810443C2semidirect
6. C2×#16810440C2semidirect
7. C2×#16810439C2semidirect
19. #16811001C22semidirect
25. #16810443C22semidirect
26. #16810440C22semidirect
27. #16810439C22semidirect
213. C2×C53#2214semidirect
241. C53C2×#2214semidirect
411. C2#20276794direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168110632
C2×#168110012
C2×#168107402
#202767942
C2×#168104432
C2×#168104402
C2×#168104392
C2×C53×#5102
C2×#221453