Finite Group Databasebeta

← C2×#66172⋊C3 #14514232

Subgroups — C2×#66172⋊C3 #14514232

Order 1920.

686 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 87 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×C5×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C2×#4150⋊C3C211yesmaximal, characteristic, complemented
3#66172⋊C3C214yesmaximal, direct product factor
4C2×#4094⋊C3C211yesmaximal, characteristic, complemented
5C2×#66172C311yesmaximal, characteristic, complemented, Hall
6C5×2−1+4⋊C3—21no—
7C5×2−1+4⋊C3C2212yescomplemented
8C2×C5×#41⋊C3C2211yescharacteristic
9C22×C5×SL(2,𝔽3)—21no—
10#4150⋊C3—21no—
11#4150⋊C3C2212yescomplemented
12C2×D5×SL(2,𝔽3)—21no—
13C2×C3×#686—41nomaximal
14#4094⋊C3C2212yescomplemented
15C2×#578⋊C3—51nomaximal, Hall
16C2×C5×2−1+4C2×C311yescharacteristic, complemented
17C2×#4150C2×C311yescharacteristic, complemented
18#66172—34no—
19#66172C2×C314yescomplemented
20C2×#4097—31no—
21C2×#4096—31no—
22C2×#4094C2×C311yescharacteristic, complemented
23C22×#689—31no—
24C2×#4088—31no—
25C2×C3×C5×D4—41no—
26C2×C3×D20—41no—
27C5×#41⋊C3D411yescharacteristic
28C5×#41⋊C3C2311yescharacteristic
29C2×C5×SL(2,𝔽3)—22no—
30C2×C5×SL(2,𝔽3)D411yescharacteristic
31D5×SL(2,𝔽3)—22no—
32C3×#686—44no—
33C2×C3×#168—41no—
34C2×2−1+4⋊C3—51no—
35C2×2−1+4⋊C3D511yescharacteristic, complemented
36#578⋊C3—54no—
37C2×#131⋊C3—51no—
38C5×2−1+4—21no—
39C5×2−1+4—32no—
40C5×2−1+4—61no—
41C5×2−1+4C22×C312yescomplemented
42C2×C5×#41—31no—
43C2×C5×#41—61no—
44C2×C5×#41C22×C311yescharacteristic
45C22×C5×Q8—21no—
46C22×C5×Q8—31no—
47#4150—21no—
48#4150—32no—
49#4150—61no—
50#4150C22×C312yescomplemented
51C2×Q8×D5—21no—
52C2×#711—61no—
53C2×#709—31no—
54C22×Dic10—31no—
55#4097—34no—
56#4097—34no—
57#4096—34no—
58#4096—34no—
59#4094—32no—
60#4094C22×C312yescomplemented
61C2×#689—32no—
62C2×#689—34no—
63C2×#689—34no—
64C2×#689—31no—
65#4088—34no—
66#4088—34no—
67C2×#688—31no—
68C2×#687—31no—
69C2×#686A411yescharacteristic
70C2×#681—31no—
71C22×#168—31no—
72C2×#578—51noSylow
73C23×C3×C5—81no—
74C22×C3×D5—81no—
75C3×C5×D4—42no—
76C3×C5×D4—81no—
77C2×C3×C4×C5—41no—
78C3×D20—42no—
79C3×D20—81no—
80C5×SL(2,𝔽3)C2×D411yescharacteristic
81C3×#168—42no—
822−1+4⋊C3—21no—
832−1+4⋊C3—52no—
842−1+4⋊C3—101no—
852−1+4⋊C3C2×D512yescomplemented
86C2×#41⋊C3C2×D511yescharacteristic
87C22×SL(2,𝔽3)—21no—
88C22×SL(2,𝔽3)—101no—
89C2×C3×D8—201no—
90#131⋊C3—52no—
91C5×#41—31no—
92C5×#41—31no—
93C5×#41—32no—
94C5×#41—32no—
95C5×#41—61no—
96C5×#41—61no—
97C5×#41—64no—
98C5×#41—62no—
99C5×#41—62no—
100C5×#41C3×D411yescharacteristic
101C5×#41C23×C311yescharacteristic
102C2×C5×Q8—22no—
103C2×C5×Q8—32no—
104C2×C5×Q8—32no—
105C2×C5×Q8—31no—
106C2×C5×Q8—61no—
107C2×C5×Q8—61no—
108C2×C5×Q8—61no—
109C2×C5×Q8—62no—
110C2×C5×Q8C3×D411yescharacteristic
111C2×C5×D4—31no—
112C2×C5×D4—61no—
113C2×C5×D4C2×A411yescharacteristic
114C22×C4×C5—31no—
115C22×C4×C5—61no—
116C22×C4×C5—61no—
117C2×C22⋊D5—61no—
118C22×Dic5—61no—
119Q8×D5—22no—
120Q8×D5—62no—
121#711—64no—
122#711—62no—
123#711—62no—
124#709—32no—
125#709—32no—
126#709—61no—
127#709—61no—
128C2×D20C2×A411yescharacteristic
129C2×C4×D5—61no—
130C2×Dic10—32no—
131C2×Dic10—31no—
132C2×Dic10—32no—
133C2×Dic10—61no—
134C2×Dic10—61no—
135C2×Dic10—61no—
136#689—38no—
137#689—34no—
138#688—34no—
139#687—34no—
140#686C2×A414yes—
141#681—32no—
142#681—32no—
143C2×#168—31no—
144C2×#168—32no—
145C2×#168—32no—
146C2×#168C2×A411yescharacteristic
147C2×2−1+4—51no—
148C2×2−1+4C3×D511yescharacteristic, complemented
149#578—54no—
150#578—154no—
151C2×#137—151no—
152C2×#137—151no—
153C2×#135—151no—
154C22×Q16—151no—
155C2×#131—51no—
156C22×C3×C5—41no—
157C22×C3×C5—82no—
158C22×C3×C5—162no—
159C2×C3×D5—82no—
160C2×C3×D5—162no—
161C3×C4×C5—41no—
162C3×C4×C5—41no—
163C2×C3×D4—41no—
164C2×C3×D4—201no—
165#41⋊C3C22⋊D511yescharacteristic
166#41⋊C3C22×D511yescharacteristic
167C2×SL(2,𝔽3)—22no—
168C2×SL(2,𝔽3)—102no—
169C2×SL(2,𝔽3)C22⋊D511yescharacteristic
170C3×D8—204no—
171C2×C3×C8—201no—
172C23×C5—21no—
173C23×C5—31no—
174C22×D5—21no—
175C5×Q8—31no—
176C5×Q8—32no—
177C5×Q8—62no—
178C5×Q8—62no—
179C5×Q8—61no—
180C5×Q8C2×C3×D411yescharacteristic
181C5×D4—21no—
182C5×D4—31no—
183C5×D4—31no—
184C5×D4—32no—
185C5×D4—62no—
186C5×D4—62no—
187C5×D4C22×A412yes—
188C2×C4×C5—31no—
189C2×C4×C5—31no—
190C2×C4×C5—31no—
191C2×C4×C5—31no—
192C2×C4×C5—31no—
193C2×C4×C5—62no—
194C2×C4×C5—62no—
195C2×C4×C5—62no—
196C2×C4×C5—61no—
197C2×C4×C5—62no—
198C2×C4×C5C22×A411yescharacteristic
199C22⋊D5—62no—
200C22⋊D5—62no—
201C2×Dic5—62no—
202C2×Dic5—61no—
203C2×Dic5—62no—
204D20—21no—
205D20C22×A412yes—
206C4×D5—62no—
207C4×D5—62no—
208Dic10—32no—
209Dic10—62no—
210Dic10—61no—
211Dic10—62no—
212#168—32no—
213#168C22×A412yes—
2142−1+4—21no—
2152−1+4—32no—
2162−1+4—52no—
2172−1+4—61no—
2182−1+4—101no—
2192−1+4—152no—
2202−1+4—301no—
2212−1+4C2×C3×D512yescomplemented
222C2×#41—31no—
223C2×#41—61no—
224C2×#41—151no—
225C2×#41—301no—
226C2×#41C2×C3×D511yescharacteristic
227C22×Q8—21no—
228C22×Q8—31no—
229C22×Q8—101no—
230C22×Q8—151no—
231#137—154no—
232#137—154no—
233#137—154no—
234#137—154no—
235#135—154no—
236#135—154no—
237C2×Q16—151no—
238C2×Q16—152no—
239C2×Q16—154no—
240C2×Q16—154no—
241C2×SD16—151no—
242C2×SD16—151no—
243C2×D8—51no—
244#131—52no—
245#131—152no—
246C2×M16—151no—
247C22×C8—151no—
248C2×C3×C5—41no—
249C2×C3×C5—42no—
250C2×C3×C5—162no—
251C3×D5—162no—
252C23×C3—81no—
253C23×C3—401no—
254C3×D4—42no—
255C3×D4—81no—
256C3×D4—202no—
257C3×D4—401no—
258C2×C3×C4—41no—
259SL(2,𝔽3)C2×C22⋊D511yescharacteristic
260C3×C8—202no—
261C22×C5—22no—
262C22×C5—31no—
263C22×C5—31no—
264C22×C5—42no—
265C22×C5—62no—
266C22×C5D4×A411yescharacteristic
267C2×D5—22no—
268C2×D5—42no—
269C4×C5—31no—
270C4×C5—31no—
271C4×C5—62no—
272C4×C5D4×A411yescharacteristic
273C4×C5C23×A411yescharacteristic
274Dic5—62no—
275#41—31no—
276#41—31no—
277#41—32no—
278#41—32no—
279#41—61no—
280#41—61no—
281#41—64no—
282#41—62no—
283#41—62no—
284#41—152no—
285#41—152no—
286#41—301no—
287#41—301no—
288#41—304no—
289#41—302no—
290#41—302no—
291#41C3×C22⋊D511yescharacteristic
292#41C22×C3×D511yescharacteristic
293C2×Q8—22no—
294C2×Q8—32no—
295C2×Q8—32no—
296C2×Q8—31no—
297C2×Q8—61no—
298C2×Q8—61no—
299C2×Q8—61no—
300C2×Q8—62no—
301C2×Q8—102no—
302C2×Q8—152no—
303C2×Q8—152no—
304C2×Q8—151no—
305C2×Q8—301no—
306C2×Q8—301no—
307C2×Q8—301no—
308C2×Q8—302no—
309C2×Q8C3×C22⋊D511yescharacteristic
310C2×D4—31no—
311C2×D4—51no—
312C2×D4—61no—
313C2×D4—301no—
314C2×D4D5×A411yescharacteristic
315C22×C4—31no—
316C22×C4—61no—
317C22×C4—61no—
318C22×C4—301no—
319C22×C4—301no—
320Q16—154no—
321Q16—158no—
322SD16—154no—
323SD16—154no—
324D8—54no—
325M16—152no—
326M16—152no—
327C2×C8—51no—
328C2×C8—152no—
329C2×C8—152no—
330C2×C8—151no—
331C3×C5—41noHall
332C22×C3—41no—
333C22×C3—82no—
334C22×C3—162no—
335C22×C3—402no—
336C22×C3—802no—
337C3×C4—41no—
338C3×C4—41no—
339C2×C5—42no—
340C2×C5—61no—
341C2×C5—61no—
342C2×C5#578⋊C312yescomplemented
343C2×C5C2×D4×A411yescharacteristic
344D5—42no—
345C23—21no—
346C23—31no—
347C23—101no—
348Q8—31no—
349Q8—32no—
350Q8—62no—
351Q8—62no—
352Q8—61no—
353Q8—152no—
354Q8—302no—
355Q8—302no—
356Q8—301no—
357Q8C2×C3×C22⋊D511yescharacteristic
358D4—21no—
359D4—31no—
360D4—31no—
361D4—32no—
362D4—52no—
363D4—62no—
364D4—62no—
365D4—101no—
366D4—302no—
367D4—302no—
368D4C2×D5×A412yes—
369C2×C4—31no—
370C2×C4—31no—
371C2×C4—31no—
372C2×C4—31no—
373C2×C4—31no—
374C2×C4—62no—
375C2×C4—62no—
376C2×C4—62no—
377C2×C4—61no—
378C2×C4—62no—
379C2×C4—302no—
380C2×C4—302no—
381C2×C4—302no—
382C2×C4—301no—
383C2×C4—302no—
384C2×C4C2×D5×A411yescharacteristic
385C8—52no—
386C8—152no—
387C2×C3—42no—
388C2×C3—41no—
389C2×C3—162no—
390C2×C3—802no—
391C5C2×#578⋊C311yescharacteristic, complemented, Sylow
392C22—22no—
393C22—31no—
394C22—31no—
395C22—42no—
396C22—62no—
397C22—102no—
398C22—202no—
399C22A4×C22⋊D511yescharacteristic
400C4—31no—
401C4—31no—
402C4—62no—
403C4—302no—
404C4A4×C22⋊D511yescharacteristic
405C4C22×D5×A411yescharacteristic
406C3—41noSylow
407C2—42no—
408C2—61no—
409C2—61no—
410C2—202no—
411C2#66172⋊C312yesdirect product factor
412C2C2×A4×C22⋊D511yescharacteristic
subgroupG/H
centerC22A4×C22⋊D5
commutator, cosocleC5×#41C23×C3
FrattiniC4C22×D5×A4
FittingC2×C5×2−1+4C2×C3
socleC22×C5D4×A4
hypercenterC2×D4D5×A4
hypocenterC5×Q8C2×C3×D4
LCSG ▷ C5×#41 ▷ C5×Q8
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C5×2−1+4⋊C3C2semidirect
2. C2×#4150⋊C3C2semidirect
3. #66172⋊C3C2direct (N × H)
4. C2×#4094⋊C3C2semidirect
5. C2×#66172C3semidirect
7. C5×2−1+4⋊C3C22semidirect
11. #4150⋊C3C22semidirect
14. #4094⋊C3C22semidirect
16. C2×C5×2−1+4C2×C3semidirect
17. C2×#4150C2×C3semidirect
19. #66172C2×C3semidirect
22. C2×#4094C2×C3semidirect
35. C2×2−1+4⋊C3D5semidirect
41. C5×2−1+4C22×C3semidirect
50. #4150C22×C3semidirect
60. #4094C22×C3semidirect
85. 2−1+4⋊C3C2×D5semidirect
148. C2×2−1+4C3×D5semidirect
221. 2−1+4C2×C3×D5semidirect
342. C2×C5#578⋊C3semidirect
391. C5C2×#578⋊C3semidirect
411. C2#66172⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×2−1+4⋊C32
C2×#4150⋊C32
#66172⋊C32
C2×#4094⋊C32
C2×#661723
C2×C3×#6864
C2×#578⋊C35