Finite Group Databasebeta

← C22×#11763878 #17981038

Subgroups — C22×#11763878 #17981038

Order 4352.

12270 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763878C2148yesmaximal, direct product factor
2C22×Q8×#595C212yesmaximal, complemented
3C22×D4×#595C216yesmaximal, complemented
4C22×#10587677C213yesmaximal, complemented
5C23×C4⋊#595C213yesmaximal, complemented
6C22×#41×D17C211yesmaximal, characteristic
7C24×#595C2216yes—
8#11763878C221256yesdirect product factor
9C2×Q8×#595C22148yescomplemented
10C2×D4×#595C221144yescomplemented
11C2×#10587677C22172yescomplemented
12C22×C4⋊#595C2213yes—
13C22×C4⋊#595C2216yes—
14C22×C4⋊#595C2211yescharacteristic
15C22×C4⋊#595C22172yescomplemented
16C22×C22⋊#595C2216yes—
17C22×C4×#595C2216yes—
18C22×#63256C411yescharacteristic, complemented
19C22×#63252C413yescomplemented
20C22×#63254C413yescomplemented
21C2×#41×D17C4124yescomplemented
22C2×#41×D17C22124yes—
23C22×Q8×D17C2211yescharacteristic
24C22×D4×D17C2213yes—
25C23×C4×D17C2213yes—
26C22×#41×C17C411yescharacteristic, complemented
27C24×D17C2313yes—
28C23×#595—424no—
29C23×#595C2312yes—
30C23×#595C2316yes—
31C23×#595C23172yes—
32C2×#41×C17C2×C4124yescomplemented
33C22×Q8×C17C2×C411yescharacteristic, complemented
34C22×D4×C17C2×C413yescomplemented
35C23×C4×C17C2×C413yescomplemented
36C22×C22⋊D17C2×C416yescomplemented
37C23×Dic17C2×C413yescomplemented
38#41×D17C2×C41192yescomplemented
39#41×D17C23164yes—
40C2×#63256C2×C4124yescomplemented
41C2×Q8×D17C2×C4112yescomplemented
42C2×Q8×D17C23112yes—
43C2×#63254C2×C4172yescomplemented
44C2×D4×D17C2×C4136yescomplemented
45C2×D4×D17C23136yes—
46C2×#63252C2×C4172yescomplemented
47C22×D68C2×C413yescomplemented
48C22×C4×D17C2×C4136yescomplemented
49C22×C4×D17C2×C413yescomplemented
50C22×C4×D17C2×C413yescomplemented
51C22×C4×D17C2313yes—
52C22×C4×D17C2311yescharacteristic
53C22×C4×D17C23136yes—
54C22×Dic34C2×C413yescomplemented
55C2×C22⋊#595C23172yes—
56Q8×#595C231128yescomplemented
57D4×#595C231384yescomplemented
58#10587677C231192yescomplemented
59C2×C4⋊#595C23136yes—
60C2×C4⋊#595C23172yes—
61C2×C4⋊#595C23112yes—
62C2×C4⋊#595C231192yescomplemented
63C2×C4×#595C23172yes—
64C24×C17C22×C413yes—
65C23×D17—212no—
66C23×D17—212no—
67C23×D17C22×C413yes—
68C23×D17C22×C4118yes—
69C23×D17C24118yes—
70C23×D17C2411yescharacteristic
71C22×#595—224no—
72C22×#595—28no—
73C22×#595—4144no—
74C22×#595C24112yes—
75C22×#595C24136yes—
76C22×#595C24196yes—
77#41×C17C22×C4164yescomplemented
78C2×Q8×C17C22×C4112yescomplemented
79C2×D4×C17C22×C4136yescomplemented
80C22×C4×C17C22×C413yes—
81C22×C4×C17C22×C411yescharacteristic
82C22×C4×C17C22×C4136yescomplemented
83C2×C22⋊D17C22×C4172yescomplemented
84C22×Dic17C22×C413yes—
85C22×Dic17C22×C411yescharacteristic
86C22×Dic17C22×C4136yescomplemented
87#63256C22×C4164yescomplemented
88Q8×D17C22×C4148yescomplemented
89Q8×D17C24116yes—
90#63254C22×C41192yescomplemented
91D4×D17C22×C41144yescomplemented
92D4×D17C24148yes—
93#63252C22×C41192yescomplemented
94C2×D68C22×C4136yescomplemented
95C2×C4×D17C22×C416yes—
96C2×C4×D17C22×C4118yes—
97C2×C4×D17C22×C41144yescomplemented
98C2×C4×D17C22×C4136yescomplemented
99C2×C4×D17C22×C4136yescomplemented
100C2×C4×D17C24118yes—
101C2×C4×D17C2416yes—
102C2×C4×D17C24148yes—
103C2×Dic34C22×C4136yescomplemented
104C22⋊#595C24196yes—
105C4⋊#595C24148yes—
106C4⋊#595C24196yes—
107C4⋊#595C24116yes—
108C4×#595C24196yes—
109C22×#520—171nomaximal, Sylow
110C23×C17—212no—
111C23×C17C23×C4118yes—
112C23×C17C23×C411yescharacteristic
113C22×D17—212no—
114C22×D17—272no—
115C22×D17—236no—
116C22×D17—236no—
117C22×D17C23×C4118yes—
118C22×D17C23×C4136yes—
119C22×D17C23×C413yes—
120C22×D17H(5,𝔽2)14yes—
121C22×D172−1+414yes—
122C22×D17C25112yes—
123C22×D17C2513yes—
124C2×#595—272no—
125C2×#595—224no—
126C2×#595—496no—
127C2×#595C2518yes—
128C2×#595C25124yes—
129Q8×C17C23×C4116yescomplemented
130D4×C17C23×C4148yescomplemented
131C2×C4×C17C23×C4118yes—
132C2×C4×C17C23×C416yes—
133C2×C4×C17C23×C4148yescomplemented
134C22⋊D17C23×C4196yescomplemented
135C2×Dic17C23×C4118yes—
136C2×Dic17C23×C416yes—
137C2×Dic17C23×C4148yescomplemented
138D68C23×C4148yescomplemented
139C4×D17C23×C4112yes—
140C4×D17C23×C4136yes—
141C4×D17C23×C4148yescomplemented
142C4×D17C23×C4148yescomplemented
143C4×D17C25112yes—
144C4×D17C2514yes—
145Dic34C23×C4148yescomplemented
146C23×#41—171no—
147C2×#520—1748no—
148C22×C4×Q8—172no—
149C22×C4×D4—176no—
150C22×#117—173no—
151C23×C4⋊C4—173no—
152C22×C17—236no—
153C22×C17#52014yescomplemented
154C22×C17C24×C4112yes—
155C22×C17C24×C413yes—
156C2×D17—236no—
157C2×D17—212no—
158C2×D17—212no—
159C2×D17—272no—
160C2×D17#520112yescomplemented
161C2×D17C24×C4112yes—
162C2×D17C24×C413yes—
163C2×D17C2×H(5,𝔽2)16yes—
164C2×D17C2×2−1+416yes—
165C2×D17C2611yescharacteristic
166#595—224no—
167#595—28no—
168C4×C17C24×C4112yes—
169C4×C17C24×C414yes—
170Dic17C24×C4112yes—
171Dic17C24×C414yes—
172C22×#41—1724no—
173C22×#41—1724no—
174C22×#41—171no—
175C22×#41—173no—
176C22×#41—173no—
177C22×#41#59511yescharacteristic, complemented
178C23×Q8—171no—
179C23×D4—173no—
180C24×C4—173no—
181C24×C4—176no—
182#520—17256no—
183C2×C4×Q8—1748no—
184C2×C4×D4—17144no—
185C2×#117—1772no—
186C22×C4⋊C4—173no—
187C22×C4⋊C4—176no—
188C22×C4⋊C4—1772no—
189C22×C4⋊C4—171no—
190C22×C22⋊C4—176no—
191C22×C42—176no—
192C2×C17—212no—
193C2×C17C2×#52016yescomplemented
194C2×C17C25×C411yescharacteristic
195D17—212no—
196D17C2×#52016yescomplemented
197D17C22×H(5,𝔽2)11yescharacteristic
198D17C22×2−1+411yescharacteristic
199C25—173no—
200C2×#41—1764no—
201C2×#41—17192no—
202C2×#41—1724no—
203C2×#41—1772no—
204C2×#41—1772no—
205C2×#41C2×#595124yescomplemented
206C22×Q8—1712no—
207C22×Q8—1712no—
208C22×Q8—173no—
209C22×Q8C2×#59511yescharacteristic, complemented
210C22×D4—176no—
211C22×D4—1736no—
212C22×D4—1736no—
213C22×D4—173no—
214C22×D4C2×#59513yescomplemented
215C23×C4—173no—
216C23×C4—171no—
217C23×C4—173no—
218C23×C4—173no—
219C23×C4—173no—
220C23×C4—1736no—
221C23×C4—1736no—
222C23×C4—172no—
223C23×C4—176no—
224C23×C4—1772no—
225C23×C4—6824no—
226C23×C4C2×#59513yescomplemented
227C4×Q8—17128no—
228C4×D4—17384no—
229#117—17192no—
230C2×C4⋊C4—1736no—
231C2×C4⋊C4—1772no—
232C2×C4⋊C4—17192no—
233C2×C4⋊C4—1712no—
234C2×C22⋊C4—1772no—
235C2×C42—1772no—
236C17C22×#52011yescharacteristic, complemented, Sylow
237C24—171no—
238C24—1718no—
239C24—1718no—
240C24—173no—
241C24—3412no—
242C24—3412no—
243C24C22×#59513yes—
244#41—1764no—
245#41—17192no—
246#41—17192no—
247#41C22×#595164yescomplemented
248C2×Q8—1748no—
249C2×Q8—1716no—
250C2×Q8—1736no—
251C2×Q8C22×#595112yescomplemented
252C2×D4—1772no—
253C2×D4—17144no—
254C2×D4—1748no—
255C2×D4—1736no—
256C2×D4C22×#595136yescomplemented
257C22×C4—1736no—
258C22×C4—171no—
259C22×C4—173no—
260C22×C4—176no—
261C22×C4—1718no—
262C22×C4—1736no—
263C22×C4—1736no—
264C22×C4—1718no—
265C22×C4—176no—
266C22×C4—17144no—
267C22×C4—1748no—
268C22×C4—1712no—
269C22×C4—1736no—
270C22×C4—1796no—
271C22×C4—348no—
272C22×C4—3424no—
273C22×C4—68144no—
274C22×C4C22×#59511yescharacteristic
275C22×C4C22×#59513yes—
276C22×C4C22×#595136yescomplemented
277C4⋊C4—1748no—
278C4⋊C4—1796no—
279C4⋊C4—1716no—
280C22⋊C4—1796no—
281C42—1796no—
282C23—212no—
283C23—174no—
284C23—174no—
285C23—173no—
286C23—173no—
287C23—1712no—
288C23—1736no—
289C23—1718no—
290C23—3436no—
291C23—3436no—
292C23—3472no—
293C23—3412no—
294C23C23×#595118yes—
295C23C23×#59511yescharacteristic
296Q8—1748no—
297Q8C23×#595116yescomplemented
298D4—1796no—
299D4—1748no—
300D4C23×#595148yescomplemented
301C2×C4—176no—
302C2×C4—1718no—
303C2×C4—1748no—
304C2×C4—1712no—
305C2×C4—174no—
306C2×C4—1712no—
307C2×C4—1736no—
308C2×C4—1748no—
309C2×C4—1748no—
310C2×C4—178no—
311C2×C4—1724no—
312C2×C4—3472no—
313C2×C4—3424no—
314C2×C4—6896no—
315C2×C4C23×#59516yes—
316C2×C4C23×#595118yes—
317C2×C4C23×#595148yescomplemented
318C22—236no—
319C22—176no—
320C22—176no—
321C22—1712no—
322C22—171no—
323C22—173no—
324C22—1712no—
325C22—3412no—
326C22—3412no—
327C22—3472no—
328C22—3436no—
329C22#1176387814yesdirect product factor
330C22C24×#595112yes—
331C22C24×#59513yes—
332C4—174no—
333C4—1712no—
334C4—3424no—
335C4—348no—
336C4C24×#59514yes—
337C4C24×#595112yes—
338C2—212no—
339C2—171no—
340C2—171no—
341C2—176no—
342C2—3412no—
343C2C2×#1176387816yesdirect product factor
344C2C25×#59511yescharacteristic
subgroupG/H
centerC23C23×#595
commutatorC2×C17C25×C4
FrattiniC2C25×#595
FittingC22×#41×C17C4
socleC23×C17C23×C4
prime radicalC22×#41×D17C2
cosocleC2×D17C26
hypercenterC22×#41#595
hypocenterC17C22×#520
LCSG ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#11763878C2direct (N × H)
2. C22×Q8×#595C2semidirect
3. C22×D4×#595C2semidirect
4. C22×#10587677C2semidirect
5. C23×C4⋊#595C2semidirect
8. #11763878C22direct (N × H)
9. C2×Q8×#595C22semidirect
10. C2×D4×#595C22semidirect
11. C2×#10587677C22semidirect
15. C22×C4⋊#595C22semidirect
18. C22×#63256C4semidirect
19. C22×#63252C4semidirect
20. C22×#63254C4semidirect
21. C2×#41×D17C4semidirect
26. C22×#41×C17C4semidirect
32. C2×#41×C17C2×C4semidirect
33. C22×Q8×C17C2×C4semidirect
34. C22×D4×C17C2×C4semidirect
35. C23×C4×C17C2×C4semidirect
36. C22×C22⋊D17C2×C4semidirect
37. C23×Dic17C2×C4semidirect
38. #41×D17C2×C4semidirect
40. C2×#63256C2×C4semidirect
41. C2×Q8×D17C2×C4semidirect
43. C2×#63254C2×C4semidirect
44. C2×D4×D17C2×C4semidirect
46. C2×#63252C2×C4semidirect
47. C22×D68C2×C4semidirect
48. C22×C4×D17C2×C4semidirect
49. C22×C4×D17C2×C4semidirect
50. C22×C4×D17C2×C4semidirect
54. C22×Dic34C2×C4semidirect
56. Q8×#595C23semidirect
57. D4×#595C23semidirect
58. #10587677C23semidirect
62. C2×C4⋊#595C23semidirect
77. #41×C17C22×C4semidirect
78. C2×Q8×C17C22×C4semidirect
79. C2×D4×C17C22×C4semidirect
82. C22×C4×C17C22×C4semidirect
83. C2×C22⋊D17C22×C4semidirect
86. C22×Dic17C22×C4semidirect
87. #63256C22×C4semidirect
88. Q8×D17C22×C4semidirect
90. #63254C22×C4semidirect
91. D4×D17C22×C4semidirect
93. #63252C22×C4semidirect
94. C2×D68C22×C4semidirect
97. C2×C4×D17C22×C4semidirect
98. C2×C4×D17C22×C4semidirect
99. C2×C4×D17C22×C4semidirect
103. C2×Dic34C22×C4semidirect
129. Q8×C17C23×C4semidirect
130. D4×C17C23×C4semidirect
133. C2×C4×C17C23×C4semidirect
134. C22⋊D17C23×C4semidirect
137. C2×Dic17C23×C4semidirect
138. D68C23×C4semidirect
141. C4×D17C23×C4semidirect
142. C4×D17C23×C4semidirect
145. Dic34C23×C4semidirect
153. C22×C17#520semidirect
160. C2×D17#520semidirect
177. C22×#41#595semidirect
193. C2×C17C2×#520semidirect
196. D17C2×#520semidirect
205. C2×#41C2×#595semidirect
209. C22×Q8C2×#595semidirect
214. C22×D4C2×#595semidirect
226. C23×C4C2×#595semidirect
236. C17C22×#520semidirect
247. #41C22×#595semidirect
251. C2×Q8C22×#595semidirect
256. C2×D4C22×#595semidirect
276. C22×C4C22×#595semidirect
297. Q8C23×#595semidirect
300. D4C23×#595semidirect
317. C2×C4C23×#595semidirect
329. C22#11763878direct (N × H)
343. C2C2×#11763878direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117638782
C22×Q8×#5952
C22×D4×#5952
C22×#105876772
C23×C4⋊#5952
C22×#41×D172
C22×#52017