Finite Group Databasebeta

← C2×#14545224 #17978862

Subgroups — C2×#14545224 #17978862

Order 4352.

3654 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×#595C212yesmaximal
2#14545224C2116yesmaximal, direct product factor
3C2×C4⋊C4×#595C216yesmaximal
4C2×C4×C4⋊#595C213yesmaximal
5C2×#11763745C213yesmaximal
6C2×C4×Q8×D17C211yesmaximal, characteristic
7C2×Q8×#595C4132yescomplemented
8C2×Q8×#595C22116yes—
9C22×C4⋊#595C2216yes—
10C22×C4×#595C2216yes—
11C22×C4×#595C2216yes—
12C4⋊C4×#595C22148yes—
13C4×C4⋊#595C22124yes—
14#11763745C22124yes—
15C2×(C2×#595)⋊C4C2211yescharacteristic
16C2×(C2×#595)⋊C4C2213yes—
17C2×(C2×#595)⋊C4C2216yes—
18C22×Q8×D17C2211yescharacteristic
19C2×Q8×Dic17C411yescharacteristic, complemented
20C2×C4×Dic34C413yescomplemented
21C2×#10587573C413yescomplemented
22C4×Q8×D17C418yescomplemented
23C4×Q8×D17C2218yes—
24C2×C4⋊C4×D17C2213yes—
25C2×C42×D17C2213yes—
26C2×C4×Q8×C17C411yescharacteristic, complemented
27C23×#595C2312yes—
28C23×#595C2316yes—
29C22×Q8×C17C2×C411yescharacteristic
30C2×Q8×D17C2×C4116yes—
31C2×Q8×D17C2×C414yes—
32C2×Q8×D17C2314yes—
33C22×C4×D17C2311yescharacteristic
34C22×C4×D17C2313yes—
35C22×C4×D17C2313yes—
36C22×Dic34C2×C413yes—
37Q8×#595C2×C41128yescomplemented
38C2×C4⋊#595C2×C4148yes—
39C2×C4⋊#595C23124yes—
40C2×C4×#595—412no—
41C2×C4×#595C2×C4148yes—
42C2×C4×#595C23124yes—
43C2×C4×#595C23124yes—
44C4×Q8×C17C2×C418yescomplemented
45C2×C4⋊C4×C17C2×C413yes—
46C2×C42×C17C2×C413yes—
47Q8×Dic17C2×C418yescomplemented
48C2×C4⋊Dic17C2×C413yes—
49C2×#63225C2×C416yes—
50C2×C4×Dic17C2×C413yes—
51C2×C4×Dic17C2×C413yes—
52C2×C4×Dic17C2×C413yes—
53C4⋊C4×D17C2×C4112yes—
54C4⋊C4×D17C23112yes—
55#10587573C2×C4124yescomplemented
56C42×D17C2×C4112yes—
57C42×D17C23112yes—
58C4×Dic34C2×C4124yescomplemented
59(C2×#595)⋊C4C2314yes—
60(C2×#595)⋊C4C23112yes—
61(C2×#595)⋊C4C23124yes—
62C23×D17C2411yescharacteristic
63C22×#595—212no—
64C22×#595—24no—
65C22×#595—24no—
66C22×#595—212no—
67C22×#595C22×C418yes—
68C22×#595C22×C4124yes—
69C22×#595C24112yes—
70C22×#595C2414yes—
71C2×Q8×C17C4218yescomplemented
72C2×Q8×C17C22×C414yes—
73C22×C4×C17C22×C411yescharacteristic
74C22×C4×C17C22×C413yes—
75C22×C4×C17C22×C413yes—
76C22×Dic17C22×C411yescharacteristic
77C22×Dic17C22×C413yes—
78C22×Dic17C22×C413yes—
79Q8×D17C42132yescomplemented
80Q8×D17C22×C4132yes—
81C2×C4×D17—26no—
82C2×C4×D17—24no—
83C2×C4×D17—26no—
84C2×C4×D17C22×C412yes—
85C2×C4×D17C22×C416yes—
86C2×C4×D17C22×C416yes—
87C2×C4×D17C22×C4124yes—
88C2×C4×D17C2416yes—
89C2×C4×D17C2412yes—
90C2×C4×D17C2416yes—
91C2×Dic34C42124yescomplemented
92C2×Dic34C22×C4112yes—
93C4⋊#595C22×C4196yes—
94C4×#595—424no—
95C4×#595C22×C4196yes—
96C4⋊C4×C17C22×C4112yes—
97C42×C17C22×C4112yes—
98C4⋊Dic17C22×C4112yes—
99#63225C22×C4124yes—
100C4×Dic17C22×C4112yes—
101C4×Dic17C22×C4112yes—
102C4×Dic17C22×C4112yes—
103C2×#2276—171nomaximal, Sylow
104C23×C17C23×C411yescharacteristic
105C22×D17C23×C414yes—
106C22×D17C23×C411yescharacteristic
107C22×D17H(5,𝔽2)12yes—
108C22×D172−1+412yes—
109C22×D172−1+414yes—
110C22×D17C2511yescharacteristic
111C2×#595—216no—
112C2×#595—248no—
113C2×#595—212no—
114C2×#595—24no—
115C2×#595—24no—
116C2×#595—212no—
117C2×#595C23×C418yes—
118C2×#595C23×C4124yes—
119Q8×C17C2×C42116yescomplemented
120C2×C4×C17—22no—
121C2×C4×C17—26no—
122C2×C4×C17C2×C42112yes—
123C2×C4×C17C23×C412yes—
124C2×C4×C17C23×C416yes—
125C2×C4×C17C23×C416yes—
126C2×Dic17—22no—
127C2×Dic17—26no—
128C2×Dic17C2×C42112yes—
129C2×Dic17C23×C412yes—
130C2×Dic17C23×C416yes—
131C2×Dic17C23×C416yes—
132C4×D17—24no—
133C4×D17—212no—
134C4×D17—26no—
135C4×D17—24no—
136C4×D17—26no—
137C4×D17C2×C42124yes—
138C4×D17C23×C4124yes—
139Dic34C2×C42148yescomplemented
140C22×C4×Q8—172no—
141C22×C4×Q8—171no—
142#2276—1716no—
143C2×C4×C4⋊C4—173no—
144C2×C4×C4⋊C4—176no—
145C2×#376—173no—
146C22×C17C22×C4212yes—
147C22×C17#51912yes—
148C22×C17#52012yes—
149C22×C17C24×C411yescharacteristic
150C2×D17C22×C4212yes—
151C2×D17#51918yes—
152C2×D17#51912yes—
153C2×D17#52018yes—
154C2×D17#52012yes—
155C2×D17C24×C412yes—
156C2×D17C2×H(5,𝔽2)11yescharacteristic
157C2×D17C2×2−1+411yescharacteristic
158C2×D17C2×2−1+412yes—
159#595—28no—
160#595—224no—
161C4×C17—22no—
162C4×C17—26no—
163C4×C17C22×C42112yes—
164Dic17—22no—
165Dic17—26no—
166Dic17C22×C42112yes—
167C23×Q8—171no—
168C2×C4×Q8—1732no—
169C2×C4×Q8—1716no—
170C2×C4×Q8—171no—
171C2×C4×Q8—178no—
172C2×C4×Q8—178no—
173C2×C4×Q8—173no—
174C2×C4×Q8—173no—
175C2×C4×Q8#59511yescharacteristic, complemented
176C22×C4⋊C4—176no—
177C22×C4⋊C4—173no—
178C22×C42—176no—
179C22×C42—173no—
180C22×C42—176no—
181C4×C4⋊C4—1724no—
182C4×C4⋊C4—1748no—
183#376—1724no—
184C2×#95—171no—
185C2×#95—173no—
186C2×#95—176no—
187C2×C17#227614yescomplemented
188C2×C17C23×C4211yescharacteristic
189C2×C17C2×#51911yescharacteristic
190C2×C17C2×#52011yescharacteristic
191D17#227614yescomplemented
192D17C2×#51912yes—
193D17C2×#52012yes—
194C22×Q8—1716no—
195C22×Q8—174no—
196C22×Q8—174no—
197C22×Q8—173no—
198C22×Q8C2×#59511yescharacteristic
199C23×C4—173no—
200C23×C4—176no—
201C23×C4—172no—
202C23×C4—173no—
203C23×C4—171no—
204C4×Q8—17128no—
205C4×Q8—178no—
206C4×Q8—1724no—
207C4×Q8—1724no—
208C4×Q8C2×#59518yescomplemented
209C2×C4⋊C4—1748no—
210C2×C4⋊C4—1724no—
211C2×C4⋊C4—176no—
212C2×C4⋊C4—173no—
213C2×C4⋊C4—1712no—
214C2×C4⋊C4—1712no—
215C2×C4⋊C4C2×#59513yes—
216C2×C42—1748no—
217C2×C42—1724no—
218C2×C42—173no—
219C2×C42—173no—
220C2×C42—173no—
221C2×C42—1712no—
222C2×C42—1712no—
223C2×C42—1724no—
224C2×C42—6812no—
225C2×C42C2×#59513yes—
226#95—174no—
227#95—1724no—
228#95—1712no—
229C17C2×#227611yescharacteristic, complemented, Sylow
230C24—171no—
231C2×Q8—1732no—
232C2×Q8—1732no—
233C2×Q8—1724no—
234C2×Q8—1712no—
235C2×Q8C4×#59518yescomplemented
236C2×Q8C22×#59514yes—
237C22×C4—173no—
238C22×C4—1724no—
239C22×C4—176no—
240C22×C4—176no—
241C22×C4—178no—
242C22×C4—1724no—
243C22×C4—174no—
244C22×C4—1712no—
245C22×C4—171no—
246C22×C4—173no—
247C22×C4—172no—
248C22×C4—176no—
249C22×C4—172no—
250C22×C4—176no—
251C22×C4—346no—
252C22×C4—344no—
253C22×C4—346no—
254C22×C4—3412no—
255C22×C4—3412no—
256C22×C4—344no—
257C22×C4—344no—
258C22×C4C22×#59511yescharacteristic
259C22×C4C22×#59513yes—
260C22×C4C22×#59513yes—
261C4⋊C4—1796no—
262C4⋊C4—1724no—
263C4⋊C4—1712no—
264C4⋊C4C22×#595112yes—
265C42—1796no—
266C42—1712no—
267C42—1712no—
268C42—1712no—
269C42—6824no—
270C42C22×#595112yes—
271C23—174no—
272C23—174no—
273C23—172no—
274C23—172no—
275C23—171no—
276C23—171no—
277C23C23×#59511yescharacteristic
278Q8—1748no—
279Q8C2×C4×#595116yescomplemented
280C2×C4—22no—
281C2×C4—26no—
282C2×C4—1712no—
283C2×C4—176no—
284C2×C4—1724no—
285C2×C4—1724no—
286C2×C4—178no—
287C2×C4—1724no—
288C2×C4—172no—
289C2×C4—176no—
290C2×C4—346no—
291C2×C4—344no—
292C2×C4—344no—
293C2×C4—346no—
294C2×C4—3412no—
295C2×C4—3412no—
296C2×C4—3448no—
297C2×C4—3416no—
298C2×C4—344no—
299C2×C4—3412no—
300C2×C4—344no—
301C2×C4—342no—
302C2×C4—346no—
303C2×C4C2×C4×#595112yes—
304C2×C4C23×#59512yes—
305C2×C4C23×#59516yes—
306C2×C4C23×#59516yes—
307C22—178no—
308C22—178no—
309C22—172no—
310C22—172no—
311C22—172no—
312C22—172no—
313C22—171no—
314C22—171no—
315C22—172no—
316C22C22×C4×#59512yes—
317C22#1176387712yes—
318C22#1176387812yes—
319C22C24×#59511yescharacteristic
320C4—22no—
321C4—26no—
322C4—1712no—
323C4—3424no—
324C4—348no—
325C4—342no—
326C4—346no—
327C4C22×C4×#595112yes—
328C2—172no—
329C2—172no—
330C2—174no—
331C2#1454522414yesdirect product factor
332C2C23×C4×#59511yescharacteristic
333C2C2×#1176387711yescharacteristic
334C2C2×#1176387811yescharacteristic
subgroupG/H
centerC23C23×#595
commutatorC2×C17C23×C42
FrattiniC22C24×#595
FittingC2×C4×Q8×C17C4
socleC23×C17C23×C4
prime radicalC23×D17C24
cosocleC22×D17C25
hypercenterC2×C4×Q8#595
hypocenterC17C2×#2276
LCSG ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #14545224C2direct (N × H)
7. C2×Q8×#595C4semidirect
19. C2×Q8×Dic17C4semidirect
20. C2×C4×Dic34C4semidirect
21. C2×#10587573C4semidirect
22. C4×Q8×D17C4semidirect
26. C2×C4×Q8×C17C4semidirect
37. Q8×#595C2×C4semidirect
44. C4×Q8×C17C2×C4semidirect
47. Q8×Dic17C2×C4semidirect
55. #10587573C2×C4semidirect
58. C4×Dic34C2×C4semidirect
71. C2×Q8×C17C42semidirect
79. Q8×D17C42semidirect
91. C2×Dic34C42semidirect
119. Q8×C17C2×C42semidirect
139. Dic34C2×C42semidirect
175. C2×C4×Q8#595semidirect
187. C2×C17#2276semidirect
191. D17#2276semidirect
208. C4×Q8C2×#595semidirect
229. C17C2×#2276semidirect
235. C2×Q8C4×#595semidirect
279. Q8C2×C4×#595semidirect
331. C2#14545224direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×#5952
#145452242
C2×C4⋊C4×#5952
C2×C4×C4⋊#5952
C2×#117637452
C2×C4×Q8×D172
C2×#227617