Finite Group Databasebeta

← C2×#14545136 #17978434

Subgroups — C2×#14545136 #17978434

Order 4352.

718 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763819C211yesmaximal, characteristic
2C22×#10587534C211yesmaximal, characteristic
3#14545136C218yesmaximal, direct product factor
4C2×#11763736C212yesmaximal, complemented
5C2×#11763735C212yesmaximal, complemented
6C2×M32×D17C211yesmaximal, characteristic
7C2×#10587677—21no—
8C22×C4⋊#595—21no—
9#11763819—22no—
10#11763819C2214yes—
11C2×#10587534—21no—
12C2×#10587534—22no—
13C2×#10587534C2212yes—
14C2×#10587534C2214yes—
15C2×#10587534C2211yescharacteristic
16C2×#10587534C2211yescharacteristic
17C2×#10587533—21no—
18#11763736C2218yescomplemented
19#11763735C2218yescomplemented
20C22×C8×D17C2211yescharacteristic
21C2×#10587483C411yescharacteristic, complemented
22C2×#10587469C412yescomplemented
23M32×D17C414yescomplemented
24M32×D17C2214yes—
25C2×C16×D17C2212yes—
26C2×C17×M32C411yescharacteristic, complemented
27C23×#595—41no—
28C22×C4×D17D411yescharacteristic
29C2×C22⋊#595—41no—
30#10587677—24no—
31#10587677—42no—
32C2×C4⋊#595—21no—
33C2×C4⋊#595—21no—
34C2×C4⋊#595—22no—
35C2×C4⋊#595—24no—
36C2×C4⋊#595—41no—
37C2×C4⋊#595—42no—
38C2×C4×#595—41no—
39C22×C8×C17C2×C411yescharacteristic
40C22×#3622Q811yescharacteristic
41C2×C8×D17C2×C412yes—
42C2×C8×D17C2×C411yescharacteristic
43C2×C8×D17D411yescharacteristic
44C2×C8×D17D412yes—
45C2×C8×D17Q811yescharacteristic
46C2×C8×D17Q812yes—
47C2×C8×D17C2311yescharacteristic
48C2×C8×D17C2312yes—
49#10587534—24no—
50#10587534—22no—
51#10587534—21no—
52#10587534—21no—
53#10587534C2312yes—
54#10587534C2314yes—
55#10587534C2312yes—
56#10587533—24no—
57C17×M32C2×C414yescomplemented
58C2×C16×C17C2×C412yescomplemented
59#10587483C2×C414yescomplemented
60C2×#63214C2×C412yescomplemented
61#10587469C2×C418yescomplemented
62C16×D17C2×C414yescomplemented
63C16×D17C2314yes—
64C23×D17D811yescharacteristic
65C22×#595—41no—
66C22×#595—41no—
67C22×#595—44no—
68C22×#595—42no—
69C22×#595—84no—
70C22×C4×C17C4⋊C411yescharacteristic
71C22×Dic17Q1611yescharacteristic
72C2×C4×D17C4⋊C412yes—
73C2×C4×D17C4⋊C411yescharacteristic
74C2×C4×D17D811yescharacteristic
75C2×C4×D17D812yes—
76C2×C4×D17Q1612yes—
77C2×C4×D17Q1611yescharacteristic
78C2×C4×D17C2×D411yescharacteristic
79C2×C4×D17C2×D412yes—
80C22⋊#595—44no—
81C4⋊#595—24no—
82C4⋊#595—22no—
83C4⋊#595—22no—
84C4⋊#595—44no—
85C4⋊#595—42no—
86C4⋊#595—41no—
87C4⋊#595—41no—
88C4×#595—44no—
89C2×C8×C17C4⋊C411yescharacteristic
90C2×C8×C17C4⋊C412yes—
91C2×C8×C17C22×C411yescharacteristic
92C2×C8×C17C22×C412yes—
93C2×#3622C4⋊C412yes—
94C2×#3622C4⋊C411yescharacteristic
95C2×#3622C2×Q811yescharacteristic
96C2×#3622C2×Q812yes—
97C8×D17C4⋊C412yes—
98C8×D17C4⋊C412yes—
99C8×D17C4⋊C412yes—
100C8×D17C22×C411yescharacteristic
101C8×D17C22×C412yes—
102C8×D17C2×D412yes—
103C8×D17C2×D411yescharacteristic
104C8×D17C2×Q811yescharacteristic
105C8×D17C2×Q812yes—
106C8×D17C2411yescharacteristic
107C16×C17C22×C414yescomplemented
108#63214C22×C414yescomplemented
109C2×#2159—171nomaximal, Sylow
110C23×C17#10711yescharacteristic
111C22×D17—22no—
112C22×D17—22no—
113C22×D17#10712yes—
114C22×D17#10711yescharacteristic
115C22×D17C2×D812yes—
116C22×D17C2×D811yescharacteristic
117C2×#595—42no—
118C2×#595—42no—
119C2×#595—44no—
120C2×#595—88no—
121C2×#595—82no—
122C2×#595—84no—
123C2×#595—82no—
124C2×C4×C17#10712yes—
125C2×C4×C17#10711yescharacteristic
126C2×C4×C17C2×C4⋊C412yes—
127C2×C4×C17C2×C4⋊C411yescharacteristic
128C2×Dic17#10712yes—
129C2×Dic17#10711yescharacteristic
130C2×Dic17C2×Q1612yes—
131C2×Dic17C2×Q1611yescharacteristic
132C4×D17#10712yes—
133C4×D17#10712yes—
134C4×D17#10712yes—
135C4×D17C2×C4⋊C412yes—
136C4×D17C2×C4⋊C411yescharacteristic
137C4×D17C2×D812yes—
138C4×D17C2×D811yescharacteristic
139C4×D17C2×Q1612yes—
140C4×D17C2×Q1611yescharacteristic
141C4×D17C22×D411yescharacteristic
142C8×C17C2×C4⋊C411yescharacteristic
143C8×C17C2×C4⋊C412yes—
144C8×C17C23×C411yescharacteristic
145#3622C2×C4⋊C411yescharacteristic
146#3622C2×C4⋊C412yes—
147#3622C22×Q811yescharacteristic
148C22×M32—171no—
149C2×#427—171no—
150C22×#107—171no—
151#2159—178no—
152C2×#367—172no—
153C2×#366—172no—
154C22×C17—22no—
155C22×C17C2×#10712yes—
156C22×C17C2×#10711yescharacteristic
157C2×D17—22no—
158C2×D17—22no—
159C2×D17—22no—
160C2×D17—24no—
161C2×D17C2×#10712yes—
162C2×D17C2×#10711yescharacteristic
163C2×D17#50912yes—
164C2×D17#51012yes—
165C2×D17C22×D811yescharacteristic
166#595—82no—
167#595—84no—
168#595—82no—
169C4×C17C2×#10712yes—
170C4×C17C2×#10711yescharacteristic
171C4×C17C22×C4⋊C411yescharacteristic
172Dic17C2×#10712yes—
173Dic17C2×#10711yescharacteristic
174Dic17C22×Q1611yescharacteristic
175C23×C8—171no—
176C2×#117—341no—
177C22×C4⋊C4—341no—
178C2×M32—171no—
179C2×M32—174no—
180C2×M32—174no—
181C2×M32—172no—
182C2×M32#59511yescharacteristic, complemented
183C22×C16—172no—
184#427—174no—
185#427—342no—
186C2×#107—172no—
187C2×#107—174no—
188C2×#107—171no—
189C2×#107—171no—
190C2×#107—341no—
191C2×#107—342no—
192C2×#106—341no—
193#367—178no—
194#366—178no—
195C2×C17—22no—
196C2×C17#215912yescomplemented
197C2×C17C22×#10711yescharacteristic
198D17—22no—
199D17#215912yescomplemented
200D17C2×#50911yescharacteristic
201D17C2×#51011yescharacteristic
202C23×C4—171no—
203C23×C4—681no—
204C22×C8—171no—
205C22×C8—171no—
206C22×C8—172no—
207C22×C8—172no—
208C22×C8—171no—
209C22×C8—171no—
210C22×C8—172no—
211C22×C8—172no—
212C22×C8—171no—
213C22×C8C2×#59511yescharacteristic
214#117—344no—
215#117—682no—
216C2×C4⋊C4—342no—
217C2×C4⋊C4—344no—
218C2×C4⋊C4—341no—
219C2×C4⋊C4—341no—
220C2×C4⋊C4—681no—
221C2×C4⋊C4—682no—
222C2×C22⋊C4—681no—
223C2×C42—681no—
224M32—174no—
225M32—178no—
226M32C2×#59514yescomplemented
227C2×C16—172no—
228C2×C16—174no—
229C2×C16—174no—
230C2×C16C2×#59512yescomplemented
231#107—172no—
232#107—174no—
233#107—172no—
234#107—344no—
235#107—342no—
236#107—341no—
237#107—341no—
238#106—344no—
239C17C2×#215911yescharacteristic, complemented, Sylow
240C24—171no—
241C22×C4—171no—
242C22×C4—172no—
243C22×C4—171no—
244C22×C4—171no—
245C22×C4—172no—
246C22×C4—171no—
247C22×C4—172no—
248C22×C4—172no—
249C22×C4—171no—
250C22×C4—681no—
251C22×C4—681no—
252C22×C4—682no—
253C22×C4—684no—
254C22×C4—1364no—
255C22×C4C4⋊#59511yescharacteristic
256C2×C8—172no—
257C2×C8—171no—
258C2×C8—172no—
259C2×C8—171no—
260C2×C8—172no—
261C2×C8—171no—
262C2×C8—172no—
263C2×C8—172no—
264C2×C8—172no—
265C2×C8—171no—
266C2×C8—172no—
267C2×C8—171no—
268C2×C8—171no—
269C2×C8—172no—
270C2×C8C4⋊#59512yes—
271C2×C8C4⋊#59511yescharacteristic
272C2×C8C22×#59511yescharacteristic
273C2×C8C22×#59512yes—
274C4⋊C4—342no—
275C4⋊C4—342no—
276C4⋊C4—344no—
277C4⋊C4—684no—
278C4⋊C4—682no—
279C4⋊C4—681no—
280C4⋊C4—681no—
281C22⋊C4—684no—
282C42—684no—
283C16—174no—
284C16C22×#59514yescomplemented
285C23—171no—
286C23—172no—
287C23—172no—
288C23—171no—
289C23—342no—
290C23—342no—
291C23#1058753411yescharacteristic
292C2×C4—172no—
293C2×C4—172no—
294C2×C4—171no—
295C2×C4—171no—
296C2×C4—172no—
297C2×C4—171no—
298C2×C4—172no—
299C2×C4—172no—
300C2×C4—172no—
301C2×C4—171no—
302C2×C4—172no—
303C2×C4—171no—
304C2×C4—171no—
305C2×C4—172no—
306C2×C4—682no—
307C2×C4—682no—
308C2×C4—684no—
309C2×C4—1364no—
310C2×C4—1362no—
311C2×C4—1362no—
312C2×C4—1368no—
313C2×C4#1058753412yes—
314C2×C4#1058753411yescharacteristic
315C2×C4C2×C4⋊#59512yes—
316C2×C4C2×C4⋊#59511yescharacteristic
317C8—172no—
318C8—171no—
319C8—171no—
320C8C2×C4⋊#59512yes—
321C8C2×C4⋊#59511yescharacteristic
322C8C23×#59511yescharacteristic
323C22—22no—
324C22—171no—
325C22—171no—
326C22—172no—
327C22—172no—
328C22—172no—
329C22—342no—
330C22—342no—
331C22—344no—
332C22—342no—
333C22C2×#1058753412yes—
334C22C2×#1058753411yescharacteristic
335C4—171no—
336C4—171no—
337C4—172no—
338C4—1364no—
339C4—1362no—
340C4—1362no—
341C4C2×#1058753412yes—
342C4C2×#1058753411yescharacteristic
343C4C22×C4⋊#59511yescharacteristic
344C2—22no—
345C2—171no—
346C2—171no—
347C2—172no—
348C2—342no—
349C2#1454513612yesdirect product factor
350C2C22×#1058753411yescharacteristic
subgroupG/H
centerC22C2×#10587534
commutatorC8×C17C23×C4
FrattiniC8C23×#595
FittingC2×C17×M32C4
socleC22×C17C2×#107
prime radicalC23×D17D8
cosocleC8×D17C24
hypercenterC2×M32#595
hypocenterC17C2×#2159
LCSG ▷ C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×M32

Complements

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

NH ≅ G/Ntype
3. #14545136C2direct (N × H)
4. C2×#11763736C2semidirect
5. C2×#11763735C2semidirect
18. #11763736C22semidirect
19. #11763735C22semidirect
21. C2×#10587483C4semidirect
22. C2×#10587469C4semidirect
23. M32×D17C4semidirect
26. C2×C17×M32C4semidirect
57. C17×M32C2×C4semidirect
58. C2×C16×C17C2×C4semidirect
59. #10587483C2×C4semidirect
60. C2×#63214C2×C4semidirect
61. #10587469C2×C4semidirect
62. C16×D17C2×C4semidirect
107. C16×C17C22×C4semidirect
108. #63214C22×C4semidirect
182. C2×M32#595semidirect
196. C2×C17#2159semidirect
199. D17#2159semidirect
226. M32C2×#595semidirect
230. C2×C16C2×#595semidirect
239. C17C2×#2159semidirect
284. C16C22×#595semidirect
349. C2#14545136direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117638192
C22×#105875342
#145451362
C2×#117637362
C2×#117637352
C2×M32×D172
C2×#215917