Finite Group Databasebeta

← C2×#22594171 #32164520

Subgroups — C2×#22594171 #32164520

Order 18176.

2614 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556085C211yesmaximal, characteristic
2#22594171C2116yesmaximal, direct product factor
3C2×#17985932C214yesmaximal, complemented
4C2×#17985921C214yesmaximal
5C2×#17985895C212yesmaximal, complemented
6C2×#17985869C211yesmaximal, characteristic, complemented
7C2×#17985856C212yesmaximal
8C22×#125×C71C211yesmaximal, characteristic, complemented
9C23×C4×D71C2211yescharacteristic
10C2×#14556085—28no—
11C2×#14556085C2218yes—
12C22×C4⋊Dic71C2212yes—
13C22×C4⋊Dic71C2214yes—
14C22×Dic71⋊C4C2212yes—
15C22×Dic71⋊C4C2214yes—
16C22×(C2×D71)⋊C4C2212yes—
17C22×(C2×D71)⋊C4C2214yes—
18C22×C4×Dic71C2211yescharacteristic
19#17985932C22132yescomplemented
20#17985921C22132yes—
21#17985895C22116yescomplemented
22#17985869C2218yescomplemented
23#17985856C22116yes—
24C2×C71⋊#95C2214yes—
25C2×C71⋊#95C2214yes—
26C2×#125×C71—28no—
27C2×#125×C71C2218yescomplemented
28C22×C4⋊C4×C71C2212yes—
29C22×C4⋊C4×C71C2214yes—
30C22×C42×C71C2211yescharacteristic
31C24×D71C2311yescharacteristic
32C23×Dic71C2312yes—
33C23×Dic71C2314yes—
34C23×Dic71C2311yescharacteristic
35C22×C4×D71—24no—
36C22×C4×D71—24no—
37C22×C4×D71Q818yes—
38C22×C4×D71C2314yes—
39#14556085—232no—
40C2×C4⋊Dic71—28no—
41C2×C4⋊Dic71—48no—
42C2×C4⋊Dic71C2318yes—
43C2×C4⋊Dic71C23116yes—
44C2×Dic71⋊C4—28no—
45C2×Dic71⋊C4—48no—
46C2×Dic71⋊C4C2318yes—
47C2×Dic71⋊C4C23116yes—
48C2×(C2×D71)⋊C4—28no—
49C2×(C2×D71)⋊C4—48no—
50C2×(C2×D71)⋊C4C2318yes—
51C2×(C2×D71)⋊C4C23116yes—
52C2×C4×Dic71—24no—
53C2×C4×Dic71C2314yes—
54C71⋊#95C23116yes—
55C71⋊#95C23116yes—
56C23×C4×C71C2314yes—
57C23×C4×C71C2312yes—
58C23×C4×C71C2311yescharacteristic
59#125×C71—232no—
60C2×C4⋊C4×C71—216no—
61C2×C4⋊C4×C71—28no—
62C2×C4⋊C4×C71C2318yes—
63C2×C4⋊C4×C71C23116yes—
64C2×C42×C71D418yes—
65C2×C42×C71C2314yes—
66C23×D71—22no—
67C23×D71—22no—
68C23×D71—24no—
69C23×D71—24no—
70C23×D71C2×Q814yes—
71C23×D71C2412yes—
72C22×Dic71—22no—
73C22×Dic71—22no—
74C22×Dic71—24no—
75C22×Dic71—28no—
76C22×Dic71—28no—
77C22×Dic71—24no—
78C22×Dic71—28no—
79C22×Dic71—24no—
80C22×Dic71C2×Q814yes—
81C22×Dic71C2414yes—
82C22×Dic71C2418yes—
83C22×Dic71C2412yes—
84C2×C4×D71—232no—
85C2×C4×D71—28no—
86C2×C4×D71—28no—
87C2×C4×D71C2×Q8116yes—
88C4⋊Dic71—216no—
89C4⋊Dic71—416no—
90Dic71⋊C4—216no—
91Dic71⋊C4—416no—
92(C2×D71)⋊C4—216no—
93(C2×D71)⋊C4—416no—
94C4×Dic71—28no—
95C24×C71C2411yescharacteristic
96C22×C4×C71—24no—
97C22×C4×C71—24no—
98C22×C4×C71—28no—
99C22×C4×C71—28no—
100C22×C4×C71—28no—
101C22×C4×C71C2×D418yes—
102C22×C4×C71C2×D414yes—
103C22×C4×C71C2×Q814yes—
104C22×C4×C71#4114yes—
105C22×C4×C71C2418yes—
106C22×C4×C71C2414yes—
107C22×C4×C71C2412yes—
108C4⋊C4×C71—232no—
109C4⋊C4×C71—216no—
110C42×C71C2×D4116yes—
111C23×C71C22×D412yes—
112C23×C71C22×Q812yes—
113C23×C71C2×#4112yes—
114C23×C71H(5,𝔽2)14yes—
115C23×C712−1+414yes—
116C23×C71C2511yescharacteristic
117C22×D71—28no—
118C22×D71—216no—
119C22×D71—22no—
120C22×D71—22no—
121C22×D71—24no—
122C22×D71—24no—
123C22×D71—48no—
124C22×D71—44no—
125C22×D71—44no—
126C22×D71C22×Q814yes—
127C2×C4×C71—216no—
128C2×C4×C71—24no—
129C2×C4×C71—24no—
130C2×C4×C71—28no—
131C2×C4×C71—28no—
132C2×C4×C71—28no—
133C2×C4×C71—416no—
134C2×C4×C71C22⋊Q8116yes—
135C2×C4×C71C22×D418yes—
136C2×C4×C71C22×D414yes—
137C2×C4×C71C22×Q814yes—
138C2×C4×C71C2×#4114yes—
139C2×Dic71—28no—
140C2×Dic71—22no—
141C2×Dic71—22no—
142C2×Dic71—24no—
143C2×Dic71—28no—
144C2×Dic71—28no—
145C2×Dic71—24no—
146C2×Dic71—28no—
147C2×Dic71—24no—
148C2×Dic71—48no—
149C2×Dic71—416no—
150C2×Dic71C22×Q814yes—
151C4×D71—232no—
152C22×C71C2×C22⋊Q814yes—
153C22×C71#53718yes—
154C22×C71#54014yes—
155C22×C71#54114yes—
156C22×C71#54418yes—
157C22×C71C23×D411yescharacteristic
158C22×C71C23×Q811yescharacteristic
159C22×C71C22×#4111yescharacteristic
160C22×C71C2×H(5,𝔽2)12yes—
161C22×C71C2×2−1+412yes—
162C2×D71—24no—
163C2×D71—28no—
164C2×D71—416no—
165C2×D71—44no—
166C2×D71—42no—
167C2×D71—42no—
168C4×C71—28no—
169C4×C71—48no—
170C4×C71C2×C22⋊Q818yes—
171Dic71—24no—
172Dic71—44no—
173Dic71—48no—
174C2×#2629—711nomaximal, Sylow
175C2×C71#262918yescomplemented
176C2×C71C22×C22⋊Q811yescharacteristic
177C2×C71C2×#53712yes—
178C2×C71C2×#54011yescharacteristic
179C2×C71C2×#54111yescharacteristic
180C2×C71C2×#54412yes—
181D71—44no—
182C22×#125D7111yescharacteristic, complemented
183C22×#117—711no—
184#2629—7116no—
185C2×#398—714no—
186C2×#396—714no—
187C2×#389—712no—
188C2×#384—711no—
189C2×#380—712no—
190C71C2×#262911yescharacteristic, complemented, Sylow
191C24×C4—711no—
192C2×#125—28no—
193C2×#125C2×D7118yescomplemented
194C2×#117—718no—
195C2×#117—1428no—
196C22×C4⋊C4—714no—
197C22×C4⋊C4—714no—
198C22×C4⋊C4—712no—
199C22×C4⋊C4—712no—
200C22×C4⋊C4C2×D7114yes—
201C22×C4⋊C4C2×D7112yes—
202C22×C22⋊C4—714no—
203C22×C22⋊C4—712no—
204C22×C42—711no—
205C22×C42C2×D7111yescharacteristic
206#398—7132no—
207#396—7132no—
208#389—7116no—
209#384—718no—
210#380—7116no—
211C2×#95—714no—
212C2×#95—714no—
213C25—711no—
214C23×C4—711no—
215C23×C4—718no—
216C23×C4—714no—
217C23×C4—714no—
218C23×C4—712no—
219C23×C4—1424no—
220C23×C4—1424no—
221C23×C4C22×D7114yes—
222C23×C4C22×D7112yes—
223C23×C4C22×D7111yescharacteristic
224#125—232no—
225#117—14232no—
226C2×C4⋊C4—216no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—2848no—
229C2×C4⋊C4—2848no—
230C2×C4⋊C4—7116no—
231C2×C4⋊C4—7116no—
232C2×C4⋊C4—718no—
233C2×C4⋊C4—718no—
234C2×C4⋊C4—1428no—
235C2×C4⋊C4—1428no—
236C2×C4⋊C4C22×D7118yes—
237C2×C4⋊C4C22×D71116yes—
238C2×C22⋊C4—2848no—
239C2×C22⋊C4—7116no—
240C2×C22⋊C4—718no—
241C2×C22⋊C4—1428no—
242C2×C42—714no—
243C2×C42—1424no—
244C2×C42C22⋊D7118yes—
245C2×C42C22×D7114yes—
246#95—7116no—
247#95—7116no—
248C24—714no—
249C24—712no—
250C24—1422no—
251C24—1422no—
252C24—1424no—
253C24—1424no—
254C24C23×D7111yescharacteristic
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—28no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—714no—
261C22×C4—712no—
262C22×C4—7116no—
263C22×C4—714no—
264C22×C4—718no—
265C22×C4—1424no—
266C22×C4—1428no—
267C22×C4—1428no—
268C22×C4—1424no—
269C22×C4—1428no—
270C22×C4—1424no—
271C22×C4—1422no—
272C22×C4—1422no—
273C22×C4—1428no—
274C22×C4—1428no—
275C22×C4—14232no—
276C22×C4C2×C22⋊D7118yes—
277C22×C4C2×C22⋊D7114yes—
278C22×C4Q8×D7114yes—
279C22×C4#1176586614yes—
280C22×C4C23×D7118yes—
281C22×C4C23×D7114yes—
282C22×C4C23×D7112yes—
283C4⋊C4—232no—
284C4⋊C4—216no—
285C4⋊C4—28416no—
286C4⋊C4—28416no—
287C4⋊C4—14216no—
288C4⋊C4—14216no—
289C22⋊C4—28416no—
290C22⋊C4—14216no—
291C42—1428no—
292C42C2×C22⋊D71116yes—
293C23—2848no—
294C23—2844no—
295C23—2844no—
296C23—714no—
297C23—1422no—
298C23—1422no—
299C23—1424no—
300C23—1424no—
301C23—14216no—
302C23—1428no—
303C23C22×C22⋊D7112yes—
304C23C2×Q8×D7112yes—
305C23C2×#1176586612yes—
306C23#1455617414yes—
307C23#1455617514yes—
308C23C24×D7111yescharacteristic
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—216no—
312C2×C4—28no—
313C2×C4—28no—
314C2×C4—28no—
315C2×C4—416no—
316C2×C4—2848no—
317C2×C4—28416no—
318C2×C4—714no—
319C2×C4—1424no—
320C2×C4—1428no—
321C2×C4—1428no—
322C2×C4—1424no—
323C2×C4—1428no—
324C2×C4—1424no—
325C2×C4—1422no—
326C2×C4—1422no—
327C2×C4—1428no—
328C2×C4—14232no—
329C2×C4#14556104116yes—
330C2×C4C22×C22⋊D7118yes—
331C2×C4C22×C22⋊D7114yes—
332C2×C4C2×Q8×D7114yes—
333C2×C4C2×#1176586614yes—
334C22—28416no—
335C22—2844no—
336C22—2842no—
337C22—2842no—
338C22—1428no—
339C22—1424no—
340C22C2×#1455610414yes—
341C22#1798657718yes—
342C22#1798660514yes—
343C22#1798661214yes—
344C22#1798662918yes—
345C22C23×C22⋊D7111yescharacteristic
346C22C22×Q8×D7111yescharacteristic
347C22C22×#1176586611yescharacteristic
348C22C2×#1455617412yes—
349C22C2×#1455617512yes—
350C4—28no—
351C4—48no—
352C4—2844no—
353C4—2848no—
354C4—1424no—
355C4C2×#1455610418yes—
356C2—2844no—
357C2#2259417118yesdirect product factor
358C2C22×#1455610411yescharacteristic
359C2C2×#1798657712yes—
360C2C2×#1798660511yescharacteristic
361C2C2×#1798661211yescharacteristic
362C2C2×#1798662912yes—
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC22×#125×C71C2
socleC24×C71C24
prime radicalC24×D71C23
hypercenterC22×#125D71
hypocenterC71C2×#2629
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
2. #22594171C2direct (N × H)
3. C2×#17985932C2semidirect
5. C2×#17985895C2semidirect
6. C2×#17985869C2semidirect
8. C22×#125×C71C2semidirect
19. #17985932C22semidirect
21. #17985895C22semidirect
22. #17985869C22semidirect
27. C2×#125×C71C22semidirect
175. C2×C71#2629semidirect
182. C22×#125D71semidirect
190. C71C2×#2629semidirect
193. C2×#125C2×D71semidirect
357. C2#22594171direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145560852
#225941712
C2×#179859322
C2×#179859212
C2×#179858952
C2×#179858692
C2×#179858562
C22×#125×C712
C2×#262971