Finite Group Databasebeta

← C2×#22594170 #32164519

Subgroups — C2×#22594170 #32164519

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×#14556120C211yesmaximal, characteristic, complemented
2#22594170C2116yesmaximal, direct product factor
3C2×#17985932C214yesmaximal, complemented
4C2×#17985922C214yesmaximal
5C2×#17985897C212yesmaximal, complemented
6C2×#17985869C211yesmaximal, characteristic, complemented
7C2×#17985859C212yesmaximal
8C22×#117×C71C211yesmaximal, characteristic
9C2×#14556120—28no—
10C2×#14556120C2218yescomplemented
11C22×C4⋊Dic71C2212yes—
12C22×C4⋊Dic71C2212yes—
13C22×C4⋊Dic71C2214yes—
14C22×Dic71⋊C4C2214yes—
15C22×Dic71⋊C4C2214yes—
16C22×C22⋊Dic71C2214yes—
17#17985932C22132yescomplemented
18#17985922C22132yes—
19#17985897C22116yescomplemented
20#17985869C2218yescomplemented
21#17985859C22116yes—
22C2×C71⋊#95C2214yes—
23C2×C71⋊#95C2214yes—
24C24×C4×C71C2211yescharacteristic
25C2×#117×C71—28no—
26C2×#117×C71C2218yes—
27C22×C4⋊C4×C71C2212yes—
28C22×C22⋊C4×C71C2212yes—
29C22×C42×C71C2211yescharacteristic
30C22×C42×C71C2211yescharacteristic
31C23×Dic71C2314yes—
32C23×Dic71C2314yes—
33#14556120—232no—
34C2×C4⋊Dic71—28no—
35C2×C4⋊Dic71—28no—
36C2×C4⋊Dic71—48no—
37C2×C4⋊Dic71C2318yes—
38C2×C4⋊Dic71C2318yes—
39C2×C4⋊Dic71C23116yes—
40C2×Dic71⋊C4—216no—
41C2×Dic71⋊C4—48no—
42C2×Dic71⋊C4C23116yes—
43C2×Dic71⋊C4C23116yes—
44C2×C22⋊Dic71—48no—
45C2×C22⋊Dic71C23116yes—
46C71⋊#95C23116yes—
47C71⋊#95C23116yes—
48C25×C71C2311yescharacteristic
49C23×C4×C71—24no—
50C23×C4×C71—24no—
51C23×C4×C71Q818yes—
52C23×C4×C71C2312yes—
53C23×C4×C71C2312yes—
54C23×C4×C71C2311yescharacteristic
55C23×C4×C71C2314yes—
56C23×C4×C71C2311yescharacteristic
57#117×C71—232no—
58C2×C4⋊C4×C71—28no—
59C2×C4⋊C4×C71C2318yes—
60C2×C22⋊C4×C71—28no—
61C2×C22⋊C4×C71C2318yes—
62C2×C42×C71—24no—
63C2×C42×C71D418yes—
64C2×C42×C71C2314yes—
65C2×C42×C71C2314yes—
66C22×Dic71—28no—
67C22×Dic71—28no—
68C22×Dic71—28no—
69C22×Dic71—28no—
70C22×Dic71—28no—
71C22×Dic71—28no—
72C22×Dic71C2418yes—
73C22×Dic71C2418yes—
74C4⋊Dic71—216no—
75C4⋊Dic71—216no—
76C4⋊Dic71—416no—
77Dic71⋊C4—232no—
78Dic71⋊C4—416no—
79C22⋊Dic71—416no—
80C24×C71—22no—
81C24×C71—22no—
82C24×C71—24no—
83C24×C71—24no—
84C24×C71C2×Q814yes—
85C24×C71C2412yes—
86C24×C71C2411yescharacteristic
87C22×C4×C71—22no—
88C22×C4×C71—22no—
89C22×C4×C71—24no—
90C22×C4×C71—24no—
91C22×C4×C71—232no—
92C22×C4×C71—28no—
93C22×C4×C71—28no—
94C22×C4×C71—24no—
95C22×C4×C71—24no—
96C22×C4×C71—24no—
97C22×C4×C71C2×D418yes—
98C22×C4×C71C2×D414yes—
99C22×C4×C71C2×Q814yes—
100C22×C4×C71C2×Q8116yes—
101C22×C4×C71C2×Q814yes—
102C22×C4×C71#4114yes—
103C22×C4×C71C2414yes—
104C22×C4×C71C2414yes—
105C22×C4×C71C2412yes—
106C22×C4×C71C2412yes—
107C4⋊C4×C71—216no—
108C22⋊C4×C71—216no—
109C42×C71—28no—
110C42×C71C2×D4116yes—
111C23×C71—28no—
112C23×C71—216no—
113C23×C71—22no—
114C23×C71—22no—
115C23×C71—24no—
116C23×C71—24no—
117C23×C71—48no—
118C23×C71—44no—
119C23×C71—44no—
120C23×C71C22×D412yes—
121C23×C71C22×Q814yes—
122C23×C71C22×Q812yes—
123C23×C71C2×#4112yes—
124C23×C71H(5,𝔽2)14yes—
125C23×C712−1+414yes—
126C23×C71C2511yescharacteristic
127C2×C4×C71—28no—
128C2×C4×C71—216no—
129C2×C4×C71—22no—
130C2×C4×C71—22no—
131C2×C4×C71—24no—
132C2×C4×C71—24no—
133C2×C4×C71—232no—
134C2×C4×C71—24no—
135C2×C4×C71—24no—
136C2×C4×C71—24no—
137C2×C4×C71—48no—
138C2×C4×C71C22⋊Q8116yes—
139C2×C4×C71C22×D418yes—
140C2×C4×C71C22×D414yes—
141C2×C4×C71C22×Q814yes—
142C2×C4×C71C22×Q814yes—
143C2×C4×C71C2×#4114yes—
144C2×Dic71—28no—
145C2×Dic71—28no—
146C2×Dic71—28no—
147C2×Dic71—28no—
148C2×Dic71—28no—
149C2×Dic71—28no—
150C2×Dic71—416no—
151C2×Dic71—416no—
152C22×C71—24no—
153C22×C71—28no—
154C22×C71—416no—
155C22×C71—44no—
156C22×C71—42no—
157C22×C71—42no—
158C22×C71C2×C22⋊Q814yes—
159C22×C71#53718yes—
160C22×C71#54014yes—
161C22×C71#54114yes—
162C22×C71#54418yes—
163C22×C71C23×D411yescharacteristic
164C22×C71C23×Q811yescharacteristic
165C22×C71C22×#4111yescharacteristic
166C22×C71C2×H(5,𝔽2)12yes—
167C22×C71C2×2−1+412yes—
168C4×C71—24no—
169C4×C71—28no—
170C4×C71—44no—
171C4×C71C2×C22⋊Q818yes—
172Dic71—48no—
173Dic71—48no—
174C2×#2629—711nomaximal, Sylow
175C2×C71—44no—
176C2×C71#262918yescomplemented
177C2×C71C22×C22⋊Q811yescharacteristic
178C2×C71C2×#53712yes—
179C2×C71C2×#54011yescharacteristic
180C2×C71C2×#54111yescharacteristic
181C2×C71C2×#54412yes—
182C22×#125—711no—
183C22×#117D7111yescharacteristic
184#2629—7116no—
185C2×#398—714no—
186C2×#396—714no—
187C2×#389—712no—
188C2×#384—711no—
189C2×#380—712no—
190C71C2×#262911yescharacteristic, complemented, Sylow
191C24×C4C2×D7111yescharacteristic
192C2×#125—718no—
193C2×#125—1428no—
194C2×#117—28no—
195C2×#117C2×D7118yes—
196C22×C4⋊C4—714no—
197C22×C4⋊C4—714no—
198C22×C4⋊C4—714no—
199C22×C4⋊C4—712no—
200C22×C4⋊C4—712no—
201C22×C4⋊C4C2×D7112yes—
202C22×C22⋊C4—714no—
203C22×C22⋊C4C2×D7112yes—
204C22×C42C2×D7111yescharacteristic
205C22×C42C2×D7111yescharacteristic
206#398—7132no—
207#396—7132no—
208#389—7116no—
209#384—718no—
210#380—7116no—
211C2×#95—714no—
212C2×#95—714no—
213C25C22×D7111yescharacteristic
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—714no—
217C23×C4—714no—
218C23×C4Dic14218yes—
219C23×C4C22×D7112yes—
220C23×C4C22×D7112yes—
221C23×C4C22×D7114yes—
222C23×C4C22×D7111yescharacteristic
223C23×C4C22×D7111yescharacteristic
224#125—14232no—
225#117—232no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—2848no—
228C2×C4⋊C4—2848no—
229C2×C4⋊C4—7116no—
230C2×C4⋊C4—7116no—
231C2×C4⋊C4—718no—
232C2×C4⋊C4—7116no—
233C2×C4⋊C4—718no—
234C2×C4⋊C4—14216no—
235C2×C4⋊C4—1428no—
236C2×C4⋊C4—1428no—
237C2×C4⋊C4C22×D7118yes—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—2848no—
240C2×C22⋊C4—7116no—
241C2×C22⋊C4C22×D7118yes—
242C2×C42—24no—
243C2×C42C22⋊D7118yes—
244C2×C42C22×D7114yes—
245C2×C42C22×D7114yes—
246#95—7116no—
247#95—7116no—
248C24—22no—
249C24—22no—
250C24—24no—
251C24—24no—
252C24C2×Dic14214yes—
253C24C23×D7112yes—
254C24C23×D7111yescharacteristic
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—28no—
260C22×C4—28no—
261C22×C4—232no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—718no—
266C22×C4—718no—
267C22×C4—1428no—
268C22×C4—1428no—
269C22×C4—1428no—
270C22×C4—1428no—
271C22×C4—1428no—
272C22×C4—1428no—
273C22×C4C2×C22⋊D7118yes—
274C22×C4C2×C22⋊D7114yes—
275C22×C4C2×Dic14214yes—
276C22×C4C2×Dic142116yes—
277C22×C4C2×Dic14214yes—
278C22×C4#1176586514yes—
279C22×C4C23×D7114yes—
280C22×C4C23×D7114yes—
281C22×C4C23×D7112yes—
282C22×C4C23×D7112yes—
283C4⋊C4—216no—
284C4⋊C4—28416no—
285C4⋊C4—28416no—
286C4⋊C4—14232no—
287C4⋊C4—14216no—
288C4⋊C4—14216no—
289C22⋊C4—216no—
290C22⋊C4—28416no—
291C42—28no—
292C42C2×C22⋊D71116yes—
293C23—22no—
294C23—22no—
295C23—24no—
296C23—24no—
297C23—216no—
298C23—28no—
299C23—48no—
300C23—44no—
301C23—44no—
302C23C22×C22⋊D7112yes—
303C23C22×Dic14214yes—
304C23C22×Dic14212yes—
305C23C2×#1176586512yes—
306C23#1455617414yes—
307C23#1455617514yes—
308C23C24×D7111yescharacteristic
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—216no—
314C2×C4—28no—
315C2×C4—232no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—48no—
320C2×C4—28416no—
321C2×C4—28416no—
322C2×C4—1428no—
323C2×C4—1428no—
324C2×C4—1428no—
325C2×C4—1428no—
326C2×C4—1428no—
327C2×C4—1428no—
328C2×C4#14556107116yes—
329C2×C4C22×C22⋊D7118yes—
330C2×C4C22×C22⋊D7114yes—
331C2×C4C22×Dic14214yes—
332C2×C4C22×Dic14214yes—
333C2×C4C2×#1176586514yes—
334C22—28no—
335C22—24no—
336C22—416no—
337C22—44no—
338C22—42no—
339C22—42no—
340C22C2×#1455610714yes—
341C22#1798657718yes—
342C22#1798660414yes—
343C22#1798661114yes—
344C22#1798662718yes—
345C22C23×C22⋊D7111yescharacteristic
346C22C23×Dic14211yescharacteristic
347C22C22×#1176586511yescharacteristic
348C22C2×#1455617412yes—
349C22C2×#1455617512yes—
350C4—24no—
351C4—28no—
352C4—44no—
353C4—2848no—
354C4—2848no—
355C4C2×#1455610718yes—
356C2—44no—
357C2#2259417018yesdirect product factor
358C2C22×#1455610711yescharacteristic
359C2C2×#1798657712yes—
360C2C2×#1798660411yescharacteristic
361C2C2×#1798661111yescharacteristic
362C2C2×#1798662712yes—
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC22×#117×C71C2
socleC24×C71C24
prime radicalC25×C71C23
hypercenterC22×#117D71
hypocenterC71C2×#2629
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
1. C22×#14556120C2semidirect
2. #22594170C2direct (N × H)
3. C2×#17985932C2semidirect
5. C2×#17985897C2semidirect
6. C2×#17985869C2semidirect
10. C2×#14556120C22semidirect
17. #17985932C22semidirect
19. #17985897C22semidirect
20. #17985869C22semidirect
176. C2×C71#2629semidirect
190. C71C2×#2629semidirect
357. C2#22594170direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561202
#225941702
C2×#179859322
C2×#179859222
C2×#179858972
C2×#179858692
C2×#179858592
C22×#117×C712
C2×#262971