Finite Group Databasebeta

← C22×#17986219 #32864258

Subgroups — C22×#17986219 #32864258

Order 18176.

2918 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11765855C211yesmaximal, characteristic, complemented
2C22×#14556115C211yesmaximal, characteristic
3C22×Q8×Dic71C211yesmaximal, characteristic, complemented
4C2×#17986219C2124yesmaximal, direct product factor
5C22×Dic71⋊C8C211yesmaximal, characteristic, complemented
6C22×#14556053C211yesmaximal, characteristic, complemented
7C22×#14556049C211yesmaximal, characteristic
8C23×SD16×C71C211yesmaximal, characteristic
9C23×Dic142C2211yescharacteristic
10C22×#11765855—28no—
11C22×#11765855C22112yescomplemented
12C23×C71⋊C8C2211yescharacteristic
13C2×#14556115—26no—
14C2×#14556115C22112yes—
15C2×Q8×Dic71—26no—
16C2×Q8×Dic71C22112yescomplemented
17C22×C4⋊Dic71—21no—
18C22×C4⋊Dic71C2211yescharacteristic
19C22×C22⋊Dic71—21no—
20C22×C22⋊Dic71—21no—
21C22×C4×Dic71—21no—
22C22×C4×Dic71C2211yescharacteristic
23#17986219C22164yesdirect product factor
24C2×Dic71⋊C8C22112yescomplemented
25C2×#14556053C22112yescomplemented
26C2×#14556049C22112yes—
27C23×Q8×C71C2211yescharacteristic
28C23×D4×C71C2211yescharacteristic
29C22×SD16×C71—28no—
30C22×SD16×C71C22112yes—
31C23×C8×C71C2211yescharacteristic
32C22×Dic142—23no—
33C22×Dic142—24no—
34C22×Dic142—24no—
35C22×Dic142C2316yes—
36C23×Dic71—21no—
37C23×Dic71—21no—
38C23×Dic71D411yescharacteristic
39C23×Dic71D411yescharacteristic
40C2×#11765855—248no—
41C2×#11765855C23116yescomplemented
42C22×C71⋊C8—24no—
43C22×C71⋊C8C2316yes—
44#14556115—224no—
45#14556115C23116yes—
46Q8×Dic71—224no—
47Q8×Dic71C23116yescomplemented
48C2×C4⋊Dic71—212no—
49C2×C4⋊Dic71—23no—
50C2×C4⋊Dic71C2316yes—
51C2×C22⋊Dic71—212no—
52C2×C22⋊Dic71—212no—
53C2×C4×Dic71—212no—
54C2×C4×Dic71D416yes—
55C2×C4×Dic71C2316yes—
56Dic71⋊C8C23116yescomplemented
57#14556053C23116yescomplemented
58#14556049C23116yes—
59C25×C71—21no—
60C22×Q8×C71—23no—
61C22×Q8×C71—24no—
62C22×Q8×C71D418yes—
63C22×Q8×C71C2316yes—
64C22×D4×C71—24no—
65C22×D4×C71—23no—
66C22×D4×C71—24no—
67C22×D4×C71C2316yes—
68C23×C4×C71—21no—
69C23×C4×C71C2311yescharacteristic
70C2×SD16×C71—248no—
71C2×SD16×C71C23116yes—
72C22×C8×C71—24no—
73C22×C8×C71C2316yes—
74C2×Dic142—212no—
75C2×Dic142—212no—
76C2×Dic142—26no—
77C2×Dic142—412no—
78C2×Dic142C2414yes—
79C22×Dic71—24no—
80C22×Dic71—24no—
81C22×Dic71—26no—
82C22×Dic71—26no—
83C22×Dic71—44no—
84C22×Dic71—44no—
85C22×Dic71C2×D416yes—
86C22×Dic71C2×D416yes—
87#11765855—232no—
88C2×C71⋊C8—212no—
89C2×C71⋊C8C2414yes—
90C4⋊Dic71—26no—
91C4⋊Dic71—216no—
92C4⋊Dic71C2414yes—
93C22⋊Dic71—216no—
94C22⋊Dic71—216no—
95C4×Dic71—216no—
96C4×Dic71C2×D4112yes—
97C4×Dic71C2414yes—
98C24×C71—26no—
99C24×C71—44no—
100C24×C71—44no—
101C24×C71—44no—
102C24×C71C2×D411yescharacteristic
103C2×Q8×C71—224no—
104C2×Q8×C71—26no—
105C2×Q8×C71—212no—
106C2×Q8×C71C2×D4124yes—
107C2×Q8×C71C2414yes—
108C2×D4×C71—212no—
109C2×D4×C71—26no—
110C2×D4×C71—212no—
111C2×D4×C71—412no—
112C2×D4×C71C2414yes—
113C22×C4×C71—28no—
114C22×C4×C71—26no—
115C22×C4×C71C2×D413yes—
116C22×C4×C71C2×D414yes—
117C22×C4×C71#4114yes—
118C22×C4×C71C2413yes—
119SD16×C71—232no—
120C2×C8×C71—212no—
121C2×C8×C71C2414yes—
122C23×C71—24no—
123C23×C71—412no—
124C23×C71—412no—
125C23×C71—412no—
126C23×C71—816no—
127C23×C71#12114yes—
128C23×C71#13514yes—
129C23×C71#13714yes—
130C23×C71C22×D413yes—
131Q8×C71—224no—
132Q8×C71—24no—
133Q8×C71C22×D418yes—
134D4×C71—24no—
135D4×C71—24no—
136D4×C71—412no—
137C2×C4×C71—224no—
138C2×C4×C71—24no—
139C2×C4×C71#121112yes—
140C2×C4×C71C22×D413yes—
141C2×C4×C71C22×D416yes—
142C2×C4×C71C2×#4116yes—
143C2×C4×C71C2511yescharacteristic
144C2×Dic71—212no—
145C2×Dic71—212no—
146C2×Dic71—24no—
147C2×Dic71—24no—
148C2×Dic71—412no—
149C2×Dic71—412no—
150C2×Dic71C22×D414yes—
151C2×Dic71C22×D414yes—
152Dic142—24no—
153Dic142—24no—
154Dic142—412no—
155C8×C71—24no—
156C71⋊C8—24no—
157C22×C71—44no—
158C22×C71—44no—
159C22×C71—44no—
160C22×C71—824no—
161C22×C71#464116yescomplemented
162C22×C71C2×#12116yes—
163C22×C71C2×#13516yes—
164C22×C71C2×#13716yes—
165C22×C71C23×D411yescharacteristic
166C4×C71—28no—
167C4×C71C2×#12116yes—
168C4×C71C23×D411yescharacteristic
169C4×C71C22×#4111yescharacteristic
170Dic71—24no—
171Dic71—24no—
172Dic71—44no—
173Dic71—44no—
174C22×#464—711nomaximal, Sylow
175C2×C71—84no—
176C2×C71C2×#464112yescomplemented
177C2×C71C22×#12111yescharacteristic
178C2×C71C22×#13511yescharacteristic
179C2×C71C22×#13711yescharacteristic
180C23×Q16—711no—
181C23×SD16D7111yescharacteristic
182C22×#124—711no—
183C22×C4×Q8—711no—
184C2×#464—7124no—
185C22×C4⋊C8—711no—
186C22×#103—711no—
187C22×#102—711no—
188C71C22×#46411yescharacteristic, complemented, Sylow
189C23×Q8—711no—
190C23×Q8C2×D7111yescharacteristic
191C23×D4C2×D7111yescharacteristic
192C22×Q16—7112no—
193C22×Q16—1428no—
194C22×SD16—28no—
195C22×SD16C2×D71112yes—
196C23×C8—711no—
197C23×C8C2×D7111yescharacteristic
198C2×#124—7112no—
199C2×#124—1426no—
200C2×C4×Q8—7112no—
201C2×C4×Q8—1426no—
202C22×C4⋊C4—711no—
203C22×C4⋊C4—1421no—
204C22×C22⋊C4—1421no—
205C22×C22⋊C4—1421no—
206C22×C42—711no—
207C22×C42—1421no—
208#464—7164no—
209C2×C4⋊C8—7112no—
210C2×#103—7112no—
211C2×#102—7112no—
212C25—21no—
213C22×Q8—23no—
214C22×Q8—24no—
215C22×Q8—716no—
216C22×Q8—1424no—
217C22×Q8—1424no—
218C22×Q8—1423no—
219C22×Q8C22⋊D7118yes—
220C22×Q8C22×D7116yes—
221C22×D4—23no—
222C22×D4—24no—
223C22×D4—24no—
224C22×D4C22×D7116yes—
225C23×C4—21no—
226C23×C4—711no—
227C23×C4—711no—
228C23×C4—1421no—
229C23×C4—1421no—
230C23×C4C22×D7111yescharacteristic
231C2×Q16—7116no—
232C2×Q16—14248no—
233C2×SD16—248no—
234C2×SD16C22×D71116yes—
235C22×C8—24no—
236C22×C8—716no—
237C22×C8—1424no—
238C22×C8C22×D7116yes—
239#124—7116no—
240#124—14224no—
241C4×Q8—7116no—
242C4×Q8—14224no—
243C2×C4⋊C4—716no—
244C2×C4⋊C4—14212no—
245C2×C4⋊C4—1423no—
246C2×C22⋊C4—14212no—
247C2×C22⋊C4—14212no—
248C2×C42—716no—
249C2×C42—716no—
250C2×C42—14212no—
251C4⋊C8—7116no—
252#103—7116no—
253#102—7116no—
254C24—26no—
255C24—44no—
256C24—44no—
257C24—44no—
258C24D4×D7111yescharacteristic
259C2×Q8—26no—
260C2×Q8—212no—
261C2×Q8—224no—
262C2×Q8—28412no—
263C2×Q8—714no—
264C2×Q8—14212no—
265C2×Q8—14212no—
266C2×Q8—1426no—
267C2×Q8C2×C22⋊D71124yes—
268C2×Q8C23×D7114yes—
269C2×D4—26no—
270C2×D4—212no—
271C2×D4—212no—
272C2×D4—412no—
273C2×D4C23×D7114yes—
274C22×C4—26no—
275C22×C4—28no—
276C22×C4—2844no—
277C22×C4—2844no—
278C22×C4—716no—
279C22×C4—716no—
280C22×C4—1426no—
281C22×C4—1426no—
282C22×C4—1424no—
283C22×C4—1424no—
284C22×C4D4×D7113yes—
285C22×C4C2×C22⋊D7114yes—
286C22×C4#1176586414yes—
287C22×C4C23×D7113yes—
288Q16—14232no—
289SD16—232no—
290C2×C8—212no—
291C2×C8—714no—
292C2×C8—14212no—
293C2×C8C23×D7114yes—
294C4⋊C4—714no—
295C4⋊C4—14216no—
296C4⋊C4—1426no—
297C22⋊C4—14216no—
298C22⋊C4—14216no—
299C42—7112no—
300C42—714no—
301C42—14216no—
302C23—24no—
303C23—412no—
304C23—412no—
305C23—412no—
306C23—816no—
307C23#1455610014yes—
308C23#1455614914yes—
309C23#1455615714yes—
310C23C2×D4×D7113yes—
311Q8—24no—
312Q8—224no—
313Q8—28412no—
314Q8—1424no—
315Q8—1424no—
316Q8C22×C22⋊D7118yes—
317D4—24no—
318D4—24no—
319D4—412no—
320C2×C4—24no—
321C2×C4—224no—
322C2×C4—28412no—
323C2×C4—28412no—
324C2×C4—714no—
325C2×C4—714no—
326C2×C4—1424no—
327C2×C4—1424no—
328C2×C4—14212no—
329C2×C4—14212no—
330C2×C4#14556100112yes—
331C2×C4C2×D4×D7113yes—
332C2×C4C22×C22⋊D7116yes—
333C2×C4C2×#1176586416yes—
334C2×C4C24×D7111yescharacteristic
335C8—24no—
336C8—1424no—
337C22—44no—
338C22—44no—
339C22—44no—
340C22—824no—
341C22#17986219116yesdirect product factor
342C22C2×#1455610016yes—
343C22C2×#1455614916yes—
344C22C2×#1455615716yes—
345C22C22×D4×D7111yescharacteristic
346C4—28no—
347C4—2844no—
348C4—2844no—
349C4—1424no—
350C4—1424no—
351C4C2×#1455610016yes—
352C4C23×C22⋊D7111yescharacteristic
353C4C22×#1176586411yescharacteristic
354C2—84no—
355C2C2×#17986219112yesdirect product factor
356C2C22×#1455610011yescharacteristic
357C2C22×#1455614911yescharacteristic
358C2C22×#1455615711yescharacteristic
subgroupG/H
centerC24D4×D71
commutator, cosocleC2×C4×C71C25
FrattiniC2×C4C24×D71
FittingC23×SD16×C71C2
socleC24×C71C2×D4
prime radicalC23×D4×C71C22
hypercenterC23×SD16D71
hypocenterC71C22×#464
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C24 ◁ C23×C4 ◁ C23×SD16

Complements

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

NH ≅ G/Ntype
1. C23×#11765855C2semidirect
3. C22×Q8×Dic71C2semidirect
4. C2×#17986219C2direct (N × H)
5. C22×Dic71⋊C8C2semidirect
6. C22×#14556053C2semidirect
11. C22×#11765855C22semidirect
16. C2×Q8×Dic71C22semidirect
23. #17986219C22direct (N × H)
24. C2×Dic71⋊C8C22semidirect
25. C2×#14556053C22semidirect
41. C2×#11765855C23semidirect
47. Q8×Dic71C23semidirect
56. Dic71⋊C8C23semidirect
57. #14556053C23semidirect
161. C22×C71#464semidirect
176. C2×C71C2×#464semidirect
188. C71C22×#464semidirect
341. C22#17986219direct (N × H)
355. C2C2×#17986219direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117658552
C22×#145561152
C22×Q8×Dic712
C2×#179862192
C22×Dic71⋊C82
C22×#145560532
C22×#145560492
C23×SD16×C712
C22×#46471