Finite Group Databasebeta

← C2×#22596603 #32203862

Subgroups — C2×#22596603 #32203862

Order 18176.

3294 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
1#22596603C2116yesmaximal, direct product factor
2C2×#17985936C211yesmaximal, characteristic, complemented
3C2×#17985934C213yesmaximal, complemented
4C2×#17985923C211yesmaximal, characteristic, complemented
5C2×#17985922C213yesmaximal, complemented
6C2×#17985921C213yesmaximal, complemented
7C2×#17985854C213yesmaximal
8C2×#396×C71C211yesmaximal, characteristic, complemented
9C25×D71C2211yescharacteristic
10C22×C4⋊Dic71C2213yescomplemented
11C22×Dic71⋊C4C2213yescomplemented
12C22×Dic71⋊C4C2213yescomplemented
13C22×C22⋊Dic71C2213yes—
14C22×(C2×D71)⋊C4C2213yes—
15C22×(C2×D71)⋊C4C2213yes—
16C22×(C2×D71)⋊C4C2213yes—
17C22×(C2×D71)⋊C4C2213yes—
18#17985936C2218yescomplemented
19#17985934C22124yescomplemented
20#17985923C2218yescomplemented
21#17985922C22124yescomplemented
22#17985921C22124yescomplemented
23#17985854C22124yes—
24C2×C71⋊#95C2213yescomplemented
25C22×C4⋊C4×C71C2213yescomplemented
26C22×C22⋊C4×C71C2213yes—
27#396×C71C2218yescomplemented
28C2×#95×C71C2211yescharacteristic, complemented
29C24×D71—46no—
30C24×D71—46no—
31C24×D71Q818yes—
32C24×D71C2311yescharacteristic
33C24×D71C2311yescharacteristic
34C24×D71C2314yes—
35C23×Dic71C2313yes—
36C23×Dic71C2313yes—
37C2×C4⋊Dic71—46no—
38C2×C4⋊Dic71C23112yescomplemented
39C2×Dic71⋊C4—46no—
40C2×Dic71⋊C4—46no—
41C2×Dic71⋊C4C23112yescomplemented
42C2×Dic71⋊C4C23112yescomplemented
43C2×C22⋊Dic71—46no—
44C2×C22⋊Dic71C23112yes—
45C2×(C2×D71)⋊C4—46no—
46C2×(C2×D71)⋊C4—46no—
47C2×(C2×D71)⋊C4—46no—
48C2×(C2×D71)⋊C4—46no—
49C2×(C2×D71)⋊C4C23112yes—
50C2×(C2×D71)⋊C4C23112yes—
51C2×(C2×D71)⋊C4C23112yes—
52C2×(C2×D71)⋊C4C23112yes—
53C71⋊#95C23112yescomplemented
54C25×C71C2311yescharacteristic
55C23×C4×C71C2313yes—
56C23×C4×C71C2313yes—
57C2×C4⋊C4×C71—46no—
58C2×C4⋊C4×C71C23112yescomplemented
59C2×C22⋊C4×C71—46no—
60C2×C22⋊C4×C71C23112yes—
61#95×C71C2314yescomplemented
62C23×D71—26no—
63C23×D71—26no—
64C23×D71—26no—
65C23×D71—26no—
66C23×D71—448no—
67C23×D71—412no—
68C23×D71—412no—
69C23×D71—448no—
70C23×D71—416no—
71C23×D71C2×Q814yes—
72C23×D71C2×Q814yes—
73C23×D71C2×Q8116yes—
74C23×D71C2412yes—
75C23×D71C2412yes—
76C22×Dic71—26no—
77C22×Dic71—26no—
78C22×Dic71—26no—
79C22×Dic71—26no—
80C22×Dic71—26no—
81C22×Dic71—26no—
82C22×Dic71C2416yes—
83C22×Dic71C2416yes—
84C4⋊Dic71—412no—
85Dic71⋊C4—412no—
86Dic71⋊C4—412no—
87C22⋊Dic71—412no—
88(C2×D71)⋊C4—412no—
89(C2×D71)⋊C4—412no—
90(C2×D71)⋊C4—412no—
91(C2×D71)⋊C4—412no—
92C24×C71—26no—
93C24×C71—26no—
94C24×C71C2×Q814yes—
95C24×C71C2412yes—
96C24×C71C2411yescharacteristic
97C22×C4×C71—26no—
98C22×C4×C71—26no—
99C22×C4×C71—26no—
100C22×C4×C71—26no—
101C22×C4×C71—26no—
102C22×C4×C71—26no—
103C22×C4×C71C2416yes—
104C22×C4×C71C2416yes—
105C4⋊C4×C71—412no—
106C22⋊C4×C71—412no—
107C23×C71—224no—
108C23×C71—26no—
109C23×C71—26no—
110C23×C71—412no—
111C23×C71—44no—
112C23×C71C22×Q814yes—
113C23×C71C22×Q812yes—
114C23×C71H(5,𝔽2)16yes—
115C23×C71H(5,𝔽2)16yes—
116C23×C71C2511yescharacteristic
117C22×D71—224no—
118C22×D71—224no—
119C22×D71—26no—
120C22×D71—26no—
121C22×D71—26no—
122C22×D71—26no—
123C22×D71—448no—
124C22×D71—4128no—
125C22×D71—448no—
126C22×D71—416no—
127C22×D71—412no—
128C22×D71—412no—
129C22×D71—44no—
130C22×D71—44no—
131C22×D71C22×Q814yes—
132C22×D71C22×Q814yes—
133C2×C4×C71—26no—
134C2×C4×C71—26no—
135C2×C4×C71—26no—
136C2×C4×C71—26no—
137C2×C4×C71—26no—
138C2×C4×C71—26no—
139C2×C4×C71—412no—
140C2×C4×C71—412no—
141C2×Dic71—26no—
142C2×Dic71—26no—
143C2×Dic71—26no—
144C2×Dic71—26no—
145C2×Dic71—26no—
146C2×Dic71—26no—
147C2×Dic71—412no—
148C2×Dic71—412no—
149C22×C71—212no—
150C22×C71—416no—
151C22×C71—46no—
152C22×C71—42no—
153C22×C71#543112yes—
154C22×C71#560112yes—
155C22×C71#56014yes—
156C22×C71C23×Q811yescharacteristic
157C22×C71C2×H(5,𝔽2)13yes—
158C22×C71C2×H(5,𝔽2)13yes—
159C2×D71—212no—
160C2×D71—212no—
161C2×D71—464no—
162C2×D71—416no—
163C2×D71—416no—
164C2×D71—46no—
165C2×D71—46no—
166C2×D71—42no—
167C2×D71—42no—
168C4×C71—46no—
169C4×C71—46no—
170Dic71—46no—
171Dic71—46no—
172C2×#2842—711nomaximal, Sylow
173C2×C71—44no—
174C2×C71#284218yescomplemented
175C2×C71C2×#54313yes—
176C2×C71C2×#56013yes—
177C2×C71C2×#56011yescharacteristic
178D71—44no—
179D71—44no—
180#2842—7116no—
181C2×#399—711no—
182C2×#399—713no—
183C2×#396—711no—
184C2×#396—713no—
185C2×#396—713no—
186C2×#396D7111yescharacteristic, complemented
187C2×#379—713no—
188C71C2×#284211yescharacteristic, complemented, Sylow
189C26—711no—
190C22×C4⋊C4—713no—
191C22×C4⋊C4—713no—
192C22×C4⋊C4—713no—
193C22×C4⋊C4C2×D7113yescomplemented
194C22×C22⋊C4—713no—
195C22×C22⋊C4—713no—
196C22×C22⋊C4—713no—
197C22×C22⋊C4—713no—
198C22×C22⋊C4—713no—
199C22×C22⋊C4C2×D7113yes—
200#399—7124no—
201#399—718no—
202#396—7124no—
203#396—7124no—
204#396—718no—
205#396C2×D7118yescomplemented
206#379—7124no—
207C2×#95—713no—
208C2×#95C2×D7111yescharacteristic, complemented
209C25—2846no—
210C25—2846no—
211C25—718no—
212C25—714no—
213C25—711no—
214C25—711no—
215C25C22×D7111yescharacteristic
216C23×C4—713no—
217C23×C4—713no—
218C23×C4C22×D7113yes—
219C23×C4C22×D7113yes—
220C2×C4⋊C4—46no—
221C2×C4⋊C4—2846no—
222C2×C4⋊C4—2846no—
223C2×C4⋊C4—2846no—
224C2×C4⋊C4—7112no—
225C2×C4⋊C4—7112no—
226C2×C4⋊C4—7112no—
227C2×C4⋊C4C22×D71112yescomplemented
228C2×C22⋊C4—46no—
229C2×C22⋊C4—2846no—
230C2×C22⋊C4—2846no—
231C2×C22⋊C4—2846no—
232C2×C22⋊C4—2846no—
233C2×C22⋊C4—2846no—
234C2×C22⋊C4—7112no—
235C2×C22⋊C4—7112no—
236C2×C22⋊C4—7112no—
237C2×C22⋊C4—7112no—
238C2×C22⋊C4—7112no—
239C2×C22⋊C4C22×D71112yes—
240#95—7112no—
241#95C22×D7114yescomplemented
242C24—26no—
243C24—26no—
244C24—28412no—
245C24—28412no—
246C24—28448no—
247C24—28448no—
248C24—28416no—
249C24—7116no—
250C24—714no—
251C24—714no—
252C24—712no—
253C24—712no—
254C24—1426no—
255C24—1426no—
256C24—1426no—
257C24—1426no—
258C24Q8×D7114yes—
259C24C23×D7112yes—
260C24C23×D7111yescharacteristic
261C22×C4—26no—
262C22×C4—26no—
263C22×C4—26no—
264C22×C4—26no—
265C22×C4—26no—
266C22×C4—26no—
267C22×C4—716no—
268C22×C4—716no—
269C22×C4—1426no—
270C22×C4—1426no—
271C22×C4—1426no—
272C22×C4—1426no—
273C22×C4—1426no—
274C22×C4—1426no—
275C22×C4C23×D7116yes—
276C22×C4C23×D7116yes—
277C4⋊C4—412no—
278C4⋊C4—28412no—
279C4⋊C4—28412no—
280C4⋊C4—28412no—
281C22⋊C4—412no—
282C22⋊C4—28412no—
283C22⋊C4—28412no—
284C22⋊C4—28412no—
285C22⋊C4—28412no—
286C22⋊C4—28412no—
287C23—26no—
288C23—26no—
289C23—224no—
290C23—412no—
291C23—44no—
292C23—28448no—
293C23—28448no—
294C23—28416no—
295C23—284128no—
296C23—28412no—
297C23—28412no—
298C23—2844no—
299C23—2844no—
300C23—714no—
301C23—714no—
302C23—1426no—
303C23—1426no—
304C23—1426no—
305C23—1426no—
306C23—14224no—
307C23—14224no—
308C23C2×Q8×D7114yes—
309C23C2×Q8×D7112yes—
310C23#1455617316yes—
311C23#1455617416yes—
312C23C24×D7111yescharacteristic
313C2×C4—26no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—26no—
317C2×C4—26no—
318C2×C4—26no—
319C2×C4—412no—
320C2×C4—412no—
321C2×C4—28412no—
322C2×C4—28412no—
323C2×C4—1426no—
324C2×C4—1426no—
325C2×C4—1426no—
326C2×C4—1426no—
327C2×C4—1426no—
328C2×C4—1426no—
329C22—212no—
330C22—46no—
331C22—42no—
332C22—416no—
333C22—28464no—
334C22—2846no—
335C22—2846no—
336C22—2842no—
337C22—2842no—
338C22—28416no—
339C22—28416no—
340C22—14212no—
341C22—14212no—
342C22#17986624112yes—
343C22#17986737112yes—
344C22#1798674014yes—
345C22C22×Q8×D7111yescharacteristic
346C22C2×#1455617313yes—
347C22C2×#1455617413yes—
348C4—46no—
349C4—46no—
350C4—2846no—
351C4—2846no—
352C2—44no—
353C2—2844no—
354C2—2844no—
355C2#2259660318yesdirect product factor
356C2C2×#1798662413yes—
357C2C2×#1798673713yes—
358C2C2×#1798674011yescharacteristic
subgroupG/H
centerC24C23×D71
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC2×#396×C71C2
socleC24×C71C24
prime radicalC25×D71C22
hypercenterC2×#396D71
hypocenterC71C2×#2842
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#396

Complements

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

NH ≅ G/Ntype
1. #22596603C2direct (N × H)
2. C2×#17985936C2semidirect
3. C2×#17985934C2semidirect
4. C2×#17985923C2semidirect
5. C2×#17985922C2semidirect
6. C2×#17985921C2semidirect
8. C2×#396×C71C2semidirect
10. C22×C4⋊Dic71C22semidirect
11. C22×Dic71⋊C4C22semidirect
12. C22×Dic71⋊C4C22semidirect
18. #17985936C22semidirect
19. #17985934C22semidirect
20. #17985923C22semidirect
21. #17985922C22semidirect
22. #17985921C22semidirect
24. C2×C71⋊#95C22semidirect
25. C22×C4⋊C4×C71C22semidirect
27. #396×C71C22semidirect
28. C2×#95×C71C22semidirect
38. C2×C4⋊Dic71C23semidirect
41. C2×Dic71⋊C4C23semidirect
42. C2×Dic71⋊C4C23semidirect
53. C71⋊#95C23semidirect
58. C2×C4⋊C4×C71C23semidirect
61. #95×C71C23semidirect
174. C2×C71#2842semidirect
186. C2×#396D71semidirect
188. C71C2×#2842semidirect
193. C22×C4⋊C4C2×D71semidirect
205. #396C2×D71semidirect
208. C2×#95C2×D71semidirect
227. C2×C4⋊C4C22×D71semidirect
241. #95C22×D71semidirect
355. C2#22596603direct (N × H)

Maximal subgroups

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

subgroupindex
#225966032
C2×#179859362
C2×#179859342
C2×#179859232
C2×#179859222
C2×#179859212
C2×#179858542
C2×#396×C712
C2×#284271