Finite Group Databasebeta

← C2×#22586325 #31907593

Subgroups — C2×#22586325 #31907593

Order 18176.

496 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985781C211yesmaximal, characteristic, complemented
2C2×#17985756C211yesmaximal, characteristic, complemented
3#22586325C214yesmaximal, direct product factor
4C2×#469×C71C211yesmaximal, characteristic
5C2×#14556087—21no—
6C2×#14556068—21no—
7#17985781—21no—
8#17985781C2212yescomplemented
9#17985756—21no—
10#17985756C2212yescomplemented
11C22×D8×C71C411yescharacteristic, complemented
12C22×M16×C71C2211yescharacteristic
13C2×#121×C71—21no—
14#469×C71—21no—
15#469×C71—21no—
16#469×C71C412yescomplemented
17#469×C71C2212yes—
18C2×#106×C71C411yescharacteristic, complemented
19C2×#102×C71—21no—
20#14556087—22no—
21#14556087—22no—
22#14556087—41no—
23#14556087—41no—
24C2×C4⋊Dic71—41no—
25C2×C22⋊Dic71—41no—
26C2×C4×Dic71—22no—
27#14556068—22no—
28#14556068—41no—
29C2×C71⋊C16—41no—
30C22×D4×C71—21no—
31C22×D4×C71—41no—
32C23×C4×C71D411yescharacteristic
33C2×D8×C71—21no—
34C2×D8×C71—41no—
35C2×D8×C71—41no—
36C2×D8×C71C2×C412yescomplemented
37C2×M16×C71—41no—
38C2×M16×C71—41no—
39C2×M16×C71C2×C411yescharacteristic
40C2×M16×C71D411yescharacteristic
41C2×M16×C71D411yescharacteristic
42C2×M16×C71C2311yescharacteristic
43C22×C8×C71C2×C411yescharacteristic
44C22×C8×C71D411yescharacteristic
45#121×C71—22no—
46#121×C71—22no—
47#121×C71—41no—
48#121×C71—41no—
49C2×C4⋊C4×C71—21no—
50C2×C22⋊C4×C71—41no—
51#106×C71—21no—
52#106×C71C2×C412yescomplemented
53#102×C71—22no—
54#102×C71—22no—
55C22×Dic71—42no—
56C4⋊Dic71—42no—
57C4⋊Dic71—42no—
58C22⋊Dic71—42no—
59C22⋊Dic71—42no—
60C4×Dic71—24no—
61C4×Dic71—42no—
62C71⋊C16—42no—
63C24×C71—41no—
64C24×C71D811yescharacteristic
65C2×D4×C71—22no—
66C2×D4×C71—42no—
67C2×D4×C71—42no—
68C2×D4×C71—41no—
69C2×D4×C71—41no—
70C2×D4×C71—41no—
71C2×D4×C71—41no—
72C2×D4×C71—41no—
73C2×D4×C71—84no—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—41no—
79C22×C4×C71C22⋊C411yescharacteristic
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71D811yescharacteristic
82C22×C4×C71SD1611yescharacteristic
83C22×C4×C71SD1611yescharacteristic
84C22×C4×C71C2×D411yescharacteristic
85D8×C71—42no—
86D8×C71—42no—
87M16×C71—41no—
88M16×C71—41no—
89M16×C71—41no—
90M16×C71—41no—
91C2×C8×C71—21no—
92C2×C8×C71—21no—
93C2×C8×C71—21no—
94C2×C8×C71—21no—
95C2×C8×C71C22⋊C411yescharacteristic
96C2×C8×C71C22⋊C411yescharacteristic
97C2×C8×C71C22×C411yescharacteristic
98C2×C8×C71C2×D411yescharacteristic
99C4⋊C4×C71—22no—
100C4⋊C4×C71—41no—
101C22⋊C4×C71—42no—
102C22⋊C4×C71—42no—
103C23×C71—21no—
104C23×C71—21no—
105C23×C71—42no—
106C23×C71—42no—
107C23×C71—82no—
108C23×C71—82no—
109C23×C71—81no—
110C23×C71—81no—
111C23×C71#10211yescharacteristic
112C23×C71#10211yescharacteristic
113C23×C71C2×D811yescharacteristic
114D4×C71—42no—
115D4×C71—42no—
116D4×C71—84no—
117D4×C71—84no—
118D4×C71—81no—
119D4×C71—81no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—42no—
131C2×C4×C71—82no—
132C2×C4×C71#9911yescharacteristic
133C2×C4×C71#9911yescharacteristic
134C2×C4×C71#10211yescharacteristic
135C2×C4×C71#10211yescharacteristic
136C2×C4×C71#10411yescharacteristic
137C2×C4×C71#10411yescharacteristic
138C2×C4×C71C2×C22⋊C411yescharacteristic
139C2×C4×C71C2×SD1611yescharacteristic
140C2×Dic71—44no—
141C2×Dic71—84no—
142C8×C71—21no—
143C8×C71—21no—
144C8×C71—21no—
145C8×C71—21no—
146C22×C71—22no—
147C22×C71—21no—
148C22×C71—21no—
149C22×C71—41no—
150C22×C71—41no—
151C22×C71—42no—
152C22×C71—81no—
153C22×C71—82no—
154C22×C71—82no—
155C22×C71—82no—
156C22×C71—162no—
157C22×C71—162no—
158C22×C71#32711yescharacteristic
159C22×C71#32711yescharacteristic
160C22×C71C2×#10211yescharacteristic
161C4×C71—21no—
162C4×C71—21no—
163C4×C71—82no—
164C4×C71#32711yescharacteristic
165C4×C71#32711yescharacteristic
166C4×C71C2×#9911yescharacteristic
167C4×C71C2×#10411yescharacteristic
168Dic71—84no—
169C2×#1339—711nomaximal, Sylow
170C2×C71—21no—
171C2×C71—21no—
172C2×C71—41no—
173C2×C71—41no—
174C2×C71—162no—
175C2×C71#133912yescomplemented
176C2×C71C2×#32711yescharacteristic
177C2×#469D7111yescharacteristic
178C2×#349—711no—
179C2×H(3,ℤ/4)—711no—
180#1339—714no—
181C71C2×#133911yescharacteristic, complemented, Sylow
182C22×D8Dic7111yescharacteristic, complemented
183C22×M16C2×D7111yescharacteristic
184C2×#121—21no—
185C2×#117—1421no—
186C2×M32—1421no—
187#469—21no—
188#469—21no—
189#469Dic7112yescomplemented
190#469C2×D7112yes—
191C2×#106Dic7111yescharacteristic, complemented
192C2×#102—21no—
193#349—712no—
194#349—1421no—
195H(3,ℤ/4)—712no—
196H(3,ℤ/4)—1421no—
197C22×D4—21no—
198C22×D4—41no—
199C23×C4C22⋊D7111yescharacteristic
200C2×D8—21no—
201C2×D8—41no—
202C2×D8—41no—
203C2×D8C2×Dic7112yescomplemented
204C2×M16—41no—
205C2×M16—41no—
206C2×M16C2×Dic7111yescharacteristic
207C2×M16C22⋊D7111yescharacteristic
208C2×M16C22⋊D7111yescharacteristic
209C2×M16C22×D7111yescharacteristic
210C22×C8C2×Dic7111yescharacteristic
211C22×C8C22⋊D7111yescharacteristic
212#121—22no—
213#121—22no—
214#121—41no—
215#121—41no—
216#117—2841no—
217#117—2841no—
218#117—1422no—
219#117—1422no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—2841no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—2841no—
224C2×C42—1422no—
225M32—2841no—
226M32—1422no—
227C2×C16—2841no—
228#106—21no—
229#106C2×Dic7112yescomplemented
230#102—22no—
231#102—22no—
232C24—41no—
233C24#1176585011yescharacteristic
234C2×D4—22no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—84no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—41no—
248C22×C4—2842no—
249C22×C4C22⋊Dic7111yescharacteristic
250C22×C4C22⋊Dic7111yescharacteristic
251C22×C4#1176585011yescharacteristic
252C22×C4#1176585311yescharacteristic
253C22×C4#1176585311yescharacteristic
254C22×C4C2×C22⋊D7111yescharacteristic
255D8—42no—
256D8—42no—
257M16—41no—
258M16—41no—
259M16—41no—
260M16—41no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8C22⋊Dic7111yescharacteristic
266C2×C8C22⋊Dic7111yescharacteristic
267C2×C8C22×Dic7111yescharacteristic
268C2×C8C2×C22⋊D7111yescharacteristic
269C4⋊C4—22no—
270C4⋊C4—41no—
271C4⋊C4—2842no—
272C4⋊C4—2842no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—2842no—
276C22⋊C4—2842no—
277C42—2842no—
278C42—1424no—
279C16—2842no—
280C23—21no—
281C23—21no—
282C23—42no—
283C23—42no—
284C23—82no—
285C23—82no—
286C23—81no—
287C23—81no—
288C23#1455604911yescharacteristic
289C23#1455604911yescharacteristic
290C23C2×#1176585011yescharacteristic
291D4—42no—
292D4—42no—
293D4—84no—
294D4—84no—
295D4—81no—
296D4—81no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—82no—
309C2×C4—2844no—
310C2×C4—5684no—
311C2×C4#1455604311yescharacteristic
312C2×C4#1455604311yescharacteristic
313C2×C4#1455604911yescharacteristic
314C2×C4#1455604911yescharacteristic
315C2×C4#1455605511yescharacteristic
316C2×C4#1455605511yescharacteristic
317C2×C4C2×C22⋊Dic7111yescharacteristic
318C2×C4C2×#1176585311yescharacteristic
319C8—21no—
320C8—21no—
321C8—21no—
322C8—21no—
323C22—22no—
324C22—21no—
325C22—21no—
326C22—41no—
327C22—41no—
328C22—42no—
329C22—81no—
330C22—82no—
331C22—82no—
332C22—82no—
333C22—162no—
334C22—162no—
335C22#1798573311yescharacteristic
336C22#1798573311yescharacteristic
337C22C2×#1455604911yescharacteristic
338C4—21no—
339C4—21no—
340C4—82no—
341C4—5684no—
342C4#1798573311yescharacteristic
343C4#1798573311yescharacteristic
344C4C2×#1455604311yescharacteristic
345C4C2×#1455605511yescharacteristic
346C2—21no—
347C2—21no—
348C2—41no—
349C2—41no—
350C2—162no—
351C2#2258632512yesdirect product factor
352C2C2×#1798573311yescharacteristic
subgroupG/H
centerC22#17985733
commutatorC2×C8×C71C22×C4
FrattiniC2×M16C22×D71
Fitting, prime radicalC2×#469×C71C2
socleC22×C71#327
cosocleC2×M16×C71C23
hypercenterC2×#469D71
hypocenterC71C2×#1339
LCSG ▷ C2×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ C2×#469

Complements

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

NH ≅ G/Ntype
1. C2×#17985781C2semidirect
2. C2×#17985756C2semidirect
3. #22586325C2direct (N × H)
8. #17985781C22semidirect
10. #17985756C22semidirect
11. C22×D8×C71C4semidirect
16. #469×C71C4semidirect
18. C2×#106×C71C4semidirect
36. C2×D8×C71C2×C4semidirect
52. #106×C71C2×C4semidirect
175. C2×C71#1339semidirect
181. C71C2×#1339semidirect
182. C22×D8Dic71semidirect
189. #469Dic71semidirect
191. C2×#106Dic71semidirect
203. C2×D8C2×Dic71semidirect
229. #106C2×Dic71semidirect
351. C2#22586325direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179857812
C2×#179857562
#225863252
C2×#469×C712
C2×#133971