Finite Group Databasebeta

← C2×#22587065 #31965988

Subgroups — C2×#22587065 #31965988

Order 18176.

1182 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556128C211yesmaximal, characteristic
2C2×#17986056C211yesmaximal, characteristic
3#22587065C218yesmaximal, direct product factor
4C2×#17985747C214yesmaximal, complemented
5C2×#432×C71C211yesmaximal, characteristic
6C23×Dic142C412yes—
7C2×#14556128—22no—
8C2×#14556128—21no—
9C2×#14556128—22no—
10C2×#14556128—21no—
11C2×#14556128C412yes—
12C2×#14556128C414yes—
13C2×#14556128C2212yes—
14C2×#14556128C2214yes—
15C22×C4⋊Dic71—21no—
16C22×C4⋊Dic71—21no—
17#17986056—22no—
18#17986056C2214yes—
19C2×C4⋊(C71⋊C8)C2212yes—
20C2×C22⋊(C71⋊C8)—21no—
21C2×#14556039—21no—
22#17985747C22116yescomplemented
23C22×C42×C71C2211yescharacteristic
24#432×C71—22no—
25#432×C71C2214yes—
26C2×C4⋊C8×C71C2212yes—
27C2×C22⋊C8×C71—21no—
28C2×#97×C71—21no—
29C22×Dic142—24no—
30C22×Dic142—22no—
31C22×Dic142—48no—
32C22×Dic142C2×C414yes—
33C22×Dic142C2×C418yes—
34C23×Dic71—22no—
35C22×C71⋊C8—22no—
36#14556128—24no—
37#14556128—24no—
38#14556128—24no—
39#14556128—24no—
40#14556128—22no—
41#14556128—24no—
42#14556128—22no—
43#14556128C2×C414yes—
44#14556128C2×C418yes—
45#14556128C2314yes—
46C2×C4⋊Dic71—24no—
47C2×C4⋊Dic71—24no—
48C2×C4⋊Dic71—22no—
49C2×C4⋊Dic71—22no—
50C2×C4⋊Dic71—24no—
51C2×C4⋊Dic71—22no—
52C2×C4⋊Dic71—24no—
53C2×C4⋊Dic71—22no—
54C4⋊(C71⋊C8)—22no—
55C4⋊(C71⋊C8)C2314yes—
56C22⋊(C71⋊C8)—24no—
57#14556039—24no—
58C23×C4×C71C2×C411yescharacteristic
59C23×C4×C71D411yescharacteristic
60C23×C4×C71D411yescharacteristic
61C22×C8×C71—22no—
62C2×C42×C71C2×C411yescharacteristic
63C2×C42×C71C2×C412yes—
64C2×C42×C71D412yes—
65C2×C42×C71D411yescharacteristic
66C2×C42×C71D412yes—
67C2×C42×C71D411yescharacteristic
68C2×C42×C71C2312yes—
69C2×C42×C71C2311yescharacteristic
70C4⋊C8×C71—22no—
71C4⋊C8×C71C2314yes—
72C22⋊C8×C71—24no—
73#97×C71—24no—
74C2×Dic142—28no—
75C2×Dic142—24no—
76C2×Dic142—416no—
77C2×Dic142—48no—
78C2×Dic142—416no—
79C2×Dic142—48no—
80C2×Dic142C22×C418yes—
81C22×Dic71—28no—
82C22×Dic71—24no—
83C22×Dic71—48no—
84C2×C71⋊C8—24no—
85C2×C71⋊C8—44no—
86C4⋊Dic71—28no—
87C4⋊Dic71—28no—
88C4⋊Dic71—24no—
89C4⋊Dic71—24no—
90C4⋊Dic71—24no—
91C4⋊Dic71—24no—
92C24×C71C22⋊C411yescharacteristic
93C22×C4×C71—24no—
94C22×C4×C71—24no—
95C22×C4×C71—24no—
96C22×C4×C71C22⋊C412yes—
97C22×C4×C71C22⋊C411yescharacteristic
98C22×C4×C71C22⋊C412yes—
99C22×C4×C71C22⋊C412yes—
100C22×C4×C71C22⋊C411yescharacteristic
101C22×C4×C71C22⋊C411yescharacteristic
102C22×C4×C71C22×C412yes—
103C22×C4×C71C22×C411yescharacteristic
104C22×C4×C71C2×D412yes—
105C22×C4×C71C2×D411yescharacteristic
106C22×C4×C71C2×D412yes—
107C22×C4×C71C2×D411yescharacteristic
108C2×C8×C71—24no—
109C2×C8×C71—44no—
110C42×C71C22⋊C412yes—
111C42×C71C22⋊C412yes—
112C42×C71C22⋊C412yes—
113C42×C71C22×C412yes—
114C42×C71C22×C411yescharacteristic
115C42×C71C2×D412yes—
116C42×C71C2×D411yescharacteristic
117C42×C71C2×D412yes—
118C42×C71C2×D411yescharacteristic
119C42×C71C2411yescharacteristic
120C23×C71—22no—
121C23×C71—22no—
122C23×C71#10014yes—
123C23×C71C2×C22⋊C412yes—
124C23×C71C2×C22⋊C411yescharacteristic
125C2×C4×C71—24no—
126C2×C4×C71—22no—
127C2×C4×C71—24no—
128C2×C4×C71—22no—
129C2×C4×C71—22no—
130C2×C4×C71—22no—
131C2×C4×C71—24no—
132C2×C4×C71—24no—
133C2×C4×C71—24no—
134C2×C4×C71—24no—
135C2×C4×C71C2×C22⋊C412yes—
136C2×C4×C71C2×C22⋊C412yes—
137C2×C4×C71C2×C22⋊C412yes—
138C2×C4×C71C2×C22⋊C411yescharacteristic
139C2×C4×C71C2×C22⋊C411yescharacteristic
140C2×C4×C71C2×C22⋊C411yescharacteristic
141C2×C4×C71#13714yes—
142C2×C4×C71#13714yes—
143C2×C4×C71C23×C411yescharacteristic
144C2×C4×C71C22×D411yescharacteristic
145C2×C4×C71C22×D411yescharacteristic
146C2×Dic71—28no—
147C2×Dic71—416no—
148C2×Dic71—48no—
149Dic142—48no—
150Dic142—416no—
151Dic142—48no—
152C8×C71—44no—
153C71⋊C8—44no—
154C22×C71—28no—
155C22×C71—22no—
156C22×C71—22no—
157C22×C71C2×#10014yes—
158C22×C71C2×#10012yes—
159C22×C71#41912yes—
160C22×C71#41912yes—
161C22×C71C22×C22⋊C411yescharacteristic
162C4×C71—21no—
163C4×C71—22no—
164C4×C71—21no—
165C4×C71—22no—
166C4×C71—22no—
167C4×C71#41912yes—
168C4×C71#41912yes—
169C4×C71C2×#13712yes—
170C4×C71C2×#13712yes—
171Dic71—48no—
172C2×#1558—711nomaximal, Sylow
173C2×C71—24no—
174C2×C71#155814yescomplemented
175C2×C71C22×#10011yescharacteristic
176C2×C71C2×#41911yescharacteristic
177C2×C71C2×#41911yescharacteristic
178C22×C4⋊Q8—711no—
179C2×#432—711no—
180C2×#432D7111yescharacteristic
181#1558—718no—
182C2×#333—714no—
183C71C2×#155811yescharacteristic, complemented, Sylow
184C23×Q8—712no—
185C2×C4⋊Q8—714no—
186C2×C4⋊Q8—712no—
187C2×C4⋊Q8—712no—
188C2×C4⋊Q8—714no—
189C2×C4⋊Q8—1421no—
190C2×C4⋊Q8—1422no—
191C2×C4⋊Q8—1421no—
192C2×C4⋊Q8—1422no—
193C22×C4⋊C4—1421no—
194C22×C4⋊C4—1421no—
195C22×C42C2×D7111yescharacteristic
196#432—22no—
197#432—714no—
198#432—1422no—
199#432C2×D7114yes—
200C2×C4⋊C8—712no—
201C2×C4⋊C8C2×D7112yes—
202C2×C22⋊C8—21no—
203C2×C22⋊C8—1421no—
204C2×#97—21no—
205C2×#97—1421no—
206#333—7116no—
207C22×Q8—2848no—
208C22×Q8—718no—
209C22×Q8—714no—
210C22×Q8—1424no—
211C22×Q8—1422no—
212C23×C4—1422no—
213C23×C4C4×D7111yescharacteristic
214C23×C4D28411yescharacteristic
215C23×C4C22⋊D7111yescharacteristic
216C22×C8—22no—
217C22×C8—1422no—
218C4⋊Q8—714no—
219C4⋊Q8—718no—
220C4⋊Q8—714no—
221C4⋊Q8—1424no—
222C4⋊Q8—1424no—
223C4⋊Q8—1424no—
224C4⋊Q8—1424no—
225C4⋊Q8—1422no—
226C4⋊Q8—1424no—
227C4⋊Q8—1422no—
228C2×C4⋊C4—1424no—
229C2×C4⋊C4—1424no—
230C2×C4⋊C4—1422no—
231C2×C4⋊C4—1422no—
232C2×C4⋊C4—1424no—
233C2×C4⋊C4—1422no—
234C2×C4⋊C4—1424no—
235C2×C4⋊C4—1422no—
236C2×C42C4×D7112yes—
237C2×C42C4×D7111yescharacteristic
238C2×C42D28411yescharacteristic
239C2×C42D28412yes—
240C2×C42C22⋊D7111yescharacteristic
241C2×C42C22⋊D7112yes—
242C2×C42C22×D7111yescharacteristic
243C2×C42C22×D7112yes—
244C4⋊C8—22no—
245C4⋊C8—714no—
246C4⋊C8—1422no—
247C4⋊C8C22×D7114yes—
248C22⋊C8—24no—
249C22⋊C8—1424no—
250#97—24no—
251#97—1424no—
252C24(C2×D71)⋊C411yescharacteristic
253C2×Q8—28416no—
254C2×Q8—2848no—
255C2×Q8—28416no—
256C2×Q8—2848no—
257C2×Q8—718no—
258C2×Q8—1428no—
259C2×Q8—1424no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—2848no—
264C22×C4—1428no—
265C22×C4—1424no—
266C22×C4(C2×D71)⋊C412yes—
267C22×C4(C2×D71)⋊C412yes—
268C22×C4(C2×D71)⋊C412yes—
269C22×C4(C2×D71)⋊C411yescharacteristic
270C22×C4(C2×D71)⋊C411yescharacteristic
271C22×C4(C2×D71)⋊C411yescharacteristic
272C22×C4C2×C4×D7112yes—
273C22×C4C2×C4×D7111yescharacteristic
274C22×C4C2×C22⋊D7112yes—
275C22×C4C2×C22⋊D7111yescharacteristic
276C22×C4C2×D28412yes—
277C22×C4C2×D28411yescharacteristic
278C2×C8—24no—
279C2×C8—44no—
280C2×C8—2844no—
281C2×C8—1424no—
282C4⋊C4—1424no—
283C4⋊C4—1424no—
284C4⋊C4—1428no—
285C4⋊C4—1428no—
286C4⋊C4—1424no—
287C4⋊C4—1424no—
288C42(C2×D71)⋊C412yes—
289C42(C2×D71)⋊C412yes—
290C42(C2×D71)⋊C412yes—
291C42C2×C4×D7112yes—
292C42C2×C4×D7111yescharacteristic
293C42C2×C22⋊D7112yes—
294C42C2×C22⋊D7111yescharacteristic
295C42C2×D28412yes—
296C42C2×D28411yescharacteristic
297C42C23×D7111yescharacteristic
298C23—22no—
299C23—22no—
300C23#1455604414yes—
301C23C2×(C2×D71)⋊C412yes—
302C23C2×(C2×D71)⋊C411yescharacteristic
303Q8—2848no—
304Q8—2848no—
305Q8—28416no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—28416no—
317C2×C4—2848no—
318C2×C4—1428no—
319C2×C4C2×(C2×D71)⋊C412yes—
320C2×C4C2×(C2×D71)⋊C412yes—
321C2×C4C2×(C2×D71)⋊C412yes—
322C2×C4C2×(C2×D71)⋊C411yescharacteristic
323C2×C4C2×(C2×D71)⋊C411yescharacteristic
324C2×C4C2×(C2×D71)⋊C411yescharacteristic
325C2×C4#1455615814yes—
326C2×C4#1455616114yes—
327C2×C4C22×C4×D7111yescharacteristic
328C2×C4C22×C22⋊D7111yescharacteristic
329C2×C4C22×D28411yescharacteristic
330C8—44no—
331C8—2844no—
332C22—28no—
333C22—22no—
334C22—22no—
335C22C2×#1455604414yes—
336C22C2×#1455604412yes—
337C22#1798600712yes—
338C22#1798600912yes—
339C22C22×(C2×D71)⋊C411yescharacteristic
340C4—21no—
341C4—21no—
342C4—22no—
343C4—22no—
344C4—22no—
345C4—2848no—
346C4#1798600712yes—
347C4#1798600912yes—
348C4C2×#1455615812yes—
349C4C2×#1455616112yes—
350C2—24no—
351C2#2258706514yesdirect product factor
352C2C22×#1455604411yescharacteristic
353C2C2×#1798600711yescharacteristic
354C2C2×#1798600911yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC2×#432×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22⋊C4
cosocleC42×C71C24
hypercenterC2×#432D71
hypocenterC71C2×#1558
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#432

Complements

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

NH ≅ G/Ntype
3. #22587065C2direct (N × H)
4. C2×#17985747C2semidirect
22. #17985747C22semidirect
174. C2×C71#1558semidirect
183. C71C2×#1558semidirect
351. C2#22587065direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561282
C2×#179860562
#225870652
C2×#179857472
C2×#432×C712
C2×#155871