Finite Group Databasebeta

← C2×#22587119 #31966518

Subgroups — C2×#22587119 #31966518

Order 18176.

670 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
1C2×#17986055C211yesmaximal, characteristic
2C2×#17986051C211yesmaximal, characteristic
3C2×#17986050C211yesmaximal, characteristic
4#22587119C218yesmaximal, direct product factor
5C2×#17985752C211yesmaximal, characteristic, complemented
6C2×#17985749C211yesmaximal, characteristic, complemented
7C2×#17985749C211yesmaximal, characteristic, complemented
8C2×#335×C71C211yesmaximal, characteristic, complemented
9C2×C42×D71C2211yescharacteristic
10#17986055—22no—
11#17986055C2214yes—
12#17986051—22no—
13#17986051C2214yes—
14#17986050C414yes—
15#17986050C2214yes—
16C2×C4⋊(C71⋊C8)C2211yescharacteristic
17C2×C4⋊(C71⋊C8)C2211yescharacteristic
18C2×Dic71⋊C8C411yescharacteristic
19C2×Dic71⋊C8C411yescharacteristic
20C2×(C2×D71)⋊C8—21no—
21C2×(C2×D71)⋊C8—21no—
22C2×(C2×D71)⋊C8C411yescharacteristic
23C2×(C2×D71)⋊C8C411yescharacteristic
24C2×#14556038—21no—
25C2×C4×C71⋊C8C2211yescharacteristic
26C2×C8×Dic71—21no—
27#17985752C2214yescomplemented
28#17985749C2214yescomplemented
29#17985749C2214yescomplemented
30C2×C4⋊C8×C71C2211yescharacteristic
31C2×C4⋊C8×C71C2211yescharacteristic
32C2×C4×C8×C71C2211yescharacteristic
33#335×C71C2214yescomplemented
34C22×C4×D71C2×C411yescharacteristic
35C22×C4×D71D411yescharacteristic
36C22×C4×D71Q811yescharacteristic
37C22×C71⋊C8—21no—
38C22×C71⋊C8—21no—
39C22×C71⋊C8C2×C411yescharacteristic
40C22×C71⋊C8C2×C411yescharacteristic
41C2×C4×Dic71C2×C411yescharacteristic
42C2×C4×Dic71D411yescharacteristic
43C2×C4×Dic71Q811yescharacteristic
44C42×D71C2×C412yes—
45C42×D71D412yes—
46C42×D71Q812yes—
47C42×D71C2312yes—
48C4⋊(C71⋊C8)—21no—
49C4⋊(C71⋊C8)—21no—
50C4⋊(C71⋊C8)C2312yes—
51C4⋊(C71⋊C8)C2312yes—
52Dic71⋊C8C2×C414yes—
53Dic71⋊C8C2×C414yes—
54(C2×D71)⋊C8—24no—
55(C2×D71)⋊C8—24no—
56(C2×D71)⋊C8C2×C414yes—
57(C2×D71)⋊C8C2×C414yes—
58#14556038—24no—
59C4×C71⋊C8C2×C412yes—
60C4×C71⋊C8C2312yes—
61C8×Dic71—24no—
62C22×C8×C71—21no—
63C22×C8×C71—21no—
64C22×C8×C71C2×C411yescharacteristic
65C22×C8×C71C2×C411yescharacteristic
66C2×C42×C71C2311yescharacteristic
67C4⋊C8×C71—21no—
68C4⋊C8×C71—21no—
69C4⋊C8×C71C2312yes—
70C4⋊C8×C71C2312yes—
71C4×C8×C71C2×C412yes—
72C4×C8×C71C2312yes—
73C23×D71C4⋊C411yescharacteristic
74C22×Dic71C4⋊C411yescharacteristic
75C22×Dic71C4⋊C411yescharacteristic
76C22×Dic71C4⋊C411yescharacteristic
77C2×C4×D71—22no—
78C2×C4×D71—22no—
79C2×C4×D71—22no—
80C2×C4×D71—22no—
81C2×C4×D71—42no—
82C2×C4×D71C4⋊C412yes—
83C2×C4×D71C4⋊C412yes—
84C2×C4×D71C4⋊C412yes—
85C2×C4×D71C22×C412yes—
86C2×C4×D71C2×D412yes—
87C2×C4×D71C2×Q812yes—
88C2×C71⋊C8—22no—
89C2×C71⋊C8—22no—
90C2×C71⋊C8—22no—
91C2×C71⋊C8—22no—
92C2×C71⋊C8—42no—
93C2×C71⋊C8—42no—
94C2×C71⋊C8C22×C412yes—
95C2×C71⋊C8C22×C412yes—
96C4×Dic71C4⋊C412yes—
97C4×Dic71C4⋊C412yes—
98C4×Dic71C4⋊C412yes—
99C4×Dic71C22×C412yes—
100C4×Dic71C2×D412yes—
101C4×Dic71C2×Q812yes—
102C22×C4×C71C22×C411yescharacteristic
103C22×C4×C71C2×D411yescharacteristic
104C22×C4×C71C2×Q811yescharacteristic
105C2×C8×C71—22no—
106C2×C8×C71—22no—
107C2×C8×C71—22no—
108C2×C8×C71—22no—
109C2×C8×C71—42no—
110C2×C8×C71—42no—
111C2×C8×C71C22×C412yes—
112C2×C8×C71C22×C412yes—
113C42×C71C22×C411yescharacteristic
114C42×C71C2×D411yescharacteristic
115C42×C71C2×Q811yescharacteristic
116C42×C71C2411yescharacteristic
117C23×C71C2×C4⋊C411yescharacteristic
118C22×D71—22no—
119C22×D71—22no—
120C22×D71—22no—
121C22×D71C2×C4⋊C412yes—
122C2×C4×C71—22no—
123C2×C4×C71C2×C4⋊C411yescharacteristic
124C2×C4×C71C2×C4⋊C411yescharacteristic
125C2×C4×C71C2×C4⋊C411yescharacteristic
126C2×C4×C71#13112yes—
127C2×C4×C71#13112yes—
128C2×C4×C71#13512yes—
129C2×C4×C71#13512yes—
130C2×C4×C71C23×C411yescharacteristic
131C2×C4×C71C22×D411yescharacteristic
132C2×C4×C71C22×Q811yescharacteristic
133C2×Dic71—22no—
134C2×Dic71—22no—
135C2×Dic71—22no—
136C2×Dic71C2×C4⋊C412yes—
137C2×Dic71C2×C4⋊C412yes—
138C2×Dic71C2×C4⋊C412yes—
139C4×D71—22no—
140C4×D71—22no—
141C4×D71—22no—
142C4×D71—22no—
143C4×D71—24no—
144C4×D71—24no—
145C4×D71—42no—
146C4×D71—42no—
147C8×C71—22no—
148C8×C71—22no—
149C8×C71—42no—
150C8×C71—42no—
151C71⋊C8—22no—
152C71⋊C8—22no—
153C71⋊C8—42no—
154C71⋊C8—42no—
155C22×C71#42412yes—
156C22×C71#42712yes—
157C22×C71#43012yes—
158C22×C71C22×C4⋊C411yescharacteristic
159C2×D71—22no—
160C2×D71—22no—
161C2×D71—22no—
162C2×D71—44no—
163C4×C71—21no—
164C4×C71—21no—
165C4×C71#42412yes—
166C4×C71#42712yes—
167C4×C71C2×#13111yescharacteristic
168C4×C71C2×#13111yescharacteristic
169C4×C71C2×#13511yescharacteristic
170C4×C71C2×#13511yescharacteristic
171Dic71—22no—
172Dic71—22no—
173Dic71—22no—
174C2×#1572—711nomaximal, Sylow
175C2×C71#157214yescomplemented
176C2×C71C2×#42411yescharacteristic
177C2×C71C2×#42711yescharacteristic
178C2×C71C2×#43011yescharacteristic
179D71—42no—
180C2×#432—711no—
181C2×#431—711no—
182C2×#431—711no—
183#1572—718no—
184C2×#335—711no—
185C2×#335D7111yescharacteristic, complemented
186C2×#334—711no—
187C2×#334—711no—
188C71C2×#157211yescharacteristic, complemented, Sylow
189C22×C42—711no—
190#432—714no—
191#432—1422no—
192#431—714no—
193#431—714no—
194#431—714no—
195#431—1422no—
196C2×C4⋊C8—711no—
197C2×C4⋊C8—711no—
198C2×C4⋊C8—711no—
199C2×C4⋊C8—711no—
200C2×C4⋊C8C2×D7111yescharacteristic
201C2×C4⋊C8C2×D7111yescharacteristic
202C2×C22⋊C8—711no—
203C2×C22⋊C8—711no—
204C2×C22⋊C8—1421no—
205C2×C22⋊C8—1421no—
206C2×#97—1421no—
207C2×C4×C8—711no—
208C2×C4×C8—1421no—
209C2×C4×C8C2×D7111yescharacteristic
210#335—714no—
211#335C2×D7114yescomplemented
212#334—714no—
213#334—714no—
214C23×C4—711no—
215C23×C4—711no—
216C23×C4—711no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—711no—
220C22×C8—711no—
221C22×C8—1421no—
222C22×C8—1421no—
223C22×C8C4×D7111yescharacteristic
224C22×C8C4×D7111yescharacteristic
225C2×C42—711no—
226C2×C42—711no—
227C2×C42—711no—
228C2×C42—712no—
229C2×C42—712no—
230C2×C42—712no—
231C2×C42—712no—
232C2×C42C22×D7111yescharacteristic
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—714no—
236C4⋊C8—714no—
237C4⋊C8—712no—
238C4⋊C8—712no—
239C4⋊C8—1421no—
240C4⋊C8—1421no—
241C4⋊C8C22×D7112yes—
242C4⋊C8C22×D7112yes—
243C22⋊C8—714no—
244C22⋊C8—714no—
245C22⋊C8—1424no—
246C22⋊C8—1424no—
247#97—1424no—
248C4×C8—712no—
249C4×C8—712no—
250C4×C8—1424no—
251C4×C8C4×D7112yes—
252C4×C8C22×D7112yes—
253C24—711no—
254C22×C4—2842no—
255C22×C4—712no—
256C22×C4—712no—
257C22×C4—711no—
258C22×C4—711no—
259C22×C4—711no—
260C22×C4—712no—
261C22×C4—712no—
262C22×C4—712no—
263C22×C4—712no—
264C22×C4—1422no—
265C22×C4—1422no—
266C22×C4—1422no—
267C22×C4—1422no—
268C22×C4C2×C4×D7111yescharacteristic
269C22×C4D4×D7111yescharacteristic
270C22×C4Q8×D7111yescharacteristic
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—2842no—
278C2×C8—2842no—
279C2×C8—712no—
280C2×C8—712no—
281C2×C8—1422no—
282C2×C8—1422no—
283C2×C8—1422no—
284C2×C8—1422no—
285C2×C8C2×C4×D7112yes—
286C2×C8C2×C4×D7112yes—
287C42—712no—
288C42—712no—
289C42—712no—
290C42—712no—
291C42—712no—
292C42—712no—
293C42C2×C4×D7111yescharacteristic
294C42D4×D7111yescharacteristic
295C42Q8×D7111yescharacteristic
296C42C23×D7111yescharacteristic
297C23—712no—
298C23—1422no—
299C23—1422no—
300C23—1422no—
301C23C4⋊C4×D7111yescharacteristic
302C2×C4—22no—
303C2×C4—2842no—
304C2×C4—2842no—
305C2×C4—712no—
306C2×C4—712no—
307C2×C4—712no—
308C2×C4—1422no—
309C2×C4—1422no—
310C2×C4—1422no—
311C2×C4—1422no—
312C2×C4—1422no—
313C2×C4—1422no—
314C2×C4—1422no—
315C2×C4—1424no—
316C2×C4—1424no—
317C2×C4C4⋊C4×D7111yescharacteristic
318C2×C4C4⋊C4×D7111yescharacteristic
319C2×C4C4⋊C4×D7111yescharacteristic
320C2×C4#1455613412yes—
321C2×C4#1455613612yes—
322C2×C4#1455614712yes—
323C2×C4#1455615112yes—
324C2×C4C22×C4×D7111yescharacteristic
325C2×C4C2×D4×D7111yescharacteristic
326C2×C4C2×Q8×D7111yescharacteristic
327C8—22no—
328C8—22no—
329C8—42no—
330C8—42no—
331C8—2842no—
332C8—2842no—
333C8—1422no—
334C8—1422no—
335C22—2844no—
336C22—1422no—
337C22—1422no—
338C22—1422no—
339C22#1798602412yes—
340C22#1798603612yes—
341C22#1798604612yes—
342C22C2×C4⋊C4×D7111yescharacteristic
343C4—21no—
344C4—21no—
345C4—1422no—
346C4—1422no—
347C4—1422no—
348C4#1798602412yes—
349C4#1798603612yes—
350C4C2×#1455613411yescharacteristic
351C4C2×#1455613611yescharacteristic
352C4C2×#1455614711yescharacteristic
353C4C2×#1455615111yescharacteristic
354C2—2842no—
355C2#2258711914yesdirect product factor
356C2C2×#1798602411yescharacteristic
357C2C2×#1798603611yescharacteristic
358C2C2×#1798604611yescharacteristic
subgroupG/H
centerC22×C4D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC2×#335×C71C2
socleC23×C71C2×C4⋊C4
prime radicalC23×D71C4⋊C4
cosocleC42×C71C24
hypercenterC2×#335D71
hypocenterC71C2×#1572
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×#335

Complements

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

NH ≅ G/Ntype
4. #22587119C2direct (N × H)
5. C2×#17985752C2semidirect
6. C2×#17985749C2semidirect
7. C2×#17985749C2semidirect
8. C2×#335×C71C2semidirect
27. #17985752C22semidirect
28. #17985749C22semidirect
29. #17985749C22semidirect
33. #335×C71C22semidirect
175. C2×C71#1572semidirect
185. C2×#335D71semidirect
188. C71C2×#1572semidirect
211. #335C2×D71semidirect
355. C2#22587119direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179860552
C2×#179860512
C2×#179860502
#225871192
C2×#179857522
C2×#179857492
C2×#335×C712
C2×#157271