Finite Group Databasebeta

← C2×#22590111 #32150462

Subgroups — C2×#22590111 #32150462

Order 18176.

658 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M32×D71C211yesmaximal, characteristic
2C2×#108×D71C211yesmaximal, characteristic
3C2×#17986034C211yesmaximal, characteristic
4#22590111C218yesmaximal, direct product factor
5C2×#17985823C211yesmaximal, characteristic, complemented
6C2×#17985822C211yesmaximal, characteristic, complemented
7C2×#17985821C211yesmaximal, characteristic, complemented
8C2×#365×C71C211yesmaximal, characteristic, complemented
9C2×M16×D71—21no—
10C22×C8×D71C2211yescharacteristic
11C2×#14556086—21no—
12C2×#14556068C2211yescharacteristic
13C2×#14556067C412yescomplemented
14M32×D71C414yescomplemented
15M32×D71C2214yes—
16C2×C16×D71C412yescomplemented
17C2×#14556064C2211yescharacteristic
18C2×#14556063—21no—
19#108×D71—22no—
20#108×D71C2214yes—
21#17986034—22no—
22#17986034C2214yes—
23C2×#14556061—21no—
24C2×#14556060C2211yescharacteristic
25#17985823C2214yescomplemented
26#17985822C2214yescomplemented
27#17985821C2214yescomplemented
28C2×M32×C71C2211yescharacteristic
29C2×#108×C71C2211yescharacteristic
30C2×#106×C71C2211yescharacteristic
31#365×C71C2214yescomplemented
32C22×C4×D71D411yescharacteristic
33C2×#11765849—21no—
34C2×#11765848—41no—
35M16×D71—24no—
36M16×D71—42no—
37C22×C71⋊C8D411yescharacteristic
38C2×C8×D71—41no—
39C2×C8×D71C2×C411yescharacteristic
40C2×C8×D71C2×C412yes—
41C2×C8×D71C2×C411yescharacteristic
42C2×C8×D71D412yes—
43C2×C8×D71Q812yes—
44C2×C8×D71Q811yescharacteristic
45C2×C8×D71Q811yescharacteristic
46C2×C8×D71C2312yes—
47#14556086—24no—
48#14556086—42no—
49C2×C4⋊Dic71—21no—
50C2×(C2×D71)⋊C4—41no—
51C2×C4×Dic71—41no—
52#14556068C2×C412yes—
53#14556068C2312yes—
54#14556067C2×C418yescomplemented
55C2×C71⋊C16C2×C412yes—
56C16×D71C2×C418yescomplemented
57#14556064—21no—
58#14556064C2312yes—
59#14556063—24no—
60#14556061—24no—
61#14556060—21no—
62#14556060C2312yes—
63C2×M16×C71—21no—
64C22×C8×C71C2311yescharacteristic
65C2×C4⋊C4×C71—21no—
66M32×C71C2×C412yes—
67M32×C71C2312yes—
68C2×C16×C71C2×C412yes—
69#108×C71—21no—
70#108×C71C2312yes—
71#106×C71—21no—
72#106×C71C2312yes—
73C23×D71SD1611yescharacteristic
74C22×Dic71—41no—
75C22×Dic71SD1611yescharacteristic
76C2×C4×D71C4⋊C411yescharacteristic
77C2×C4×D71C4⋊C411yescharacteristic
78C2×C4×D71C4⋊C412yes—
79C2×C4×D71SD1612yes—
80C2×C4×D71SD1611yescharacteristic
81C2×C4×D71SD1612yes—
82C2×C4×D71SD1611yescharacteristic
83C2×C4×D71C2×D412yes—
84#11765849—22no—
85#11765849—41no—
86#11765848—44no—
87C2×C71⋊C8—41no—
88C2×C71⋊C8C4⋊C411yescharacteristic
89C2×C71⋊C8C4⋊C412yes—
90C2×C71⋊C8C4⋊C411yescharacteristic
91C2×C71⋊C8C2×D412yes—
92C8×D71—44no—
93C8×D71C4⋊C412yes—
94C8×D71C4⋊C412yes—
95C8×D71C4⋊C412yes—
96C8×D71C4⋊C412yes—
97C8×D71C22×C412yes—
98C8×D71C22×C412yes—
99C8×D71C2×Q812yes—
100C8×D71C2×Q812yes—
101C4⋊Dic71—22no—
102C4⋊Dic71—41no—
103(C2×D71)⋊C4—44no—
104C4×Dic71—44no—
105C71⋊C16C22×C414yes—
106C22×C4×C71—41no—
107C22×C4×C71C2×D411yescharacteristic
108M16×C71—22no—
109M16×C71—41no—
110C2×C8×C71—41no—
111C2×C8×C71C22×C411yescharacteristic
112C2×C8×C71C22×C411yescharacteristic
113C2×C8×C71C2×D411yescharacteristic
114C2×C8×C71C2×Q811yescharacteristic
115C2×C8×C71C2×Q811yescharacteristic
116C2×C8×C71C2411yescharacteristic
117C4⋊C4×C71—22no—
118C4⋊C4×C71—41no—
119C16×C71C22×C414yes—
120C23×C71C2×SD1611yescharacteristic
121C22×D71—42no—
122C22×D71#10611yescharacteristic
123C22×D71#10611yescharacteristic
124C22×D71#10612yes—
125C22×D71C2×SD1612yes—
126C2×C4×C71—42no—
127C2×C4×C71—82no—
128C2×C4×C71C2×C4⋊C411yescharacteristic
129C2×C4×C71C2×C4⋊C411yescharacteristic
130C2×C4×C71C2×SD1611yescharacteristic
131C2×C4×C71C2×SD1611yescharacteristic
132C2×C4×C71C2×SD1611yescharacteristic
133C2×C4×C71C22×D411yescharacteristic
134C2×Dic71—42no—
135C2×Dic71—82no—
136C2×Dic71#10612yes—
137C2×Dic71#10611yescharacteristic
138C2×Dic71#10611yescharacteristic
139C2×Dic71C2×SD1612yes—
140C4×D71#10612yes—
141C4×D71#10612yes—
142C4×D71#10612yes—
143C4×D71#10612yes—
144C4×D71C2×C4⋊C412yes—
145C4×D71C2×C4⋊C412yes—
146C4×D71C2×SD1612yes—
147C4×D71C2×SD1612yes—
148C8×C71—42no—
149C8×C71C2×C4⋊C411yescharacteristic
150C8×C71C2×C4⋊C411yescharacteristic
151C8×C71C23×C411yescharacteristic
152C8×C71C22×Q811yescharacteristic
153C71⋊C8—42no—
154C71⋊C8C2×C4⋊C412yes—
155C71⋊C8C2×C4⋊C412yes—
156C22×C71—22no—
157C22×C71C2×#10611yescharacteristic
158C22×C71C2×#10611yescharacteristic
159C22×C71C22×SD1611yescharacteristic
160C2×D71—22no—
161C2×D71—22no—
162C2×D71—42no—
163C2×D71—42no—
164C2×D71C2×#10612yes—
165C2×D71C2×#10612yes—
166C4×C71—82no—
167C4×C71C2×#10611yescharacteristic
168C4×C71C2×#10611yescharacteristic
169C4×C71C22×C4⋊C411yescharacteristic
170C4×C71C22×SD1611yescharacteristic
171Dic71—82no—
172Dic71C2×#10612yes—
173Dic71C2×#10612yes—
174C2×#2155—711nomaximal, Sylow
175C2×C71—21no—
176C2×C71—21no—
177C2×C71#215512yescomplemented
178C2×C71C22×#10611yescharacteristic
179D71—22no—
180D71—22no—
181C22×M32—711no—
182C22×#108—711no—
183C2×#427—711no—
184#2155—718no—
185C2×#365—711no—
186C2×#365—711no—
187C2×#365—711no—
188C2×#365D7111yescharacteristic, complemented
189C71C2×#215511yescharacteristic, complemented, Sylow
190C22×M16—1421no—
191C23×C8—711no—
192C2×#117—1421no—
193C2×M32—711no—
194C2×M32—714no—
195C2×M32—714no—
196C2×M32—712no—
197C2×M32C2×D7111yescharacteristic
198C22×C16—712no—
199C2×#108—711no—
200C2×#108—714no—
201C2×#108—1422no—
202C2×#108—1421no—
203C2×#108C2×D7111yescharacteristic
204#427—714no—
205#427—1422no—
206C2×#107—1421no—
207C2×#106—711no—
208C2×#106C2×D7111yescharacteristic
209#365—714no—
210#365—714no—
211#365—714no—
212#365C2×D7114yescomplemented
213C23×C4—711no—
214C2×M16—21no—
215C2×M16—2841no—
216C2×M16—2842no—
217C2×M16—1424no—
218C2×M16—1421no—
219C22×C8—2841no—
220C22×C8—711no—
221C22×C8—711no—
222C22×C8—712no—
223C22×C8—711no—
224C22×C8—712no—
225C22×C8—712no—
226C22×C8—711no—
227C22×C8—711no—
228C22×C8—712no—
229C22×C8C22×D7111yescharacteristic
230#117—2842no—
231#117—1424no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—1421no—
234C2×C22⋊C4—2841no—
235C2×C42—2841no—
236M32—712no—
237M32—712no—
238M32—718no—
239M32C4×D7112yes—
240M32C22×D7112yes—
241C2×C16—712no—
242C2×C16—718no—
243C2×C16C4×D7112yes—
244#108—21no—
245#108—712no—
246#108—1424no—
247#108—1421no—
248#108C22×D7112yes—
249#107—1424no—
250#106—21no—
251#106—712no—
252#106—1421no—
253#106C22×D7112yes—
254C24—711no—
255C22×C4—41no—
256C22×C4—2841no—
257C22×C4—711no—
258C22×C4—712no—
259C22×C4—712no—
260C22×C4—711no—
261C22×C4—711no—
262C22×C4—712no—
263C22×C4—711no—
264C22×C4—712no—
265C22×C4—711no—
266C22×C4D4×D7111yescharacteristic
267M16—22no—
268M16—41no—
269M16—2844no—
270M16—2841no—
271M16—1422no—
272C2×C8—41no—
273C2×C8—2844no—
274C2×C8—2841no—
275C2×C8—712no—
276C2×C8—712no—
277C2×C8—711no—
278C2×C8—711no—
279C2×C8—712no—
280C2×C8—712no—
281C2×C8—712no—
282C2×C8—712no—
283C2×C8—712no—
284C2×C8—712no—
285C2×C8—712no—
286C2×C8—712no—
287C2×C8C2×C4×D7111yescharacteristic
288C2×C8C2×C4×D7111yescharacteristic
289C2×C8D4×D7111yescharacteristic
290C2×C8Q8×D7111yescharacteristic
291C2×C8Q8×D7111yescharacteristic
292C2×C8C23×D7111yescharacteristic
293C4⋊C4—22no—
294C4⋊C4—41no—
295C4⋊C4—2841no—
296C4⋊C4—1422no—
297C22⋊C4—2844no—
298C42—2844no—
299C16—714no—
300C16C2×C4×D7114yes—
301C23—2842no—
302C23—712no—
303C23—712no—
304C23—711no—
305C23—711no—
306C23SD16×D7111yescharacteristic
307C2×C4—42no—
308C2×C4—82no—
309C2×C4—2842no—
310C2×C4—5682no—
311C2×C4—712no—
312C2×C4—712no—
313C2×C4—711no—
314C2×C4—711no—
315C2×C4—712no—
316C2×C4—712no—
317C2×C4—712no—
318C2×C4—712no—
319C2×C4—712no—
320C2×C4—712no—
321C2×C4—712no—
322C2×C4—712no—
323C2×C4C4⋊C4×D7111yescharacteristic
324C2×C4C4⋊C4×D7111yescharacteristic
325C2×C4SD16×D7111yescharacteristic
326C2×C4SD16×D7111yescharacteristic
327C2×C4SD16×D7111yescharacteristic
328C2×C4C2×D4×D7111yescharacteristic
329C8—42no—
330C8—2842no—
331C8—712no—
332C8—712no—
333C8C4⋊C4×D7111yescharacteristic
334C8C4⋊C4×D7111yescharacteristic
335C8C22×C4×D7111yescharacteristic
336C8C2×Q8×D7111yescharacteristic
337C22—22no—
338C22—2842no—
339C22—2842no—
340C22—712no—
341C22—712no—
342C22—1422no—
343C22—1422no—
344C22#106×D7111yescharacteristic
345C22#106×D7111yescharacteristic
346C22C2×SD16×D7111yescharacteristic
347C4—82no—
348C4—5682no—
349C4—712no—
350C4—712no—
351C4#106×D7111yescharacteristic
352C4#106×D7111yescharacteristic
353C4C2×C4⋊C4×D7111yescharacteristic
354C4C2×SD16×D7111yescharacteristic
355C2—21no—
356C2—21no—
357C2—1422no—
358C2—1422no—
359C2#2259011112yesdirect product factor
360C2C2×#106×D7111yescharacteristic
subgroupG/H
centerC22#106×D71
commutatorC8×C71C23×C4
FrattiniC2×C8C23×D71
FittingC2×#365×C71C2
socleC22×C71C2×#106
prime radicalC23×D71SD16
cosocleC2×C8×C71C24
hypercenterC2×#365D71
hypocenterC71C2×#2155
LCSG ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#365

Complements

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

NH ≅ G/Ntype
4. #22590111C2direct (N × H)
5. C2×#17985823C2semidirect
6. C2×#17985822C2semidirect
7. C2×#17985821C2semidirect
8. C2×#365×C71C2semidirect
13. C2×#14556067C4semidirect
14. M32×D71C4semidirect
16. C2×C16×D71C4semidirect
25. #17985823C22semidirect
26. #17985822C22semidirect
27. #17985821C22semidirect
31. #365×C71C22semidirect
54. #14556067C2×C4semidirect
56. C16×D71C2×C4semidirect
177. C2×C71#2155semidirect
188. C2×#365D71semidirect
189. C71C2×#2155semidirect
212. #365C2×D71semidirect
359. C2#22590111direct (N × H)

Maximal subgroups

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

subgroupindex
C2×M32×D712
C2×#108×D712
C2×#179860342
#225901112
C2×#179858232
C2×#179858222
C2×#179858212
C2×#365×C712
C2×#215571