Finite Group Databasebeta

← C2×#22587842 #31979976

Subgroups — C2×#22587842 #31979976

Order 18176.

726 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986395C211yesmaximal, characteristic, complemented
2C2×#17986286C211yesmaximal, characteristic
3C2×#17986205C211yesmaximal, characteristic
4C2×#17986076C211yesmaximal, characteristic
5#22587842C218yesmaximal, direct product factor
6C2×#17985751C211yesmaximal, characteristic, complemented
7C2×#17985744C211yesmaximal, characteristic, complemented
8C2×#333×C71C211yesmaximal, characteristic, complemented
9C22×#11765854—21no—
10C22×#11765848—21no—
11C2×#14556128C2211yescharacteristic
12C2×C4×D284C2211yescharacteristic
13#17986395—22no—
14#17986395C2214yescomplemented
15#17986286—22no—
16#17986286C2214yes—
17#17986205—22no—
18#17986205C2214yes—
19#17986076—22no—
20#17986076C2214yes—
21C2×#14556059—21no—
22C2×#14556059—21no—
23C2×#14556059—21no—
24C2×C4⋊(C71⋊C8)C2211yescharacteristic
25C2×#14556053—21no—
26C2×#14556048—21no—
27C2×(C2×D71)⋊C8—21no—
28C2×C4×C71⋊C8C2211yescharacteristic
29#17985751C2214yescomplemented
30#17985744C2214yescomplemented
31C2×C4⋊Q8×C71C2211yescharacteristic
32C2×C4⋊C8×C71C2211yescharacteristic
33C2×C4⋊C8×C71C2211yescharacteristic
34#333×C71C2214yescomplemented
35C22×Dic142—21no—
36C22×D284D411yescharacteristic
37C22×C4×D71—21no—
38C2×#11765854—24no—
39C2×#11765854—44no—
40C2×#11765848—24no—
41C2×#11765848—44no—
42C22×C71⋊C8—21no—
43C22×C71⋊C8D411yescharacteristic
44C22×C71⋊C8D411yescharacteristic
45#14556128—21no—
46#14556128—21no—
47#14556128—21no—
48#14556128C2312yes—
49C4×D284—21no—
50C4×D284—21no—
51C4×D284D412yes—
52C4×D284C2312yes—
53C2×C4⋊Dic71—21no—
54C2×C4⋊Dic71—21no—
55C2×C4⋊Dic71D411yescharacteristic
56C2×(C2×D71)⋊C4—21no—
57#14556059—24no—
58#14556059—24no—
59#14556059—24no—
60C4⋊(C71⋊C8)—21no—
61C4⋊(C71⋊C8)C2312yes—
62#14556053—24no—
63#14556048—24no—
64(C2×D71)⋊C8—24no—
65C4×C71⋊C8D412yes—
66C4×C71⋊C8C2312yes—
67C22×Q8×C71—21no—
68C22×C8×C71—21no—
69C22×C8×C71—21no—
70C4⋊Q8×C71—21no—
71C4⋊Q8×C71—21no—
72C4⋊Q8×C71—21no—
73C4⋊Q8×C71C2312yes—
74C2×C4⋊C4×C71—21no—
75C2×C4⋊C4×C71—21no—
76C2×C42×C71C2311yescharacteristic
77C4⋊C8×C71—21no—
78C4⋊C8×C71—21no—
79C4⋊C8×C71C2312yes—
80C4⋊C8×C71C2312yes—
81C23×D71—21no—
82C2×Dic142—22no—
83C2×Dic142—42no—
84C2×Dic142—42no—
85C2×Dic142—41no—
86C2×D284—22no—
87C2×D284—22no—
88C2×D284—21no—
89C2×D284C2×D412yes—
90C22×Dic71—21no—
91C22×Dic71—41no—
92C2×C4×D71—22no—
93C2×C4×D71—22no—
94C2×C4×D71—42no—
95C2×C4×D71—42no—
96#11765854—48no—
97#11765848—48no—
98C2×C71⋊C8—22no—
99C2×C71⋊C8—22no—
100C2×C71⋊C8—42no—
101C2×C71⋊C8SD1614yescomplemented
102C2×C71⋊C8C2×D412yes—
103C2×C71⋊C8C2×D412yes—
104C4⋊Dic71—21no—
105C4⋊Dic71—22no—
106C4⋊Dic71—22no—
107C4⋊Dic71—41no—
108C4⋊Dic71—41no—
109C4⋊Dic71C2×D412yes—
110(C2×D71)⋊C4—22no—
111(C2×D71)⋊C4—22no—
112C2×Q8×C71—22no—
113C2×Q8×C71—41no—
114C2×Q8×C71—82no—
115C22×C4×C71—41no—
116C22×C4×C71C2×D411yescharacteristic
117C22×C4×C71C2×D411yescharacteristic
118C22×C4×C71#4111yescharacteristic
119C2×C8×C71—22no—
120C2×C8×C71—22no—
121C2×C8×C71—42no—
122C2×C8×C71—42no—
123C4⋊C4×C71—22no—
124C4⋊C4×C71—22no—
125C4⋊C4×C71—41no—
126C4⋊C4×C71—41no—
127C42×C71C2×D411yescharacteristic
128C42×C71C2×D411yescharacteristic
129C42×C71#4111yescharacteristic
130C42×C71C2411yescharacteristic
131C23×C71#12111yescharacteristic
132C22×D71—22no—
133C22×D71—42no—
134C22×D71—42no—
135C22×D71—42no—
136Q8×C71—82no—
137Q8×C71—82no—
138C2×C4×C71—22no—
139C2×C4×C71—42no—
140C2×C4×C71—82no—
141C2×C4×C71#12111yescharacteristic
142C2×C4×C71#12111yescharacteristic
143C2×C4×C71#12111yescharacteristic
144C2×C4×C71C2×SD1612yes—
145C2×C4×C71AGL(1,ℤ/8)12yes—
146C2×C4×C71#13712yes—
147C2×C4×C71#13712yes—
148C2×C4×C71C22×D411yescharacteristic
149C2×C4×C71C22×D411yescharacteristic
150C2×C4×C71C2×#4111yescharacteristic
151C2×Dic71—22no—
152C2×Dic71—42no—
153C2×Dic71—42no—
154C2×Dic71—82no—
155D284—22no—
156D284—22no—
157D284—42no—
158C4×D71—42no—
159C4×D71—42no—
160C4×D71—44no—
161Dic142—42no—
162Dic142—42no—
163Dic142—82no—
164C8×C71—42no—
165C8×C71—42no—
166C71⋊C8—22no—
167C71⋊C8—42no—
168C71⋊C8C2×SD1614yescomplemented
169C22×C71#46112yes—
170C22×C71#46812yes—
171C22×C71#47112yes—
172C22×C71C2×#12111yescharacteristic
173C2×D71—42no—
174C2×D71—42no—
175C2×D71—42no—
176C2×D71—84no—
177C4×C71—21no—
178C4×C71—21no—
179C4×C71—82no—
180C4×C71#46112yes—
181C4×C71#46812yes—
182C4×C71C22×SD1611yescharacteristic
183C4×C71C2×AGL(1,ℤ/8)11yescharacteristic
184C4×C71C2×#13711yescharacteristic
185C4×C71C2×#13711yescharacteristic
186Dic71—42no—
187Dic71—82no—
188C2×#1691—711nomaximal, Sylow
189C2×C71#169114yescomplemented
190C2×C71C2×#46111yescharacteristic
191C2×C71C2×#46811yescharacteristic
192C2×C71C2×#47111yescharacteristic
193D71—82no—
194C2×#498—711no—
195C2×#476—711no—
196C2×#461—711no—
197C2×#436—711no—
198#1691—718no—
199C2×#335—711no—
200C2×#333D7111yescharacteristic, complemented
201C2×#332—711no—
202C71C2×#169111yescharacteristic, complemented, Sylow
203C22×SD16—1421no—
204C22×M16—1421no—
205C2×C4⋊Q8—711no—
206C2×C4⋊Q8C2×D7111yescharacteristic
207C2×C4×D4—711no—
208#498—714no—
209#498—1422no—
210#476—714no—
211#476—1422no—
212#461—714no—
213#461—1422no—
214#436—714no—
215#436—1422no—
216C2×#106—1421no—
217C2×#106—1421no—
218C2×#106—1421no—
219C2×C4⋊C8—711no—
220C2×C4⋊C8C2×D7111yescharacteristic
221C2×C4⋊C8C2×D7111yescharacteristic
222C2×#103—1421no—
223C2×#102—1421no—
224C2×C22⋊C8—1421no—
225C2×C4×C8—711no—
226#335—714no—
227#333C2×D7114yescomplemented
228#332—714no—
229C22×Q8—21no—
230C22×Q8—1421no—
231C22×D4—711no—
232C23×C4—1421no—
233C2×SD16—2844no—
234C2×SD16—1424no—
235C2×M16—2844no—
236C2×M16—1424no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—711no—
240C22×C8—711no—
241C22×C8—1421no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—712no—
246C4⋊Q8—1421no—
247C4⋊Q8—1421no—
248C4⋊Q8—1421no—
249C4⋊Q8C22×D7112yes—
250C4×D4—712no—
251C4×D4—712no—
252C4×D4—1421no—
253C4×D4—1421no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—711no—
257C2×C4⋊C4—1421no—
258C2×C4⋊C4—1421no—
259C2×C22⋊C4—1421no—
260C2×C42C22×D7111yescharacteristic
261#106—1424no—
262#106—1424no—
263#106—1424no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—712no—
267C4⋊C8—1421no—
268C4⋊C8C22×D7112yes—
269C4⋊C8C22×D7112yes—
270#103—1424no—
271#102—1424no—
272C22⋊C8—1424no—
273C4×C8—712no—
274C4×C8—712no—
275C24—1421no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—82no—
279C2×Q8—2841no—
280C2×Q8—2842no—
281C2×Q8—2842no—
282C2×Q8—1422no—
283C2×D4—712no—
284C2×D4—1422no—
285C2×D4—1421no—
286C2×D4—1422no—
287C22×C4—41no—
288C22×C4—2842no—
289C22×C4—2842no—
290C22×C4—2841no—
291C22×C4—1421no—
292C22×C4—1422no—
293C22×C4—1422no—
294C22×C4D4×D7111yescharacteristic
295C22×C4D4×D7111yescharacteristic
296C22×C4#1176586511yescharacteristic
297SD16—2848no—
298M16—2848no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—2842no—
304C2×C8—712no—
305C2×C8—714no—
306C2×C8—712no—
307C2×C8—1422no—
308C2×C8—1422no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—2841no—
314C4⋊C4—2841no—
315C4⋊C4—712no—
316C4⋊C4—1422no—
317C4⋊C4—1422no—
318C4⋊C4—1421no—
319C22⋊C4—1422no—
320C22⋊C4—1422no—
321C42D4×D7111yescharacteristic
322C42D4×D7111yescharacteristic
323C42#1176586511yescharacteristic
324C42C23×D7111yescharacteristic
325C23—2842no—
326C23—2842no—
327C23—2842no—
328C23—1422no—
329C23#1455610111yescharacteristic
330Q8—82no—
331Q8—82no—
332Q8—2842no—
333Q8—2842no—
334Q8—5682no—
335D4—2842no—
336D4—1422no—
337D4—1422no—
338C2×C4—22no—
339C2×C4—42no—
340C2×C4—82no—
341C2×C4—2842no—
342C2×C4—2844no—
343C2×C4—2842no—
344C2×C4—2842no—
345C2×C4—2842no—
346C2×C4—5682no—
347C2×C4—1422no—
348C2×C4#1455610111yescharacteristic
349C2×C4#1455610111yescharacteristic
350C2×C4#1455610111yescharacteristic
351C2×C4SD16×D7112yes—
352C2×C4#1455615512yes—
353C2×C4#1455616012yes—
354C2×C4#1455616012yes—
355C2×C4C2×D4×D7111yescharacteristic
356C2×C4C2×D4×D7111yescharacteristic
357C2×C4C2×#1176586511yescharacteristic
358C8—42no—
359C8—42no—
360C8—2842no—
361C8—714no—
362C8—1422no—
363C22—2842no—
364C22—2842no—
365C22—2842no—
366C22—5684no—
367C22#1798620312yes—
368C22#1798624812yes—
369C22#1798626112yes—
370C22C2×#1455610111yescharacteristic
371C4—21no—
372C4—21no—
373C4—82no—
374C4—2842no—
375C4—5682no—
376C4#1798620312yes—
377C4#1798624812yes—
378C4C2×SD16×D7111yescharacteristic
379C4C2×#1455615511yescharacteristic
380C4C2×#1455616011yescharacteristic
381C4C2×#1455616011yescharacteristic
382C2—5682no—
383C2#2258784214yesdirect product factor
384C2C2×#1798620311yescharacteristic
385C2C2×#1798624811yescharacteristic
386C2C2×#1798626111yescharacteristic
subgroupG/H
centerC23#14556101
commutator, cosocleC42×C71C24
FrattiniC42C23×D71
FittingC2×#333×C71C2
socleC23×C71#121
prime radicalC22×D284D4
hypercenterC2×#333D71
hypocenterC71C2×#1691
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
1. C2×#17986395C2semidirect
5. #22587842C2direct (N × H)
6. C2×#17985751C2semidirect
7. C2×#17985744C2semidirect
8. C2×#333×C71C2semidirect
14. #17986395C22semidirect
29. #17985751C22semidirect
30. #17985744C22semidirect
34. #333×C71C22semidirect
101. C2×C71⋊C8SD16semidirect
168. C71⋊C8C2×SD16semidirect
189. C2×C71#1691semidirect
200. C2×#333D71semidirect
202. C71C2×#1691semidirect
227. #333C2×D71semidirect
383. C2#22587842direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179863952
C2×#179862862
C2×#179862052
C2×#179860762
#225878422
C2×#179857512
C2×#179857442
C2×#333×C712
C2×#169171