Finite Group Databasebeta

← C2×#22587977 #31985190

Subgroups — C2×#22587977 #31985190

Order 18176.

966 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986375C211yesmaximal, characteristic
2C2×#17986351C211yesmaximal, characteristic, complemented
3C2×#17986347C211yesmaximal, characteristic, complemented
4C2×#17986346C211yesmaximal, characteristic, complemented
5C2×#17986345C211yesmaximal, characteristic, complemented
6#22587977C218yesmaximal, direct product factor
7C2×#17985721C211yesmaximal, characteristic, complemented
8C2×#492×C71C211yesmaximal, characteristic, complemented
9C22×#11765850—22no—
10C2×#14556128C2211yescharacteristic
11C2×#14556125C2211yescharacteristic
12#17986375—22no—
13#17986375C2214yes—
14#17986351—22no—
15#17986351C2214yescomplemented
16#17986347—22no—
17#17986347C2214yescomplemented
18#17986346—22no—
19#17986346C2214yescomplemented
20#17986345—22no—
21#17986345C2214yescomplemented
22C2×#14556051—22no—
23C2×#14556050—22no—
24C2×#14556049—22no—
25C2×#14556048—22no—
26C2×C4×C71⋊C8C2211yescharacteristic
27C2×C4×C71⋊C8C2211yescharacteristic
28#17985721C2214yescomplemented
29C22×SD16×C71—22no—
30C2×C4⋊Q8×C71C2211yescharacteristic
31C2×#127×C71C2211yescharacteristic
32#492×C71—22no—
33#492×C71C2214yescomplemented
34C2×C4×C8×C71C2211yescharacteristic
35C22×Dic142—21no—
36C22×D284—21no—
37C22×D284—21no—
38C22×D284—21no—
39C2×#11765850—28no—
40C2×#11765850—48no—
41C22×C71⋊C8D412yes—
42C22×C71⋊C8D412yes—
43#14556128—21no—
44#14556128—21no—
45#14556128—21no—
46#14556128C2312yes—
47#14556125—21no—
48#14556125—21no—
49#14556125—21no—
50#14556125C2312yes—
51C2×C4⋊Dic71—21no—
52C2×C4⋊Dic71—21no—
53#14556051—28no—
54#14556050—28no—
55#14556049—28no—
56#14556048—28no—
57C4×C71⋊C8D412yes—
58C4×C71⋊C8D412yes—
59C4×C71⋊C8C2312yes—
60C4×C71⋊C8C2312yes—
61C22×Q8×C71—21no—
62C22×D4×C71—21no—
63C22×D4×C71—21no—
64C22×D4×C71—21no—
65C2×SD16×C71—28no—
66C2×SD16×C71—48no—
67C22×C8×C71D412yes—
68C4⋊Q8×C71—21no—
69C4⋊Q8×C71—21no—
70C4⋊Q8×C71—21no—
71C4⋊Q8×C71C2312yes—
72#127×C71—21no—
73#127×C71—21no—
74#127×C71—21no—
75#127×C71C2312yes—
76C2×C4⋊C4×C71—21no—
77C2×C4⋊C4×C71—21no—
78C2×C42×C71C2311yescharacteristic
79C4×C8×C71D412yes—
80C4×C8×C71C2312yes—
81C23×D71—41no—
82C2×Dic142—22no—
83C2×Dic142—44no—
84C2×Dic142—41no—
85C2×D284—22no—
86C2×D284—22no—
87C2×D284—22no—
88C2×D284—44no—
89C2×D284—44no—
90C2×D284—44no—
91C2×D284—41no—
92C2×D284—41no—
93C2×D284—41no—
94C22×Dic71—41no—
95#11765850—416no—
96C2×C71⋊C8—24no—
97C2×C71⋊C8—24no—
98C2×C71⋊C8C2×D414yes—
99C2×C71⋊C8C2×D414yes—
100C4⋊Dic71—22no—
101C4⋊Dic71—22no—
102C4⋊Dic71—41no—
103C4⋊Dic71—41no—
104C24×C71—41no—
105C2×Q8×C71—22no—
106C2×Q8×C71—41no—
107C2×Q8×C71—44no—
108C2×D4×C71—22no—
109C2×D4×C71—22no—
110C2×D4×C71—22no—
111C2×D4×C71—41no—
112C2×D4×C71—41no—
113C2×D4×C71—41no—
114C2×D4×C71—44no—
115C2×D4×C71—44no—
116C2×D4×C71—44no—
117C22×C4×C71—41no—
118C22×C4×C71C2×D411yescharacteristic
119C22×C4×C71C2×D411yescharacteristic
120C22×C4×C71C2×D411yescharacteristic
121SD16×C71—416no—
122C2×C8×C71—24no—
123C2×C8×C71C2×D414yes—
124C4⋊C4×C71—22no—
125C4⋊C4×C71—22no—
126C4⋊C4×C71—41no—
127C4⋊C4×C71—41no—
128C42×C71C2×D411yescharacteristic
129C42×C71C2×D411yescharacteristic
130C42×C71C2×D411yescharacteristic
131C42×C71C2411yescharacteristic
132C23×C71—42no—
133C23×C71—82no—
134C23×C71—82no—
135C23×C71—82no—
136C23×C71#12711yescharacteristic
137C22×D71—42no—
138C22×D71—82no—
139C22×D71—82no—
140C22×D71—82no—
141Q8×C71—44no—
142Q8×C71—82no—
143D4×C71—44no—
144D4×C71—44no—
145D4×C71—44no—
146D4×C71—82no—
147D4×C71—82no—
148D4×C71—82no—
149C2×C4×C71—42no—
150C2×C4×C71—82no—
151C2×C4×C71#12711yescharacteristic
152C2×C4×C71#12711yescharacteristic
153C2×C4×C71#12711yescharacteristic
154C2×C4×C71#13514yes—
155C2×C4×C71AGL(1,ℤ/8)14yes—
156C2×C4×C71AGL(1,ℤ/8)14yes—
157C2×C4×C71C22×D411yescharacteristic
158C2×C4×C71C22×D411yescharacteristic
159C2×C4×C71C22×D411yescharacteristic
160C2×Dic71—42no—
161C2×Dic71—82no—
162D284—44no—
163D284—44no—
164D284—44no—
165D284—82no—
166D284—82no—
167D284—82no—
168Dic142—44no—
169Dic142—82no—
170C8×C71—24no—
171C71⋊C8—24no—
172C71⋊C8—24no—
173C22×C71—82no—
174C22×C71—82no—
175C22×C71—82no—
176C22×C71—164no—
177C22×C71#49512yes—
178C22×C71#49612yes—
179C22×C71#49612yes—
180C22×C71C2×#12711yescharacteristic
181C2×D71—82no—
182C2×D71—82no—
183C2×D71—82no—
184C2×D71—164no—
185C4×C71—82no—
186C4×C71#49512yes—
187C4×C71#49612yes—
188C4×C71#49612yes—
189C4×C71C2×#13512yes—
190C4×C71C2×AGL(1,ℤ/8)12yes—
191C4×C71C2×AGL(1,ℤ/8)12yes—
192Dic71—82no—
193C2×#1717—711nomaximal, Sylow
194C2×C71—162no—
195C2×C71#171714yescomplemented
196C2×C71C2×#49511yescharacteristic
197C2×C71C2×#49611yescharacteristic
198C2×C71C2×#49611yescharacteristic
199D71—162no—
200C2×#493—711no—
201C2×#492D7111yescharacteristic, complemented
202C2×#487—711no—
203C2×#486—711no—
204C2×#486—711no—
205C2×#486—711no—
206#1717—718no—
207C2×#322—711no—
208C71C2×#171711yescharacteristic, complemented, Sylow
209C22×SD16—22no—
210C22×D8—1422no—
211C2×C4⋊Q8—711no—
212C2×C4⋊Q8C2×D7111yescharacteristic
213C2×#127—711no—
214C2×#127C2×D7111yescharacteristic
215#493—714no—
216#493—1422no—
217#492—22no—
218#492C2×D7114yescomplemented
219#487—714no—
220#487—1422no—
221#486—714no—
222#486—714no—
223#486—714no—
224#486—1422no—
225#486—1422no—
226#486—1422no—
227C2×#103—1422no—
228C2×#102—1422no—
229C2×#102—1422no—
230C2×#102—1422no—
231C2×C4×C8—711no—
232C2×C4×C8—711no—
233C2×C4×C8C2×D7111yescharacteristic
234#322—714no—
235C22×Q8—21no—
236C22×Q8—1421no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4—1421no—
241C22×D4—1421no—
242C22×D4—1421no—
243C2×SD16—28no—
244C2×SD16—48no—
245C2×D8—2848no—
246C2×D8—1428no—
247C22×C8—712no—
248C22×C8—712no—
249C22×C8C22⋊D7112yes—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—712no—
254C4⋊Q8—1421no—
255C4⋊Q8—1421no—
256C4⋊Q8—1421no—
257C4⋊Q8C22×D7112yes—
258#127—21no—
259#127—21no—
260#127—21no—
261#127—712no—
262#127—1421no—
263#127—1421no—
264#127—1421no—
265#127C22×D7112yes—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—1421no—
269C2×C4⋊C4—1421no—
270C2×C42C22×D7111yescharacteristic
271#103—1428no—
272#102—1428no—
273#102—1428no—
274#102—1428no—
275C4×C8—712no—
276C4×C8—712no—
277C4×C8—712no—
278C4×C8—712no—
279C4×C8C22⋊D7112yes—
280C4×C8C22×D7112yes—
281C24—41no—
282C24—2841no—
283C2×Q8—22no—
284C2×Q8—41no—
285C2×Q8—44no—
286C2×Q8—2841no—
287C2×Q8—2844no—
288C2×Q8—1422no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—2841no—
299C2×D4—2841no—
300C2×D4—2841no—
301C2×D4—2844no—
302C2×D4—2844no—
303C2×D4—2844no—
304C2×D4—1422no—
305C2×D4—1422no—
306C2×D4—1422no—
307C22×C4—41no—
308C22×C4—2841no—
309C22×C4D4×D7111yescharacteristic
310C22×C4D4×D7111yescharacteristic
311C22×C4C2×C22⋊D7111yescharacteristic
312SD16—416no—
313D8—28416no—
314C2×C8—24no—
315C2×C8—714no—
316C2×C8—714no—
317C2×C8—1424no—
318C2×C8—1424no—
319C2×C8C2×C22⋊D7114yes—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—2841no—
325C4⋊C4—2841no—
326C4⋊C4—1422no—
327C4⋊C4—1422no—
328C42D4×D7111yescharacteristic
329C42D4×D7111yescharacteristic
330C42C2×C22⋊D7111yescharacteristic
331C42C23×D7111yescharacteristic
332C23—42no—
333C23—82no—
334C23—82no—
335C23—82no—
336C23—2842no—
337C23—5682no—
338C23—5682no—
339C23—5682no—
340C23#1455612411yescharacteristic
341Q8—44no—
342Q8—82no—
343Q8—2844no—
344Q8—5682no—
345D4—44no—
346D4—44no—
347D4—44no—
348D4—82no—
349D4—82no—
350D4—82no—
351D4—2844no—
352D4—2844no—
353D4—2844no—
354D4—5682no—
355D4—5682no—
356D4—5682no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4—2842no—
360C2×C4—5682no—
361C2×C4#1455612411yescharacteristic
362C2×C4#1455612411yescharacteristic
363C2×C4#1455612411yescharacteristic
364C2×C4#1455614914yes—
365C2×C4#1455615414yes—
366C2×C4#1455615514yes—
367C2×C4C2×D4×D7111yescharacteristic
368C2×C4C2×D4×D7111yescharacteristic
369C2×C4C22×C22⋊D7111yescharacteristic
370C8—24no—
371C8—1424no—
372C8—1424no—
373C22—82no—
374C22—82no—
375C22—82no—
376C22—164no—
377C22—5682no—
378C22—5682no—
379C22—5682no—
380C22—11364no—
381C22#1798638012yes—
382C22#1798638512yes—
383C22#1798638612yes—
384C22C2×#1455612411yescharacteristic
385C4—82no—
386C4—5682no—
387C4#1798638012yes—
388C4#1798638512yes—
389C4#1798638612yes—
390C4C2×#1455614912yes—
391C4C2×#1455615412yes—
392C4C2×#1455615512yes—
393C2—162no—
394C2—11362no—
395C2#2258797714yesdirect product factor
396C2C2×#1798638011yescharacteristic
397C2C2×#1798638511yescharacteristic
398C2C2×#1798638611yescharacteristic
subgroupG/H
centerC23#14556124
commutator, cosocleC42×C71C24
FrattiniC42C23×D71
FittingC2×#492×C71C2
socleC23×C71#127
prime radicalC2×#17986375C2
hypercenterC2×#492D71
hypocenterC71C2×#1717
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#492

Complements

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

NH ≅ G/Ntype
2. C2×#17986351C2semidirect
3. C2×#17986347C2semidirect
4. C2×#17986346C2semidirect
5. C2×#17986345C2semidirect
6. #22587977C2direct (N × H)
7. C2×#17985721C2semidirect
8. C2×#492×C71C2semidirect
15. #17986351C22semidirect
17. #17986347C22semidirect
19. #17986346C22semidirect
21. #17986345C22semidirect
28. #17985721C22semidirect
33. #492×C71C22semidirect
195. C2×C71#1717semidirect
201. C2×#492D71semidirect
208. C71C2×#1717semidirect
218. #492C2×D71semidirect
395. C2#22587977direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179863752
C2×#179863512
C2×#179863472
C2×#179863462
C2×#179863452
#225879772
C2×#179857212
C2×#492×C712
C2×#171771