Finite Group Databasebeta

← C2×#22587878 #31980098

Subgroups — C2×#22587878 #31980098

Order 18176.

702 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986345C211yesmaximal, characteristic
2C2×#17986294C211yesmaximal, characteristic, complemented
3C2×#17986130C211yesmaximal, characteristic, complemented
4#22587878C218yesmaximal, direct product factor
5C2×#17985751C211yesmaximal, characteristic, complemented
6C2×#17985747C211yesmaximal, characteristic, complemented
7C2×#17985743C211yesmaximal, characteristic
8C2×#460×C71C211yesmaximal, characteristic
9C2×#14556128C2211yescharacteristic
10#17986345—22no—
11#17986345C2214yes—
12#17986294—22no—
13#17986294C2214yescomplemented
14#17986130—22no—
15#17986130C2214yescomplemented
16C2×#14556061—21no—
17C2×C4⋊(C71⋊C8)—21no—
18C2×C4⋊(C71⋊C8)C2211yescharacteristic
19C2×C4⋊(C71⋊C8)C2211yescharacteristic
20C2×#14556052—21no—
21C2×#14556049—21no—
22C2×#14556049—21no—
23C2×#14556039—21no—
24C2×C4×C71⋊C8C2211yescharacteristic
25#17985751C2214yescomplemented
26#17985747C2214yescomplemented
27#17985743C2214yes—
28C22×SD16×C71—21no—
29C2×#127×C71C2211yescharacteristic
30C2×C4×Q8×C71C2211yescharacteristic
31#460×C71—22no—
32#460×C71C2214yes—
33C2×C4⋊C8×C71C2211yescharacteristic
34C2×#102×C71—21no—
35C22×Dic142—21no—
36C22×C71⋊C8—21no—
37C22×C71⋊C8—21no—
38C22×C71⋊C8D411yescharacteristic
39C22×C71⋊C8D411yescharacteristic
40#14556128—21no—
41#14556128—21no—
42#14556128—21no—
43#14556128C2312yes—
44C2×C4⋊Dic71—21no—
45C2×C4⋊Dic71—21no—
46#14556061—24no—
47C4⋊(C71⋊C8)—24no—
48C4⋊(C71⋊C8)—21no—
49C4⋊(C71⋊C8)—21no—
50C4⋊(C71⋊C8)C2312yes—
51C4⋊(C71⋊C8)C2312yes—
52#14556052—24no—
53#14556049—24no—
54#14556049—24no—
55#14556039—24no—
56C4×C71⋊C8D412yes—
57C4×C71⋊C8C2312yes—
58C22×Q8×C71D411yescharacteristic
59C22×D4×C71—21no—
60C22×D4×C71—21no—
61C22×D4×C71—21no—
62C2×SD16×C71—24no—
63C2×SD16×C71—44no—
64C22×C8×C71—21no—
65#127×C71—21no—
66#127×C71—21no—
67#127×C71—21no—
68#127×C71C2312yes—
69C4×Q8×C71—21no—
70C4×Q8×C71—21no—
71C4×Q8×C71D412yes—
72C4×Q8×C71C2312yes—
73C2×C4⋊C4×C71—21no—
74C2×C4⋊C4×C71D411yescharacteristic
75C2×C42×C71—21no—
76C2×C42×C71C2311yescharacteristic
77C4⋊C8×C71—21no—
78C4⋊C8×C71C2312yes—
79#102×C71—24no—
80C2×Dic142—22no—
81C2×Dic142—41no—
82C2×Dic142—82no—
83C22×Dic71—41no—
84C2×C71⋊C8—22no—
85C2×C71⋊C8—22no—
86C2×C71⋊C8—22no—
87C2×C71⋊C8—22no—
88C2×C71⋊C8—42no—
89C2×C71⋊C8—42no—
90C2×C71⋊C8C2×D412yes—
91C2×C71⋊C8C2×D412yes—
92C4⋊Dic71—22no—
93C4⋊Dic71—22no—
94C4⋊Dic71—41no—
95C4⋊Dic71—41no—
96C24×C71—41no—
97C2×Q8×C71—21no—
98C2×Q8×C71—22no—
99C2×Q8×C71—22no—
100C2×Q8×C71C2×D412yes—
101C2×D4×C71—22no—
102C2×D4×C71—22no—
103C2×D4×C71—22no—
104C2×D4×C71—41no—
105C2×D4×C71—41no—
106C2×D4×C71—41no—
107C2×D4×C71—42no—
108C2×D4×C71—42no—
109C2×D4×C71—42no—
110C2×D4×C71—42no—
111C2×D4×C71—82no—
112C22×C4×C71—21no—
113C22×C4×C71—21no—
114C22×C4×C71C2×D411yescharacteristic
115C22×C4×C71C2×D411yescharacteristic
116C22×C4×C71#4111yescharacteristic
117SD16×C71—48no—
118C2×C8×C71—22no—
119C2×C8×C71—42no—
120C4⋊C4×C71—22no—
121C4⋊C4×C71—21no—
122C4⋊C4×C71—22no—
123C4⋊C4×C71C2×D412yes—
124C42×C71—22no—
125C42×C71—22no—
126C42×C71C2×D411yescharacteristic
127C42×C71C2×D411yescharacteristic
128C42×C71#4111yescharacteristic
129C42×C71C2411yescharacteristic
130C23×C71—42no—
131C23×C71—82no—
132C23×C71—82no—
133C23×C71—82no—
134C23×C71#12111yescharacteristic
135Q8×C71—22no—
136Q8×C71—22no—
137Q8×C71—42no—
138D4×C71—42no—
139D4×C71—42no—
140D4×C71—42no—
141D4×C71—42no—
142D4×C71—82no—
143D4×C71—82no—
144D4×C71—82no—
145D4×C71—82no—
146C2×C4×C71—22no—
147C2×C4×C71—22no—
148C2×C4×C71—22no—
149C2×C4×C71—42no—
150C2×C4×C71—42no—
151C2×C4×C71#12111yescharacteristic
152C2×C4×C71#12111yescharacteristic
153C2×C4×C71#12111yescharacteristic
154C2×C4×C71#13512yes—
155C2×C4×C71AGL(1,ℤ/8)12yes—
156C2×C4×C71#13712yes—
157C2×C4×C71#13712yes—
158C2×C4×C71C22×D411yescharacteristic
159C2×C4×C71C22×D411yescharacteristic
160C2×C4×C71C2×#4111yescharacteristic
161C2×Dic71—42no—
162C2×Dic71—82no—
163Dic142—82no—
164Dic142—82no—
165C8×C71—42no—
166C71⋊C8—22no—
167C71⋊C8—22no—
168C71⋊C8—42no—
169C71⋊C8—42no—
170C22×C71—82no—
171C22×C71—82no—
172C22×C71—82no—
173C22×C71—164no—
174C22×C71#46412yes—
175C22×C71#46812yes—
176C22×C71#47212yes—
177C22×C71C2×#12111yescharacteristic
178C4×C71—21no—
179C4×C71—21no—
180C4×C71—42no—
181C4×C71—42no—
182C4×C71#46412yes—
183C4×C71#46812yes—
184C4×C71C2×#13511yescharacteristic
185C4×C71C2×AGL(1,ℤ/8)11yescharacteristic
186C4×C71C2×#13711yescharacteristic
187C4×C71C2×#13711yescharacteristic
188Dic71—82no—
189C2×#1698—711nomaximal, Sylow
190C2×C71—162no—
191C2×C71#169814yescomplemented
192C2×C71C2×#46411yescharacteristic
193C2×C71C2×#46811yescharacteristic
194C2×C71C2×#47211yescharacteristic
195C2×#486—711no—
196C2×#477—711no—
197C2×#460D7111yescharacteristic
198C2×#446—711no—
199#1698—718no—
200C2×#335—711no—
201C2×#333—711no—
202C2×#331—711no—
203C71C2×#169811yescharacteristic, complemented, Sylow
204C22×SD16—21no—
205C2×C4⋊Q8—711no—
206C2×#127C2×D7111yescharacteristic
207C2×C4×Q8C2×D7111yescharacteristic
208#486—714no—
209#486—1422no—
210#477—714no—
211#477—1422no—
212#460—22no—
213#460C2×D7114yes—
214#446—714no—
215#446—1422no—
216C2×#107—1421no—
217C2×C4⋊C8—711no—
218C2×C4⋊C8—711no—
219C2×C4⋊C8—1421no—
220C2×C4⋊C8C2×D7111yescharacteristic
221C2×#103—1421no—
222C2×#102—21no—
223C2×#102—1421no—
224C2×#102—1421no—
225C2×#97—1421no—
226C2×C4×C8—711no—
227#335—714no—
228#333—714no—
229#331—714no—
230C22×Q8—1421no—
231C22×Q8C22⋊D7111yescharacteristic
232C22×D4—21no—
233C22×D4—21no—
234C22×D4—21no—
235C2×SD16—24no—
236C2×SD16—44no—
237C22×C8—21no—
238C22×C8—711no—
239C22×C8—711no—
240C22×C8—1421no—
241C22×C8—1421no—
242C4⋊Q8—712no—
243C4⋊Q8—1421no—
244C4⋊Q8—1421no—
245C4⋊Q8—1421no—
246#127—21no—
247#127—21no—
248#127—21no—
249#127C22×D7112yes—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8C22⋊D7112yes—
253C4×Q8C22×D7112yes—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—1421no—
256C2×C4⋊C4—1421no—
257C2×C4⋊C4C22⋊D7111yescharacteristic
258C2×C42—21no—
259C2×C42C22×D7111yescharacteristic
260#107—1424no—
261C4⋊C8—21no—
262C4⋊C8—712no—
263C4⋊C8—712no—
264C4⋊C8—1424no—
265C4⋊C8—1421no—
266C4⋊C8—1421no—
267C4⋊C8C22×D7112yes—
268#103—1424no—
269#102—24no—
270#102—1424no—
271#102—1424no—
272#97—1424no—
273C4×C8—712no—
274C4×C8—712no—
275C24—41no—
276C2×Q8—21no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—2841no—
280C2×Q8—5682no—
281C2×Q8—1422no—
282C2×Q8C2×C22⋊D7112yes—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—82no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—2841no—
297C22×C4D4×D7111yescharacteristic
298C22×C4C2×C22⋊D7111yescharacteristic
299C22×C4#1176586411yescharacteristic
300SD16—48no—
301C2×C8—22no—
302C2×C8—42no—
303C2×C8—2842no—
304C2×C8—2842no—
305C2×C8—712no—
306C2×C8—712no—
307C2×C8—1422no—
308C2×C8—1422no—
309C2×C8—1422no—
310C2×C8—1422no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—2841no—
315C4⋊C4—2841no—
316C4⋊C4—1422no—
317C4⋊C4—1422no—
318C4⋊C4C2×C22⋊D7112yes—
319C42—22no—
320C42—22no—
321C42D4×D7111yescharacteristic
322C42C2×C22⋊D7111yescharacteristic
323C42#1176586411yescharacteristic
324C42C23×D7111yescharacteristic
325C23—42no—
326C23—82no—
327C23—82no—
328C23—82no—
329C23#1455610011yescharacteristic
330Q8—22no—
331Q8—22no—
332Q8—42no—
333Q8—5682no—
334Q8—5682no—
335D4—42no—
336D4—42no—
337D4—42no—
338D4—42no—
339D4—82no—
340D4—82no—
341D4—82no—
342D4—82no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—2842no—
349C2×C4—5682no—
350C2×C4#1455610011yescharacteristic
351C2×C4#1455610011yescharacteristic
352C2×C4#1455610011yescharacteristic
353C2×C4#1455614912yes—
354C2×C4#1455615412yes—
355C2×C4#1455615712yes—
356C2×C4#1455615812yes—
357C2×C4C2×D4×D7111yescharacteristic
358C2×C4C22×C22⋊D7111yescharacteristic
359C2×C4C2×#1176586411yescharacteristic
360C8—42no—
361C8—2842no—
362C8—2842no—
363C8—1422no—
364C8—1422no—
365C22—82no—
366C22—82no—
367C22—82no—
368C22—164no—
369C22#1798621912yes—
370C22#1798624312yes—
371C22#1798626712yes—
372C22C2×#1455610011yescharacteristic
373C4—21no—
374C4—21no—
375C4—42no—
376C4—42no—
377C4—5682no—
378C4#1798621912yes—
379C4#1798624312yes—
380C4C2×#1455614911yescharacteristic
381C4C2×#1455615411yescharacteristic
382C4C2×#1455615711yescharacteristic
383C4C2×#1455615811yescharacteristic
384C2—162no—
385C2#2258787814yesdirect product factor
386C2C2×#1798621911yescharacteristic
387C2C2×#1798624311yescharacteristic
388C2C2×#1798626711yescharacteristic
subgroupG/H
centerC23#14556100
commutator, cosocleC42×C71C24
FrattiniC42C23×D71
FittingC2×#460×C71C2
socleC23×C71#121
prime radicalC2×#127×C71C22
hypercenterC2×#460D71
hypocenterC71C2×#1698
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
2. C2×#17986294C2semidirect
3. C2×#17986130C2semidirect
4. #22587878C2direct (N × H)
5. C2×#17985751C2semidirect
6. C2×#17985747C2semidirect
13. #17986294C22semidirect
15. #17986130C22semidirect
25. #17985751C22semidirect
26. #17985747C22semidirect
191. C2×C71#1698semidirect
203. C71C2×#1698semidirect
385. C2#22587878direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179863452
C2×#179862942
C2×#179861302
#225878782
C2×#179857512
C2×#179857472
C2×#179857432
C2×#460×C712
C2×#169871