Finite Group Databasebeta

← C2×#22587882 #31980102

Subgroups — C2×#22587882 #31980102

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×#17986347C211yesmaximal, characteristic
2C2×#17986199C211yesmaximal, characteristic
3C2×#17986130C211yesmaximal, characteristic, complemented
4#22587882C218yesmaximal, direct product factor
5C2×#17985751C211yesmaximal, characteristic, complemented
6C2×#17985748C211yesmaximal, characteristic, complemented
7C2×#17985742C211yesmaximal, characteristic
8C2×#477×C71C211yesmaximal, characteristic, complemented
9C22×#11765852—21no—
10C2×#14556125C2211yescharacteristic
11#17986347—22no—
12#17986347C2214yes—
13#17986199—22no—
14#17986199C2214yes—
15#17986130—22no—
16#17986130C2214yescomplemented
17C2×C4⋊(C71⋊C8)—21no—
18C2×C4⋊(C71⋊C8)C2211yescharacteristic
19C2×C4⋊(C71⋊C8)C2211yescharacteristic
20C2×#14556048—21no—
21C2×#14556048—21no—
22C2×#14556048—21no—
23C2×#14556039—21no—
24C2×C4×C71⋊C8C2211yescharacteristic
25#17985751C2214yescomplemented
26#17985748C2214yescomplemented
27#17985742C2214yes—
28C2×C4⋊Q8×C71C2211yescharacteristic
29C2×C4×Q8×C71C2211yescharacteristic
30#477×C71—22no—
31#477×C71C2214yescomplemented
32C2×#107×C71—21no—
33C2×C4⋊C8×C71C2211yescharacteristic
34C2×#103×C71—21no—
35C22×D284—21no—
36C22×D284—21no—
37C22×D284—21no—
38C2×#11765852—24no—
39C2×#11765852—44no—
40C22×C71⋊C8—21no—
41C22×C71⋊C8—21no—
42C22×C71⋊C8D411yescharacteristic
43C22×C71⋊C8D411yescharacteristic
44#14556125—21no—
45#14556125—21no—
46#14556125—21no—
47#14556125C2312yes—
48C4⋊(C71⋊C8)—24no—
49C4⋊(C71⋊C8)—21no—
50C4⋊(C71⋊C8)—21no—
51C4⋊(C71⋊C8)C2312yes—
52C4⋊(C71⋊C8)C2312yes—
53#14556048—24no—
54#14556048—24no—
55#14556048—24no—
56#14556039—24no—
57C4×C71⋊C8D412yes—
58C4×C71⋊C8C2312yes—
59C22×Q8×C71—21no—
60C22×Q8×C71D411yescharacteristic
61C22×C8×C71—21no—
62C4⋊Q8×C71—21no—
63C4⋊Q8×C71—21no—
64C4⋊Q8×C71—21no—
65C4⋊Q8×C71C2312yes—
66C4×Q8×C71—21no—
67C4×Q8×C71—21no—
68C4×Q8×C71D412yes—
69C4×Q8×C71C2312yes—
70C2×C4⋊C4×C71—21no—
71C2×C4⋊C4×C71—21no—
72C2×C4⋊C4×C71—21no—
73C2×C4⋊C4×C71D411yescharacteristic
74C2×C42×C71—21no—
75C2×C42×C71C2311yescharacteristic
76#107×C71—24no—
77C4⋊C8×C71—21no—
78C4⋊C8×C71C2312yes—
79#103×C71—24no—
80C23×D71—41no—
81C2×D284—22no—
82C2×D284—22no—
83C2×D284—22no—
84C2×D284—42no—
85C2×D284—42no—
86C2×D284—42no—
87C2×D284—42no—
88C2×D284—41no—
89C2×D284—41no—
90C2×D284—41no—
91C2×D284—82no—
92#11765852—48no—
93C2×C71⋊C8—22no—
94C2×C71⋊C8—22no—
95C2×C71⋊C8—22no—
96C2×C71⋊C8—22no—
97C2×C71⋊C8—42no—
98C2×C71⋊C8—42no—
99C2×C71⋊C8C2×D412yes—
100C2×C71⋊C8C2×D412yes—
101C2×Q8×C71—21no—
102C2×Q8×C71—22no—
103C2×Q8×C71—22no—
104C2×Q8×C71—22no—
105C2×Q8×C71—41no—
106C2×Q8×C71—82no—
107C2×Q8×C71C2×D412yes—
108C22×C4×C71—21no—
109C22×C4×C71—21no—
110C22×C4×C71—41no—
111C22×C4×C71C2×D411yescharacteristic
112C22×C4×C71C2×D411yescharacteristic
113C22×C4×C71#4111yescharacteristic
114C2×C8×C71—22no—
115C2×C8×C71—42no—
116C4⋊C4×C71—22no—
117C4⋊C4×C71—21no—
118C4⋊C4×C71—22no—
119C4⋊C4×C71—22no—
120C4⋊C4×C71—22no—
121C4⋊C4×C71—41no—
122C4⋊C4×C71—41no—
123C4⋊C4×C71C2×D412yes—
124C42×C71—22no—
125C42×C71—22no—
126C42×C71C2×D411yescharacteristic
127C42×C71C2×D411yescharacteristic
128C42×C71#4111yescharacteristic
129C42×C71C2411yescharacteristic
130C23×C71#12111yescharacteristic
131C22×D71—42no—
132C22×D71—82no—
133C22×D71—82no—
134C22×D71—82no—
135Q8×C71—22no—
136Q8×C71—22no—
137Q8×C71—42no—
138Q8×C71—82no—
139Q8×C71—82no—
140C2×C4×C71—22no—
141C2×C4×C71—22no—
142C2×C4×C71—22no—
143C2×C4×C71—42no—
144C2×C4×C71—42no—
145C2×C4×C71—42no—
146C2×C4×C71—82no—
147C2×C4×C71#12111yescharacteristic
148C2×C4×C71#12111yescharacteristic
149C2×C4×C71#12111yescharacteristic
150C2×C4×C71#13512yes—
151C2×C4×C71AGL(1,ℤ/8)12yes—
152C2×C4×C71#13712yes—
153C2×C4×C71#13712yes—
154C2×C4×C71C22×D411yescharacteristic
155C2×C4×C71C22×D411yescharacteristic
156C2×C4×C71C2×#4111yescharacteristic
157D284—42no—
158D284—42no—
159D284—42no—
160D284—42no—
161D284—82no—
162D284—82no—
163D284—82no—
164D284—82no—
165C8×C71—42no—
166C71⋊C8—22no—
167C71⋊C8—22no—
168C71⋊C8—42no—
169C71⋊C8—42no—
170C22×C71#46412yes—
171C22×C71#46812yes—
172C22×C71#47212yes—
173C22×C71C2×#12111yescharacteristic
174C2×D71—82no—
175C2×D71—82no—
176C2×D71—82no—
177C2×D71—164no—
178C4×C71—21no—
179C4×C71—21no—
180C4×C71—42no—
181C4×C71—42no—
182C4×C71—82no—
183C4×C71#46412yes—
184C4×C71#46812yes—
185C4×C71C2×#13511yescharacteristic
186C4×C71C2×AGL(1,ℤ/8)11yescharacteristic
187C4×C71C2×#13711yescharacteristic
188C4×C71C2×#13711yescharacteristic
189C2×#1698—711nomaximal, Sylow
190C2×C71#169814yescomplemented
191C2×C71C2×#46411yescharacteristic
192C2×C71C2×#46811yescharacteristic
193C2×C71C2×#47211yescharacteristic
194D71—162no—
195C2×#486—711no—
196C2×#477D7111yescharacteristic, complemented
197C2×#460—711no—
198C2×#446—711no—
199#1698—718no—
200C2×#335—711no—
201C2×#333—711no—
202C2×#331—711no—
203C71C2×#169811yescharacteristic, complemented, Sylow
204C22×SD16—1421no—
205C2×C4⋊Q8C2×D7111yescharacteristic
206C2×#127—711no—
207C2×C4×Q8C2×D7111yescharacteristic
208#486—714no—
209#486—1422no—
210#477—22no—
211#477C2×D7114yescomplemented
212#460—714no—
213#460—1422no—
214#446—714no—
215#446—1422no—
216C2×#107—21no—
217C2×C4⋊C8—711no—
218C2×C4⋊C8—711no—
219C2×C4⋊C8—1421no—
220C2×C4⋊C8C2×D7111yescharacteristic
221C2×#103—21no—
222C2×#102—1421no—
223C2×#102—1421no—
224C2×#102—1421no—
225C2×#97—1421no—
226C2×C4×C8—711no—
227#335—714no—
228#333—714no—
229#331—714no—
230C22×Q8—21no—
231C22×Q8C22⋊D7111yescharacteristic
232C22×D4—1421no—
233C22×D4—1421no—
234C22×D4—1421no—
235C2×SD16—2844no—
236C2×SD16—1424no—
237C22×C8—21no—
238C22×C8—711no—
239C22×C8—711no—
240C22×C8—1421no—
241C22×C8—1421no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8C22×D7112yes—
246#127—712no—
247#127—1421no—
248#127—1421no—
249#127—1421no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8C22⋊D7112yes—
253C4×Q8C22×D7112yes—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4C22⋊D7111yescharacteristic
258C2×C42—21no—
259C2×C42C22×D7111yescharacteristic
260#107—24no—
261C4⋊C8—21no—
262C4⋊C8—712no—
263C4⋊C8—712no—
264C4⋊C8—1424no—
265C4⋊C8—1421no—
266C4⋊C8—1421no—
267C4⋊C8C22×D7112yes—
268#103—24no—
269#102—1424no—
270#102—1424no—
271#102—1424no—
272#97—1424no—
273C4×C8—712no—
274C4×C8—712no—
275C24—2841no—
276C2×Q8—21no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—41no—
281C2×Q8—82no—
282C2×Q8C2×C22⋊D7112yes—
283C2×D4—2842no—
284C2×D4—2841no—
285C2×D4—2841no—
286C2×D4—2841no—
287C2×D4—2842no—
288C2×D4—2842no—
289C2×D4—2842no—
290C2×D4—5682no—
291C2×D4—1422no—
292C2×D4—1422no—
293C2×D4—1422no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4D4×D7111yescharacteristic
298C22×C4C2×C22⋊D7111yescharacteristic
299C22×C4#1176586411yescharacteristic
300SD16—2848no—
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—22no—
315C4⋊C4—22no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4C2×C22⋊D7112yes—
319C42—22no—
320C42—22no—
321C42D4×D7111yescharacteristic
322C42C2×C22⋊D7111yescharacteristic
323C42#1176586411yescharacteristic
324C42C23×D7111yescharacteristic
325C23—2842no—
326C23—5682no—
327C23—5682no—
328C23—5682no—
329C23#1455610011yescharacteristic
330Q8—22no—
331Q8—22no—
332Q8—42no—
333Q8—82no—
334Q8—82no—
335D4—2842no—
336D4—2842no—
337D4—2842no—
338D4—2842no—
339D4—5682no—
340D4—5682no—
341D4—5682no—
342D4—5682no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—82no—
350C2×C4#1455610011yescharacteristic
351C2×C4#1455610011yescharacteristic
352C2×C4#1455610011yescharacteristic
353C2×C4#1455615112yes—
354C2×C4#1455615312yes—
355C2×C4#1455615912yes—
356C2×C4#1455616012yes—
357C2×C4C2×D4×D7111yescharacteristic
358C2×C4C22×C22⋊D7111yescharacteristic
359C2×C4C2×#1176586411yescharacteristic
360C8—42no—
361C8—2842no—
362C8—2842no—
363C8—1422no—
364C8—1422no—
365C22—5682no—
366C22—5682no—
367C22—5682no—
368C22—11364no—
369C22#1798622312yes—
370C22#1798624712yes—
371C22#1798626712yes—
372C22C2×#1455610011yescharacteristic
373C4—21no—
374C4—21no—
375C4—42no—
376C4—42no—
377C4—82no—
378C4#1798622312yes—
379C4#1798624712yes—
380C4C2×#1455615111yescharacteristic
381C4C2×#1455615311yescharacteristic
382C4C2×#1455615911yescharacteristic
383C4C2×#1455616011yescharacteristic
384C2—11362no—
385C2#2258788214yesdirect product factor
386C2C2×#1798622311yescharacteristic
387C2C2×#1798624711yescharacteristic
388C2C2×#1798626711yescharacteristic
subgroupG/H
centerC23#14556100
commutator, cosocleC42×C71C24
FrattiniC42C23×D71
FittingC2×#477×C71C2
socleC23×C71#121
prime radicalC2×#14556125C22
hypercenterC2×#477D71
hypocenterC71C2×#1698
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
3. C2×#17986130C2semidirect
4. #22587882C2direct (N × H)
5. C2×#17985751C2semidirect
6. C2×#17985748C2semidirect
8. C2×#477×C71C2semidirect
16. #17986130C22semidirect
25. #17985751C22semidirect
26. #17985748C22semidirect
31. #477×C71C22semidirect
190. C2×C71#1698semidirect
196. C2×#477D71semidirect
203. C71C2×#1698semidirect
211. #477C2×D71semidirect
385. C2#22587882direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179863472
C2×#179861992
C2×#179861302
#225878822
C2×#179857512
C2×#179857482
C2×#179857422
C2×#477×C712
C2×#169871