Finite Group Databasebeta

← C2×#22587861 #31979985

Subgroups — C2×#22587861 #31979985

Order 18176.

694 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986402C211yesmaximal, characteristic, complemented
2C2×#17986289C211yesmaximal, characteristic
3C2×#17986077C211yesmaximal, characteristic
4#22587861C218yesmaximal, direct product factor
5C2×#17985749C211yesmaximal, characteristic, complemented
6C2×#17985748C211yesmaximal, characteristic, complemented
7C2×#17985747C211yesmaximal, characteristic, complemented
8C2×#476×C71C211yesmaximal, characteristic
9C22×#11765849—21no—
10C2×#14556128C2211yescharacteristic
11#17986402—22no—
12#17986402C2214yescomplemented
13#17986289—22no—
14#17986289C2214yes—
15#17986077—22no—
16#17986077C2214yes—
17C2×#14556061—21no—
18C2×#14556061—21no—
19C2×#14556059—21no—
20C2×C4⋊(C71⋊C8)C2211yescharacteristic
21C2×C4⋊(C71⋊C8)C2211yescharacteristic
22C2×#14556049—21no—
23C2×C22⋊(C71⋊C8)—21no—
24C2×C4×C71⋊C8C2211yescharacteristic
25#17985749C2214yescomplemented
26#17985748C2214yescomplemented
27#17985747C2214yescomplemented
28C2×C4⋊Q8×C71C2211yescharacteristic
29C2×C4×D4×C71C2211yescharacteristic
30#476×C71—22no—
31#476×C71C2214yes—
32C2×#106×C71—21no—
33C2×C4⋊C8×C71C2211yescharacteristic
34C2×#102×C71—21no—
35C22×Dic142—21no—
36C2×#11765849—24no—
37C2×#11765849—44no—
38C22×C71⋊C8—21no—
39C22×C71⋊C8—21no—
40C22×C71⋊C8D411yescharacteristic
41C22×C71⋊C8D411yescharacteristic
42#14556128—21no—
43#14556128—21no—
44#14556128—21no—
45#14556128C2312yes—
46C2×C4⋊Dic71—21no—
47C2×C4⋊Dic71—21no—
48#14556061—24no—
49#14556061—24no—
50#14556059—24no—
51C4⋊(C71⋊C8)—21no—
52C4⋊(C71⋊C8)—21no—
53C4⋊(C71⋊C8)C2312yes—
54C4⋊(C71⋊C8)C2312yes—
55#14556049—24no—
56C22⋊(C71⋊C8)—24no—
57C4×C71⋊C8D412yes—
58C4×C71⋊C8C2312yes—
59C22×Q8×C71—21no—
60C22×D4×C71D411yescharacteristic
61C23×C4×C71—21no—
62C22×C8×C71—21no—
63C4⋊Q8×C71—21no—
64C4⋊Q8×C71—21no—
65C4⋊Q8×C71—21no—
66C4⋊Q8×C71C2312yes—
67C4×D4×C71—21no—
68C4×D4×C71—21no—
69C4×D4×C71D412yes—
70C4×D4×C71C2312yes—
71C2×C4⋊C4×C71—21no—
72C2×C4⋊C4×C71—21no—
73C2×C4⋊C4×C71D411yescharacteristic
74C2×C22⋊C4×C71—21no—
75C2×C42×C71C2311yescharacteristic
76#106×C71—24no—
77C4⋊C8×C71—21no—
78C4⋊C8×C71C2312yes—
79#102×C71—24no—
80C2×Dic142—22no—
81C2×Dic142—41no—
82C2×Dic142—82no—
83C22×Dic71—41no—
84#11765849—48no—
85C2×C71⋊C8—22no—
86C2×C71⋊C8—22no—
87C2×C71⋊C8—22no—
88C2×C71⋊C8—42no—
89C2×C71⋊C8—42no—
90C2×C71⋊C8SD1614yes—
91C2×C71⋊C8C2×D412yes—
92C2×C71⋊C8C2×D412yes—
93C4⋊Dic71—22no—
94C4⋊Dic71—22no—
95C4⋊Dic71—41no—
96C4⋊Dic71—41no—
97C24×C71—21no—
98C2×Q8×C71—22no—
99C2×Q8×C71—41no—
100C2×Q8×C71—82no—
101C2×D4×C71—21no—
102C2×D4×C71—22no—
103C2×D4×C71—22no—
104C2×D4×C71C2×D412yes—
105C22×C4×C71—21no—
106C22×C4×C71—22no—
107C22×C4×C71—22no—
108C22×C4×C71—42no—
109C22×C4×C71—42no—
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—21no—
117C4⋊C4×C71—22no—
118C4⋊C4×C71—22no—
119C4⋊C4×C71—41no—
120C4⋊C4×C71—41no—
121C4⋊C4×C71C2×D412yes—
122C22⋊C4×C71—22no—
123C22⋊C4×C71—22no—
124C42×C71C2×D411yescharacteristic
125C42×C71C2×D411yescharacteristic
126C42×C71#4111yescharacteristic
127C42×C71C2411yescharacteristic
128C23×C71—22no—
129C23×C71—42no—
130C23×C71—42no—
131C23×C71—42no—
132C23×C71#12111yescharacteristic
133Q8×C71—82no—
134Q8×C71—82no—
135D4×C71—22no—
136D4×C71—22no—
137D4×C71—42no—
138C2×C4×C71—22no—
139C2×C4×C71—22no—
140C2×C4×C71—42no—
141C2×C4×C71—42no—
142C2×C4×C71—42no—
143C2×C4×C71—44no—
144C2×C4×C71—42no—
145C2×C4×C71—82no—
146C2×C4×C71#12111yescharacteristic
147C2×C4×C71#12111yescharacteristic
148C2×C4×C71#12111yescharacteristic
149C2×C4×C71C2×SD1612yes—
150C2×C4×C71#13712yes—
151C2×C4×C71#13712yes—
152C2×C4×C71#13712yes—
153C2×C4×C71C22×D411yescharacteristic
154C2×C4×C71C22×D411yescharacteristic
155C2×C4×C71C2×#4111yescharacteristic
156C2×Dic71—42no—
157C2×Dic71—82no—
158Dic142—82no—
159Dic142—82no—
160C8×C71—42no—
161C71⋊C8—22no—
162C71⋊C8—42no—
163C71⋊C8—42no—
164C71⋊C8C2×SD1614yes—
165C22×C71—42no—
166C22×C71—42no—
167C22×C71—42no—
168C22×C71—84no—
169C22×C71#46112yes—
170C22×C71#47012yes—
171C22×C71#47212yes—
172C22×C71C2×#12111yescharacteristic
173C4×C71—21no—
174C4×C71—21no—
175C4×C71—42no—
176C4×C71—82no—
177C4×C71#46112yes—
178C4×C71#47012yes—
179C4×C71C22×SD1611yescharacteristic
180C4×C71C2×#13711yescharacteristic
181C4×C71C2×#13711yescharacteristic
182C4×C71C2×#13711yescharacteristic
183Dic71—82no—
184C2×#1695—711nomaximal, Sylow
185C2×C71—82no—
186C2×C71#169514yescomplemented
187C2×C71C2×#46111yescharacteristic
188C2×C71C2×#47011yescharacteristic
189C2×C71C2×#47211yescharacteristic
190C2×#500—711no—
191C2×#476—711no—
192C2×#476D7111yescharacteristic
193C2×#436—711no—
194#1695—718no—
195C2×#334—711no—
196C2×#333—711no—
197C2×#333—711no—
198C71C2×#169511yescharacteristic, complemented, Sylow
199C22×M16—1421no—
200C2×C4⋊Q8—711no—
201C2×C4⋊Q8C2×D7111yescharacteristic
202C2×C4×D4C2×D7111yescharacteristic
203#500—714no—
204#500—1422no—
205#476—22no—
206#476—714no—
207#476—1422no—
208#476C2×D7114yes—
209#436—714no—
210#436—1422no—
211C2×#107—1421no—
212C2×#107—1421no—
213C2×#106—21no—
214C2×#106—1421no—
215C2×C4⋊C8—711no—
216C2×C4⋊C8—711no—
217C2×C4⋊C8C2×D7111yescharacteristic
218C2×#102—21no—
219C2×#102—1421no—
220C2×C22⋊C8—1421no—
221C2×C4×C8—711no—
222#334—714no—
223#333—714no—
224#333—714no—
225C22×Q8—21no—
226C22×Q8—1421no—
227C22×D4C22⋊D7111yescharacteristic
228C23×C4—21no—
229C2×M16—2844no—
230C2×M16—1424no—
231C22×C8—21no—
232C22×C8—711no—
233C22×C8—711no—
234C22×C8—1421no—
235C22×C8—1421no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—712no—
240C4⋊Q8—1421no—
241C4⋊Q8—1421no—
242C4⋊Q8—1421no—
243C4⋊Q8C22×D7112yes—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4C22⋊D7112yes—
247C4×D4C22×D7112yes—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—1421no—
251C2×C4⋊C4—1421no—
252C2×C4⋊C4C22⋊D7111yescharacteristic
253C2×C22⋊C4—21no—
254C2×C42C22×D7111yescharacteristic
255#107—1424no—
256#107—1424no—
257#106—24no—
258#106—1424no—
259C4⋊C8—21no—
260C4⋊C8—712no—
261C4⋊C8—712no—
262C4⋊C8—1421no—
263C4⋊C8—1421no—
264C4⋊C8C22×D7112yes—
265#102—24no—
266#102—1424no—
267C22⋊C8—1424no—
268C4×C8—712no—
269C4×C8—712no—
270C24—21no—
271C2×Q8—22no—
272C2×Q8—41no—
273C2×Q8—82no—
274C2×Q8—2841no—
275C2×Q8—5682no—
276C2×Q8—1422no—
277C2×D4—21no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4C2×C22⋊D7112yes—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—2841no—
288C22×C4D4×D7111yescharacteristic
289C22×C4C2×C22⋊D7111yescharacteristic
290C22×C4#1176586411yescharacteristic
291M16—2848no—
292C2×C8—22no—
293C2×C8—42no—
294C2×C8—2842no—
295C2×C8—2842no—
296C2×C8—712no—
297C2×C8—714no—
298C2×C8—712no—
299C2×C8—1422no—
300C2×C8—1422no—
301C2×C8—1422no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—2841no—
308C4⋊C4—2841no—
309C4⋊C4—1422no—
310C4⋊C4—1422no—
311C4⋊C4C2×C22⋊D7112yes—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C42D4×D7111yescharacteristic
315C42C2×C22⋊D7111yescharacteristic
316C42#1176586411yescharacteristic
317C42C23×D7111yescharacteristic
318C23—22no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23#1455610011yescharacteristic
323Q8—82no—
324Q8—82no—
325Q8—5682no—
326Q8—5682no—
327D4—22no—
328D4—22no—
329D4—42no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—82no—
338C2×C4—2842no—
339C2×C4—5682no—
340C2×C4#1455610011yescharacteristic
341C2×C4#1455610011yescharacteristic
342C2×C4#1455610011yescharacteristic
343C2×C4SD16×D7112yes—
344C2×C4#1455615712yes—
345C2×C4#1455615812yes—
346C2×C4#1455615912yes—
347C2×C4C2×D4×D7111yescharacteristic
348C2×C4C22×C22⋊D7111yescharacteristic
349C2×C4C2×#1176586411yescharacteristic
350C8—42no—
351C8—2842no—
352C8—2842no—
353C8—714no—
354C8—1422no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—84no—
359C22#1798620212yes—
360C22#1798625512yes—
361C22#1798626712yes—
362C22C2×#1455610011yescharacteristic
363C4—21no—
364C4—21no—
365C4—42no—
366C4—82no—
367C4—5682no—
368C4#1798620212yes—
369C4#1798625512yes—
370C4C2×SD16×D7111yescharacteristic
371C4C2×#1455615711yescharacteristic
372C4C2×#1455615811yescharacteristic
373C4C2×#1455615911yescharacteristic
374C2—82no—
375C2#2258786114yesdirect product factor
376C2C2×#1798620211yescharacteristic
377C2C2×#1798625511yescharacteristic
378C2C2×#1798626711yescharacteristic
subgroupG/H
centerC23#14556100
commutator, cosocleC42×C71C24
FrattiniC42C23×D71
FittingC2×#476×C71C2
socleC23×C71#121
prime radicalC22×D4×C71D4
hypercenterC2×#476D71
hypocenterC71C2×#1695
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. C2×#17986402C2semidirect
4. #22587861C2direct (N × H)
5. C2×#17985749C2semidirect
6. C2×#17985748C2semidirect
7. C2×#17985747C2semidirect
12. #17986402C22semidirect
25. #17985749C22semidirect
26. #17985748C22semidirect
27. #17985747C22semidirect
186. C2×C71#1695semidirect
198. C71C2×#1695semidirect
375. C2#22587861direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179864022
C2×#179862892
C2×#179860772
#225878612
C2×#179857492
C2×#179857482
C2×#179857472
C2×#476×C712
C2×#169571