Finite Group Databasebeta

← C2×#11741520 #14217242

Subgroups — C2×#11741520 #14217242

Order 1792.

5206 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
1C23×C7×#41C211yesmaximal, characteristic
2C22×#6511C212yesmaximal, complemented
3#11741520C2116yesmaximal, direct product factor
4C2×#90006C216yesmaximal, complemented
5C2×#89979C213yesmaximal, complemented
6C22×C22⋊(C7⋊C8)C213yesmaximal, complemented
7C22×C7×#41—216no—
8C22×C7×#41C418yes—
9C22×C7×#41C411yescharacteristic
10C22×C7×#41C413yes—
11C22×C7×#41C2213yes—
12C22×C7×#41C2218yes—
13C22×C7×#41C2211yescharacteristic
14C23×C7×Q8C411yescharacteristic
15C23×C7×D4C413yes—
16C24×C4×C7C2213yes—
17C2×#6511—216no—
18C2×#6511C22116yescomplemented
19C22×#1136C2216yescomplemented
20C23×C7⋊C8C2216yescomplemented
21#90006C22148yescomplemented
22#89979C22124yescomplemented
23C2×C22⋊(C7⋊C8)C2211yescharacteristic
24C2×C22⋊(C7⋊C8)C2213yes—
25C2×C22⋊(C7⋊C8)C22124yescomplemented
26C2×C22⋊(C7⋊C8)C2216yescomplemented
27C2×C22⋊(C7⋊C8)C2216yescomplemented
28C2×#2880—71nomaximal, Sylow
29C25×C7C2×C413yes—
30C2×C7×#41—264no—
31C2×C7×#41—264no—
32C2×C7×#41—28no—
33C2×C7×#41—224no—
34C2×C7×#41—224no—
35C2×C7×#41C2×C418yes—
36C2×C7×#41C2×C4124yes—
37C2×C7×#41C2×C4112yes—
38C2×C7×#41C2×C414yes—
39C2×C7×#41D4116yes—
40C2×C7×#41C23112yes—
41C2×C7×#41C2314yes—
42C22×C7×Q8—28no—
43C22×C7×Q8C2×C418yes—
44C22×C7×Q8C2×C413yes—
45C22×C7×Q8C2×C411yescharacteristic
46C22×C7×D4—224no—
47C22×C7×D4C2×C416yes—
48C22×C7×D4C2×C4124yes—
49C22×C7×D4C2×C413yes—
50C22×C7×D4C2×C413yes—
51C23×C4×C7—224no—
52C23×C4×C7C2×C413yes—
53C23×C4×C7C2×C413yes—
54C23×C4×C7C2×C4112yes—
55C23×C4×C7C2×C413yes—
56C23×C4×C7C2313yes—
57C23×C4×C7C23112yes—
58C23×C4×C7C2311yescharacteristic
59C23×C4×C7C2313yes—
60#6511—264no—
61C2×#1136—224no—
62C2×#1136C23124yescomplemented
63C22×C7⋊C8—224no—
64C22×C7⋊C8C2312yes—
65C22×C7⋊C8C2316yes—
66C22×C7⋊C8C23124yescomplemented
67C22⋊(C7⋊C8)C2314yes—
68C22⋊(C7⋊C8)C23112yes—
69C22⋊(C7⋊C8)C23124yescomplemented
70C22⋊(C7⋊C8)C23124yescomplemented
71C23×#41D711yescharacteristic
72C22×#131—72no—
73#2880—716no—
74C2×#408—76no—
75C2×#407—73no—
76C22×C22⋊C8—73no—
77C24×C7—212no—
78C24×C7—212no—
79C24×C7—46no—
80C24×C7C22×C413yes—
81C24×C7C22×C4112yes—
82C24×C7C22×C413yes—
83C24×C7C22×C411yescharacteristic
84C7×#41—232no—
85C7×#41—296no—
86C7×#41—248no—
87C7×#41—248no—
88C7×#41C22⋊C4132yes—
89C7×#41C2×D4132yes—
90C2×C7×Q8—232no—
91C2×C7×Q8—212no—
92C2×C7×Q8C22⋊C418yes—
93C2×C7×Q8C22×C4112yes—
94C2×C7×Q8C22×C414yes—
95C2×C7×D4—224no—
96C2×C7×D4—296no—
97C2×C7×D4—212no—
98C2×C7×D4C22⋊C4124yes—
99C2×C7×D4C22×C4124yes—
100C2×C7×D4C22×C4112yes—
101C2×C7×D4C22×C4112yes—
102C22×C4×C7—212no—
103C22×C4×C7—212no—
104C22×C4×C7—212no—
105C22×C4×C7—248no—
106C22×C4×C7—24no—
107C22×C4×C7—212no—
108C22×C4×C7—248no—
109C22×C4×C7C22×C411yescharacteristic
110C22×C4×C7C22×C412yes—
111C22×C4×C7C22×C413yes—
112C22×C4×C7C22×C4112yes—
113C22×C4×C7C22×C4112yes—
114C22×C4×C7C22×C4112yes—
115C22×C4×C7C22×C416yes—
116C22×C4×C7C22×C413yes—
117C22×C4×C7C22×C416yes—
118C22×C4×C7C2×D4124yes—
119C22×C4×C7C2412yes—
120C22×C4×C7C2416yes—
121C22×C4×C7C2416yes—
122C22×C4×C7C2411yescharacteristic
123#1136—248no—
124C2×C7⋊C8—24no—
125C2×C7⋊C8—212no—
126C2×C7⋊C8—248no—
127C2×C7⋊C8C2414yes—
128C2×C7⋊C8C24112yes—
129C22×#41—216no—
130C22×#41Dic718yes—
131C22×#41Dic711yescharacteristic
132C22×#41Dic713yes—
133C22×#41C2×D713yes—
134C22×#41C2×D718yes—
135C22×#41C2×D711yescharacteristic
136C23×Q8Dic711yescharacteristic
137C23×D4Dic713yes—
138C24×C4C2×D713yes—
139C2×#131—716no—
140C2×#131—1416no—
141C22×M16—76no—
142C23×C8—76no—
143#408—748no—
144#407—724no—
145C2×C22⋊C8—71no—
146C2×C22⋊C8—73no—
147C2×C22⋊C8—76no—
148C2×C22⋊C8—724no—
149C2×C22⋊C8—76no—
150C23×C7—22no—
151C23×C7—26no—
152C23×C7—22no—
153C23×C7—26no—
154C23×C7—224no—
155C23×C7—26no—
156C23×C7—26no—
157C23×C7—224no—
158C23×C7—26no—
159C23×C7—412no—
160C23×C7—448no—
161C23×C7C2×C22⋊C4112yes—
162C23×C7#13114yescomplemented
163C23×C7C23×C416yes—
164C23×C7C23×C416yes—
165C23×C7C23×C412yes—
166C23×C7C23×C411yescharacteristic
167C7×Q8—224no—
168C7×Q8C2×C22⋊C4116yes—
169C7×D4—248no—
170C7×D4—224no—
171C7×D4C2×C22⋊C4148yes—
172C2×C4×C7—22no—
173C2×C4×C7—26no—
174C2×C4×C7—224no—
175C2×C4×C7—24no—
176C2×C4×C7—224no—
177C2×C4×C7—224no—
178C2×C4×C7—212no—
179C2×C4×C7—24no—
180C2×C4×C7—212no—
181C2×C4×C7C2×C22⋊C4112yes—
182C2×C4×C7C2×C22⋊C4124yes—
183C2×C4×C7C23×C412yes—
184C2×C4×C7C23×C416yes—
185C2×C4×C7C23×C411yescharacteristic
186C2×C4×C7C23×C416yes—
187C2×C4×C7C22×D414yes—
188C2×C4×C7C22×D4124yes—
189C2×C4×C7C2511yescharacteristic
190C7⋊C8—24no—
191C7⋊C8—212no—
192C25C2×Dic713yes—
193C2×#41—264no—
194C2×#41—264no—
195C2×#41—28no—
196C2×#41—224no—
197C2×#41—224no—
198C2×#41C2×Dic718yes—
199C2×#41C2×Dic7124yes—
200C2×#41C2×Dic7112yes—
201C2×#41C2×Dic714yes—
202C2×#41C22⋊D7116yes—
203C2×#41C22×D7112yes—
204C2×#41C22×D714yes—
205C22×Q8—28no—
206C22×Q8C2×Dic718yes—
207C22×Q8C2×Dic713yes—
208C22×Q8C2×Dic711yescharacteristic
209C22×D4—224no—
210C22×D4C2×Dic716yes—
211C22×D4C2×Dic7124yes—
212C22×D4C2×Dic713yes—
213C22×D4C2×Dic713yes—
214C23×C4—224no—
215C23×C4C2×Dic713yes—
216C23×C4C2×Dic713yes—
217C23×C4C2×Dic7112yes—
218C23×C4C2×Dic713yes—
219C23×C4C22×D7112yes—
220C23×C4C22×D711yescharacteristic
221C23×C4C22×D713yes—
222C23×C4C22×D713yes—
223#131—1464no—
224C2×M16—724no—
225C2×M16—1424no—
226C22×C8—724no—
227C22×C8—72no—
228C22×C8—76no—
229C22×C8—1424no—
230C22⋊C8—74no—
231C22⋊C8—712no—
232C22⋊C8—724no—
233C22⋊C8—724no—
234C22×C7—28no—
235C22×C7—224no—
236C22×C7—22no—
237C22×C7—26no—
238C22×C7—22no—
239C22×C7—26no—
240C22×C7—26no—
241C22×C7—26no—
242C22×C7—26no—
243C22×C7—448no—
244C22×C7—412no—
245C22×C7C22×C22⋊C4112yes—
246C22×C7C22×C22⋊C412yes—
247C22×C7C2×#13112yes—
248C22×C7C2×#13114yescomplemented
249C22×C7#56812yes—
250C22×C7C24×C411yescharacteristic
251C4×C7—22no—
252C4×C7—26no—
253C4×C7C22×C22⋊C412yes—
254C4×C7C22×C22⋊C4112yes—
255C4×C7C23×D412yes—
256C24—212no—
257C24—212no—
258C24—46no—
259C24C22×Dic713yes—
260C24C22×Dic7112yes—
261C24C22×Dic711yescharacteristic
262C24C22×Dic713yes—
263#41—232no—
264#41—296no—
265#41—248no—
266#41—248no—
267#41C22⋊Dic7132yes—
268#41C2×C22⋊D7132yes—
269C2×Q8—232no—
270C2×Q8—212no—
271C2×Q8C22⋊Dic718yes—
272C2×Q8C22×Dic7112yes—
273C2×Q8C22×Dic714yes—
274C2×D4—224no—
275C2×D4—296no—
276C2×D4—212no—
277C2×D4C22⋊Dic7124yes—
278C2×D4C22×Dic7124yes—
279C2×D4C22×Dic7112yes—
280C2×D4C22×Dic7112yes—
281C22×C4—212no—
282C22×C4—212no—
283C22×C4—212no—
284C22×C4—248no—
285C22×C4—24no—
286C22×C4—212no—
287C22×C4—248no—
288C22×C4C22×Dic711yescharacteristic
289C22×C4C22×Dic712yes—
290C22×C4C22×Dic713yes—
291C22×C4C22×Dic7112yes—
292C22×C4C22×Dic7112yes—
293C22×C4C22×Dic7112yes—
294C22×C4C22×Dic716yes—
295C22×C4C22×Dic713yes—
296C22×C4C22×Dic716yes—
297C22×C4C2×C22⋊D7124yes—
298C22×C4C23×D712yes—
299C22×C4C23×D716yes—
300C22×C4C23×D716yes—
301C22×C4C23×D711yescharacteristic
302M16—1448no—
303C2×C8—74no—
304C2×C8—712no—
305C2×C8—144no—
306C2×C8—1412no—
307C2×C8—1448no—
308C2×C7—24no—
309C2×C7—212no—
310C2×C7—46no—
311C2×C7#288014yescomplemented
312C2×C7C23×C22⋊C411yescharacteristic
313C2×C7C22×#13111yescharacteristic
314C2×C7C2×#56811yescharacteristic
315C23—26no—
316C23—22no—
317C23—26no—
318C23—22no—
319C23—26no—
320C23—224no—
321C23—26no—
322C23—224no—
323C23—26no—
324C23—412no—
325C23—448no—
326C23#651114yescomplemented
327C23C2×C22⋊Dic7112yes—
328C23C23×Dic716yes—
329C23C23×Dic712yes—
330C23C23×Dic716yes—
331C23C23×Dic711yescharacteristic
332Q8—224no—
333Q8C2×C22⋊Dic7116yes—
334D4—248no—
335D4—224no—
336D4C2×C22⋊Dic7148yes—
337C2×C4—22no—
338C2×C4—26no—
339C2×C4—224no—
340C2×C4—24no—
341C2×C4—224no—
342C2×C4—224no—
343C2×C4—212no—
344C2×C4—24no—
345C2×C4—212no—
346C2×C4C2×C22⋊Dic7112yes—
347C2×C4C2×C22⋊Dic7124yes—
348C2×C4C23×Dic712yes—
349C2×C4C23×Dic716yes—
350C2×C4C23×Dic711yescharacteristic
351C2×C4C23×Dic716yes—
352C2×C4C22×C22⋊D714yes—
353C2×C4C22×C22⋊D7124yes—
354C2×C4C24×D711yescharacteristic
355C8—144no—
356C8—1412no—
357C7C2×#288011yescharacteristic, complemented, Sylow
358C22—26no—
359C22—22no—
360C22—26no—
361C22—28no—
362C22—224no—
363C22—22no—
364C22—26no—
365C22—26no—
366C22—26no—
367C22—448no—
368C22—412no—
369C22C2×#651112yes—
370C22C2×#651114yescomplemented
371C22#9051012yes—
372C22C22×C22⋊Dic7112yes—
373C22C22×C22⋊Dic712yes—
374C22C24×Dic711yescharacteristic
375C4—22no—
376C4—26no—
377C4C22×C22⋊Dic712yes—
378C4C22×C22⋊Dic7112yes—
379C4C23×C22⋊D712yes—
380C2—24no—
381C2—212no—
382C2—46no—
383C2#1174152014yesdirect product factor
384C2C22×#651111yescharacteristic
385C2C2×#9051011yescharacteristic
386C2C23×C22⋊Dic711yescharacteristic
subgroupG/H
centerC22×C4C23×D7
commutatorC22×C7C24×C4
FrattiniC2×C4C24×D7
Fitting, prime radicalC23×C7×#41C2
socleC23×C7C23×C4
cosocleC2×C4×C7C25
hypercenterC23×#41D7
hypocenterC7C2×#2880
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C23×#41

Complements

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

NH ≅ G/Ntype
2. C22×#6511C2semidirect
3. #11741520C2direct (N × H)
4. C2×#90006C2semidirect
5. C2×#89979C2semidirect
6. C22×C22⋊(C7⋊C8)C2semidirect
18. C2×#6511C22semidirect
19. C22×#1136C22semidirect
20. C23×C7⋊C8C22semidirect
21. #90006C22semidirect
22. #89979C22semidirect
25. C2×C22⋊(C7⋊C8)C22semidirect
26. C2×C22⋊(C7⋊C8)C22semidirect
27. C2×C22⋊(C7⋊C8)C22semidirect
62. C2×#1136C23semidirect
66. C22×C7⋊C8C23semidirect
69. C22⋊(C7⋊C8)C23semidirect
70. C22⋊(C7⋊C8)C23semidirect
162. C23×C7#131semidirect
248. C22×C7C2×#131semidirect
311. C2×C7#2880semidirect
326. C23#6511semidirect
357. C7C2×#2880semidirect
370. C22C2×#6511semidirect
383. C2#11741520direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C7×#412
C22×#65112
#117415202
C2×#900062
C2×#899792
C22×C22⋊(C7⋊C8)2
C2×#28807