Finite Group Databasebeta

← C2×#11727415 #13380852

Subgroups — C2×#11727415 #13380852

Order 1792.

2254 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#1142C212yesmaximal
2C22×#6432C211yesmaximal, characteristic, complemented
3C2×C7×#390C211yesmaximal, characteristic
4C22×#6384C212yesmaximal, complemented
5C2×#89319C211yesmaximal, characteristic, complemented
6#11727415C218yesmaximal, direct product factor
7C23×C7×D4C2211yescharacteristic
8C23×Dic14C2212yes—
9C22×C7×C4⋊C4C2211yescharacteristic
10C22×C7×C22⋊C4—22no—
11C22×C42×C7C2211yescharacteristic
12C22×#1142—28no—
13C22×#1142—28no—
14C22×#1142—28no—
15C22×#1142C2218yes—
16C22×C4⋊Dic7—22no—
17C23×C7⋊C8C2212yes—
18C2×#6432—28no—
19C2×#6432—22no—
20C2×#6432C2214yescomplemented
21C7×#390—22no—
22C7×#390C2214yes—
23C2×#6384—28no—
24C2×#6384C2218yescomplemented
25#89319C2214yescomplemented
26C2×#2013—71nomaximal, Sylow
27C25×C7—21no—
28C22×C7×D4—24no—
29C22×C7×D4—24no—
30C22×C7×D4—21no—
31C22×C7×D4—22no—
32C22×C7×D4D414yes—
33C22×C7×D4C2312yes—
34C23×C4×C7—21no—
35C23×C4×C7D412yes—
36C23×C4×C7C2311yescharacteristic
37C23×Dic7—22no—
38C22×Dic14—24no—
39C22×Dic14—24no—
40C22×Dic14—24no—
41C22×Dic14—24no—
42C22×Dic14—22no—
43C22×Dic14—24no—
44C22×Dic14D418yes—
45C22×Dic14C2314yes—
46C2×C7×C4⋊C4—24no—
47C2×C7×C4⋊C4—21no—
48C2×C7×C4⋊C4C2312yes—
49C2×C7×C22⋊C4—28no—
50C2×C7×C22⋊C4—48no—
51C2×C42×C7—24no—
52C2×C42×C7D412yes—
53C2×C42×C7C2312yes—
54C2×#1142—264no—
55C2×#1142—216no—
56C2×#1142—216no—
57C2×#1142—216no—
58C2×C4⋊Dic7—216no—
59C2×C4⋊Dic7—28no—
60C22×C7⋊C8—24no—
61C22×C7⋊C8—24no—
62C22×C7⋊C8D418yes—
63C22×C7⋊C8C2314yes—
64#6432—216no—
65#6432—216no—
66#6384—216no—
67C23×SD16—72no—
68C22×C4⋊Q8—71no—
69C22×#103—72no—
70C2×#390D711yescharacteristic
71#2013—78no—
72C2×#336—71no—
73C24×C7—24no—
74C24×C7—22no—
75C24×C7—42no—
76C24×C7—42no—
77C24×C7—44no—
78C24×C7—44no—
79C24×C7C2×D411yescharacteristic
80C2×C7×D4—216no—
81C2×C7×D4—24no—
82C2×C7×D4—24no—
83C2×C7×D4—24no—
84C2×C7×D4—22no—
85C2×C7×D4—42no—
86C2×C7×D4—48no—
87C2×C7×D4—42no—
88C2×C7×D4C2×D414yes—
89C22×C4×C7—24no—
90C22×C4×C7—24no—
91C22×C4×C7—22no—
92C22×C4×C7—42no—
93C22×C4×C7—44no—
94C22×C4×C7SD1618yescomplemented
95C22×C4×C7C2×D414yes—
96C22×C4×C7C2×D412yes—
97C22×C4×C7C2×D414yes—
98C22×C4×C7C2×D411yescharacteristic
99C22×C4×C7C2×D414yes—
100C22×C4×C7#4112yes—
101C22×C4×C7C2411yescharacteristic
102C22×Dic7—28no—
103C22×Dic7—24no—
104C22×Dic7—44no—
105C22×Dic7—44no—
106C2×Dic14—216no—
107C2×Dic14—216no—
108C2×Dic14—24no—
109C2×Dic14—24no—
110C2×Dic14—24no—
111C2×Dic14—24no—
112C2×Dic14—28no—
113C2×Dic14—24no—
114C2×Dic14—48no—
115C2×Dic14—48no—
116C2×Dic14—44no—
117C2×Dic14—44no—
118C2×Dic14C2×D418yes—
119C7×C4⋊C4—24no—
120C7×C4⋊C4—42no—
121C7×C22⋊C4—416no—
122C42×C7—24no—
123C42×C7—24no—
124#1142—264no—
125C4⋊Dic7—232no—
126C2×C7⋊C8—216no—
127C2×C7⋊C8—24no—
128C2×C7⋊C8—24no—
129C2×C7⋊C8C2×D418yes—
130C23×Q8—72no—
131C23×D4C2×D711yescharacteristic
132C22×SD16—78no—
133C22×SD16—148no—
134C22×SD16—148no—
135C22×SD16—148no—
136C23×C8—72no—
137C2×C4⋊Q8—74no—
138C2×C4⋊Q8—148no—
139C2×C4⋊Q8—142no—
140C22×C4⋊C4—142no—
141C22×C4⋊C4C2×D711yescharacteristic
142C22×C22⋊C4—22no—
143C22×C42C2×D711yescharacteristic
144C2×#103—78no—
145C2×#103—148no—
146#390—22no—
147#390C2×D714yes—
148#336—74no—
149C23×C7—24no—
150C23×C7—42no—
151C23×C7—42no—
152C23×C7—44no—
153C23×C7—44no—
154C23×C7—48no—
155C23×C7—416no—
156C23×C7—88no—
157C23×C7—84no—
158C23×C7—84no—
159C23×C7#12012yes—
160C23×C7#12112yes—
161C23×C7#12712yes—
162C23×C7C2×SD1614yes—
163C23×C7#13714yes—
164C23×C7C22×D411yescharacteristic
165C7×D4—28no—
166C7×D4—44no—
167C7×D4—48no—
168C2×C4×C7—24no—
169C2×C4×C7—24no—
170C2×C4×C7—216no—
171C2×C4×C7—42no—
172C2×C4×C7—44no—
173C2×C4×C7—84no—
174C2×C4×C7#12018yes—
175C2×C4×C7#12012yes—
176C2×C4×C7#12118yes—
177C2×C4×C7#12112yes—
178C2×C4×C7#12712yes—
179C2×C4×C7C2×SD1618yescomplemented
180C2×C4×C7C22×D411yescharacteristic
181C2×C4×C7C22×D412yes—
182C2×C4×C7C22×D412yes—
183C2×C4×C7C2×#4111yescharacteristic
184C2×Dic7—28no—
185C2×Dic7—44no—
186C2×Dic7—44no—
187C2×Dic7—416no—
188Dic14—28no—
189Dic14—28no—
190Dic14—44no—
191Dic14—44no—
192Dic14—416no—
193C7⋊C8—28no—
194C25—21no—
195C22×Q8—78no—
196C22×Q8—74no—
197C22×Q8—144no—
198C22×Q8—142no—
199C22×Q8—144no—
200C22×Q8—144no—
201C22×Q8—144no—
202C22×Q8—144no—
203C22×D4—22no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—21no—
207C22×D4D2814yes—
208C22×D4C22×D712yes—
209C23×C4—21no—
210C23×C4—142no—
211C23×C4C22⋊D712yes—
212C23×C4C22×D711yescharacteristic
213C2×SD16—1416no—
214C2×SD16—1416no—
215C2×SD16—1464no—
216C2×SD16—1416no—
217C22×C8—78no—
218C22×C8—74no—
219C22×C8—144no—
220C22×C8—144no—
221C4⋊Q8—1416no—
222C4⋊Q8—1416no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—1416no—
226C2×C4⋊C4—148no—
227C2×C4⋊C4C22×D712yes—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—48no—
230C2×C42—24no—
231C2×C42C22⋊D712yes—
232C2×C42C22×D712yes—
233#103—1416no—
234C22×C7—44no—
235C22×C7—48no—
236C22×C7—82no—
237C22×C7—82no—
238C22×C7—84no—
239C22×C7—816no—
240C22×C7#39214yes—
241C22×C7#44818yes—
242C22×C7#46118yes—
243C22×C7#49214yes—
244C22×C7#49714yes—
245C22×C7C2×#12011yescharacteristic
246C22×C7C2×#12111yescharacteristic
247C22×C7C2×#12711yescharacteristic
248C22×C7C22×SD1612yes—
249C22×C7C2×#13712yes—
250C4×C7—28no—
251C4×C7—82no—
252C4×C7#39214yes—
253C4×C7C2×#12012yes—
254C4×C7C2×#12112yes—
255Dic7—48no—
256C24—24no—
257C24—22no—
258C24—44no—
259C24—42no—
260C24—42no—
261C24—44no—
262C24C2×C22⋊D711yescharacteristic
263C2×Q8—78no—
264C2×Q8—148no—
265C2×Q8—144no—
266C2×Q8—1416no—
267C2×Q8—1416no—
268C2×Q8—144no—
269C2×Q8—144no—
270C2×Q8—144no—
271C2×Q8—144no—
272C2×Q8—288no—
273C2×Q8—288no—
274C2×Q8—284no—
275C2×Q8—284no—
276C2×D4—216no—
277C2×D4—24no—
278C2×D4—22no—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—48no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4C2×D2814yes—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—148no—
291C22×C4—144no—
292C22×C4—284no—
293C22×C4—284no—
294C22×C4#114218yescomplemented
295C22×C4C2×D2812yes—
296C22×C4#115712yes—
297C22×C4D4×D714yes—
298C22×C4D4×D714yes—
299C22×C4C2×C22⋊D714yes—
300C22×C4C2×C22⋊D711yescharacteristic
301C22×C4C23×D711yescharacteristic
302SD16—1464no—
303C2×C8—78no—
304C2×C8—1416no—
305C2×C8—144no—
306C2×C8—144no—
307C4⋊C4—24no—
308C4⋊C4—42no—
309C4⋊C4—1432no—
310C22⋊C4—416no—
311C42—24no—
312C42—24no—
313C2×C7—84no—
314C2×C7#201318yescomplemented
315C2×C7C2×#39211yescharacteristic
316C2×C7C2×#44812yes—
317C2×C7C2×#46112yes—
318C2×C7C2×#49211yescharacteristic
319C2×C7C2×#49711yescharacteristic
320C23—24no—
321C23—44no—
322C23—42no—
323C23—42no—
324C23—48no—
325C23—44no—
326C23—416no—
327C23—88no—
328C23—84no—
329C23—84no—
330C23#649312yes—
331C23C2×#114214yes—
332C23#650012yes—
333C23#650312yes—
334C23#651414yes—
335C23C22×C22⋊D711yescharacteristic
336Q8—148no—
337Q8—148no—
338Q8—284no—
339Q8—284no—
340Q8—2816no—
341D4—28no—
342D4—44no—
343D4—48no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—216no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—84no—
350C2×C4—148no—
351C2×C4—284no—
352C2×C4—284no—
353C2×C4—2816no—
354C2×C4#644518yes—
355C2×C4#644718yes—
356C2×C4#649312yes—
357C2×C4C2×#114218yescomplemented
358C2×C4#650012yes—
359C2×C4#650312yes—
360C2×C4C22×D2811yescharacteristic
361C2×C4C2×#115711yescharacteristic
362C2×C4C2×D4×D712yes—
363C2×C4C2×D4×D712yes—
364C8—148no—
365C7C2×#201311yescharacteristic, complemented, Sylow
366C22—48no—
367C22—44no—
368C22—816no—
369C22—84no—
370C22—82no—
371C22—82no—
372C22#8973214yes—
373C22#8979118yes—
374C22#8981218yes—
375C22#8983214yes—
376C22#8984814yes—
377C22C2×#649311yescharacteristic
378C22C22×#114212yes—
379C22C2×#650011yescharacteristic
380C22C2×#650311yescharacteristic
381C22C2×#651412yes—
382C4—28no—
383C4—82no—
384C4—288no—
385C4#8973214yes—
386C4C2×#644512yes—
387C4C2×#644712yes—
388C2—84no—
389C2#1172741518yesdirect product factor
390C2C2×#8973211yescharacteristic
391C2C2×#8979112yes—
392C2C2×#8981212yes—
393C2C2×#8983211yescharacteristic
394C2C2×#8984811yescharacteristic
subgroupG/H
centerC24C2×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#390C2
socleC24×C7C2×D4
prime radicalC23×C7×D4C22
hypercenterC2×#390D7
hypocenterC7C2×#2013
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C24 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C22×#6432C2semidirect
4. C22×#6384C2semidirect
5. C2×#89319C2semidirect
6. #11727415C2direct (N × H)
20. C2×#6432C22semidirect
24. C2×#6384C22semidirect
25. #89319C22semidirect
94. C22×C4×C7SD16semidirect
179. C2×C4×C7C2×SD16semidirect
294. C22×C4#1142semidirect
314. C2×C7#2013semidirect
357. C2×C4C2×#1142semidirect
365. C7C2×#2013semidirect
389. C2#11727415direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#11422
C22×#64322
C2×C7×#3902
C22×#63842
C2×#893192
#117274152
C2×#20137