Finite Group Databasebeta

← C2×#11731396 #13520867

Subgroups — C2×#11731396 #13520867

Order 1792.

2590 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6487C211yesmaximal, characteristic
2C22×#6489C211yesmaximal, characteristic
3C2×C7×#380C211yesmaximal, characteristic
4C2×#89746C212yesmaximal, complemented
5C2×#89757C212yesmaximal, complemented
6C2×#89725C214yesmaximal
7#11731396C2116yesmaximal, direct product factor
8C2×#89431C212yesmaximal, complemented
9C2×#89433C212yesmaximal, complemented
10C24×C4×C7C2211yescharacteristic
11C23×Dic14C2211yescharacteristic
12C22×C7×C4⋊C4C2212yes—
13C22×C7×C22⋊C4C2212yes—
14C22×C22⋊Dic7C2212yes—
15C22×C22⋊Dic7C2212yes—
16C2×#6489—28no—
17C2×#6489C2218yes—
18C22×C4⋊Dic7C2211yescharacteristic
19C22×C4⋊Dic7C2211yescharacteristic
20C22×C4⋊Dic7C2211yescharacteristic
21C22×C4⋊Dic7C2214yes—
22C2×#6487—28no—
23C2×#6487C2218yes—
24C22×Dic7⋊C4C2212yes—
25C22×C4×Dic7C2212yes—
26C7×#380C2218yes—
27C2×C7×#95C2212yes—
28C2×C7⋊#95C2214yes—
29C2×C7⋊#95C2214yes—
30C2×C7⋊#95C2214yes—
31#89757C22116yescomplemented
32#89746C22116yescomplemented
33#89725C22132yes—
34#89433C22116yescomplemented
35#89431C22116yescomplemented
36C2×#2625—71nomaximal, Sylow
37C25×C7C2311yescharacteristic
38C23×C4×C7—28no—
39C23×C4×C7D418yes—
40C23×C4×C7C2314yes—
41C23×C4×C7C2311yescharacteristic
42C23×C4×C7C2314yes—
43C23×C4×C7C2311yescharacteristic
44C23×Dic7C2312yes—
45C23×Dic7C2312yes—
46C23×Dic7C2312yes—
47C23×Dic7C2312yes—
48C22×Dic14—24no—
49C22×Dic14—28no—
50C22×Dic14C2314yes—
51C2×C7×C4⋊C4—28no—
52C2×C7×C4⋊C4C2318yes—
53C2×C7×C22⋊C4—44no—
54C2×C7×C22⋊C4C2318yes—
55C2×C22⋊Dic7—28no—
56C2×C22⋊Dic7—28no—
57C2×C22⋊Dic7C2318yes—
58C2×C22⋊Dic7C2318yes—
59#6489—232no—
60C2×C4⋊Dic7—24no—
61C2×C4⋊Dic7—24no—
62C2×C4⋊Dic7—24no—
63C2×C4⋊Dic7—48no—
64C2×C4⋊Dic7C2314yes—
65C2×C4⋊Dic7C2314yes—
66C2×C4⋊Dic7C2314yes—
67C2×C4⋊Dic7C23116yes—
68#6487—232no—
69C2×Dic7⋊C4—28no—
70C2×Dic7⋊C4C2318yes—
71C2×C4×Dic7—28no—
72C2×C4×Dic7C2318yes—
73C7×#95C2318yes—
74C7⋊#95C23116yes—
75C7⋊#95C23116yes—
76C7⋊#95C23116yes—
77C22×C22⋊Q8—71no—
78C22×#117—71no—
79#2625—716no—
80C2×#400—72no—
81C2×#398—72no—
82C2×#397—74no—
83C2×#391—72no—
84C2×#389—72no—
85C2×#380D711yescharacteristic
86C24×C7—24no—
87C24×C7—22no—
88C24×C7—24no—
89C24×C7—22no—
90C24×C7C2×D414yes—
91C24×C7C2412yes—
92C24×C7C2411yescharacteristic
93C22×C4×C7—24no—
94C22×C4×C7—232no—
95C22×C4×C7—216no—
96C22×C4×C7—28no—
97C22×C4×C7—28no—
98C22×C4×C7—28no—
99C22×C4×C7C2×D414yes—
100C22×C4×C7C2×D4116yes—
101C22×C4×C7C2×D414yes—
102C22×C4×C7#4118yes—
103C22×C4×C7C2418yes—
104C22×C4×C7C2412yes—
105C22×C4×C7C2412yes—
106C22×Dic7—24no—
107C22×Dic7—24no—
108C22×Dic7—24no—
109C22×Dic7—24no—
110C22×Dic7—28no—
111C22×Dic7—24no—
112C22×Dic7—24no—
113C22×Dic7—28no—
114C22×Dic7—24no—
115C22×Dic7—24no—
116C22×Dic7C2414yes—
117C22×Dic7C2414yes—
118C22×Dic7C2414yes—
119C22×Dic7C2414yes—
120C2×Dic14—28no—
121C2×Dic14—216no—
122C2×Dic14—416no—
123C7×C4⋊C4—216no—
124C7×C22⋊C4—48no—
125C22⋊Dic7—216no—
126C22⋊Dic7—216no—
127C4⋊Dic7—28no—
128C4⋊Dic7—28no—
129C4⋊Dic7—28no—
130C4⋊Dic7—416no—
131Dic7⋊C4—216no—
132C4×Dic7—216no—
133C23×Q8—71no—
134C24×C4C2×D711yescharacteristic
135C2×C22⋊Q8—78no—
136C2×C22⋊Q8—148no—
137C2×#117—78no—
138C2×#117—148no—
139C22×C4⋊C4—71no—
140C22×C4⋊C4—72no—
141C22×C4⋊C4—71no—
142C22×C4⋊C4—71no—
143C22×C4⋊C4—74no—
144C22×C4⋊C4C2×D712yes—
145C22×C22⋊C4—72no—
146C22×C22⋊C4—72no—
147C22×C22⋊C4C2×D712yes—
148C22×C42—72no—
149#400—716no—
150#398—716no—
151#397—732no—
152#391—716no—
153#389—716no—
154#380C2×D718yes—
155C2×#95—74no—
156C2×#95—74no—
157C2×#95—74no—
158C2×#95C2×D712yes—
159C23×C7—28no—
160C23×C7—28no—
161C23×C7—28no—
162C23×C7—24no—
163C23×C7—22no—
164C23×C7—22no—
165C23×C7—24no—
166C23×C7—48no—
167C23×C7—48no—
168C23×C7C22×D414yes—
169C23×C7C22×D412yes—
170C23×C7C2×#4114yes—
171C23×C7H(5,𝔽2)12yes—
172C23×C72−1+412yes—
173C23×C72−1+414yes—
174C23×C7C2511yescharacteristic
175C2×C4×C7—28no—
176C2×C4×C7—24no—
177C2×C4×C7—232no—
178C2×C4×C7—28no—
179C2×C4×C7—28no—
180C2×C4×C7—28no—
181C2×C4×C7—416no—
182C2×C4×C7#123116yes—
183C2×C4×C7C22×D414yes—
184C2×C4×C7C22×D414yes—
185C2×C4×C7C2×#4118yes—
186C2×Dic7—24no—
187C2×Dic7—24no—
188C2×Dic7—24no—
189C2×Dic7—24no—
190C2×Dic7—28no—
191C2×Dic7—24no—
192C2×Dic7—24no—
193C2×Dic7—24no—
194C2×Dic7—24no—
195C2×Dic7—28no—
196C2×Dic7—48no—
197C2×Dic7—48no—
198C2×Dic7—48no—
199C2×Dic7—48no—
200Dic14—416no—
201C25C22×D711yescharacteristic
202C22×Q8—74no—
203C22×Q8—144no—
204C22×Q8—148no—
205C23×C4—28no—
206C23×C4—72no—
207C23×C4—72no—
208C23×C4—72no—
209C23×C4—72no—
210C23×C4D2818yes—
211C23×C4C22×D714yes—
212C23×C4C22×D714yes—
213C23×C4C22×D711yescharacteristic
214C23×C4C22×D711yescharacteristic
215C22⋊Q8—1432no—
216#117—1432no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—74no—
219C2×C4⋊C4—74no—
220C2×C4⋊C4—78no—
221C2×C4⋊C4—74no—
222C2×C4⋊C4—716no—
223C2×C4⋊C4—144no—
224C2×C4⋊C4—144no—
225C2×C4⋊C4—148no—
226C2×C4⋊C4—144no—
227C2×C4⋊C4—288no—
228C2×C4⋊C4C22×D718yes—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—78no—
231C2×C22⋊C4—78no—
232C2×C22⋊C4—148no—
233C2×C22⋊C4—148no—
234C2×C22⋊C4C22×D718yes—
235C2×C42—78no—
236C2×C42—148no—
237#95—716no—
238#95—716no—
239#95—716no—
240#95C22×D718yes—
241C22×C7—24no—
242C22×C7—24no—
243C22×C7—24no—
244C22×C7—416no—
245C22×C7—44no—
246C22×C7—44no—
247C22×C7C2×#12314yes—
248C22×C7#53614yes—
249C22×C7#53714yes—
250C22×C7#54118yes—
251C22×C7#54218yes—
252C22×C7C23×D411yescharacteristic
253C22×C7C22×#4112yes—
254C22×C7C2×H(5,𝔽2)11yescharacteristic
255C22×C7C2×2−1+411yescharacteristic
256C22×C7C2×2−1+412yes—
257C4×C7—24no—
258C4×C7—48no—
259C4×C7C2×#12318yes—
260Dic7—44no—
261Dic7—44no—
262Dic7—44no—
263Dic7—44no—
264C24—24no—
265C24—22no—
266C24—22no—
267C24—24no—
268C24C2×D2814yes—
269C24C23×D712yes—
270C24C23×D711yescharacteristic
271C2×Q8—148no—
272C2×Q8—1416no—
273C2×Q8—2816no—
274C22×C4—24no—
275C22×C4—232no—
276C22×C4—216no—
277C22×C4—28no—
278C22×C4—28no—
279C22×C4—28no—
280C22×C4—74no—
281C22×C4—74no—
282C22×C4—74no—
283C22×C4—74no—
284C22×C4—144no—
285C22×C4—144no—
286C22×C4—144no—
287C22×C4—144no—
288C22×C4—148no—
289C22×C4—144no—
290C22×C4—144no—
291C22×C4—144no—
292C22×C4—144no—
293C22×C4—148no—
294C22×C4C2×D2814yes—
295C22×C4C2×D28116yes—
296C22×C4C2×D2814yes—
297C22×C4#115918yes—
298C22×C4C23×D718yes—
299C22×C4C23×D712yes—
300C22×C4C23×D712yes—
301C4⋊C4—216no—
302C4⋊C4—1416no—
303C4⋊C4—148no—
304C4⋊C4—148no—
305C4⋊C4—148no—
306C4⋊C4—2816no—
307C22⋊C4—48no—
308C22⋊C4—1416no—
309C22⋊C4—1416no—
310C42—1416no—
311C2×C7—44no—
312C2×C7#262518yescomplemented
313C2×C7C22×#12311yescharacteristic
314C2×C7C2×#53611yescharacteristic
315C2×C7C2×#53711yescharacteristic
316C2×C7C2×#54112yes—
317C2×C7C2×#54212yes—
318C23—28no—
319C23—28no—
320C23—28no—
321C23—24no—
322C23—22no—
323C23—22no—
324C23—24no—
325C23—48no—
326C23—48no—
327C23C22×D2814yes—
328C23C22×D2812yes—
329C23C2×#115914yes—
330C23#654812yes—
331C23#655112yes—
332C23#655414yes—
333C23C24×D711yescharacteristic
334Q8—2816no—
335C2×C4—28no—
336C2×C4—24no—
337C2×C4—232no—
338C2×C4—28no—
339C2×C4—28no—
340C2×C4—28no—
341C2×C4—416no—
342C2×C4—144no—
343C2×C4—144no—
344C2×C4—144no—
345C2×C4—144no—
346C2×C4—148no—
347C2×C4—144no—
348C2×C4—144no—
349C2×C4—144no—
350C2×C4—144no—
351C2×C4—148no—
352C2×C4—288no—
353C2×C4—288no—
354C2×C4—288no—
355C2×C4—288no—
356C2×C4#6449116yes—
357C2×C4C22×D2814yes—
358C2×C4C22×D2814yes—
359C2×C4C2×#115918yes—
360C7C2×#262511yescharacteristic, complemented, Sylow
361C22—24no—
362C22—24no—
363C22—24no—
364C22—44no—
365C22—44no—
366C22—416no—
367C22C2×#644914yes—
368C22#9020114yes—
369C22#9021714yes—
370C22#9029418yes—
371C22#9031618yes—
372C22C23×D2811yescharacteristic
373C22C22×#115912yes—
374C22C2×#654811yescharacteristic
375C22C2×#655111yescharacteristic
376C22C2×#655412yes—
377C4—24no—
378C4—48no—
379C4—284no—
380C4—284no—
381C4—284no—
382C4—284no—
383C4C2×#644918yes—
384C2—44no—
385C2#1173139618yesdirect product factor
386C2C22×#644911yescharacteristic
387C2C2×#9020111yescharacteristic
388C2C2×#9021711yescharacteristic
389C2C2×#9029412yes—
390C2C2×#9031612yes—
subgroupG/H
centerC24C23×D7
commutator, cosocleC23×C7C25
FrattiniC23C24×D7
FittingC2×C7×#380C2
socleC24×C7C24
prime radicalC25×C7C23
hypercenterC2×#380D7
hypocenterC7C2×#2625
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
4. C2×#89746C2semidirect
5. C2×#89757C2semidirect
7. #11731396C2direct (N × H)
8. C2×#89431C2semidirect
9. C2×#89433C2semidirect
31. #89757C22semidirect
32. #89746C22semidirect
34. #89433C22semidirect
35. #89431C22semidirect
312. C2×C7#2625semidirect
360. C7C2×#2625semidirect
385. C2#11731396direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#64872
C22×#64892
C2×C7×#3802
C2×#897462
C2×#897572
C2×#897252
#117313962
C2×#894312
C2×#894332
C2×#26257