Finite Group Databasebeta

← C2×#11725956 #13329423

Subgroups — C2×#11725956 #13329423

Order 1792.

1054 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90234C211yesmaximal, characteristic
2C2×#89780C211yesmaximal, characteristic
3C2×#89622C211yesmaximal, characteristic, complemented
4C2×#89471C211yesmaximal, characteristic, complemented
5C2×C7×D4⋊C8C211yesmaximal, characteristic
6C2×#89276C211yesmaximal, characteristic, complemented
7C2×#89251C211yesmaximal, characteristic, complemented
8#11725956C218yesmaximal, direct product factor
9C2×C4×C7×D4C2211yescharacteristic
10C2×D4×Dic7—21no—
11C22×C4⋊Dic7—21no—
12C2×#6487—21no—
13C22×Dic28—22no—
14#90234—22no—
15#90234—22no—
16#90234—22no—
17#90234C2214yes—
18C2×#6453—21no—
19C2×#6441—22no—
20C2×#6432C2211yescharacteristic
21C2×#6432C2211yescharacteristic
22C2×C4×Dic14C2211yescharacteristic
23C2×C7×C4⋊C8C2211yescharacteristic
24C2×C4×C7×C8C2211yescharacteristic
25C2×#6406—21no—
26C2×#6392—21no—
27C2×#6390—21no—
28#89780—22no—
29#89780C2214yes—
30C2×#6382—21no—
31C2×C4⋊(C7⋊C8)C2211yescharacteristic
32#89622—22no—
33#89622C2214yescomplemented
34#89471—22no—
35#89471C2214yescomplemented
36C7×D4⋊C8C2214yes—
37#89276C2214yescomplemented
38#89251C2214yescomplemented
39C2×#1632—71nomaximal, Sylow
40C22×C7×D4D411yescharacteristic
41C23×C4×C7—21no—
42C23×Dic7—22no—
43C22×Dic14—21no—
44C22×Dic14D411yescharacteristic
45C22×Dic14D412yes—
46C22×C7×C8—21no—
47C22×C7×C8—21no—
48C4×C7×D4—21no—
49C4×C7×D4—21no—
50C4×C7×D4D412yes—
51C4×C7×D4C2312yes—
52C2×C7×C4⋊C4D411yescharacteristic
53C2×C7×C22⋊C4—21no—
54C2×C42×C7C2311yescharacteristic
55C2×C22⋊Dic7—22no—
56D4×Dic7—24no—
57D4×Dic7—24no—
58C2×C4⋊Dic7—22no—
59C2×C4⋊Dic7—24no—
60C2×C4⋊Dic7—21no—
61C2×C4⋊Dic7—21no—
62C2×C4⋊Dic7—24no—
63C2×C4⋊Dic7—21no—
64C2×C4⋊Dic7—21no—
65C2×C4⋊Dic7D411yescharacteristic
66#6487—24no—
67#6487—24no—
68C2×Dic7⋊C4—22no—
69C2×Dic7⋊C4—21no—
70C2×C4×Dic7—21no—
71C22×C7⋊C8—21no—
72C2×Dic28—28no—
73C2×Dic28—48no—
74#6453—24no—
75#6453—24no—
76#6441—28no—
77#6441—28no—
78#6432—21no—
79#6432—21no—
80#6432—21no—
81#6432—21no—
82#6432—21no—
83#6432D412yes—
84#6432C2312yes—
85#6432C2312yes—
86C4×Dic14—21no—
87C4×Dic14—21no—
88C4×Dic14D412yes—
89C4×Dic14C2312yes—
90C7×C4⋊C8—21no—
91C7×C4⋊C8C2312yes—
92C4×C7×C8—21no—
93C4×C7×C8C2312yes—
94#6406—24no—
95#6392—24no—
96#6390—24no—
97#6382—24no—
98C4⋊(C7⋊C8)—21no—
99C4⋊(C7⋊C8)C2312yes—
100C2×#554—71no—
101C2×#494—71no—
102C2×#477—71no—
103C2×#476—71no—
104#1632—78no—
105C2×#333—71no—
106C2×Q8⋊C8—71no—
107C2×D4⋊C8D711yescharacteristic
108C24×C7—21no—
109C2×C7×D4—21no—
110C2×C7×D4—22no—
111C2×C7×D4Q1614yescomplemented
112C2×C7×D4C2×D412yes—
113C22×C4×C7—21no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—42no—
117C22×C4×C7—42no—
118C22×C4×C7C2×D411yescharacteristic
119C22×C4×C7C2×D411yescharacteristic
120C22×C4×C7C2×D411yescharacteristic
121C22×Dic7—21no—
122C22×Dic7—21no—
123C22×Dic7—22no—
124C22×Dic7—28no—
125C22×Dic7—41no—
126C22×Dic7—84no—
127C2×Dic14—22no—
128C2×Dic14—22no—
129C2×Dic14—21no—
130C2×Dic14—22no—
131C2×Dic14—22no—
132C2×Dic14—44no—
133C2×Dic14—44no—
134C2×Dic14—41no—
135C2×Dic14C2×D412yes—
136C2×Dic14C2×D414yes—
137C2×C7×C8—24no—
138C2×C7×C8—22no—
139C2×C7×C8—22no—
140C2×C7×C8—42no—
141C7×C4⋊C4—21no—
142C7×C4⋊C4C2×D412yes—
143C7×C22⋊C4—22no—
144C7×C22⋊C4—22no—
145C42×C7C2×D411yescharacteristic
146C42×C7C2×D411yescharacteristic
147C42×C7C2×D411yescharacteristic
148C42×C7C2411yescharacteristic
149C22⋊Dic7—28no—
150C4⋊Dic7—28no—
151C4⋊Dic7—21no—
152C4⋊Dic7—22no—
153C4⋊Dic7—22no—
154C4⋊Dic7—22no—
155C4⋊Dic7—22no—
156C4⋊Dic7—22no—
157C4⋊Dic7—22no—
158C4⋊Dic7—41no—
159C4⋊Dic7—41no—
160C4⋊Dic7C2×D412yes—
161Dic7⋊C4—28no—
162Dic7⋊C4—22no—
163Dic7⋊C4—22no—
164C4×Dic7—22no—
165C4×Dic7—22no—
166C2×C7⋊C8—22no—
167C2×C7⋊C8—42no—
168Dic28—416no—
169C22×Q16—142no—
170#554—74no—
171#554—142no—
172#554—142no—
173#554—142no—
174C2×C4⋊Q8—71no—
175C2×C4⋊Q8—71no—
176C2×#125—141no—
177C2×C22⋊Q8—142no—
178C2×C22⋊Q8—141no—
179C2×C4×Q8—71no—
180C2×C4×D4—141no—
181C2×C4×D4C2×D711yescharacteristic
182C22×C4⋊C4—141no—
183#494—74no—
184#494—142no—
185#477—74no—
186#477—142no—
187#476—74no—
188#476—142no—
189C2×#107—141no—
190C2×#106—141no—
191C2×C4⋊C8—71no—
192C2×C4⋊C8C2×D711yescharacteristic
193C2×#103—141no—
194C2×#102—141no—
195C2×C4×C8C2×D711yescharacteristic
196#333—74no—
197Q8⋊C8—74no—
198D4⋊C8C2×D714yes—
199C23×C7—22no—
200C23×C7—24no—
201C23×C7—42no—
202C23×C7—42no—
203C23×C7#12011yescharacteristic
204C7×D4—22no—
205C7×D4—24no—
206C7×D4C2×Q1614yescomplemented
207C2×C4×C7—22no—
208C2×C4×C7—22no—
209C2×C4×C7—42no—
210C2×C4×C7—42no—
211C2×C4×C7—42no—
212C2×C4×C7—44no—
213C2×C4×C7#12011yescharacteristic
214C2×C4×C7#12011yescharacteristic
215C2×C4×C7#12011yescharacteristic
216C2×C4×C7C2×Q1612yes—
217C2×C4×C7#13512yes—
218C2×C4×C7#13712yes—
219C2×C4×C7#13712yes—
220C2×C4×C7C22×D411yescharacteristic
221C2×C4×C7C22×D411yescharacteristic
222C2×C4×C7C22×D411yescharacteristic
223C2×Dic7—22no—
224C2×Dic7—22no—
225C2×Dic7—24no—
226C2×Dic7—42no—
227C2×Dic7—44no—
228C2×Dic7—42no—
229C2×Dic7—42no—
230C2×Dic7—88no—
231C2×Dic7—82no—
232Dic14—22no—
233Dic14—22no—
234Dic14—42no—
235Dic14—44no—
236Dic14—44no—
237Dic14—82no—
238Dic14—82no—
239C7×C8—24no—
240C7×C8—42no—
241C7⋊C8—42no—
242C22×Q8—71no—
243C22×Q8—72no—
244C22×Q8—141no—
245C22×D4D2811yescharacteristic
246C23×C4—21no—
247C23×C4—142no—
248C2×Q16—148no—
249C2×Q16—288no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—141no—
253C4⋊Q8—72no—
254C4⋊Q8—72no—
255C4⋊Q8—72no—
256C4⋊Q8—141no—
257C4⋊Q8—141no—
258C4⋊Q8—141no—
259C4⋊Q8—141no—
260C4⋊Q8—141no—
261#125—144no—
262#125—144no—
263C22⋊Q8—148no—
264C22⋊Q8—148no—
265C22⋊Q8—144no—
266C22⋊Q8—144no—
267C4×Q8—72no—
268C4×Q8—72no—
269C4×Q8—141no—
270C4×Q8—141no—
271C4×D4—21no—
272C4×D4—21no—
273C4×D4—144no—
274C4×D4—144no—
275C4×D4D2812yes—
276C4×D4C22×D712yes—
277C2×C4⋊C4—71no—
278C2×C4⋊C4—144no—
279C2×C4⋊C4—142no—
280C2×C4⋊C4—144no—
281C2×C4⋊C4—142no—
282C2×C4⋊C4—141no—
283C2×C4⋊C4—141no—
284C2×C4⋊C4—141no—
285C2×C4⋊C4—141no—
286C2×C4⋊C4—141no—
287C2×C4⋊C4D2811yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—142no—
290C2×C42—141no—
291C2×C42C22×D711yescharacteristic
292#107—144no—
293#106—144no—
294C4⋊C8—21no—
295C4⋊C8—72no—
296C4⋊C8—141no—
297C4⋊C8C22×D712yes—
298#103—144no—
299#102—144no—
300C4×C8—21no—
301C4×C8C22×D712yes—
302C22×C7—24no—
303C22×C7—48no—
304C22×C7—42no—
305C22×C7—42no—
306C22×C7#45112yes—
307C22×C7#45212yes—
308C22×C7#45312yes—
309C22×C7C2×#12011yescharacteristic
310C4×C7—21no—
311C4×C7—21no—
312C4×C7—42no—
313C4×C7#45112yes—
314C4×C7#45212yes—
315C4×C7C22×Q1611yescharacteristic
316C4×C7C2×#13511yescharacteristic
317C4×C7C2×#13711yescharacteristic
318C4×C7C2×#13711yescharacteristic
319Dic7—42no—
320Dic7—44no—
321Dic7—42no—
322Dic7—82no—
323C24—21no—
324C2×Q8—72no—
325C2×Q8—74no—
326C2×Q8—141no—
327C2×Q8—142no—
328C2×Q8—142no—
329C2×Q8—142no—
330C2×Q8—142no—
331C2×Q8—284no—
332C2×Q8—284no—
333C2×Q8—281no—
334C2×D4—21no—
335C2×D4—22no—
336C2×D4Dic2814yescomplemented
337C2×D4C2×D2812yes—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—141no—
344C22×C4—141no—
345C22×C4—142no—
346C22×C4—148no—
347C22×C4—281no—
348C22×C4—564no—
349C22×C4C2×D2811yescharacteristic
350C22×C4D4×D711yescharacteristic
351C22×C4D4×D711yescharacteristic
352Q16—2816no—
353C2×C8—24no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—42no—
357C2×C8—142no—
358C2×C8—282no—
359C4⋊C4—21no—
360C4⋊C4—72no—
361C4⋊C4—148no—
362C4⋊C4—148no—
363C4⋊C4—142no—
364C4⋊C4—141no—
365C4⋊C4—142no—
366C4⋊C4—142no—
367C4⋊C4—142no—
368C4⋊C4—142no—
369C4⋊C4—142no—
370C4⋊C4—142no—
371C4⋊C4—142no—
372C4⋊C4—281no—
373C4⋊C4—281no—
374C4⋊C4C2×D2812yes—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—148no—
378C42—142no—
379C42—142no—
380C42C2×D2811yescharacteristic
381C42D4×D711yescharacteristic
382C42D4×D711yescharacteristic
383C42C23×D711yescharacteristic
384C2×C7—44no—
385C2×C7#163214yescomplemented
386C2×C7C2×#45111yescharacteristic
387C2×C7C2×#45211yescharacteristic
388C2×C7C2×#45311yescharacteristic
389C23—22no—
390C23—24no—
391C23—42no—
392C23—42no—
393C23#644511yescharacteristic
394Q8—142no—
395Q8—142no—
396Q8—284no—
397Q8—284no—
398Q8—282no—
399Q8—562no—
400Q8—562no—
401D4—22no—
402D4—24no—
403D4C2×Dic2814yescomplemented
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—142no—
411C2×C4—142no—
412C2×C4—144no—
413C2×C4—282no—
414C2×C4—282no—
415C2×C4—284no—
416C2×C4—282no—
417C2×C4—568no—
418C2×C4—562no—
419C2×C4#644511yescharacteristic
420C2×C4#644511yescharacteristic
421C2×C4#644511yescharacteristic
422C2×C4C2×Dic2812yes—
423C2×C4#647212yes—
424C2×C4#647512yes—
425C2×C4#647812yes—
426C2×C4C22×D2811yescharacteristic
427C2×C4C2×D4×D711yescharacteristic
428C2×C4C2×D4×D711yescharacteristic
429C8—24no—
430C8—42no—
431C8—282no—
432C7C2×#163211yescharacteristic, complemented, Sylow
433C22—24no—
434C22—42no—
435C22—42no—
436C22—48no—
437C22#8951112yes—
438C22#8955512yes—
439C22#8959412yes—
440C22C2×#644511yescharacteristic
441C4—21no—
442C4—21no—
443C4—42no—
444C4—282no—
445C4—284no—
446C4—282no—
447C4—562no—
448C4#8951112yes—
449C4#8955512yes—
450C4C22×Dic2811yescharacteristic
451C4C2×#647211yescharacteristic
452C4C2×#647511yescharacteristic
453C4C2×#647811yescharacteristic
454C2—44no—
455C2#1172595614yesdirect product factor
456C2C2×#8951111yescharacteristic
457C2C2×#8955511yescharacteristic
458C2C2×#8959411yescharacteristic
subgroupG/H
centerC23#6445
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×D4⋊C8C2
socleC23×C7#120
prime radicalC22×C7×D4D4
hypercenterC2×D4⋊C8D7
hypocenterC7C2×#1632
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
3. C2×#89622C2semidirect
4. C2×#89471C2semidirect
6. C2×#89276C2semidirect
7. C2×#89251C2semidirect
8. #11725956C2direct (N × H)
33. #89622C22semidirect
35. #89471C22semidirect
37. #89276C22semidirect
38. #89251C22semidirect
111. C2×C7×D4Q16semidirect
206. C7×D4C2×Q16semidirect
336. C2×D4Dic28semidirect
385. C2×C7#1632semidirect
403. D4C2×Dic28semidirect
432. C7C2×#1632semidirect
455. C2#11725956direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#902342
C2×#897802
C2×#896222
C2×#894712
C2×C7×D4⋊C82
C2×#892762
C2×#892512
#117259562
C2×#16327