Finite Group Databasebeta

← C2×#11737080 #14234295

Subgroups — C2×#11737080 #14234295

Order 1792.

2006 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90261C211yesmaximal, characteristic, complemented
2C2×D4×Dic14C211yesmaximal, characteristic
3C2×C4×C7×SD16C211yesmaximal, characteristic
4C2×#89936C212yesmaximal
5C2×#89792C211yesmaximal, characteristic, complemented
6C2×#89779C211yesmaximal, characteristic
7#11737080C2116yesmaximal, direct product factor
8C2×#89628C212yesmaximal, complemented
9C2×#89566C212yesmaximal, complemented
10C2×#89538C212yesmaximal
11C2×#89451C211yesmaximal, characteristic, complemented
12C2×#89454C211yesmaximal, characteristic, complemented
13C23×Dic14—21no—
14C22×C7×SD16C2211yescharacteristic
15C2×C4×C7×Q8C2211yescharacteristic
16C2×C4×C7×D4C2211yescharacteristic
17C2×Q8×Dic7C2212yes—
18C2×D4×Dic7C2212yes—
19C2×#6487—21no—
20#90261—24no—
21#90261C2218yescomplemented
22D4×Dic14—24no—
23D4×Dic14C2218yes—
24C2×#6453—22no—
25C2×#6453C2212yes—
26C2×#6451C2212yes—
27C2×#6441—22no—
28C2×#6432—21no—
29C2×#6432C2211yescharacteristic
30C2×#6432C2211yescharacteristic
31C2×C4×Dic14—21no—
32C2×C4×Dic14C2211yescharacteristic
33C4×C7×SD16C2218yes—
34C2×C7×#106C2211yescharacteristic
35C2×C7×#103C2211yescharacteristic
36C2×C7×#102C2211yescharacteristic
37C2×C4×C7×C8C2211yescharacteristic
38C2×#6409C2212yes—
39C2×#6406C2212yes—
40#89936C22116yes—
41C2×#6392C2212yes—
42C2×#6392C2212yes—
43C2×#6389C2212yes—
44C2×Dic7⋊C8C2212yes—
45#89792C2218yescomplemented
46#89779C2218yes—
47C2×#6382C2212yes—
48C2×#6381C2212yes—
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50#89628C22116yescomplemented
51#89566C22116yescomplemented
52#89538C22116yes—
53#89454C2218yescomplemented
54#89451C2218yescomplemented
55C2×#3386—71nomaximal, Sylow
56C22×C7×Q8C2311yescharacteristic
57C22×C7×D4C2311yescharacteristic
58C23×C4×C7—21no—
59C23×Dic7—22no—
60C22×Dic14—21no—
61C22×Dic14—21no—
62C22×Dic14—22no—
63C22×Dic14—28no—
64C22×Dic14—48no—
65C22×Dic14D411yescharacteristic
66C2×C7×SD16Q818yescomplemented
67C2×C7×SD16C2314yes—
68C22×C7×C8C2311yescharacteristic
69C22×C7×C8C2311yescharacteristic
70C4×C7×Q8—22no—
71C4×C7×Q8C2314yes—
72C4×C7×D4—22no—
73C4×C7×D4C2314yes—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C22⋊C4—21no—
78C2×C42×C7—21no—
79C2×C42×C7C2311yescharacteristic
80C2×C22⋊Dic7—22no—
81Q8×Dic7—24no—
82Q8×Dic7C2318yes—
83D4×Dic7—24no—
84D4×Dic7C2318yes—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7—22no—
88C2×C4⋊Dic7—22no—
89C2×C4⋊Dic7D411yescharacteristic
90C2×C4⋊Dic7C2312yes—
91C2×C4⋊Dic7C2312yes—
92#6487—28no—
93C2×Dic7⋊C4—22no—
94C2×Dic7⋊C4—22no—
95C2×Dic7⋊C4D412yes—
96C2×C4×Dic7—22no—
97C2×C4×Dic7C2312yes—
98C22×C7⋊C8C2312yes—
99#6453—216no—
100#6453—24no—
101#6453C2318yes—
102#6451—24no—
103#6451C2318yes—
104#6441—216no—
105#6432—28no—
106#6432—22no—
107#6432—22no—
108#6432C2314yes—
109#6432C2314yes—
110C4×Dic14—28no—
111C4×Dic14D414yes—
112C4×Dic14C2314yes—
113C7×#106C2314yes—
114C7×#103C2314yes—
115C7×#102C2314yes—
116C4×C7×C8C2314yes—
117#6409C2318yes—
118#6406C2318yes—
119#6392C2318yes—
120#6392C2318yes—
121#6389C2318yes—
122Dic7⋊C8C2318yes—
123#6382C2318yes—
124#6381C2318yes—
125C4⋊(C7⋊C8)C2314yes—
126C2×#557—71no—
127C2×D4×Q8—71no—
128#3386—716no—
129C2×#501—72no—
130C2×#500—71no—
131C2×#477—72no—
132C2×#475—71no—
133C2×#478—72no—
134C2×#474—71no—
135C2×#446—71no—
136C2×#440—72no—
137C2×C4×SD16D711yescharacteristic
138C24×C7—21no—
139C2×C7×Q8—21no—
140C2×C7×Q8—22no—
141C2×C7×Q8C2×Q814yescomplemented
142C2×C7×Q8C2412yes—
143C2×C7×D4—21no—
144C2×C7×D4—22no—
145C2×C7×D4C2×Q814yes—
146C2×C7×D4C2412yes—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—24no—
151C22×C4×C7—44no—
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2411yescharacteristic
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—28no—
158C22×Dic7—48no—
159C22×Dic7C2×D412yes—
160C22×Dic7C2×D412yes—
161C2×Dic14—24no—
162C2×Dic14—24no—
163C2×Dic14—24no—
164C2×Dic14—28no—
165C2×Dic14—432no—
166C2×Dic14—48no—
167C2×Dic14—44no—
168C2×Dic14—44no—
169C2×Dic14C2×D414yes—
170C7×SD16C2×Q8116yescomplemented
171C2×C7×C8—22no—
172C2×C7×C8C2×Q814yescomplemented
173C2×C7×C8C2412yes—
174C2×C7×C8C2412yes—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—24no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4C2412yes—
179C7×C4⋊C4C2412yes—
180C7×C22⋊C4—24no—
181C42×C7—24no—
182C42×C7C2×D412yes—
183C42×C7C2412yes—
184C22⋊Dic7—28no—
185C4⋊Dic7—22no—
186C4⋊Dic7—22no—
187C4⋊Dic7—24no—
188C4⋊Dic7—24no—
189C4⋊Dic7—28no—
190C4⋊Dic7—28no—
191C4⋊Dic7C2×D414yes—
192C4⋊Dic7C2414yes—
193C4⋊Dic7C2414yes—
194Dic7⋊C4—28no—
195Dic7⋊C4—28no—
196Dic7⋊C4C2×D418yes—
197C4×Dic7—28no—
198C4×Dic7C2×D414yes—
199C4×Dic7C2414yes—
200C2×C7⋊C8—24no—
201C2×C7⋊C8C2414yes—
202C23×Q8—141no—
203C22×SD16C2×D711yescharacteristic
204#557—78no—
205#557—144no—
206D4×Q8—78no—
207D4×Q8—144no—
208C2×C4⋊Q8—72no—
209C2×C4⋊Q8—71no—
210C2×C4⋊Q8—71no—
211C2×C4⋊Q8—141no—
212C2×#125—72no—
213C2×#125—142no—
214C2×C22⋊Q8—142no—
215C2×C22⋊Q8—141no—
216C2×C4×Q8—71no—
217C2×C4×Q8—72no—
218C2×C4×Q8—141no—
219C2×C4×Q8C2×D711yescharacteristic
220C2×C4×D4—72no—
221C2×C4×D4C2×D711yescharacteristic
222#501—716no—
223#500—78no—
224#478—716no—
225#477—716no—
226#475—78no—
227#474—78no—
228#446—78no—
229#440—716no—
230C4×SD16C2×D718yes—
231C2×#107—72no—
232C2×#107—72no—
233C2×#107—72no—
234C2×#106—72no—
235C2×#106C2×D711yescharacteristic
236C2×C4⋊C8—72no—
237C2×C4⋊C8—71no—
238C2×#103—72no—
239C2×#103C2×D711yescharacteristic
240C2×#102—72no—
241C2×#102C2×D711yescharacteristic
242C2×#97—72no—
243C2×C4×C8C2×D711yescharacteristic
244C23×C7—24no—
245C23×C7—22no—
246C23×C7—42no—
247C23×C7—42no—
248C23×C7C22×D411yescharacteristic
249C7×Q8—24no—
250C7×Q8—22no—
251C7×Q8C22×Q814yescomplemented
252C7×D4—24no—
253C7×D4—22no—
254C7×D4C22×Q814yes—
255C2×C4×C7—24no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—48no—
263C2×C4×C7AGL(1,ℤ/8)14yes—
264C2×C4×C7C22×D412yes—
265C2×C4×C7C22×D411yescharacteristic
266C2×C4×C7C22×D412yes—
267C2×C4×C7C22×Q812yes—
268C2×C4×C72−1+412yes—
269C2×C4×C7C2511yescharacteristic
270C2×Dic7—24no—
271C2×Dic7—24no—
272C2×Dic7—24no—
273C2×Dic7—24no—
274C2×Dic7—416no—
275C2×Dic7—44no—
276C2×Dic7—44no—
277C2×Dic7C22×D414yes—
278C2×Dic7C22×D414yes—
279Dic14—28no—
280Dic14—416no—
281Dic14—48no—
282Dic14—48no—
283C7×C8—22no—
284C7×C8C22×Q814yescomplemented
285C7⋊C8—24no—
286C22×Q8—71no—
287C22×Q8—142no—
288C22×Q8—148no—
289C22×Q8—141no—
290C22×Q8—141no—
291C22×Q8—288no—
292C22×Q8C22×D711yescharacteristic
293C22×D4C22×D711yescharacteristic
294C23×C4—21no—
295C23×C4—142no—
296C2×SD16Dic1418yescomplemented
297C2×SD16C22×D714yes—
298C22×C8—72no—
299C22×C8C22×D711yescharacteristic
300C22×C8C22×D711yescharacteristic
301C4⋊Q8—78no—
302C4⋊Q8—74no—
303C4⋊Q8—74no—
304C4⋊Q8—148no—
305C4⋊Q8—144no—
306C4⋊Q8—142no—
307C4⋊Q8—142no—
308#125—78no—
309#125—144no—
310#125—1416no—
311C22⋊Q8—1416no—
312C22⋊Q8—148no—
313C4×Q8—22no—
314C4×Q8—74no—
315C4×Q8—74no—
316C4×Q8—78no—
317C4×Q8—148no—
318C4×Q8—144no—
319C4×Q8C22×D714yes—
320C4×D4—22no—
321C4×D4—78no—
322C4×D4—144no—
323C4×D4C22×D714yes—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—71no—
326C2×C4⋊C4—72no—
327C2×C4⋊C4—72no—
328C2×C4⋊C4—72no—
329C2×C4⋊C4—142no—
330C2×C4⋊C4—142no—
331C2×C4⋊C4—142no—
332C2×C4⋊C4—142no—
333C2×C4⋊C4—141no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4C22×D711yescharacteristic
336C2×C4⋊C4C22×D711yescharacteristic
337C2×C22⋊C4—21no—
338C2×C22⋊C4—142no—
339C2×C42—21no—
340C2×C42—72no—
341C2×C42—142no—
342C2×C42C22×D711yescharacteristic
343#107—78no—
344#107—78no—
345#107—78no—
346#106—78no—
347#106C22×D714yes—
348C4⋊C8—78no—
349C4⋊C8—74no—
350#103—78no—
351#103C22×D714yes—
352#102—78no—
353#102C22×D714yes—
354#97—78no—
355C4×C8C22×D714yes—
356C22×C7—24no—
357C22×C7—48no—
358C22×C7—42no—
359C22×C7—42no—
360C22×C7D4×Q812yes—
361C22×C7C2×AGL(1,ℤ/8)12yes—
362C22×C7#57812yes—
363C22×C7C23×D411yescharacteristic
364C4×C7—24no—
365C4×C7—22no—
366C4×C7—42no—
367C4×C7—42no—
368C4×C7D4×Q812yes—
369C4×C7C2×AGL(1,ℤ/8)14yes—
370C4×C7C23×Q811yescharacteristic
371C4×C7C2×2−1+411yescharacteristic
372Dic7—24no—
373Dic7—24no—
374Dic7—44no—
375Dic7—44no—
376C24—21no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—74no—
380C2×Q8—148no—
381C2×Q8—144no—
382C2×Q8—144no—
383C2×Q8—144no—
384C2×Q8—288no—
385C2×Q8—2832no—
386C2×Q8—284no—
387C2×Q8—284no—
388C2×Q8C2×Dic1414yescomplemented
389C2×Q8C23×D712yes—
390C2×D4—21no—
391C2×D4—22no—
392C2×D4C2×Dic1414yes—
393C2×D4C23×D712yes—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—44no—
399C22×C4—72no—
400C22×C4—72no—
401C22×C4—148no—
402C22×C4—142no—
403C22×C4—142no—
404C22×C4—288no—
405C22×C4D4×D711yescharacteristic
406C22×C4D4×D711yescharacteristic
407C22×C4C23×D711yescharacteristic
408SD16C2×Dic14116yescomplemented
409C2×C8—22no—
410C2×C8—74no—
411C2×C8—144no—
412C2×C8C2×Dic1414yescomplemented
413C2×C8C23×D712yes—
414C2×C8C23×D712yes—
415C4⋊C4—24no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—78no—
419C4⋊C4—74no—
420C4⋊C4—74no—
421C4⋊C4—74no—
422C4⋊C4—148no—
423C4⋊C4—148no—
424C4⋊C4—148no—
425C4⋊C4—148no—
426C4⋊C4—142no—
427C4⋊C4—142no—
428C4⋊C4—144no—
429C4⋊C4—144no—
430C4⋊C4C23×D712yes—
431C4⋊C4C23×D712yes—
432C22⋊C4—24no—
433C22⋊C4—148no—
434C42—24no—
435C42—74no—
436C42—74no—
437C42—148no—
438C42D4×D712yes—
439C42C23×D712yes—
440C2×C7—44no—
441C2×C7#338614yescomplemented
442C2×C7C2×D4×Q811yescharacteristic
443C2×C7C22×AGL(1,ℤ/8)11yescharacteristic
444C2×C7C2×#57811yescharacteristic
445C23—24no—
446C23—22no—
447C23—42no—
448C23—42no—
449C23C2×D4×D711yescharacteristic
450Q8—24no—
451Q8—22no—
452Q8—148no—
453Q8—2816no—
454Q8—288no—
455Q8—288no—
456Q8C22×Dic1414yescomplemented
457D4—24no—
458D4—22no—
459D4C22×Dic1414yes—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—48no—
468C2×C4—74no—
469C2×C4—74no—
470C2×C4—144no—
471C2×C4—144no—
472C2×C4—144no—
473C2×C4—144no—
474C2×C4—2816no—
475C2×C4—284no—
476C2×C4—284no—
477C2×C4#647714yes—
478C2×C4C22×Dic1412yes—
479C2×C4C2×D4×D712yes—
480C2×C4C2×D4×D712yes—
481C2×C4C2×D4×D711yescharacteristic
482C2×C4#655412yes—
483C2×C4C24×D711yescharacteristic
484C8—22no—
485C8—144no—
486C8C22×Dic1414yescomplemented
487C7C2×#338611yescharacteristic, complemented, Sylow
488C22—24no—
489C22—42no—
490C22—48no—
491C22—42no—
492C22D4×Dic1412yes—
493C22C2×#647712yes—
494C22#9046212yes—
495C22C22×D4×D711yescharacteristic
496C4—24no—
497C4—22no—
498C4—42no—
499C4—42no—
500C4—144no—
501C4—144no—
502C4—284no—
503C4—284no—
504C4D4×Dic1412yes—
505C4C2×#647714yes—
506C4C23×Dic1411yescharacteristic
507C4C2×#655411yescharacteristic
508C2—44no—
509C2#1173708014yesdirect product factor
510C2C2×D4×Dic1411yescharacteristic
511C2C22×#647711yescharacteristic
512C2C2×#9046211yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C4×C7×SD16C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenterC2×C4×SD16D7
hypocenterC7C2×#3386
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#90261C2semidirect
5. C2×#89792C2semidirect
7. #11737080C2direct (N × H)
8. C2×#89628C2semidirect
9. C2×#89566C2semidirect
11. C2×#89451C2semidirect
12. C2×#89454C2semidirect
21. #90261C22semidirect
45. #89792C22semidirect
50. #89628C22semidirect
51. #89566C22semidirect
53. #89454C22semidirect
54. #89451C22semidirect
66. C2×C7×SD16Q8semidirect
141. C2×C7×Q8C2×Q8semidirect
170. C7×SD16C2×Q8semidirect
172. C2×C7×C8C2×Q8semidirect
251. C7×Q8C22×Q8semidirect
284. C7×C8C22×Q8semidirect
296. C2×SD16Dic14semidirect
388. C2×Q8C2×Dic14semidirect
408. SD16C2×Dic14semidirect
412. C2×C8C2×Dic14semidirect
441. C2×C7#3386semidirect
456. Q8C22×Dic14semidirect
486. C8C22×Dic14semidirect
487. C7C2×#3386semidirect
509. C2#11737080direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#902612
C2×D4×Dic142
C2×C4×C7×SD162
C2×#899362
C2×#897922
C2×#897792
#117370802
C2×#896282
C2×#895662
C2×#895382
C2×#894512
C2×#894542
C2×#33867