Finite Group Databasebeta

← C2×#11741068 #14232253

Subgroups — C2×#11741068 #14232253

Order 1792.

1942 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#554C211yesmaximal, characteristic
2C2×#90234C211yesmaximal, characteristic
3#11741068C2116yesmaximal, direct product factor
4C2×#89863C211yesmaximal, characteristic, complemented
5C2×#89850C211yesmaximal, characteristic, complemented
6C2×#89823C212yesmaximal, complemented
7C2×#89816C212yesmaximal, complemented
8C2×#89793C211yesmaximal, characteristic, complemented
9C2×C4×#1144C211yesmaximal, characteristic, complemented
10C2×D4×C7⋊C8C211yesmaximal, characteristic
11C2×#89742C212yesmaximal, complemented
12C22×#6381C212yesmaximal, complemented
13C7×#554—24no—
14C7×#554C2218yes—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#125—21no—
17C2×C7×C22⋊Q8—22no—
18C2×C7×C22⋊Q8C2212yes—
19C2×C4×C7×Q8C2211yescharacteristic
20C2×C4×C7×D4—21no—
21C2×C4×C7×D4C2211yescharacteristic
22C22×C7×C4⋊C4C2212yes—
23C22×#1144C2211yescharacteristic
24C2×D4×Dic7—21no—
25C22×C4⋊Dic7C2212yes—
26C2×#6487C2212yes—
27C23×C7⋊C8C2212yes—
28#90234—24no—
29#90234C2218yes—
30C2×#6453—21no—
31C2×#6441—22no—
32C2×#6432C2211yescharacteristic
33C2×C4×Dic14C2211yescharacteristic
34C2×#6409C2211yescharacteristic
35C2×#6409C2212yes—
36C2×C22⋊(C7⋊C8)C2212yes—
37#89863C2218yescomplemented
38#89850C2218yescomplemented
39#89823C22116yescomplemented
40#89816C22116yescomplemented
41C4×#1144C2218yescomplemented
42#89793C2218yescomplemented
43D4×C7⋊C8C2218yes—
44#89742C22116yescomplemented
45C2×#6384C2212yes—
46C2×#6384C2211yescharacteristic
47C2×#6381C2212yes—
48C2×#6381C2212yes—
49C2×#6381C2212yes—
50C2×#6381C2211yescharacteristic
51C2×#6381C2212yes—
52C2×#6381C22116yescomplemented
53C2×C4⋊(C7⋊C8)C2211yescharacteristic
54C2×C4×C7⋊C8C2211yescharacteristic
55C2×#3338—71nomaximal, Sylow
56C22×C7×Q8—21no—
57C22×C7×Q8C2311yescharacteristic
58C22×C7×D4D411yescharacteristic
59C23×C4×C7—22no—
60C23×C4×C7C2312yes—
61C23×Dic7—22no—
62C22×Dic14—21no—
63C22×Dic14C2311yescharacteristic
64C7×C4⋊Q8—22no—
65C7×C4⋊Q8C2314yes—
66C7×#125—28no—
67C7×C22⋊Q8—24no—
68C7×C22⋊Q8—216no—
69C7×C22⋊Q8C2318yes—
70C4×C7×Q8—22no—
71C4×C7×Q8C2314yes—
72C4×C7×D4—28no—
73C4×C7×D4D414yes—
74C4×C7×D4C2314yes—
75C2×C7×C4⋊C4—22no—
76C2×C7×C4⋊C4—24no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4—22no—
80C2×C7×C4⋊C4D411yescharacteristic
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C7×C4⋊C4C2312yes—
83C2×C7×C4⋊C4C2318yes—
84C2×C7×C22⋊C4—22no—
85C2×C7×C22⋊C4D412yes—
86C2×C42×C7—21no—
87C2×C42×C7C2311yescharacteristic
88C2×C22⋊Dic7—22no—
89C2×#1144—24no—
90C2×#1144C2314yes—
91D4×Dic7—28no—
92C2×C4⋊Dic7—21no—
93C2×C4⋊Dic7—22no—
94C2×C4⋊Dic7—24no—
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2312yes—
97C2×C4⋊Dic7C2318yes—
98#6487—24no—
99#6487C2318yes—
100C2×Dic7⋊C4—21no—
101C2×Dic7⋊C4—22no—
102C2×C4×Dic7—21no—
103C22×C7⋊C8—28no—
104C22×C7⋊C8C2311yescharacteristic
105C22×C7⋊C8C2312yes—
106C22×C7⋊C8C2318yes—
107C22×C7⋊C8C2311yescharacteristic
108#6453—28no—
109#6441—216no—
110#6432—22no—
111#6432C2314yes—
112C4×Dic14—22no—
113C4×Dic14C2314yes—
114#6409C2318yes—
115#6409C2314yes—
116C22⋊(C7⋊C8)C2318yes—
117#6384C2318yes—
118#6384C2314yes—
119#6381C2318yes—
120#6381C2314yes—
121#6381C2318yes—
122#6381C2318yes—
123#6381C2318yes—
124C4⋊(C7⋊C8)C2314yes—
125C4×C7⋊C8C2314yes—
126C2×#554—71no—
127C2×#554D711yescharacteristic
128#3338—716no—
129C2×#500—71no—
130C2×#484—72no—
131C2×#484—72no—
132C2×#477—71no—
133C2×#477—71no—
134C2×#467—72no—
135C2×C4×Q16—71no—
136C2×C8×D4—71no—
137C22×#107—72no—
138C24×C7C2×D412yes—
139C2×C7×Q8—21no—
140C2×C7×Q8—24no—
141C2×C7×Q8—22no—
142C2×C7×Q8—22no—
143C2×C7×Q8—82no—
144C2×C7×Q8C2412yes—
145C2×C7×D4Q1618yescomplemented
146C2×C7×D4C2×D414yes—
147C22×C4×C7—24no—
148C22×C4×C7—24no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—22no—
152C22×C4×C7—28no—
153C22×C4×C7—84no—
154C22×C4×C7C2×D412yes—
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D414yes—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2414yes—
159C22×C4×C7C2411yescharacteristic
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—22no—
163C22×Dic7—28no—
164C22×Dic7—84no—
165C2×Dic14—22no—
166C2×Dic14—22no—
167C2×Dic14—21no—
168C2×Dic14—24no—
169C2×Dic14—82no—
170C2×Dic14C2412yes—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—28no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—24no—
175C7×C4⋊C4—24no—
176C7×C4⋊C4—28no—
177C7×C4⋊C4C2×D414yes—
178C7×C4⋊C4C2412yes—
179C7×C4⋊C4C2414yes—
180C7×C22⋊C4—28no—
181C7×C22⋊C4C2×D418yes—
182C42×C7—24no—
183C42×C7C2×D412yes—
184C42×C7C2412yes—
185C22⋊Dic7—28no—
186#1144—28no—
187C4⋊Dic7—21no—
188C4⋊Dic7—22no—
189C4⋊Dic7—24no—
190C4⋊Dic7—28no—
191C4⋊Dic7C2412yes—
192C4⋊Dic7C2414yes—
193Dic7⋊C4—24no—
194Dic7⋊C4—28no—
195C4×Dic7—24no—
196C2×C7⋊C8—24no—
197C2×C7⋊C8—22no—
198C2×C7⋊C8—216no—
199C2×C7⋊C8#4114yes—
200C2×C7⋊C8C2412yes—
201C2×C7⋊C8C2414yes—
202C2×C7⋊C8C2412yes—
203C22×Q16—71no—
204C23×C8—72no—
205#554—24no—
206#554—78no—
207#554—144no—
208#554C2×D718yes—
209C2×C4⋊Q8—71no—
210C2×C4⋊Q8C2×D711yescharacteristic
211C2×#125—21no—
212C2×#125—141no—
213C2×C22⋊Q8—22no—
214C2×C22⋊Q8—72no—
215C2×C22⋊Q8—142no—
216C2×C22⋊Q8C2×D712yes—
217C2×C4×Q8—71no—
218C2×C4×Q8C2×D711yescharacteristic
219C2×C4×D4—21no—
220C2×C4×D4—141no—
221C2×C4×D4C2×D711yescharacteristic
222C22×C4⋊C4—72no—
223C22×C4⋊C4C2×D712yes—
224#500—78no—
225#484—716no—
226#484—716no—
227#477—78no—
228#477—78no—
229#467—716no—
230C4×Q16—78no—
231C8×D4—78no—
232C2×#107—72no—
233C2×#107—72no—
234C2×#107—716no—
235C2×#107—72no—
236C2×#107—72no—
237C2×#107—71no—
238C2×C4⋊C8—71no—
239C2×#103—72no—
240C2×#103—71no—
241C2×#103—72no—
242C2×#103—71no—
243C2×C22⋊C8—72no—
244C2×C4×C8—71no—
245C23×C7—24no—
246C23×C7—24no—
247C23×C7C2×Q1618yes—
248C23×C7C22×D414yes—
249C23×C7C22×D411yescharacteristic
250C7×Q8—22no—
251C7×Q8—22no—
252C7×Q8—42no—
253C7×Q8—84no—
254C7×D4C2×Q16116yescomplemented
255C2×C4×C7—24no—
256C2×C4×C7—22no—
257C2×C4×C7—28no—
258C2×C4×C7—24no—
259C2×C4×C7—24no—
260C2×C4×C7—22no—
261C2×C4×C7—22no—
262C2×C4×C7—24no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—44no—
266C2×C4×C7—88no—
267C2×C4×C7C2×Q1614yes—
268C2×C4×C7C22×D414yes—
269C2×C4×C7C22×D412yes—
270C2×C4×C7C22×D411yescharacteristic
271C2×C4×C7C22×D412yes—
272C2×C4×C7C2×#4112yes—
273C2×C4×C72−1+412yes—
274C2×C4×C7C2511yescharacteristic
275C2×Dic7—22no—
276C2×Dic7—22no—
277C2×Dic7—24no—
278C2×Dic7—42no—
279C2×Dic7—44no—
280C2×Dic7—42no—
281C2×Dic7—88no—
282Dic14—22no—
283Dic14—22no—
284Dic14—42no—
285Dic14—84no—
286C7⋊C8—24no—
287C7⋊C8—22no—
288C7⋊C8C2×#4114yes—
289C22×Q8—21no—
290C22×Q8—71no—
291C22×Q8—141no—
292C22×Q8C22×D711yescharacteristic
293C22×D4C22⋊D711yescharacteristic
294C23×C4—22no—
295C23×C4—142no—
296C23×C4C22×D712yes—
297C2×Q16—74no—
298C2×Q16—144no—
299C22×C8—71no—
300C22×C8—72no—
301C22×C8—78no—
302C22×C8—71no—
303C22×C8—148no—
304C4⋊Q8—22no—
305C4⋊Q8—74no—
306C4⋊Q8—142no—
307C4⋊Q8C22×D714yes—
308#125—28no—
309#125—148no—
310C22⋊Q8—24no—
311C22⋊Q8—216no—
312C22⋊Q8—78no—
313C22⋊Q8—144no—
314C22⋊Q8—1416no—
315C22⋊Q8C22×D718yes—
316C4×Q8—22no—
317C4×Q8—74no—
318C4×Q8—142no—
319C4×Q8C22×D714yes—
320C4×D4—28no—
321C4×D4—148no—
322C4×D4C22⋊D714yes—
323C4×D4C22×D714yes—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—72no—
331C2×C4⋊C4—78no—
332C2×C4⋊C4—142no—
333C2×C4⋊C4—144no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4—142no—
337C2×C4⋊C4C22⋊D711yescharacteristic
338C2×C4⋊C4C22×D711yescharacteristic
339C2×C4⋊C4C22×D712yes—
340C2×C4⋊C4C22×D718yes—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—142no—
343C2×C22⋊C4C22⋊D712yes—
344C2×C42—21no—
345C2×C42—141no—
346C2×C42C22×D711yescharacteristic
347#107—74no—
348#107—78no—
349#107—78no—
350#107—78no—
351#107—78no—
352C4⋊C8—74no—
353#103—74no—
354#103—78no—
355#103—74no—
356#103—78no—
357C22⋊C8—78no—
358C4×C8—74no—
359C22×C7—216no—
360C22×C7—24no—
361C22×C7—24no—
362C22×C7#54712yes—
363C22×C7C22×Q1618yes—
364C22×C7C22×Q1612yes—
365C22×C7#57612yes—
366C22×C7C23×D411yescharacteristic
367C4×C7—24no—
368C4×C7—22no—
369C4×C7—42no—
370C4×C7—42no—
371C4×C7—44no—
372C4×C7#54712yes—
373C4×C7C22×Q1614yes—
374C4×C7C22×#4111yescharacteristic
375C4×C7C2×2−1+411yescharacteristic
376Dic7—42no—
377Dic7—44no—
378Dic7—42no—
379C24C2×C22⋊D712yes—
380C2×Q8—21no—
381C2×Q8—24no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—72no—
385C2×Q8—82no—
386C2×Q8—141no—
387C2×Q8—144no—
388C2×Q8—142no—
389C2×Q8—142no—
390C2×Q8—562no—
391C2×Q8C23×D712yes—
392C2×D4#114418yescomplemented
393C2×D4C2×C22⋊D714yes—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—28no—
400C22×C4—84no—
401C22×C4—141no—
402C22×C4—141no—
403C22×C4—142no—
404C22×C4—148no—
405C22×C4—564no—
406C22×C4C2×C22⋊D714yes—
407C22×C4C2×C22⋊D711yescharacteristic
408C22×C4C2×C22⋊D712yes—
409C22×C4C2×C22⋊D711yescharacteristic
410C22×C4C23×D714yes—
411C22×C4C23×D711yescharacteristic
412Q16—148no—
413C2×C8—74no—
414C2×C8—72no—
415C2×C8—72no—
416C2×C8—74no—
417C2×C8—1416no—
418C2×C8—144no—
419C2×C8—142no—
420C4⋊C4—21no—
421C4⋊C4—28no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—28no—
426C4⋊C4—72no—
427C4⋊C4—74no—
428C4⋊C4—144no—
429C4⋊C4—148no—
430C4⋊C4—141no—
431C4⋊C4—148no—
432C4⋊C4—142no—
433C4⋊C4—144no—
434C4⋊C4C2×C22⋊D714yes—
435C4⋊C4C23×D712yes—
436C4⋊C4C23×D714yes—
437C22⋊C4—28no—
438C22⋊C4—148no—
439C22⋊C4C2×C22⋊D718yes—
440C42—24no—
441C42—144no—
442C42C2×C22⋊D712yes—
443C42C23×D712yes—
444C2×C7—28no—
445C2×C7#333814yescomplemented
446C2×C7C2×#54711yescharacteristic
447C2×C7C23×Q1611yescharacteristic
448C2×C7C2×#57611yescharacteristic
449C23—24no—
450C23—24no—
451C23C2×#114418yes—
452C23C22×C22⋊D714yes—
453C23C22×C22⋊D711yescharacteristic
454Q8—22no—
455Q8—22no—
456Q8—42no—
457Q8—84no—
458Q8—142no—
459Q8—142no—
460Q8—282no—
461Q8—564no—
462D4C2×#1144116yescomplemented
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—28no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—88no—
475C2×C4—142no—
476C2×C4—142no—
477C2×C4—144no—
478C2×C4—282no—
479C2×C4—282no—
480C2×C4—284no—
481C2×C4—568no—
482C2×C4C2×#114414yes—
483C2×C4D7×#4112yes—
484C2×C4#655412yes—
485C2×C4C22×C22⋊D712yes—
486C2×C4C22×C22⋊D714yes—
487C2×C4C22×C22⋊D712yes—
488C2×C4C22×C22⋊D711yescharacteristic
489C2×C4C24×D711yescharacteristic
490C8—74no—
491C8—144no—
492C8—142no—
493C7C2×#333811yescharacteristic, complemented, Sylow
494C22—24no—
495C22—24no—
496C22—216no—
497C22C22×#114418yes—
498C22C22×#114412yes—
499C22#9051512yes—
500C22#9052612yes—
501C22C23×C22⋊D711yescharacteristic
502C4—24no—
503C4—22no—
504C4—42no—
505C4—42no—
506C4—44no—
507C4—282no—
508C4—282no—
509C4—284no—
510C4C22×#114414yes—
511C4#9051512yes—
512C4C2×D7×#4111yescharacteristic
513C4C2×#655411yescharacteristic
514C2—28no—
515C2#1174106814yesdirect product factor
516C2C23×#114411yescharacteristic
517C2C2×#9051511yescharacteristic
518C2C2×#9052611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#554C2
socleC23×C7C22×D4
prime radicalC22×C7×D4D4
hypercenterC2×#554D7
hypocenterC7C2×#3338
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
3. #11741068C2direct (N × H)
4. C2×#89863C2semidirect
5. C2×#89850C2semidirect
6. C2×#89823C2semidirect
7. C2×#89816C2semidirect
8. C2×#89793C2semidirect
9. C2×C4×#1144C2semidirect
11. C2×#89742C2semidirect
12. C22×#6381C2semidirect
37. #89863C22semidirect
38. #89850C22semidirect
39. #89823C22semidirect
40. #89816C22semidirect
41. C4×#1144C22semidirect
42. #89793C22semidirect
44. #89742C22semidirect
52. C2×#6381C22semidirect
145. C2×C7×D4Q16semidirect
254. C7×D4C2×Q16semidirect
392. C2×D4#1144semidirect
445. C2×C7#3338semidirect
462. D4C2×#1144semidirect
493. C7C2×#3338semidirect
515. C2#11741068direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5542
C2×#902342
#117410682
C2×#898632
C2×#898502
C2×#898232
C2×#898162
C2×#897932
C2×C4×#11442
C2×D4×C7⋊C82
C2×#897422
C22×#63812
C2×#33387