Finite Group Databasebeta

← #13358448

Subgroups — #13358448

Order 1792.

1142 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90314C211yesmaximal, characteristic
2#11731314C211yesmaximal, characteristic
3#11730880C211yesmaximal, characteristic
4C7×#1996C211yesmaximal, characteristic
5#11729455C211yesmaximal, characteristic
6#11727283C211yesmaximal, characteristic
7#11727211C211yesmaximal, characteristic
8#11726709C214yesmaximal
9#11726705C214yesmaximal
10C2×C7×C22⋊Q8C2211yescharacteristic
11C2×#6507C412yescomplemented
12C2×#6487C2211yescharacteristic
13C22×Dic7⋊C4C2211yescharacteristic
14#90314—22no—
15#90314—22no—
16#90314—22no—
17#90314—22no—
18C2×#6452C412yescomplemented
19C2×#6452C412yescomplemented
20C2×#6451C412yescomplemented
21C2×#6441C2212yes—
22C2×#6440C2212yes—
23C2×#6440C2212yes—
24C2×#6440C2212yes—
25C7×#380C2211yescharacteristic
26C4×C7×C22⋊C4C2211yescharacteristic
27#89984C2211yescharacteristic
28C4×C22⋊Dic7C2211yescharacteristic
29#89744—22no—
30#89714C2212yes—
31C22⋊C4×Dic7C2212yes—
32C4×Dic7⋊C4—22no—
33#89421—22no—
34#89419—22no—
35C7×#342C2214yes—
36#89322C2214yes—
37#89322C2214yes—
38#89321C2214yes—
39#21334—71nomaximal, Sylow
40C22×C7×Q8C2×C411yescharacteristic
41C23×C4×C7C2311yescharacteristic
42C23×Dic7C2311yescharacteristic
43C23×Dic7C2311yescharacteristic
44C22×Dic14C2×C412yes—
45C22×Dic14C2×C411yescharacteristic
46C7×C22⋊Q8—21no—
47C7×C22⋊Q8—21no—
48C7×C22⋊Q8—21no—
49C7×C22⋊Q8—21no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4C2×C412yes—
52C2×C7×C4⋊C4C2×C411yescharacteristic
53C2×C7×C22⋊C4C2312yes—
54C2×C7×C22⋊C4C2312yes—
55C2×C7×C22⋊C4C2312yes—
56C2×C42×C7—21no—
57C2×C22⋊Dic7C2312yes—
58C2×C22⋊Dic7C2312yes—
59C2×C22⋊Dic7C2312yes—
60#6507—24no—
61#6507—24no—
62C2×C4⋊Dic7—21no—
63C2×C4⋊Dic7C2×C412yes—
64C2×C4⋊Dic7C2×C411yescharacteristic
65#6487—21no—
66#6487—21no—
67#6487—21no—
68#6487—21no—
69C2×Dic7⋊C4—22no—
70C2×Dic7⋊C4—22no—
71C2×Dic7⋊C4—22no—
72C2×Dic7⋊C4C2×C412yes—
73C2×Dic7⋊C4C2×C412yes—
74C2×Dic7⋊C4C2×C412yes—
75C2×Dic7⋊C4C2×C412yes—
76C2×Dic7⋊C4C2×C412yes—
77C2×Dic7⋊C4C2×C412yes—
78C2×Dic7⋊C4C2×C412yes—
79C2×Dic7⋊C4D412yes—
80C2×Dic7⋊C4D412yes—
81C2×Dic7⋊C4D412yes—
82C2×Dic7⋊C4D412yes—
83C2×C4×Dic7—22no—
84C2×C4×Dic7—22no—
85C2×C4×Dic7—21no—
86C2×C4×Dic7—22no—
87C2×C4×Dic7C2×C412yes—
88C2×C4×Dic7C2×C412yes—
89#6452—24no—
90#6452—24no—
91#6452—24no—
92#6452—24no—
93#6451—24no—
94#6451—24no—
95#6441—24no—
96#6441—24no—
97#6440—24no—
98#6440—24no—
99#6440—24no—
100#6440—28no—
101#6440Q818yescomplemented
102C7×#95—21no—
103C7×#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—22no—
107C7⋊#95—22no—
108C2×#544—71no—
109#2316—71no—
110#2274—71no—
111#1996—71no—
112#1996—71no—
113#1996—74no—
114#1996—71no—
115#1996D711yescharacteristic
116#1740—74no—
117C24×C7C2411yescharacteristic
118C2×C7×Q8—21no—
119C2×C7×Q8—21no—
120C2×C7×Q8—81no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—81no—
124C22×C4×C7C22×C412yes—
125C22×C4×C7C22×C412yes—
126C22×C4×C7C22×C411yescharacteristic
127C22×C4×C7C22×C411yescharacteristic
128C22×C4×C7C2×D411yescharacteristic
129C22×C4×C7C2×D411yescharacteristic
130C22×C4×C7C2×D411yescharacteristic
131C22×C4×C7C2×D411yescharacteristic
132C22×Dic7—22no—
133C22×Dic7—22no—
134C22×Dic7—81no—
135C22×Dic7C22×C412yes—
136C22×Dic7C22×C412yes—
137C22×Dic7C22×C412yes—
138C22×Dic7C22×C412yes—
139C22×Dic7C2×D412yes—
140C22×Dic7C2×D412yes—
141C22×Dic7C2×D412yes—
142C22×Dic7C2×D412yes—
143C22×Dic7C2×Q814yes—
144C22×Dic7#4114yes—
145C2×Dic14—21no—
146C2×Dic14—21no—
147C2×Dic14—24no—
148C2×Dic14—82no—
149C2×Dic14—81no—
150C7×C4⋊C4—22no—
151C7×C4⋊C4—22no—
152C7×C4⋊C4—21no—
153C7×C4⋊C4—21no—
154C7×C4⋊C4—81no—
155C7×C22⋊C4—22no—
156C7×C22⋊C4—22no—
157C7×C22⋊C4—22no—
158C7×C22⋊C4—24no—
159C7×C22⋊C4C2×Q814yes—
160C42×C7—81no—
161C22⋊Dic7—22no—
162C22⋊Dic7—22no—
163C22⋊Dic7—22no—
164C22⋊Dic7—24no—
165C22⋊Dic7C2×Q814yes—
166C4⋊Dic7—21no—
167C4⋊Dic7—21no—
168C4⋊Dic7—24no—
169C4⋊Dic7—81no—
170Dic7⋊C4—22no—
171Dic7⋊C4—22no—
172Dic7⋊C4—24no—
173Dic7⋊C4—24no—
174Dic7⋊C4—24no—
175Dic7⋊C4—24no—
176Dic7⋊C4—28no—
177Dic7⋊C4—44no—
178Dic7⋊C4—82no—
179Dic7⋊C4C22⋊C414yes—
180Dic7⋊C4C22⋊C414yes—
181Dic7⋊C4C22⋊C414yes—
182Dic7⋊C4C22⋊C414yes—
183C4×Dic7—24no—
184C4×Dic7—24no—
185C4×Dic7—28no—
186C4×Dic7—44no—
187C4×Dic7—82no—
188C4×Dic7—81no—
189#544—142no—
190#544—142no—
191#544—142no—
192#544—142no—
193C2×C4⋊Q8—72no—
194C2×C4⋊Q8—72no—
195C2×#125—72no—
196C2×#125—72no—
197C2×C22⋊Q8—71no—
198C2×C22⋊Q8—72no—
199C2×C22⋊Q8C2×D711yescharacteristic
200C2×#117—72no—
201C2×#117—72no—
202C2×#117—72no—
203C22×C4⋊C4—71no—
204#389—142no—
205#389—142no—
206#380—71no—
207#380—72no—
208#380C2×D711yescharacteristic
209C4×C4⋊C4—142no—
210C4×C4⋊C4—142no—
211C4×C22⋊C4—71no—
212C4×C22⋊C4—72no—
213C4×C22⋊C4C2×D711yescharacteristic
214#342—74no—
215#342—74no—
216#342—74no—
217#342C2×D714yes—
218C23×C7—41no—
219C23×C7—41no—
220C23×C7C23×C411yescharacteristic
221C23×C7C22×D411yescharacteristic
222C23×C7C22×D411yescharacteristic
223C23×C7C22×Q812yes—
224C23×C7C2×#4112yes—
225C7×Q8—81no—
226C7×Q8—81no—
227C2×C4×C7—22no—
228C2×C4×C7—22no—
229C2×C4×C7—24no—
230C2×C4×C7—41no—
231C2×C4×C7—41no—
232C2×C4×C7—41no—
233C2×C4×C7—41no—
234C2×C4×C7—41no—
235C2×C4×C7—41no—
236C2×C4×C7—42no—
237C2×C4×C7—42no—
238C2×C4×C7—42no—
239C2×C4×C7—42no—
240C2×C4×C7—42no—
241C2×C4×C7—42no—
242C2×C4×C7—42no—
243C2×C4×C7C2×C22⋊C411yescharacteristic
244C2×C4×C7C2×C22⋊C411yescharacteristic
245C2×C4×C7C2×C22⋊C411yescharacteristic
246C2×C4×C7C2×C22⋊C411yescharacteristic
247C2×Dic7—22no—
248C2×Dic7—22no—
249C2×Dic7—24no—
250C2×Dic7—42no—
251C2×Dic7—42no—
252C2×Dic7—42no—
253C2×Dic7—42no—
254C2×Dic7—42no—
255C2×Dic7—42no—
256C2×Dic7—42no—
257C2×Dic7—42no—
258C2×Dic7—42no—
259C2×Dic7—42no—
260C2×Dic7C2×C22⋊C414yes—
261C2×Dic7C2×C22⋊C414yes—
262C2×Dic7C4×Q818yes—
263C2×Dic7C22⋊Q814yes—
264C2×Dic7C22⋊Q814yes—
265C2×Dic7#12414yes—
266C2×Dic7C4⋊Q814yes—
267Dic14—84no—
268Dic14—81no—
269Dic14—81no—
270C22×Q8—72no—
271C22×Q8—71no—
272C22×Q8C4×D711yescharacteristic
273C23×C4—71no—
274C23×C4—71no—
275C23×C4C22×D711yescharacteristic
276C4⋊Q8—144no—
277C4⋊Q8—144no—
278C4⋊Q8—144no—
279C4⋊Q8—144no—
280#125—144no—
281#125—144no—
282#125—144no—
283#125—144no—
284C22⋊Q8—21no—
285C22⋊Q8—21no—
286C22⋊Q8—21no—
287C22⋊Q8—21no—
288C22⋊Q8—144no—
289C22⋊Q8—144no—
290C22⋊Q8—141no—
291C22⋊Q8—141no—
292C22⋊Q8—141no—
293C22⋊Q8—141no—
294#117—78no—
295#117—144no—
296#117—144no—
297#117—144no—
298#117—148no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—72no—
301C2×C4⋊C4—72no—
302C2×C4⋊C4—72no—
303C2×C4⋊C4—72no—
304C2×C4⋊C4—72no—
305C2×C4⋊C4—72no—
306C2×C4⋊C4—72no—
307C2×C4⋊C4—72no—
308C2×C4⋊C4—72no—
309C2×C4⋊C4—71no—
310C2×C4⋊C4—72no—
311C2×C4⋊C4—72no—
312C2×C4⋊C4—72no—
313C2×C4⋊C4—142no—
314C2×C4⋊C4—142no—
315C2×C4⋊C4—142no—
316C2×C4⋊C4—141no—
317C2×C4⋊C4C4×D711yescharacteristic
318C2×C4⋊C4C4×D712yes—
319C2×C22⋊C4—72no—
320C2×C22⋊C4—72no—
321C2×C22⋊C4—72no—
322C2×C22⋊C4C22×D712yes—
323C2×C22⋊C4C22×D712yes—
324C2×C22⋊C4C22×D712yes—
325C2×C42—21no—
326C2×C42—72no—
327C2×C42—72no—
328C2×C42—142no—
329C2×C42—142no—
330C2×C42—142no—
331C2×C42—141no—
332#95—21no—
333#95—21no—
334#95—142no—
335#95—142no—
336#95—141no—
337#95—141no—
338C22×C7—21no—
339C22×C7—21no—
340C22×C7—21no—
341C22×C7—21no—
342C22×C7—21no—
343C22×C7—21no—
344C22×C7—42no—
345C22×C7—42no—
346C22×C7C22×C22⋊C411yescharacteristic
347C22×C7C2×C4×Q812yes—
348C22×C7C2×C22⋊Q812yes—
349C22×C7C2×#12411yescharacteristic
350C22×C7C2×C4⋊Q811yescharacteristic
351C4×C7—21no—
352C4×C7—21no—
353C4×C7—21no—
354C4×C7—21no—
355C4×C7—44no—
356C4×C7—42no—
357C4×C7—42no—
358C4×C7—82no—
359Dic7—24no—
360Dic7—44no—
361Dic7—42no—
362Dic7—42no—
363Dic7—82no—
364Dic7#39118yes—
365C24C23×D711yescharacteristic
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—81no—
369C2×Q8—144no—
370C2×Q8—141no—
371C2×Q8—141no—
372C2×Q8—561no—
373C2×Q8—562no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—72no—
377C22×C4—72no—
378C22×C4—72no—
379C22×C4—74no—
380C22×C4—74no—
381C22×C4—72no—
382C22×C4—72no—
383C22×C4—72no—
384C22×C4—72no—
385C22×C4—72no—
386C22×C4—81no—
387C22×C4—142no—
388C22×C4—142no—
389C22×C4—561no—
390C22×C4C2×C4×D712yes—
391C22×C4C2×C4×D712yes—
392C22×C4C2×C4×D711yescharacteristic
393C22×C4C2×C4×D711yescharacteristic
394C22×C4D4×D711yescharacteristic
395C22×C4D4×D711yescharacteristic
396C22×C4D4×D711yescharacteristic
397C22×C4D4×D711yescharacteristic
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—74no—
403C4⋊C4—74no—
404C4⋊C4—74no—
405C4⋊C4—74no—
406C4⋊C4—81no—
407C4⋊C4—144no—
408C4⋊C4—144no—
409C4⋊C4—144no—
410C4⋊C4—144no—
411C4⋊C4—148no—
412C4⋊C4—144no—
413C4⋊C4—142no—
414C4⋊C4—142no—
415C4⋊C4—141no—
416C4⋊C4—141no—
417C4⋊C4—284no—
418C4⋊C4—562no—
419C4⋊C4—561no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—24no—
424C22⋊C4—74no—
425C22⋊C4—142no—
426C22⋊C4—142no—
427C22⋊C4—142no—
428C22⋊C4—144no—
429C22⋊C4Q8×D714yes—
430C42—81no—
431C42—144no—
432C42—144no—
433C42—148no—
434C42—284no—
435C42—561no—
436C42—562no—
437C2×C7—22no—
438C2×C7—21no—
439C2×C7—22no—
440C2×C7—21no—
441C2×C7C2×#39111yescharacteristic
442C2×C7#288311yescharacteristic
443C2×C7#288411yescharacteristic
444C23—41no—
445C23—41no—
446C23C22×C4×D711yescharacteristic
447C23C2×D4×D711yescharacteristic
448C23C2×D4×D711yescharacteristic
449C23C2×Q8×D712yes—
450C23D7×#4112yes—
451Q8—81no—
452Q8—81no—
453Q8—561no—
454Q8—561no—
455Q8—564no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—74no—
473C2×C4—74no—
474C2×C4—78no—
475C2×C4—74no—
476C2×C4—74no—
477C2×C4—74no—
478C2×C4—74no—
479C2×C4—142no—
480C2×C4—142no—
481C2×C4—144no—
482C2×C4—282no—
483C2×C4—282no—
484C2×C4—282no—
485C2×C4—282no—
486C2×C4—282no—
487C2×C4—282no—
488C2×C4—282no—
489C2×C4—282no—
490C2×C4—282no—
491C2×C4—282no—
492C2×C4D7×C22⋊C411yescharacteristic
493C2×C4D7×C22⋊C411yescharacteristic
494C2×C4D7×C22⋊C411yescharacteristic
495C2×C4D7×C22⋊C411yescharacteristic
496C7#2133411yescharacteristic, complemented, Sylow
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—42no—
504C22—42no—
505C22C2×D7×C22⋊C411yescharacteristic
506C22C4×Q8×D712yes—
507C22D7×C22⋊Q812yes—
508C22D7×#12411yescharacteristic
509C22D7×C4⋊Q811yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—44no—
515C4—42no—
516C4—42no—
517C4—78no—
518C4—82no—
519C4—144no—
520C4—284no—
521C4—282no—
522C4—282no—
523C4—562no—
524C2—22no—
525C2—21no—
526C2—21no—
527C2—22no—
528C2D7×#39111yescharacteristic
529C2#1173613311yescharacteristic
530C2#1173613411yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC23×C7C23×C4
FrattiniC24C23×D7
FittingC7×#1996C2
socleC22×C7C22×C22⋊C4
prime radical, cosocleC24×C7C24
hypercenter#1996D7
hypocenterC7#21334
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
11. C2×#6507C4semidirect
18. C2×#6452C4semidirect
19. C2×#6452C4semidirect
20. C2×#6451C4semidirect
101. #6440Q8semidirect
496. C7#21334semidirect

Maximal subgroups

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

subgroupindex
C2×#903142
#117313142
#117308802
C7×#19962
#117294552
#117272832
#117272112
#117267092
#117267052
#213347