Finite Group Databasebeta

← #14218819

Subgroups — #14218819

Order 1792.

1702 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#1157C211yesmaximal, characteristic
2C7×#2984C211yesmaximal, characteristic, complemented
3#11739880C211yesmaximal, characteristic, complemented
4#11737412C213yesmaximal, complemented
5#11737410C213yesmaximal, complemented
6#11737405C213yesmaximal, complemented
7#11737403C213yesmaximal, complemented
8#11737366C211yesmaximal, characteristic, complemented
9#11737365C211yesmaximal, characteristic, complemented
10Q8×#1131C211yesmaximal, characteristic, complemented
11C8×Q8×D7C211yesmaximal, characteristic, complemented
12#11736779C213yesmaximal
13#11736761C213yesmaximal, complemented
14#11736728C213yesmaximal
15#11736712C213yesmaximal, complemented
16C2×C4×C7×Q8C2211yescharacteristic
17Q8×C22⋊D7C411yescharacteristic
18Q8×C22⋊D7C411yescharacteristic
19#90270C413yes—
20#90269C2213yes—
21#90269C2213yes—
22Q8×D28C411yescharacteristic
23C4×Q8×D7C2211yescharacteristic
24C4×Q8×D7C2211yescharacteristic
25#90258C413yes—
26Q8×Dic14C411yescharacteristic
27#90202C413yes—
28#90202C413yes—
29C4×#1157C2213yes—
30#90161C2213yes—
31C7×#446C2213yes—
32C7×#446C2213yes—
33C7×C8×Q8C2211yescharacteristic
34C7×C8×Q8C2211yescharacteristic
35C7×#432C2213yes—
36C7×#431C2213yes—
37Q8×C7⋊C8C2211yescharacteristic
38Q8×C7⋊C8C2211yescharacteristic
39Q8×C7⋊C8C2213yes—
40Q8×C7⋊C8C2213yes—
41#89697C2213yes—
42#89697C2213yes—
43#89612C2213yes—
44#89612C2213yes—
45#89611C2213yes—
46#89611C2213yes—
47#89609C2213yes—
48#89607C2213yes—
49#89605C2213yes—
50D7×C4⋊C8C2213yes—
51#89603C2213yes—
52#89603C2213yes—
53#89603C2216yes—
54#89602C2213yes—
55#89602C2213yes—
56#89602C2216yes—
57#89499C2211yescharacteristic
58#89499C2213yes—
59#89492C2213yes—
60#89492C2216yes—
61#89492C2213yes—
62#89481C2216yes—
63#89480C2216yes—
64#89473C2216yes—
65#89473C2216yes—
66C4×#1131C2213yes—
67#89457C2213yes—
68#89457C2213yes—
69C4×C8×D7C2213yes—
70#89451C2213yes—
71#89451C2213yes—
72C8×Dic14C2213yes—
73C8×Dic14C2213yes—
74#60253—71nomaximal, Sylow
75C22×C7×Q8C2×C411yescharacteristic
76C2×Q8×D7C2×C411yescharacteristic
77C2×Q8×D7C2×C411yescharacteristic
78C2×#1157C2311yescharacteristic
79C4×C7×Q8C2×C411yescharacteristic
80C4×C7×Q8C2×C413yes—
81C4×C7×Q8C2311yescharacteristic
82C4×C7×Q8C2313yes—
83C2×C7×C4⋊C4C2×C413yes—
84C2×C42×C7C2313yes—
85#6509C2×C413yes—
86#6509C2×C413yes—
87Q8×Dic7C2×C411yescharacteristic
88Q8×Dic7C2×C411yescharacteristic
89#6507C2×C413yes—
90#6507C2×C413yes—
91C4×C22⋊D7C2×C413yes—
92C4×C22⋊D7C2×C413yes—
93C4×C22⋊D7C2×C413yes—
94C4×C22⋊D7C2×C413yes—
95#6489C2313yes—
96#6487C2×C416yes—
97#6487C2×C413yes—
98#6487C2×C413yes—
99C2×#1131C2313yes—
100C2×#1131C2311yescharacteristic
101C2×C8×D7C2313yes—
102C2×C8×D7C2311yescharacteristic
103#6460C2×C413yes—
104#6460C2×C413yes—
105#6459C2×C416yes—
106#6459C2×C416yes—
107D7×C4⋊C4C2×C413yes—
108D7×C4⋊C4C2×C413yes—
109#6452C2×C416yes—
110#6452C2×C416yes—
111#6451C2×C413yes—
112#6451C2×C413yes—
113#6450C2×C413yes—
114#6450C2×C413yes—
115C4×D28C2×C413yes—
116C4×D28C2×C413yes—
117#6435C2313yes—
118#6435C2313yes—
119C42×D7C2313yes—
120C42×D7C2313yes—
121#6433C2×C416yes—
122#6432C2×C413yes—
123#6432C2×C413yes—
124C4×Dic14C2×C413yes—
125C4×Dic14C2×C413yes—
126C4×Dic14C2313yes—
127C4×Dic14C2313yes—
128C4×Dic14C2316yes—
129C7×C4⋊C8C2313yes—
130C7×C4⋊C8C2313yes—
131C7×C4⋊C8C2313yes—
132C7×C4⋊C8C2313yes—
133C7×C22⋊C8C2311yescharacteristic
134C7×C22⋊C8C2313yes—
135C7×#97C2316yes—
136C4×C7×C8C2313yes—
137C4×C7×C8C2313yes—
138C22⋊(C7⋊C8)C2313yes—
139C22⋊(C7⋊C8)C2311yescharacteristic
140(C2×D7)⋊C8C2313yes—
141(C2×D7)⋊C8C2313yes—
142(C2×D7)⋊C8C2311yescharacteristic
143(C2×D7)⋊C8C2311yescharacteristic
144#6389C2316yes—
145#6389C2313yes—
146#6389C2313yes—
147Dic7⋊C8C2313yes—
148Dic7⋊C8C2313yes—
149Dic7⋊C8C2316yes—
150Dic7⋊C8C2316yes—
151Dic7⋊C8C2313yes—
152Dic7⋊C8C2313yes—
153C8×Dic7C2316yes—
154C8×Dic7C2313yes—
155C8×Dic7C2313yes—
156C4⋊(C7⋊C8)C2313yes—
157C4⋊(C7⋊C8)C2313yes—
158C4⋊(C7⋊C8)C2313yes—
159C4⋊(C7⋊C8)C2313yes—
160C4×C7⋊C8C2313yes—
161C4×C7⋊C8C2316yes—
162C4×C7⋊C8C2313yes—
163Q8×#41—71no—
164#2986—73no—
165#2986—73no—
166#2985—73no—
167#2985—73no—
168#2984—71no—
169#2984D711yescharacteristic, complemented
170#2982—71no—
171#2982—71no—
172Q8×M16—71no—
173#2962—73no—
174#2960—73no—
175C2×C8×Q8—71no—
176#2925—73no—
177#2922—73no—
178C2×C7×Q8—24no—
179C2×C7×Q8C22×C413yes—
180C2×C7×Q8C22×C411yescharacteristic
181C22×C4×C7C22×C413yes—
182C22×C4×C7C22×C413yes—
183C22×C4×C7C2411yescharacteristic
184C2×C22⋊D7C22×C411yescharacteristic
185C2×C22⋊D7C22×C411yescharacteristic
186Q8×D7—24no—
187Q8×D7—24no—
188#1157—22no—
189#1157—22no—
190C2×D28C22×C411yescharacteristic
191C2×C4×D7C22×C413yes—
192C2×C4×D7C22×C413yes—
193C2×C4×D7C22×C413yes—
194C2×C4×D7C22×C413yes—
195C2×C4×D7C2411yescharacteristic
196C2×C4×D7C2411yescharacteristic
197C2×Dic14C22×C411yescharacteristic
198C2×Dic14C22×C413yes—
199C2×Dic14C22×C413yes—
200C2×Dic14C22×C416yes—
201C2×C7×C8C2411yescharacteristic
202C2×C7×C8C2411yescharacteristic
203C2×C7×C8C2413yes—
204C2×C7×C8C2413yes—
205C7×C4⋊C4C22×C413yes—
206C7×C4⋊C4C22×C413yes—
207C7×C4⋊C4C22×C416yes—
208C42×C7C22×C413yes—
209C42×C7C22×C413yes—
210C42×C7C2413yes—
211C42×C7C2413yes—
212C22⋊Dic7C22×C413yes—
213C22⋊Dic7C22×C413yes—
214(C2×D7)⋊C4C22×C413yes—
215(C2×D7)⋊C4C22×C413yes—
216(C2×D7)⋊C4C22×C413yes—
217(C2×D7)⋊C4C22×C413yes—
218C4⋊Dic7C22×C413yes—
219C4⋊Dic7C22×C413yes—
220C4⋊Dic7C22×C413yes—
221C4⋊Dic7C22×C413yes—
222C4⋊Dic7C22×C416yes—
223Dic7⋊C4C22×C413yes—
224Dic7⋊C4C22×C413yes—
225Dic7⋊C4C22×C413yes—
226Dic7⋊C4C22×C413yes—
227Dic7⋊C4C22×C416yes—
228Dic7⋊C4C22×C416yes—
229Dic7⋊C4C22×C416yes—
230Dic7⋊C4C22×C416yes—
231C4×Dic7C22×C413yes—
232C4×Dic7C22×C413yes—
233C4×Dic7C22×C413yes—
234C4×Dic7C22×C413yes—
235C4×Dic7C2413yes—
236C4×Dic7C2413yes—
237C2×C7⋊C8C2413yes—
238C2×C7⋊C8C2411yescharacteristic
239C2×C7⋊C8C2413yes—
240C2×C7⋊C8C2411yescharacteristic
241#1131—22no—
242#1131—26no—
243C8×D7—26no—
244C8×D7—22no—
245Q82—71no—
246#557—73no—
247#554—73no—
248#554—73no—
249#554—73no—
250D4×Q8—71no—
251D4×Q8—71no—
252D4×Q8—71no—
253#533—73no—
254#533—73no—
255#533—73no—
256C4×#41—73no—
257C2×C4×Q8—71no—
258C2×C4×Q8—71no—
259C2×C4×Q8C2×D711yescharacteristic
260#446—73no—
261#446—76no—
262#446—73no—
263#446—76no—
264#446—76no—
265#446—73no—
266#446—73no—
267#446C2×D713yes—
268#446C2×D713yes—
269C8×Q8—71no—
270C8×Q8—73no—
271C8×Q8—73no—
272C8×Q8—71no—
273C8×Q8—73no—
274C8×Q8—73no—
275C8×Q8—76no—
276C8×Q8—73no—
277C8×Q8—73no—
278C8×Q8C2×D711yescharacteristic
279C8×Q8C2×D711yescharacteristic
280#433—73no—
281#433—73no—
282#433—73no—
283#433—73no—
284#433—76no—
285#433—76no—
286#432—73no—
287#432C2×D713yes—
288#431—73no—
289#431—73no—
290#431—73no—
291#431—73no—
292#431—73no—
293#431C2×D713yes—
294#424—73no—
295#424—73no—
296#423—73no—
297C2×C4⋊C8—73no—
298#408—71no—
299#408—73no—
300#405—76no—
301C4×M16—73no—
302C2×C4×C8—73no—
303C23×C7C23×C411yescharacteristic
304C22×D7C23×C411yescharacteristic
305C22×D7C23×C411yescharacteristic
306C7×Q8—26no—
307C7×Q8#13114yes—
308C2×C4×C7—26no—
309C2×C4×C7—21no—
310C2×C4×C7—21no—
311C2×C4×C7—43no—
312C2×C4×C7C23×C413yes—
313C2×C4×C7C23×C413yes—
314C2×C4×C7C23×C413yes—
315C2×C4×C7C23×C411yescharacteristic
316C2×C4×C7C23×C413yes—
317C2×C4×C7C2511yescharacteristic
318C22⋊D7—22no—
319C22⋊D7—22no—
320C2×Dic7C23×C411yescharacteristic
321C2×Dic7C23×C413yes—
322C2×Dic7C23×C413yes—
323C2×Dic7C23×C413yes—
324C2×Dic7C23×C413yes—
325C2×Dic7C23×C411yescharacteristic
326D28—22no—
327C4×D7—26no—
328C4×D7—26no—
329C4×D7—21no—
330C4×D7—21no—
331C4×D7—43no—
332C4×D7—43no—
333C4×D72−1+412yes—
334C4×D72−1+412yes—
335Dic14—26no—
336Dic14—26no—
337Dic14—22no—
338Dic14—46no—
339C7×C8—21no—
340C7×C8—23no—
341C7×C8—21no—
342C7×C8—23no—
343C7⋊C8—23no—
344C7⋊C8—21no—
345C7⋊C8—21no—
346C7⋊C8—23no—
347C2×#41—71no—
348C22×Q8—71no—
349C22×Q8—71no—
350C22×Q8C4×D711yescharacteristic
351C2×M16—71no—
352C2×M16—73no—
353C22×C8—71no—
354C22×C8—73no—
355C4⋊Q8—73no—
356C4⋊Q8—73no—
357C4⋊Q8—73no—
358C4⋊Q8—73no—
359C4⋊Q8—73no—
360C4⋊Q8—73no—
361#125—76no—
362#125—76no—
363#125—76no—
364C22⋊Q8—73no—
365C22⋊Q8—73no—
366C22⋊Q8—73no—
367C22⋊Q8—73no—
368C22⋊Q8—76no—
369C22⋊Q8—73no—
370C22⋊Q8—73no—
371C22⋊Q8—76no—
372C22⋊Q8—76no—
373C4×Q8—71no—
374C4×Q8—71no—
375C4×Q8—73no—
376C4×Q8—73no—
377C4×Q8—73no—
378C4×Q8—73no—
379C4×Q8—73no—
380C4×Q8—73no—
381C4×Q8—76no—
382C4×Q8C4×D711yescharacteristic
383C4×Q8C4×D713yes—
384C4×Q8C22×D711yescharacteristic
385C4×Q8C22×D713yes—
386C4×D4—73no—
387C4×D4—73no—
388C4×D4—73no—
389C4×D4—73no—
390C4×D4—73no—
391C4×D4—73no—
392#117—73no—
393#117—73no—
394#117—73no—
395C2×C4⋊C4—73no—
396C2×C4⋊C4—73no—
397C2×C4⋊C4C4×D713yes—
398C2×C42—73no—
399C2×C42—73no—
400C2×C42C22×D713yes—
401C4⋊C8—76no—
402C4⋊C8—76no—
403C4⋊C8—73no—
404C4⋊C8—73no—
405C4⋊C8—73no—
406C4⋊C8—73no—
407C4⋊C8—73no—
408C4⋊C8—73no—
409C4⋊C8—73no—
410C4⋊C8—73no—
411C4⋊C8C22×D713yes—
412C4⋊C8C22×D713yes—
413C4⋊C8C22×D713yes—
414C4⋊C8C22×D713yes—
415C22⋊C8—71no—
416C22⋊C8—73no—
417C22⋊C8—73no—
418C22⋊C8—71no—
419C22⋊C8—71no—
420C22⋊C8—73no—
421C22⋊C8C22×D711yescharacteristic
422C22⋊C8C22×D713yes—
423#97—73no—
424#97—73no—
425#97—76no—
426#97C22×D716yes—
427C4×C8—73no—
428C4×C8—73no—
429C4×C8—76no—
430C4×C8—73no—
431C4×C8—73no—
432C4×C8—76no—
433C4×C8C22×D713yes—
434C4×C8C22×D713yes—
435C22×C7—21no—
436C22×C7—21no—
437C22×C7—21no—
438C22×C7C24×C411yescharacteristic
439C2×D7—21no—
440C2×D7—21no—
441C2×D7—21no—
442C2×D7—21no—
443C2×D7—21no—
444C2×D7#51912yes—
445C4×C7—21no—
446C4×C7—23no—
447C4×C7—23no—
448C4×C7—23no—
449C4×C7C2×#13116yes—
450C4×C7C2×2−1+411yescharacteristic
451C4×C7C2×2−1+411yescharacteristic
452Dic7—21no—
453Dic7—23no—
454Dic7—23no—
455Dic7—23no—
456Dic7—23no—
457Dic7#51912yes—
458#41—142no—
459#41—142no—
460C2×Q8—24no—
461C2×Q8—71no—
462C2×Q8—73no—
463C2×Q8—73no—
464C2×Q8—76no—
465C2×Q8—144no—
466C2×Q8—144no—
467C2×Q8C2×C4×D713yes—
468C2×Q8C2×C4×D711yescharacteristic
469C2×D4—71no—
470C2×D4—71no—
471C2×D4—71no—
472C22×C4—71no—
473C22×C4—73no—
474C22×C4—73no—
475C22×C4—71no—
476C22×C4—73no—
477C22×C4—73no—
478C22×C4C2×C4×D713yes—
479C22×C4C2×C4×D713yes—
480C22×C4C23×D711yescharacteristic
481M16—142no—
482M16—146no—
483C2×C8—73no—
484C2×C8—71no—
485C2×C8—71no—
486C2×C8—73no—
487C2×C8—146no—
488C2×C8—142no—
489C2×C8C23×D711yescharacteristic
490C2×C8C23×D713yes—
491C2×C8C23×D713yes—
492C2×C8C23×D711yescharacteristic
493C4⋊C4—76no—
494C4⋊C4—76no—
495C4⋊C4—73no—
496C4⋊C4—73no—
497C4⋊C4—73no—
498C4⋊C4—73no—
499C4⋊C4—76no—
500C4⋊C4—76no—
501C4⋊C4—73no—
502C4⋊C4—73no—
503C4⋊C4—73no—
504C4⋊C4—73no—
505C4⋊C4—76no—
506C4⋊C4C2×C4×D713yes—
507C4⋊C4C2×C4×D716yes—
508C4⋊C4C2×C4×D713yes—
509C22⋊C4—73no—
510C22⋊C4—73no—
511C22⋊C4—73no—
512C22⋊C4—73no—
513C22⋊C4—73no—
514C22⋊C4—73no—
515C42—73no—
516C42—73no—
517C42—73no—
518C42—73no—
519C42—73no—
520C42—73no—
521C42C2×C4×D713yes—
522C42C2×C4×D713yes—
523C42C23×D713yes—
524C42C23×D713yes—
525C2×C7—41no—
526C2×C7C2×#51911yescharacteristic
527C2×C7C22×#13111yescharacteristic
528C2×C7#357311yescharacteristic
529D7—22no—
530D7—41no—
531C23—71no—
532C23—71no—
533C23C22×C4×D711yescharacteristic
534Q8—26no—
535Q8—142no—
536Q8—146no—
537Q8—146no—
538Q8—286no—
539Q8#647014yes—
540D4—142no—
541D4—142no—
542D4—142no—
543C2×C4—26no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—43no—
547C2×C4—71no—
548C2×C4—72no—
549C2×C4—72no—
550C2×C4—71no—
551C2×C4—73no—
552C2×C4—73no—
553C2×C4—73no—
554C2×C4—73no—
555C2×C4—146no—
556C2×C4—146no—
557C2×C4—141no—
558C2×C4—141no—
559C2×C4—283no—
560C2×C4—283no—
561C2×C4C22×C4×D713yes—
562C2×C4C22×C4×D711yescharacteristic
563C2×C4C22×C4×D713yes—
564C2×C4C22×C4×D713yes—
565C2×C4C22×C4×D713yes—
566C2×C4C24×D711yescharacteristic
567C8—21no—
568C8—23no—
569C8—23no—
570C8—21no—
571C8—143no—
572C8—141no—
573C8—141no—
574C8—143no—
575C7#6025311yescharacteristic, complemented, Sylow
576C22—21no—
577C22—21no—
578C22—21no—
579C22—72no—
580C22—141no—
581C22—141no—
582C22—141no—
583C22—141no—
584C22—141no—
585C22C23×C4×D711yescharacteristic
586C4—23no—
587C4—21no—
588C4—23no—
589C4—23no—
590C4—72no—
591C4—143no—
592C4—143no—
593C4—141no—
594C4—143no—
595C4—143no—
596C4C2×#647016yes—
597C4D7×2−1+411yescharacteristic
598C4D7×2−1+411yescharacteristic
599C2—41no—
600C2—142no—
601C2—281no—
602C2D7×#51911yescharacteristic
603C2C22×#647011yescharacteristic
604C2#1174422811yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D7
commutator, socleC22×C7C24×C4
FittingC7×#2984C2
prime radicalC2×#1157C23
cosocleC2×C4×C7C25
hypercenter#2984D7
hypocenterC7#60253
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ #2984

Complements

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

NH ≅ G/Ntype
2. C7×#2984C2semidirect
3. #11739880C2semidirect
4. #11737412C2semidirect
5. #11737410C2semidirect
6. #11737405C2semidirect
7. #11737403C2semidirect
8. #11737366C2semidirect
9. #11737365C2semidirect
10. Q8×#1131C2semidirect
11. C8×Q8×D7C2semidirect
13. #11736761C2semidirect
15. #11736712C2semidirect
169. #2984D7semidirect
575. C7#60253semidirect

Maximal subgroups

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

subgroupindex
Q8×#11572
C7×#29842
#117398802
#117374122
#117374102
#117374052
#117374032
#117373662
#117373652
Q8×#11312
C8×Q8×D72
#117367792
#117367612
#117367282
#117367122
#602537