Finite Group Databasebeta

← #13249154

Subgroups — #13249154

Order 1792.

670 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6509C211yesmaximal, characteristic
2C2×C4×#1131C211yesmaximal, characteristic
3C2×C7×Q8⋊C8C211yesmaximal, characteristic, complemented
4C2×Q8⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
5#11728229C211yesmaximal, characteristic
6#11725510C211yesmaximal, characteristic, complemented
7#11725461C211yesmaximal, characteristic, complemented
8C2×C4×C7×Q8C2211yescharacteristic
9C2×#6509C411yescharacteristic
10C22×#1131—21no—
11C2×C42×D7C2211yescharacteristic
12C2×C7×C4⋊C8C2211yescharacteristic
13C2×C4×C7×C8C2211yescharacteristic
14#90002C411yescharacteristic
15C2×(C2×D7)⋊C8—21no—
16C2×#6389—21no—
17#89761—21no—
18#89759C411yescharacteristic
19#89750—21no—
20#89745C411yescharacteristic
21C4×(C2×D7)⋊C4—21no—
22C4×C4⋊Dic7C2211yescharacteristic
23C4×Dic7⋊C4—21no—
24C2×C4⋊(C7⋊C8)C2211yescharacteristic
25C2×C4×C7⋊C8C2211yescharacteristic
26C4×#1131—22no—
27C4×#1131—22no—
28C7×Q8⋊C8—22no—
29#89345—21no—
30Q8⋊(C7⋊C8)—22no—
31#9077—71nomaximal, Sylow
32C22×C7×Q8C2×C411yescharacteristic
33C22×C4×D7C2×C411yescharacteristic
34C22×C4×D7D411yescharacteristic
35C22×C4×D7D411yescharacteristic
36C22×C7×C8—21no—
37C22×C7×C8—21no—
38C4×C7×Q8—21no—
39C4×C7×Q8—21no—
40C4×C7×Q8—21no—
41C4×C7×Q8—21no—
42C2×C7×C4⋊C4—21no—
43C2×C7×C4⋊C4C2×C411yescharacteristic
44C2×C42×C7—21no—
45C2×C42×C7C2311yescharacteristic
46#6509—22no—
47#6509—22no—
48C2×(C2×D7)⋊C4—21no—
49C2×(C2×D7)⋊C4—21no—
50C2×C4⋊Dic7C2×C411yescharacteristic
51C2×C4⋊Dic7C2×C411yescharacteristic
52C2×Dic7⋊C4—21no—
53C2×Dic7⋊C4—21no—
54C2×C4×Dic7—21no—
55C2×C4×Dic7C2×C411yescharacteristic
56C2×C4×Dic7D411yescharacteristic
57C2×C4×Dic7D411yescharacteristic
58C22×C7⋊C8—21no—
59C22×C7⋊C8—21no—
60C2×#1131—24no—
61C2×#1131—24no—
62C2×#1131—24no—
63C42×D7—21no—
64C42×D7—21no—
65C42×D7—21no—
66C42×D7—21no—
67C7×C4⋊C8—21no—
68C7×C4⋊C8—21no—
69C4×C7×C8—21no—
70C4×C7×C8D412yes—
71C7⋊#95—21no—
72C7⋊#95—21no—
73C7⋊#95—21no—
74C7⋊#95—21no—
75C7⋊#95—21no—
76(C2×D7)⋊C8—42no—
77#6389—24no—
78C4⋊(C7⋊C8)—21no—
79C4⋊(C7⋊C8)—21no—
80C4×C7⋊C8—21no—
81C4×C7⋊C8D412yes—
82C2×C4×M16—71no—
83C4×C22⋊Q8—71no—
84#1797—71no—
85C2×Q8⋊C8—71no—
86C2×Q8⋊C8D711yescharacteristic, complemented
87#1279—71no—
88#1279—71no—
89C23×D7C22⋊C411yescharacteristic
90C2×C7×Q8—21no—
91C2×C7×Q8—21no—
92C2×C7×Q8—21no—
93C2×C7×Q8—21no—
94C2×C7×Q8—21no—
95C2×C7×Q8M1612yescomplemented
96C22×C4×C7—21no—
97C22×C4×C7—21no—
98C22×C4×C7C22×C411yescharacteristic
99C22×C4×C7C2×D411yescharacteristic
100C22×C4×C7C2×D411yescharacteristic
101C22×Dic7—21no—
102C22×Dic7—21no—
103C22×Dic7C22⋊C411yescharacteristic
104C22×Dic7C22⋊C411yescharacteristic
105C22×Dic7C22⋊C411yescharacteristic
106C2×C4×D7—22no—
107C2×C4×D7—21no—
108C2×C4×D7—21no—
109C2×C4×D7—22no—
110C2×C4×D7—21no—
111C2×C4×D7—21no—
112C2×C4×D7—21no—
113C2×C4×D7—21no—
114C2×C4×D7—21no—
115C2×C4×D7—21no—
116C2×C4×D7—21no—
117C2×C4×D7—21no—
118C2×C4×D7—21no—
119C2×C4×D7—21no—
120C2×C4×D7—21no—
121C2×C4×D7—21no—
122C2×C7×C8—22no—
123C2×C7×C8—22no—
124C2×C7×C8—22no—
125C2×C7×C8—41no—
126C2×C7×C8—41no—
127C2×C7×C8—41no—
128C7×C4⋊C4—21no—
129C7×C4⋊C4—21no—
130C7×C4⋊C4—41no—
131C7×C4⋊C4—41no—
132C42×C7—41no—
133C42×C7—41no—
134C42×C7C2×D411yescharacteristic
135C42×C7C2×D411yescharacteristic
136C42×C7#4111yescharacteristic
137C42×C7#4111yescharacteristic
138(C2×D7)⋊C4—24no—
139(C2×D7)⋊C4—42no—
140C4⋊Dic7—21no—
141C4⋊Dic7—21no—
142C4⋊Dic7—21no—
143C4⋊Dic7M1612yescomplemented
144Dic7⋊C4—24no—
145Dic7⋊C4—42no—
146C4×Dic7—21no—
147C4×Dic7—21no—
148C4×Dic7—21no—
149C4×Dic7—21no—
150C4×Dic7—21no—
151C4×Dic7—21no—
152C4×Dic7—41no—
153C4×Dic7—41no—
154C2×C7⋊C8—22no—
155C2×C7⋊C8—22no—
156C2×C7⋊C8—22no—
157C2×C7⋊C8—41no—
158C2×C7⋊C8—41no—
159C2×C7⋊C8—41no—
160#1131—216no—
161C22×M16—141no—
162C2×C22⋊Q8—71no—
163C2×C4×Q8C2×D711yescharacteristic
164C22×C42—71no—
165C2×C4⋊C8—71no—
166C2×C4⋊C8C2×D711yescharacteristic
167C2×C22⋊C8—141no—
168C4×M16—142no—
169C4×M16—142no—
170C2×#97—141no—
171C2×C4×C8—71no—
172C2×C4×C8C2×D711yescharacteristic
173#391—71no—
174#389—71no—
175#387—141no—
176#385—141no—
177#380—71no—
178C4×C4⋊C4—71no—
179C4×C4⋊C4—141no—
180C4×C22⋊C4—141no—
181#336—141no—
182Q8⋊C8—22no—
183Q8⋊C8—142no—
184C23×C7C2×C22⋊C411yescharacteristic
185C22×D7—21no—
186C22×D7—21no—
187C22×D7—21no—
188C22×D7—21no—
189C22×D7—21no—
190C22×D7—21no—
191C22×D7#10412yes—
192C7×Q8—22no—
193C7×Q8—22no—
194C7×Q8—41no—
195C7×Q8—41no—
196C2×C4×C7—21no—
197C2×C4×C7—21no—
198C2×C4×C7—22no—
199C2×C4×C7—41no—
200C2×C4×C7—41no—
201C2×C4×C7—41no—
202C2×C4×C7—41no—
203C2×C4×C7—41no—
204C2×C4×C7C4×D411yescharacteristic
205C2×C4×C7C4×D411yescharacteristic
206C2×C4×C7#12011yescharacteristic
207C2×C4×C7#12111yescharacteristic
208C2×C4×C7#12111yescharacteristic
209C2×C4×C7#12311yescharacteristic
210C2×C4×C7C2×M1611yescharacteristic
211C2×C4×C7C2×M1611yescharacteristic
212C2×C4×C7#13111yescharacteristic
213C2×C4×C7#13111yescharacteristic
214C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
215C2×C4×C7#13711yescharacteristic
216C2×C4×C7#13711yescharacteristic
217C2×C4×C7#13711yescharacteristic
218C2×Dic7—21no—
219C2×Dic7—21no—
220C2×Dic7—21no—
221C2×Dic7—21no—
222C2×Dic7—21no—
223C2×Dic7—22no—
224C2×Dic7—21no—
225C2×Dic7—21no—
226C2×Dic7—21no—
227C2×Dic7—41no—
228C2×Dic7—41no—
229C2×Dic7—41no—
230C2×Dic7—41no—
231C2×Dic7—41no—
232C2×Dic7#10412yes—
233C4×D7—24no—
234C4×D7—24no—
235C4×D7—41no—
236C4×D7—41no—
237C4×D7—41no—
238C4×D7—41no—
239C4×D7—41no—
240C4×D7—41no—
241C4×D7—41no—
242C4×D7—41no—
243C7×C8—24no—
244C7×C8—81no—
245C7⋊C8—24no—
246C7⋊C8—81no—
247C22×Q8C4×D711yescharacteristic
248C23×C4—71no—
249C23×C4—71no—
250C23×C4—71no—
251C2×M16—144no—
252C2×M16—144no—
253C2×M16—144no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—141no—
257C22×C8—141no—
258C22⋊Q8—142no—
259C22⋊Q8—142no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×Q8—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—71no—
266C2×C4⋊C4—71no—
267C2×C4⋊C4—141no—
268C2×C4⋊C4—141no—
269C2×C4⋊C4C4×D711yescharacteristic
270C2×C22⋊C4—141no—
271C2×C22⋊C4—141no—
272C2×C42—21no—
273C2×C42—71no—
274C2×C42—71no—
275C2×C42—71no—
276C2×C42—141no—
277C2×C42—141no—
278C2×C42—141no—
279C2×C42—141no—
280C2×C42—141no—
281C2×C42C22×D711yescharacteristic
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—141no—
285C4⋊C8—141no—
286C22⋊C8—282no—
287#97—144no—
288C4×C8—21no—
289C4×C8—72no—
290C4×C8—141no—
291C4×C8C22⋊D712yes—
292#95—141no—
293#95—141no—
294#95—141no—
295#95—141no—
296#95—141no—
297C22×C7#38611yescharacteristic
298C22×C7#40711yescharacteristic
299C22×C7#40811yescharacteristic
300C22×C7#41711yescharacteristic
301C22×C7#41911yescharacteristic
302C22×C7C2×#10411yescharacteristic
303C22×C7C2×#10411yescharacteristic
304C2×D7—22no—
305C2×D7—22no—
306C2×D7—22no—
307C2×D7—41no—
308C2×D7—41no—
309C2×D7—41no—
310C2×D7—41no—
311C4×C7—21no—
312C4×C7—21no—
313C4×C7—42no—
314C4×C7—81no—
315C4×C7#43511yescharacteristic
316C4×C7#43511yescharacteristic
317C4×C7#43611yescharacteristic
318C4×C7#43611yescharacteristic
319C4×C7#46811yescharacteristic
320C4×C7#46811yescharacteristic
321C4×C7#47011yescharacteristic
322C4×C7#47011yescharacteristic
323Dic7—22no—
324Dic7—41no—
325Dic7—41no—
326Dic7—42no—
327Dic7—81no—
328C24—71no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8#113112yescomplemented
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—71no—
338C22×C4—71no—
339C22×C4—71no—
340C22×C4—142no—
341C22×C4—142no—
342C22×C4—141no—
343C22×C4—141no—
344C22×C4—141no—
345C22×C4—141no—
346C22×C4—141no—
347C22×C4—141no—
348C22×C4—141no—
349C22×C4—141no—
350C22×C4—141no—
351C22×C4—141no—
352C22×C4—141no—
353C22×C4—141no—
354C22×C4—141no—
355C22×C4—141no—
356C22×C4—141no—
357C22×C4—141no—
358C22×C4C2×C4×D711yescharacteristic
359C22×C4D4×D711yescharacteristic
360C22×C4D4×D711yescharacteristic
361M16—1416no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—142no—
369C2×C8—142no—
370C2×C8—142no—
371C2×C8—281no—
372C2×C8—281no—
373C2×C8—281no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—72no—
379C4⋊C4—144no—
380C4⋊C4—141no—
381C4⋊C4—141no—
382C4⋊C4—141no—
383C4⋊C4—282no—
384C22⋊C4—144no—
385C22⋊C4—282no—
386C42—41no—
387C42—41no—
388C42—141no—
389C42—141no—
390C42—141no—
391C42—141no—
392C42—141no—
393C42—141no—
394C42—281no—
395C42—281no—
396C42#115711yescharacteristic
397C42D4×D711yescharacteristic
398C42#115911yescharacteristic
399C42C2×C22⋊D711yescharacteristic
400C2×C7#148311yescharacteristic
401C2×C7#148511yescharacteristic
402C2×C7#148911yescharacteristic
403C2×C7#149111yescharacteristic
404C2×C7#187011yescharacteristic
405C2×C7#189411yescharacteristic
406C2×C7#189611yescharacteristic
407D7—42no—
408C23—72no—
409C23—141no—
410C23—141no—
411C23—141no—
412C23—141no—
413C23—141no—
414C23—141no—
415C23D7×C22⋊C411yescharacteristic
416Q8—22no—
417Q8—22no—
418Q8—41no—
419Q8—41no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—72no—
429C2×C4—144no—
430C2×C4—144no—
431C2×C4—141no—
432C2×C4—141no—
433C2×C4—141no—
434C2×C4—141no—
435C2×C4—142no—
436C2×C4—141no—
437C2×C4—141no—
438C2×C4—141no—
439C2×C4—141no—
440C2×C4—281no—
441C2×C4—281no—
442C2×C4—281no—
443C2×C4—281no—
444C2×C4—281no—
445C2×C4—281no—
446C2×C4—281no—
447C2×C4—281no—
448C2×C4—281no—
449C2×C4—281no—
450C2×C4—281no—
451C2×C4—281no—
452C2×C4—281no—
453C2×C4(C22⋊D7)⋊C411yescharacteristic
454C2×C4#644611yescharacteristic
455C2×C4#644711yescharacteristic
456C2×C4C2×#113111yescharacteristic
457C2×C4#646411yescharacteristic
458C2×C4D7×M1611yescharacteristic
459C2×C4#647011yescharacteristic
460C2×C4#647711yescharacteristic
461C2×C4#647811yescharacteristic
462C2×C4#648111yescharacteristic
463C2×C4#648111yescharacteristic
464C2×C4C4×C22⋊D711yescharacteristic
465C2×C4#650011yescharacteristic
466C2×C4#650111yescharacteristic
467C8—24no—
468C8—81no—
469C8—144no—
470C8—561no—
471C7#907711yescharacteristic, complemented, Sylow
472C22—142no—
473C22—142no—
474C22—142no—
475C22—281no—
476C22—281no—
477C22—281no—
478C22—281no—
479C22#8949811yescharacteristic
480C22#8949911yescharacteristic
481C22#8957411yescharacteristic
482C22#8957511yescharacteristic
483C22D7×#10411yescharacteristic
484C22D7×#10411yescharacteristic
485C22#8972811yescharacteristic
486C4—21no—
487C4—21no—
488C4—42no—
489C4—81no—
490C4—142no—
491C4—281no—
492C4—281no—
493C4—282no—
494C4—561no—
495C4#8950111yescharacteristic
496C4#8950211yescharacteristic
497C4#8958411yescharacteristic
498C4#8958811yescharacteristic
499C4#8988311yescharacteristic
500C4#8990011yescharacteristic
501C4#8994611yescharacteristic
502C4#8996111yescharacteristic
503C2—282no—
504C2#1172598811yescharacteristic
505C2#1172598911yescharacteristic
506C2#1172599211yescharacteristic
507C2#1172599311yescharacteristic
508C2#1172771211yescharacteristic
509C2#1172802511yescharacteristic
510C2#1172813611yescharacteristic
subgroupG/H
centerC22×C4D4×D7
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC2×C7×Q8⋊C8C2
socleC23×C7C2×C22⋊C4
prime radicalC23×D7C22⋊C4
cosocleC2×C42×C7C23
hypercenterC2×Q8⋊C8D7
hypocenterC7#9077
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
3. C2×C7×Q8⋊C8C2semidirect
4. C2×Q8⋊(C7⋊C8)C2semidirect
6. #11725510C2semidirect
7. #11725461C2semidirect
86. C2×Q8⋊C8D7semidirect
95. C2×C7×Q8M16semidirect
143. C4⋊Dic7M16semidirect
334. C2×Q8#1131semidirect
471. C7#9077semidirect

Maximal subgroups

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

subgroupindex
C4×#65092
C2×C4×#11312
C2×C7×Q8⋊C82
C2×Q8⋊(C7⋊C8)2
#117282292
#117255102
#117254612
#90777