Finite Group Databasebeta

← #13365836

Subgroups — #13365836

Order 1792.

848 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89618C211yesmaximal, characteristic
2C4×#6432C211yesmaximal, characteristic, complemented
3#11730744C211yesmaximal, characteristic
4C4×C7×C4⋊C8C211yesmaximal, characteristic, complemented
5#11728233C211yesmaximal, characteristic
6#11728214C211yesmaximal, characteristic, complemented
7#11728207C211yesmaximal, characteristic, complemented
8#11727524C212yesmaximal, complemented
9#11727522C212yesmaximal
10#11726379C212yesmaximal
11#11726375C212yesmaximal, complemented
12C2×C4×D28C2211yescharacteristic
13C2×#6435—21no—
14C2×#6432C2211yescharacteristic
15C2×C4×Dic14—21no—
16C2×C7×C4⋊C8C2211yescharacteristic
17C2×C7×C4⋊C8C2211yescharacteristic
18C2×C4×C7×C8C2212yes—
19C43×C7C2211yescharacteristic
20C2×#6395C2212yes—
21C2×#6395C2212yes—
22C2×#6392C2212yes—
23C2×#6392C2212yes—
24#89712—21no—
25#89712—21no—
26#89711C2211yescharacteristic
27#89711C2211yescharacteristic
28C4×(C2×D7)⋊C4—21no—
29#89707C2211yescharacteristic
30C4×C4⋊Dic7—21no—
31C4×C4⋊Dic7C2212yes—
32C4×C4⋊Dic7C2211yescharacteristic
33#89701—21no—
34#89700—21no—
35#89700C2211yescharacteristic
36#89700C2211yescharacteristic
37#89700C2212yes—
38#89618C418yescomplemented
39C7×#336C2212yes—
40#89346C2212yes—
41#89346C2212yes—
42#89346C2212yes—
43#89346C2212yes—
44#22110—71nomaximal, Sylow
45C22×D28C2311yescharacteristic
46C22×C4×D7—21no—
47C22×Dic14—21no—
48C22×C7×C8C2312yes—
49C22×C7×C8C2312yes—
50C2×C42×C7D411yescharacteristic
51C2×C42×C7D411yescharacteristic
52C2×C42×C7D411yescharacteristic
53C2×C42×C7D411yescharacteristic
54C2×C42×C7C2311yescharacteristic
55C2×C42×C7C2311yescharacteristic
56C2×C42×C7C2311yescharacteristic
57C2×(C2×D7)⋊C4—21no—
58C2×(C2×D7)⋊C4—21no—
59C2×(C2×D7)⋊C4—21no—
60C2×(C2×D7)⋊C4—21no—
61C2×(C2×D7)⋊C4—21no—
62C2×C4⋊Dic7—21no—
63C2×C4⋊Dic7—21no—
64C2×C4⋊Dic7—21no—
65C2×C4⋊Dic7C2311yescharacteristic
66C2×C4⋊Dic7C2311yescharacteristic
67C2×C4⋊Dic7C2312yes—
68C2×C4⋊Dic7C2311yescharacteristic
69C2×C4⋊Dic7C2312yes—
70C2×Dic7⋊C4—22no—
71C2×Dic7⋊C4—21no—
72C2×Dic7⋊C4—21no—
73C2×C4×Dic7—22no—
74C2×C4×Dic7—21no—
75C4×D28—22no—
76C4×D28C2×C414yes—
77#6435—44no—
78#6432—22no—
79#6432C2×C414yes—
80C4×Dic14—44no—
81C7×C4⋊C8—22no—
82C7×C4⋊C8C2×C414yes—
83C4×C7×C8—42no—
84C7⋊#95—22no—
85C7⋊#95—22no—
86C7⋊#95—21no—
87C7⋊#95—21no—
88C7⋊#95—21no—
89#6395—42no—
90#6395C2×C418yes—
91#6392—42no—
92#6392C2×C418yes—
93C2×#474—71no—
94C4×C4⋊Q8—71no—
95#2293—71no—
96#1988—71no—
97#1947—71no—
98#1942—72no—
99#1934—72no—
100#1925—72no—
101#1918—72no—
102#1802—71no—
103C4×C4⋊C8D711yescharacteristic, complemented
104C23×D7—21no—
105C22×C4×C7C2×D411yescharacteristic
106C22×C4×C7C2×D411yescharacteristic
107C22×C4×C7C2×D411yescharacteristic
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7C2×D411yescharacteristic
111C22×C4×C7C2411yescharacteristic
112C22×Dic7—22no—
113C22×Dic7—21no—
114C22×Dic7—22no—
115C22×Dic7—21no—
116C22×Dic7—21no—
117C2×D28—21no—
118C2×D28—21no—
119C2×D28—21no—
120C2×D28—21no—
121C2×D28—21no—
122C2×D28C22×C412yes—
123C2×C4×D7—24no—
124C2×C4×D7—42no—
125C2×C4×D7—42no—
126C2×Dic14—24no—
127C2×Dic14—42no—
128C2×Dic14—42no—
129C2×C7×C8—22no—
130C2×C7×C8—22no—
131C2×C7×C8—22no—
132C2×C7×C8—22no—
133C2×C7×C8—22no—
134C2×C7×C8C22×C414yes—
135C42×C7—22no—
136C42×C7—22no—
137C42×C7—21no—
138C42×C7—21no—
139C42×C7—21no—
140C42×C7—21no—
141C42×C7—41no—
142C42×C7D814yes—
143C42×C7C22×C412yes—
144C42×C7#4112yes—
145(C2×D7)⋊C4—24no—
146(C2×D7)⋊C4—42no—
147(C2×D7)⋊C4—42no—
148(C2×D7)⋊C4—42no—
149(C2×D7)⋊C4—42no—
150C4⋊Dic7—24no—
151C4⋊Dic7—22no—
152C4⋊Dic7—21no—
153C4⋊Dic7—21no—
154C4⋊Dic7—22no—
155C4⋊Dic7—21no—
156C4⋊Dic7—21no—
157C4⋊Dic7—21no—
158C4⋊Dic7—22no—
159C4⋊Dic7—42no—
160C4⋊Dic7—42no—
161C4⋊Dic7C22×C414yes—
162C4⋊Dic7C22×C412yes—
163Dic7⋊C4—44no—
164Dic7⋊C4—42no—
165Dic7⋊C4—42no—
166C4×Dic7—44no—
167C4×Dic7—42no—
168C2×C4⋊Q8—71no—
169C2×C4×Q8—141no—
170C2×C4×D4—71no—
171C2×#117—141no—
172#474—78no—
173C2×#107—72no—
174C2×#107—72no—
175C2×C4⋊C8C2×D711yescharacteristic
176C2×C4⋊C8C2×D711yescharacteristic
177C2×#102—72no—
178C2×#102—72no—
179C2×C4×C8C2×D712yes—
180#391—141no—
181#390—71no—
182#390—71no—
183#389—72no—
184#389—71no—
185#389—71no—
186#389—141no—
187#388—141no—
188#388—141no—
189#384—71no—
190C4×C4⋊C4—72no—
191C4×C4⋊C4—71no—
192C4×C4⋊C4—141no—
193C4×C22⋊C4—141no—
194C43C2×D711yescharacteristic
195#340—72no—
196#340—72no—
197#340—72no—
198#340—72no—
199#336C2×D712yes—
200C23×C7C22×D411yescharacteristic
201C22×D7—22no—
202C22×D7—41no—
203C22×D7—41no—
204C22×D7—41no—
205C22×D7—41no—
206C22×D7—41no—
207C22×D7—41no—
208C2×C4×C7—21no—
209C2×C4×C7—21no—
210C2×C4×C7—21no—
211C2×C4×C7—21no—
212C2×C4×C7—21no—
213C2×C4×C7—21no—
214C2×C4×C7—21no—
215C2×C4×C7—21no—
216C2×C4×C7—21no—
217C2×C4×C7—21no—
218C2×C4×C7—21no—
219C2×C4×C7C4×D412yes—
220C2×C4×C7C4×D412yes—
221C2×C4×C7C2×D812yes—
222C2×C4×C7C2×D812yes—
223C2×C4×C7C2×D812yes—
224C2×C4×C7#13512yes—
225C2×C4×C7#13712yes—
226C2×C4×C7C23×C411yescharacteristic
227C2×C4×C7C2×#4111yescharacteristic
228C2×C4×C7H(5,𝔽2)11yescharacteristic
229C2×C4×C72−1+411yescharacteristic
230C2×C4×C72−1+411yescharacteristic
231C2×C4×C72−1+411yescharacteristic
232C2×Dic7—24no—
233C2×Dic7—22no—
234C2×Dic7—42no—
235C2×Dic7—41no—
236C2×Dic7—42no—
237C2×Dic7—41no—
238C2×Dic7—41no—
239C2×Dic7—41no—
240C2×Dic7—42no—
241C2×Dic7—42no—
242C2×Dic7—41no—
243C2×Dic7—41no—
244C2×Dic7—41no—
245C2×Dic7—42no—
246C2×Dic7—41no—
247D28—22no—
248D28—22no—
249D28—41no—
250D28—41no—
251C4×D7—48no—
252Dic14—48no—
253C7×C8—24no—
254C7×C8—42no—
255C22×Q8—141no—
256C22×D4—71no—
257C23×C4—141no—
258C22×C8C22×D712yes—
259C22×C8C22×D712yes—
260C4⋊Q8—74no—
261C4⋊Q8—142no—
262C4×Q8—284no—
263C4×D4—74no—
264C4×D4—142no—
265#117—284no—
266C2×C4⋊C4—71no—
267C2×C4⋊C4—72no—
268C2×C4⋊C4—71no—
269C2×C4⋊C4—72no—
270C2×C4⋊C4—71no—
271C2×C4⋊C4—141no—
272C2×C4⋊C4—141no—
273C2×C4⋊C4—141no—
274C2×C4⋊C4—142no—
275C2×C4⋊C4—141no—
276C2×C4⋊C4—141no—
277C2×C22⋊C4—141no—
278C2×C22⋊C4—141no—
279C2×C22⋊C4—141no—
280C2×C22⋊C4—141no—
281C2×C22⋊C4—141no—
282C2×C42—141no—
283C2×C42—142no—
284C2×C42D2811yescharacteristic
285C2×C42D2811yescharacteristic
286C2×C42D2811yescharacteristic
287C2×C42D2811yescharacteristic
288C2×C42C22×D711yescharacteristic
289C2×C42C22×D711yescharacteristic
290C2×C42C22×D711yescharacteristic
291#107—78no—
292#107—282no—
293C4⋊C8—22no—
294C4⋊C8C4×D714yes—
295#102—78no—
296#102—282no—
297C4×C8—42no—
298#95—141no—
299#95—141no—
300#95—141no—
301#95—142no—
302#95—142no—
303C22×C7C2×C4×D411yescharacteristic
304C22×C7#53611yescharacteristic
305C22×C7#53711yescharacteristic
306C22×C7C22×D811yescharacteristic
307C22×C7C2×#13511yescharacteristic
308C22×C7C2×#13711yescharacteristic
309C22×C7#57511yescharacteristic
310C2×D7—42no—
311C2×D7—42no—
312C2×D7—42no—
313C2×D7—81no—
314C2×D7—81no—
315C2×D7—81no—
316C2×D7—81no—
317C4×C7—22no—
318C4×C7—22no—
319C4×C7—22no—
320C4×C7—22no—
321C4×C7—41no—
322C4×C7C4×D814yes—
323C4×C7#51911yescharacteristic
324C4×C7#52011yescharacteristic
325C4×C7#54111yescharacteristic
326C4×C7#54211yescharacteristic
327Dic7—44no—
328Dic7—42no—
329Dic7—81no—
330Dic7—82no—
331Dic7—81no—
332C24—141no—
333C2×Q8—144no—
334C2×Q8—282no—
335C2×Q8—282no—
336C2×D4—72no—
337C2×D4—141no—
338C2×D4—141no—
339C2×D4—141no—
340C2×D4—141no—
341C2×D4—141no—
342C22×C4—142no—
343C22×C4—141no—
344C22×C4—144no—
345C22×C4—141no—
346C22×C4—141no—
347C22×C4—142no—
348C22×C4—282no—
349C22×C4—282no—
350C22×C4C2×D2811yescharacteristic
351C22×C4C2×D2811yescharacteristic
352C22×C4C2×D2811yescharacteristic
353C22×C4C2×D2811yescharacteristic
354C22×C4C2×D2811yescharacteristic
355C22×C4C2×D2811yescharacteristic
356C22×C4C23×D711yescharacteristic
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8C2×C4×D714yes—
363C4⋊C4—74no—
364C4⋊C4—72no—
365C4⋊C4—144no—
366C4⋊C4—141no—
367C4⋊C4—142no—
368C4⋊C4—142no—
369C4⋊C4—141no—
370C4⋊C4—142no—
371C4⋊C4—141no—
372C4⋊C4—141no—
373C4⋊C4—141no—
374C4⋊C4—282no—
375C4⋊C4—282no—
376C4⋊C4—282no—
377C4⋊C4—284no—
378C4⋊C4—282no—
379C22⋊C4—144no—
380C22⋊C4—282no—
381C22⋊C4—282no—
382C22⋊C4—282no—
383C22⋊C4—282no—
384C42—22no—
385C42—22no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—41no—
391C42—282no—
392C42—284no—
393C42D5614yes—
394C42C2×C4×D712yes—
395C42#115712yes—
396C2×C7#232311yescharacteristic
397C2×C7C2×C4×D811yescharacteristic
398C2×C7C4×#13711yescharacteristic
399C2×C7#322611yescharacteristic
400C2×C7#323311yescharacteristic
401C2×C7#324511yescharacteristic
402C2×C7#324811yescharacteristic
403D7—82no—
404C23—142no—
405C23—281no—
406C23—281no—
407C23—281no—
408C23—281no—
409C23—281no—
410C23—281no—
411C23C22×D2811yescharacteristic
412Q8—288no—
413D4—142no—
414D4—142no—
415D4—281no—
416D4—281no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—142no—
429C2×C4—144no—
430C2×C4—282no—
431C2×C4—281no—
432C2×C4—281no—
433C2×C4—282no—
434C2×C4—288no—
435C2×C4—281no—
436C2×C4—281no—
437C2×C4—282no—
438C2×C4—281no—
439C2×C4—281no—
440C2×C4—282no—
441C2×C4—282no—
442C2×C4—281no—
443C2×C4—281no—
444C2×C4C4×D2812yes—
445C2×C4C4×D2812yes—
446C2×C4C2×D5612yes—
447C2×C4C2×D5612yes—
448C2×C4C2×D5612yes—
449C2×C4#646712yes—
450C2×C4#647212yes—
451C2×C4C22×C4×D711yescharacteristic
452C2×C4C2×#115711yescharacteristic
453C2×C4#654811yescharacteristic
454C2×C4#655111yescharacteristic
455C2×C4#655411yescharacteristic
456C2×C4#655411yescharacteristic
457C8—24no—
458C8—42no—
459C7#2211011yescharacteristic, complemented, Sylow
460C22—282no—
461C22—282no—
462C22—282no—
463C22—561no—
464C22—561no—
465C22—561no—
466C22—561no—
467C22C2×C4×D2811yescharacteristic
468C22#9020111yescharacteristic
469C22#9021711yescharacteristic
470C22C22×D5611yescharacteristic
471C22C2×#646711yescharacteristic
472C22C2×#647211yescharacteristic
473C22#9043911yescharacteristic
474C4—22no—
475C4—22no—
476C4—22no—
477C4—22no—
478C4—41no—
479C4—282no—
480C4—284no—
481C4—561no—
482C4—561no—
483C4—562no—
484C4C4×D5614yes—
485C4#9023711yescharacteristic
486C4#9025211yescharacteristic
487C4#9026311yescharacteristic
488C4#9027411yescharacteristic
489C2—562no—
490C2#1173120711yescharacteristic
491C2C2×C4×D5611yescharacteristic
492C2C4×#647211yescharacteristic
493C2#1173647111yescharacteristic
494C2#1173647411yescharacteristic
495C2#1173671311yescharacteristic
496C2#1173672011yescharacteristic
subgroupG/H
centerC23C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC4×C7×C4⋊C8C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenterC4×C4⋊C8D7
hypocenterC7#22110
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C43 ◁ C4×C4⋊C8

Complements

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

NH ≅ G/Ntype
2. C4×#6432C2semidirect
4. C4×C7×C4⋊C8C2semidirect
6. #11728214C2semidirect
7. #11728207C2semidirect
8. #11727524C2semidirect
11. #11726375C2semidirect
38. #89618C4semidirect
103. C4×C4⋊C8D7semidirect
459. C7#22110semidirect

Maximal subgroups

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

subgroupindex
C2×#896182
C4×#64322
#117307442
C4×C7×C4⋊C82
#117282332
#117282142
#117282072
#117275242
#117275222
#117263792
#117263752
#221107