Finite Group Databasebeta

← #11733792

Subgroups — #11733792

Order 896.

760 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6502C211yesmaximal, characteristic, complemented
2C2×#6459C211yesmaximal, characteristic, complemented
3C2×#6448C211yesmaximal, characteristic, complemented
4C2×#6447C211yesmaximal, characteristic, complemented
5C2×(C22⋊D7)⋊C4C211yesmaximal, characteristic, complemented
6C2×#6441C211yesmaximal, characteristic, complemented
7C7×#390C211yesmaximal, characteristic, complemented
8#89995C211yesmaximal, characteristic, complemented
9#89764C211yesmaximal, characteristic, complemented
10#89732C211yesmaximal, characteristic, complemented
11#89729C211yesmaximal, characteristic, complemented
12#89729C211yesmaximal, characteristic, complemented
13#89721C211yesmaximal, characteristic, complemented
14#89711C211yesmaximal, characteristic, complemented
15C4×Dic7⋊C4C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C22×C22⋊D7C2211yescharacteristic
18C22×C22⋊D7C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C22×C22⋊D7C2211yescharacteristic
21C23×Dic7C2211yescharacteristic
22C22×D28C2211yescharacteristic
23C22×C4×D7C2211yescharacteristic
24C22×Dic14C2211yescharacteristic
25C2×C7×C4⋊C4C2211yescharacteristic
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C7×C22⋊C4C2211yescharacteristic
28C2×C7×C22⋊C4C2211yescharacteristic
29C2×C7×C22⋊C4C2211yescharacteristic
30C2×C42×C7C2211yescharacteristic
31C2×C22⋊Dic7C2211yescharacteristic
32C2×C22⋊Dic7C2211yescharacteristic
33C2×C22⋊Dic7C2211yescharacteristic
34#6502—24no—
35C2×(C2×D7)⋊C4C2211yescharacteristic
36C2×(C2×D7)⋊C4C2211yescharacteristic
37C2×(C2×D7)⋊C4C2211yescharacteristic
38C2×(C2×D7)⋊C4C2211yescharacteristic
39C2×(C2×D7)⋊C4C2211yescharacteristic
40C2×(C2×D7)⋊C4C2211yescharacteristic
41C2×(C2×D7)⋊C4C2211yescharacteristic
42C2×C4⋊Dic7C2211yescharacteristic
43C2×Dic7⋊C4C2211yescharacteristic
44C2×Dic7⋊C4C2211yescharacteristic
45C2×Dic7⋊C4C2211yescharacteristic
46C2×Dic7⋊C4C2211yescharacteristic
47C2×Dic7⋊C4C2211yescharacteristic
48C2×C4×Dic7C2211yescharacteristic
49C2×C4×Dic7C2211yescharacteristic
50#6459—24no—
51#6448—24no—
52#6447—24no—
53(C22⋊D7)⋊C4—24no—
54#6441—24no—
55C7⋊#95C2211yescharacteristic
56C7⋊#95C2211yescharacteristic
57#2540—71nomaximal, Sylow
58C24×C7C2311yescharacteristic
59C24×C7C2311yescharacteristic
60C23×D7C2311yescharacteristic
61C23×D7C2311yescharacteristic
62C2×C7×D4—22no—
63C2×C7×D4—22no—
64C2×C7×D4—41no—
65C22×C4×C7C2311yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C22×C4×C7C2311yescharacteristic
68C22×C4×C7C2311yescharacteristic
69C22×C4×C7C2311yescharacteristic
70C2×C22⋊D7—22no—
71C2×C22⋊D7—22no—
72C2×C22⋊D7—22no—
73C2×C22⋊D7—22no—
74C2×C22⋊D7—22no—
75C2×C22⋊D7—22no—
76C2×C22⋊D7—22no—
77C2×C22⋊D7—22no—
78C2×C22⋊D7—22no—
79C2×C22⋊D7—22no—
80C2×C22⋊D7—41no—
81C2×C22⋊D7—41no—
82C22×Dic7—22no—
83C22×Dic7—22no—
84C22×Dic7—22no—
85C22×Dic7C2311yescharacteristic
86C22×Dic7C2311yescharacteristic
87C22×Dic7C2311yescharacteristic
88C22×Dic7C2311yescharacteristic
89C22×Dic7C2311yescharacteristic
90C22×Dic7C2311yescharacteristic
91C2×D28—22no—
92C2×D28—22no—
93C2×D28—41no—
94C2×C4×D7—22no—
95C2×C4×D7—22no—
96C2×C4×D7—22no—
97C2×Dic14—22no—
98C2×Dic14—22no—
99C2×Dic14—22no—
100C7×C4⋊C4—22no—
101C7×C22⋊C4—22no—
102C7×C22⋊C4—22no—
103C7×C22⋊C4—22no—
104C7×C22⋊C4—22no—
105C42×C7—41no—
106C22⋊Dic7—22no—
107C22⋊Dic7—22no—
108C22⋊Dic7—22no—
109(C2×D7)⋊C4—22no—
110(C2×D7)⋊C4—22no—
111(C2×D7)⋊C4—22no—
112(C2×D7)⋊C4—22no—
113(C2×D7)⋊C4—22no—
114(C2×D7)⋊C4—22no—
115(C2×D7)⋊C4—22no—
116C4⋊Dic7—22no—
117Dic7⋊C4—22no—
118Dic7⋊C4—22no—
119Dic7⋊C4—22no—
120Dic7⋊C4D414yescomplemented
121Dic7⋊C4D414yescomplemented
122C4×Dic7—41no—
123C4×Dic7D414yescomplemented
124C2×#124—71no—
125C2×C22⋊Q8—71no—
126C2×C22⋊Q8—71no—
127C2×#121—71no—
128C2×#121—71no—
129C2×C4×D4—71no—
130#399—71no—
131#399—71no—
132#392—71no—
133#392—71no—
134#390—71no—
135#390—71no—
136#390—71no—
137#390D711yescharacteristic, complemented
138C4×C4⋊C4—71no—
139C23×C7—21no—
140C23×C7—21no—
141C23×C7—21no—
142C23×C7—21no—
143C23×C7—21no—
144C23×C7—21no—
145C23×C7—21no—
146C23×C7—21no—
147C23×C7—21no—
148C23×C7—21no—
149C23×C7—21no—
150C23×C7—21no—
151C23×C7—21no—
152C23×C7#4112yes—
153C23×C7C2411yescharacteristic
154C22×D7—21no—
155C22×D7—21no—
156C22×D7—21no—
157C22×D7—21no—
158C22×D7—21no—
159C22×D7—21no—
160C22×D7—21no—
161C22×D7—21no—
162C22×D7—21no—
163C22×D7—21no—
164C22×D7—21no—
165C22×D7—21no—
166C22×D7—21no—
167C22×D7#4112yes—
168C7×D4—44no—
169C2×C4×C7—21no—
170C2×C4×C7—21no—
171C2×C4×C7—21no—
172C2×C4×C7—21no—
173C2×C4×C7—21no—
174C2×C4×C7—21no—
175C2×C4×C7—21no—
176C2×C4×C7—21no—
177C2×C4×C7—21no—
178C2×C4×C7—21no—
179C2×C4×C7—21no—
180C2×C4×C7—21no—
181C2×C4×C7C2×D412yes—
182C2×C4×C7C2×D412yes—
183C2×C4×C7C2×D412yes—
184C22⋊D7—28no—
185C22⋊D7—44no—
186C22⋊D7—44no—
187C22⋊D7—44no—
188C2×Dic7—21no—
189C2×Dic7—21no—
190C2×Dic7—21no—
191C2×Dic7—21no—
192C2×Dic7—21no—
193C2×Dic7—21no—
194C2×Dic7—21no—
195C2×Dic7—21no—
196C2×Dic7—21no—
197C2×Dic7—21no—
198C2×Dic7—21no—
199C2×Dic7—21no—
200C2×Dic7—44no—
201C2×Dic7C2×D412yes—
202C2×Dic7C2×D412yes—
203C2×Dic7C2×D412yes—
204C2×Dic7C2×D412yes—
205C2×Dic7C2×D412yes—
206C2×Dic7C2×D412yes—
207D28—44no—
208C4×D7—44no—
209Dic14—44no—
210C22×Q8—71no—
211C22×D4—71no—
212C22×D4—71no—
213C22×D4—71no—
214C22×D4—71no—
215C22×D4—71no—
216C22×D4C2×D711yescharacteristic
217C23×C4—71no—
218C23×C4—71no—
219#124—144no—
220C22⋊Q8—144no—
221C22⋊Q8—144no—
222#121—144no—
223#121—144no—
224C4×D4—144no—
225C2×C4⋊C4—71no—
226C2×C4⋊C4—71no—
227C2×C4⋊C4—71no—
228C2×C4⋊C4—71no—
229C2×C4⋊C4—71no—
230C2×C4⋊C4—71no—
231C2×C4⋊C4C2×D711yescharacteristic
232C2×C22⋊C4—71no—
233C2×C22⋊C4—71no—
234C2×C22⋊C4—71no—
235C2×C22⋊C4—71no—
236C2×C22⋊C4—71no—
237C2×C22⋊C4—71no—
238C2×C22⋊C4—71no—
239C2×C22⋊C4—71no—
240C2×C22⋊C4—71no—
241C2×C22⋊C4—71no—
242C2×C22⋊C4C2×D711yescharacteristic
243C2×C22⋊C4C2×D711yescharacteristic
244C2×C22⋊C4C2×D711yescharacteristic
245C2×C22⋊C4C2×D711yescharacteristic
246C2×C42—71no—
247C2×C42—71no—
248C2×C42C2×D711yescharacteristic
249#95—71no—
250#95—71no—
251C22×C7—22no—
252C22×C7—22no—
253C22×C7—22no—
254C22×C7—41no—
255C22×C7—41no—
256C22×C7—41no—
257C22×C7—41no—
258C22×C7—41no—
259C22×C7—41no—
260C22×C7—41no—
261C22×C7—41no—
262C22×C7—41no—
263C22×C7—41no—
264C22×C7—41no—
265C22×C7C22×D411yescharacteristic
266C22×C7C22×D411yescharacteristic
267C22×C7C22×D411yescharacteristic
268C22×C7C2×#4111yescharacteristic
269C22×C7C2×#4111yescharacteristic
270C22×C7H(5,𝔽2)11yescharacteristic
271C22×C72−1+411yescharacteristic
272C2×D7—22no—
273C2×D7—22no—
274C2×D7—22no—
275C2×D7—41no—
276C2×D7—41no—
277C2×D7—41no—
278C2×D7—41no—
279C2×D7—41no—
280C2×D7—41no—
281C2×D7—41no—
282C2×D7—41no—
283C2×D7—41no—
284C2×D7—41no—
285C2×D7—41no—
286C4×C7—22no—
287C4×C7—22no—
288C4×C7—22no—
289C4×C7—41no—
290C4×C7—41no—
291Dic7—22no—
292Dic7—22no—
293Dic7—22no—
294Dic7—41no—
295Dic7—41no—
296Dic7#12714yes—
297C24—71no—
298C24—71no—
299C24C22×D711yescharacteristic
300C24C22×D711yescharacteristic
301C2×Q8—142no—
302C2×Q8—142no—
303C2×Q8—142no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—41no—
307C2×D4—142no—
308C2×D4—142no—
309C2×D4—142no—
310C2×D4—142no—
311C2×D4—142no—
312C2×D4—142no—
313C2×D4—142no—
314C2×D4—142no—
315C2×D4—142no—
316C2×D4—142no—
317C2×D4—142no—
318C2×D4—142no—
319C2×D4—281no—
320C2×D4—281no—
321C2×D4—281no—
322C22×C4—71no—
323C22×C4—71no—
324C22×C4—71no—
325C22×C4—71no—
326C22×C4—71no—
327C22×C4—71no—
328C22×C4—142no—
329C22×C4—142no—
330C22×C4—142no—
331C22×C4—142no—
332C22×C4—142no—
333C22×C4—142no—
334C22×C4C22×D711yescharacteristic
335C22×C4C22×D711yescharacteristic
336C22×C4C22×D711yescharacteristic
337C22×C4C22×D711yescharacteristic
338C22×C4C22×D711yescharacteristic
339C4⋊C4—22no—
340C4⋊C4—74no—
341C4⋊C4—74no—
342C4⋊C4—142no—
343C4⋊C4—142no—
344C4⋊C4—142no—
345C4⋊C4—142no—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—22no—
350C22⋊C4—142no—
351C22⋊C4—142no—
352C22⋊C4—142no—
353C22⋊C4—142no—
354C22⋊C4—142no—
355C22⋊C4—142no—
356C22⋊C4—142no—
357C22⋊C4—142no—
358C22⋊C4—142no—
359C22⋊C4—142no—
360C42—41no—
361C42—74no—
362C42—281no—
363C2×C7—42no—
364C2×C7—81no—
365C2×C7#52511yescharacteristic
366C2×C7C2×#12711yescharacteristic
367C2×C7#53711yescharacteristic
368C2×C7#54611yescharacteristic
369C2×C7#54611yescharacteristic
370C2×C7#54811yescharacteristic
371C2×C7#54811yescharacteristic
372D7—42no—
373D7—81no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—72no—
388C23—141no—
389C23—141no—
390C23—141no—
391C23—141no—
392C23—141no—
393C23—141no—
394C23—141no—
395C23—141no—
396C23—141no—
397C23—141no—
398C23—141no—
399C23—141no—
400C23—141no—
401C23#115712yes—
402C23C23×D711yescharacteristic
403Q8—284no—
404D4—44no—
405D4—148no—
406D4—284no—
407D4—284no—
408D4—284no—
409D4—284no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—72no—
423C2×C4—72no—
424C2×C4—72no—
425C2×C4—72no—
426C2×C4—72no—
427C2×C4—72no—
428C2×C4—141no—
429C2×C4—141no—
430C2×C4—141no—
431C2×C4—141no—
432C2×C4—141no—
433C2×C4—141no—
434C2×C4—141no—
435C2×C4—141no—
436C2×C4—141no—
437C2×C4—141no—
438C2×C4—141no—
439C2×C4—141no—
440C2×C4—284no—
441C2×C4—284no—
442C2×C4D4×D712yes—
443C2×C4D4×D712yes—
444C2×C4D4×D712yes—
445C7#254011yescharacteristic, complemented, Sylow
446C22—22no—
447C22—22no—
448C22—22no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—142no—
461C22—142no—
462C22—142no—
463C22—281no—
464C22—281no—
465C22—281no—
466C22—281no—
467C22—281no—
468C22—281no—
469C22—281no—
470C22—281no—
471C22—281no—
472C22—281no—
473C22—281no—
474C22C2×#115711yescharacteristic
475C22C2×D4×D711yescharacteristic
476C22C2×D4×D711yescharacteristic
477C22C2×D4×D711yescharacteristic
478C22#654811yescharacteristic
479C22D7×#4111yescharacteristic
480C22#655411yescharacteristic
481C4—22no—
482C4—22no—
483C4—22no—
484C4—41no—
485C4—41no—
486C4—74no—
487C4—142no—
488C4—142no—
489C4—142no—
490C4—281no—
491C4—281no—
492C2—42no—
493C2—81no—
494C2—282no—
495C2—561no—
496C2#9018111yescharacteristic
497C2#9023811yescharacteristic
498C2#9024311yescharacteristic
499C2#9028811yescharacteristic
500C2#9030111yescharacteristic
501C2#9036811yescharacteristic
502C2D7×#12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#390C2
hypercenter#390D7
hypocenterC7#2540
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#6502C2semidirect
2. C2×#6459C2semidirect
3. C2×#6448C2semidirect
4. C2×#6447C2semidirect
5. C2×(C22⋊D7)⋊C4C2semidirect
6. C2×#6441C2semidirect
7. C7×#390C2semidirect
8. #89995C2semidirect
9. #89764C2semidirect
10. #89732C2semidirect
11. #89729C2semidirect
12. #89729C2semidirect
13. #89721C2semidirect
14. #89711C2semidirect
15. C4×Dic7⋊C4C2semidirect
120. Dic7⋊C4D4semidirect
121. Dic7⋊C4D4semidirect
123. C4×Dic7D4semidirect
137. #390D7semidirect
445. C7#2540semidirect

Maximal subgroups

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

subgroupindex
C2×#65022
C2×#64592
C2×#64482
C2×#64472
C2×(C22⋊D7)⋊C42
C2×#64412
C7×#3902
#899952
#897642
#897322
#897292
#897212
#897112
C4×Dic7⋊C42
#25407