Finite Group Databasebeta

← #11734127

Subgroups — #11734127

Order 896.

710 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6502C211yesmaximal, characteristic, complemented
2C2×#6502C211yesmaximal, characteristic, complemented
3C2×#6498C211yesmaximal, characteristic, complemented
4C2×D4×Dic7C211yesmaximal, characteristic, complemented
5C7×#392C211yesmaximal, characteristic, complemented
6#89995C211yesmaximal, characteristic, complemented
7#89992C211yesmaximal, characteristic, complemented
8#89733C211yesmaximal, characteristic, complemented
9#89730C211yesmaximal, characteristic, complemented
10#89729C211yesmaximal, characteristic, complemented
11#89718C211yesmaximal, characteristic, complemented
12#89716C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic7C211yesmaximal, characteristic, complemented
14#89441C211yesmaximal, characteristic, complemented
15#89429C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C22×C7×D4C2211yescharacteristic
18C22×C7×D4C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C22×C22⋊D7C2211yescharacteristic
21C22×C22⋊D7C2211yescharacteristic
22C23×Dic7C2211yescharacteristic
23C23×Dic7C2211yescharacteristic
24C23×Dic7C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C7×C22⋊C4C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31C2×C22⋊Dic7C2211yescharacteristic
32C2×C22⋊Dic7C2211yescharacteristic
33C2×C22⋊Dic7C2211yescharacteristic
34#6502—24no—
35#6502—24no—
36#6498—24no—
37D4×Dic7—24no—
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×C4×Dic7C2211yescharacteristic
48C2×C4×Dic7C2211yescharacteristic
49C7×#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53C7⋊#95C2211yescharacteristic
54C7⋊#95C2211yescharacteristic
55#2464—71nomaximal, Sylow
56C24×C7C2311yescharacteristic
57C24×C7C2311yescharacteristic
58C24×C7C2311yescharacteristic
59C23×D7C2311yescharacteristic
60C2×C7×D4—22no—
61C2×C7×D4—22no—
62C2×C7×D4—22no—
63C2×C7×D4—22no—
64C2×C7×D4—22no—
65C2×C7×D4—22no—
66C2×C7×D4—22no—
67C2×C7×D4—41no—
68C2×C7×D4—41no—
69C22×C4×C7C2311yescharacteristic
70C22×C4×C7C2311yescharacteristic
71C22×C4×C7C2311yescharacteristic
72C22×C4×C7C2311yescharacteristic
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—41no—
80C2×C22⋊D7—41no—
81C2×C22⋊D7—41no—
82C22×Dic7—22no—
83C22×Dic7—22no—
84C22×Dic7—22no—
85C22×Dic7—22no—
86C22×Dic7—22no—
87C22×Dic7—22no—
88C22×Dic7—41no—
89C22×Dic7—41no—
90C22×Dic7—41no—
91C22×Dic7C2311yescharacteristic
92C22×Dic7C2311yescharacteristic
93C22×Dic7C2311yescharacteristic
94C22×Dic7C2311yescharacteristic
95C22×Dic7C2311yescharacteristic
96C22×Dic7C2311yescharacteristic
97C22×Dic7C2311yescharacteristic
98C7×C22⋊C4—22no—
99C7×C22⋊C4—41no—
100C7×C22⋊C4—41no—
101C22⋊Dic7—22no—
102C22⋊Dic7—22no—
103C22⋊Dic7—22no—
104C22⋊Dic7—22no—
105C22⋊Dic7D414yescomplemented
106C22⋊Dic7D414yescomplemented
107(C2×D7)⋊C4—22no—
108(C2×D7)⋊C4—22no—
109(C2×D7)⋊C4—22no—
110(C2×D7)⋊C4—41no—
111C4⋊Dic7—22no—
112Dic7⋊C4—22no—
113Dic7⋊C4—22no—
114Dic7⋊C4—22no—
115Dic7⋊C4—22no—
116C4×Dic7—22no—
117C4×Dic7—41no—
118C2×#123—71no—
119C2×#121—71no—
120C2×#121—71no—
121C2×C4×D4—71no—
122#398—71no—
123#394—71no—
124#392—71no—
125#392D711yescharacteristic, complemented
126#390—71no—
127#388—71no—
128#388—71no—
129#386—71no—
130#386—71no—
131#385—71no—
132C4×C22⋊C4—71no—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7—21no—
136C23×C7—21no—
137C23×C7—21no—
138C23×C7—21no—
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×C7C2×D412yes—
152C23×C7C2×D412yes—
153C23×C7#4112yes—
154C23×C7C2411yescharacteristic
155C22×D7—21no—
156C22×D7—21no—
157C22×D7—21no—
158C22×D7—21no—
159C22×D7—21no—
160C22×D7—21no—
161C22×D7—21no—
162C7×D4—28no—
163C7×D4—44no—
164C7×D4—44no—
165C2×C4×C7—21no—
166C2×C4×C7—21no—
167C2×C4×C7—21no—
168C2×C4×C7—21no—
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#4112yes—
177C22⋊D7—44no—
178C22⋊D7—44no—
179C22⋊D7—44no—
180C2×Dic7—21no—
181C2×Dic7—21no—
182C2×Dic7—21no—
183C2×Dic7—21no—
184C2×Dic7—21no—
185C2×Dic7—21no—
186C2×Dic7—21no—
187C2×Dic7—21no—
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—44no—
197C2×Dic7—44no—
198C2×Dic7—44no—
199C2×Dic7C2×D412yes—
200C2×Dic7C2×D412yes—
201C2×Dic7C2×D412yes—
202C2×Dic7C2×D412yes—
203C2×Dic7#4112yes—
204C22×D4—71no—
205C22×D4—71no—
206C22×D4—71no—
207C22×D4C2×D711yescharacteristic
208C22×D4C2×D711yescharacteristic
209C22×D4C2×D711yescharacteristic
210C23×C4—71no—
211C23×C4—71no—
212C23×C4—71no—
213#123—144no—
214#121—144no—
215#121—144no—
216C4×D4—144no—
217C2×C4⋊C4—71no—
218C2×C4⋊C4—71no—
219C2×C4⋊C4—71no—
220C2×C4⋊C4—71no—
221C2×C4⋊C4—71no—
222C2×C22⋊C4—71no—
223C2×C22⋊C4—71no—
224C2×C22⋊C4—71no—
225C2×C22⋊C4—71no—
226C2×C22⋊C4—71no—
227C2×C22⋊C4—71no—
228C2×C22⋊C4—71no—
229C2×C22⋊C4—71no—
230C2×C22⋊C4—71no—
231C2×C22⋊C4—71no—
232C2×C22⋊C4C2×D711yescharacteristic
233C2×C22⋊C4C2×D711yescharacteristic
234C2×C22⋊C4C2×D711yescharacteristic
235C2×C42—71no—
236C2×C42—71no—
237#95—71no—
238#95—71no—
239#95—71no—
240#95—71no—
241#95—71no—
242#95C2×D711yescharacteristic
243C22×C7—22no—
244C22×C7—22no—
245C22×C7—22no—
246C22×C7—22no—
247C22×C7—22no—
248C22×C7—22no—
249C22×C7—22no—
250C22×C7—22no—
251C22×C7—22no—
252C22×C7—41no—
253C22×C7—41no—
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×C7C22×D411yescharacteristic
265C22×C7C22×D411yescharacteristic
266C22×C7C2×#4111yescharacteristic
267C22×C7C2×#4111yescharacteristic
268C22×C7C2×#4111yescharacteristic
269C22×C7H(5,𝔽2)11yescharacteristic
270C22×C7H(5,𝔽2)11yescharacteristic
271C2×D7—41no—
272C2×D7—41no—
273C2×D7—41no—
274C2×D7—41no—
275C2×D7—41no—
276C2×D7—41no—
277C2×D7—41no—
278C4×C7—22no—
279C4×C7—41no—
280C4×C7—41no—
281C4×C7—41no—
282Dic7—22no—
283Dic7—22no—
284Dic7—22no—
285Dic7—22no—
286Dic7—22no—
287Dic7—41no—
288Dic7—41no—
289C24—71no—
290C24C22×D711yescharacteristic
291C24C22×D711yescharacteristic
292C24C22×D711yescharacteristic
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—142no—
303C2×D4—142no—
304C2×D4—142no—
305C2×D4—142no—
306C2×D4—142no—
307C2×D4—142no—
308C2×D4—281no—
309C2×D4—281no—
310C2×D4—281no—
311C22×C4—71no—
312C22×C4—71no—
313C22×C4—71no—
314C22×C4—71no—
315C22×C4—71no—
316C22×C4—71no—
317C22×C4—71no—
318C22×C4—142no—
319C22×C4—142no—
320C22×C4—142no—
321C22×C4—142no—
322C22×C4—142no—
323C22×C4—142no—
324C22×C4—281no—
325C22×C4—281no—
326C22×C4—281no—
327C22×C4C22×D711yescharacteristic
328C22×C4C22×D711yescharacteristic
329C22×C4C22×D711yescharacteristic
330C22×C4C22×D711yescharacteristic
331C4⋊C4—142no—
332C4⋊C4—142no—
333C4⋊C4—142no—
334C4⋊C4—142no—
335C4⋊C4—142no—
336C22⋊C4—22no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—74no—
340C22⋊C4—74no—
341C22⋊C4—142no—
342C22⋊C4—142no—
343C22⋊C4—142no—
344C22⋊C4—142no—
345C22⋊C4—142no—
346C22⋊C4—142no—
347C22⋊C4—142no—
348C22⋊C4—281no—
349C42—142no—
350C42—281no—
351C2×C7—42no—
352C2×C7—42no—
353C2×C7—42no—
354C2×C7#52511yescharacteristic
355C2×C7#53211yescharacteristic
356C2×C7#54011yescharacteristic
357C2×C7D4211yescharacteristic
358C2×C7#54611yescharacteristic
359C2×C7#54611yescharacteristic
360C2×C7#55311yescharacteristic
361D7—81no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—141no—
381C23—141no—
382C23—141no—
383C23—141no—
384C23—141no—
385C23—141no—
386C23—141no—
387C23D4×D712yes—
388C23D4×D712yes—
389C23#115912yes—
390C23C23×D711yescharacteristic
391D4—28no—
392D4—44no—
393D4—44no—
394D4—284no—
395D4—284no—
396D4—284no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—72no—
409C2×C4—72no—
410C2×C4—72no—
411C2×C4—72no—
412C2×C4—72no—
413C2×C4—141no—
414C2×C4—141no—
415C2×C4—141no—
416C2×C4—141no—
417C2×C4—141no—
418C2×C4—141no—
419C2×C4—141no—
420C2×C4—141no—
421C2×C4—141no—
422C2×C4—141no—
423C2×C4—141no—
424C2×C4—141no—
425C2×C4—141no—
426C2×C4—141no—
427C2×C4—141no—
428C2×C4—141no—
429C2×C4—284no—
430C2×C4—284no—
431C2×C4—284no—
432C2×C4#115912yes—
433C7#246411yescharacteristic, complemented, Sylow
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—281no—
456C22—281no—
457C22—281no—
458C22—281no—
459C22—281no—
460C22—281no—
461C22—281no—
462C22C2×D4×D711yescharacteristic
463C22C2×D4×D711yescharacteristic
464C22C2×#115911yescharacteristic
465C22C2×#115911yescharacteristic
466C22#654811yescharacteristic
467C22#654811yescharacteristic
468C22D7×#4111yescharacteristic
469C4—22no—
470C4—41no—
471C4—41no—
472C4—41no—
473C4—142no—
474C4—142no—
475C4—142no—
476C4—142no—
477C4—142no—
478C4—281no—
479C4—281no—
480C2—42no—
481C2—42no—
482C2—42no—
483C2—561no—
484C2#9027711yescharacteristic
485C2#9028111yescharacteristic
486C2#9028211yescharacteristic
487C2#9028711yescharacteristic
488C2#9029211yescharacteristic
489C2#9030111yescharacteristic
490C2#9041011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#392C2
hypercenter#392D7
hypocenterC7#2464
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#6502C2semidirect
2. C2×#6502C2semidirect
3. C2×#6498C2semidirect
4. C2×D4×Dic7C2semidirect
5. C7×#392C2semidirect
6. #89995C2semidirect
7. #89992C2semidirect
8. #89733C2semidirect
9. #89730C2semidirect
10. #89729C2semidirect
11. #89718C2semidirect
12. #89716C2semidirect
13. C22⋊C4×Dic7C2semidirect
14. #89441C2semidirect
15. #89429C2semidirect
105. C22⋊Dic7D4semidirect
106. C22⋊Dic7D4semidirect
125. #392D7semidirect
433. C7#2464semidirect

Maximal subgroups

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

subgroupindex
C2×#65022
C2×#64982
C2×D4×Dic72
C7×#3922
#899952
#899922
#897332
#897302
#897292
#897182
#897162
C22⋊C4×Dic72
#894412
#894292
#24647