Finite Group Databasebeta

← #11733662

Subgroups — #11733662

Order 896.

666 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6458C211yesmaximal, characteristic
2C2×#6457C211yesmaximal, characteristic, complemented
3C2×#6457C211yesmaximal, characteristic, complemented
4C2×C4×D28C211yesmaximal, characteristic, complemented
5C7×#389C211yesmaximal, characteristic, complemented
6#89766C211yesmaximal, characteristic, complemented
7#89765C211yesmaximal, characteristic, complemented
8#89763C211yesmaximal, characteristic, complemented
9#89761C211yesmaximal, characteristic, complemented
10#89756C211yesmaximal, characteristic, complemented
11#89712C211yesmaximal, characteristic, complemented
12#89439C211yesmaximal, characteristic, complemented
13#89437C211yesmaximal, characteristic, complemented
14#89437C211yesmaximal, characteristic, complemented
15D7×#95C211yesmaximal, characteristic, complemented
16C22×D28C2211yescharacteristic
17C22×D28C2211yescharacteristic
18C22×D28C2211yescharacteristic
19C22×C4×D7C2211yescharacteristic
20C22×C4×D7C2211yescharacteristic
21C22×C4×D7C2211yescharacteristic
22C22×C4×D7C2211yescharacteristic
23C2×C7×C4⋊C4C2211yescharacteristic
24C2×C7×C4⋊C4C2211yescharacteristic
25C2×C7×C4⋊C4C2211yescharacteristic
26C2×C7×C4⋊C4C2211yescharacteristic
27C2×C42×C7C2211yescharacteristic
28C2×(C2×D7)⋊C4C2211yescharacteristic
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×(C2×D7)⋊C4C2211yescharacteristic
33C2×(C2×D7)⋊C4C2211yescharacteristic
34C2×(C2×D7)⋊C4C2211yescharacteristic
35C2×(C2×D7)⋊C4C2211yescharacteristic
36C2×(C2×D7)⋊C4C2211yescharacteristic
37C2×(C2×D7)⋊C4C2211yescharacteristic
38C2×(C2×D7)⋊C4C2211yescharacteristic
39C2×C4⋊Dic7C2211yescharacteristic
40C2×Dic7⋊C4C2211yescharacteristic
41C2×Dic7⋊C4C2211yescharacteristic
42C2×Dic7⋊C4C2211yescharacteristic
43#6458—24no—
44#6457—24no—
45#6457—24no—
46C4×D28—24no—
47C7×#95C2211yescharacteristic
48C7×#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53C7⋊#95C2211yescharacteristic
54C7⋊#95C2211yescharacteristic
55#2425—71nomaximal, Sylow
56C23×D7C2311yescharacteristic
57C23×D7C2311yescharacteristic
58C23×D7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C22×Dic7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C2×D28—22no—
72C2×D28—22no—
73C2×D28—22no—
74C2×D28—22no—
75C2×D28—22no—
76C2×D28—22no—
77C2×D28—22no—
78C2×D28—41no—
79C2×D28D414yes—
80C2×C4×D7—22no—
81C2×C4×D7—22no—
82C2×C4×D7—22no—
83C2×C4×D7—22no—
84C2×C4×D7—22no—
85C2×C4×D7—22no—
86C2×C4×D7—22no—
87C2×C4×D7—22no—
88C2×C4×D7—22no—
89C2×C4×D7—41no—
90C2×C4×D7—41no—
91C2×C4×D7—41no—
92C7×C4⋊C4—22no—
93C7×C4⋊C4—22no—
94C7×C4⋊C4—22no—
95C7×C4⋊C4—22no—
96C42×C7—22no—
97(C2×D7)⋊C4—22no—
98(C2×D7)⋊C4—22no—
99(C2×D7)⋊C4—22no—
100(C2×D7)⋊C4—22no—
101(C2×D7)⋊C4—22no—
102(C2×D7)⋊C4—22no—
103(C2×D7)⋊C4—22no—
104(C2×D7)⋊C4—22no—
105(C2×D7)⋊C4—22no—
106(C2×D7)⋊C4—41no—
107(C2×D7)⋊C4D414yescomplemented
108C4⋊Dic7—22no—
109Dic7⋊C4—22no—
110Dic7⋊C4—22no—
111Dic7⋊C4—41no—
112C2×#123—71no—
113C2×#123—71no—
114C2×#121—71no—
115C2×C4×D4—71no—
116#398—71no—
117#397—71no—
118#396—71no—
119#394—71no—
120#389D711yescharacteristic, complemented
121#388—71no—
122#386—71no—
123#386—71no—
124#385—71no—
125#380—71no—
126C2×#95—71no—
127C23×C7C2411yescharacteristic
128C22×D7—21no—
129C22×D7—21no—
130C22×D7—21no—
131C22×D7—21no—
132C22×D7—21no—
133C22×D7—21no—
134C22×D7—21no—
135C22×D7—21no—
136C22×D7—21no—
137C22×D7—21no—
138C22×D7—21no—
139C22×D7—21no—
140C22×D7—21no—
141C22×D7—21no—
142C22×D7—21no—
143C22×D7—21no—
144C22×D7—21no—
145C22×D7C2×D412yes—
146C22×D7C2×D412yes—
147C22×D7C2×D412yes—
148C22×D7#4112yes—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—21no—
160C2×C4×C7—21no—
161C2×C4×C7—21no—
162C2×C4×C7—21no—
163C2×C4×C7—21no—
164C2×C4×C7—21no—
165C2×C4×C7—21no—
166C2×C4×C7C2×D412yes—
167C2×C4×C7C2×D412yes—
168C2×C4×C7#4112yes—
169C2×C4×C7#4112yes—
170C2×Dic7—21no—
171C2×Dic7—21no—
172C2×Dic7—21no—
173C2×Dic7—21no—
174C2×Dic7—21no—
175C2×Dic7—21no—
176C2×Dic7—21no—
177C2×Dic7—21no—
178C2×Dic7—21no—
179C2×Dic7—21no—
180C2×Dic7—21no—
181C2×Dic7—21no—
182C2×Dic7—21no—
183C2×Dic7—21no—
184C2×Dic7C2×D412yes—
185D28—28no—
186D28—44no—
187D28—44no—
188C4×D7—44no—
189C4×D7—44no—
190C4×D7—44no—
191C4×D7—44no—
192C22×D4—71no—
193C22×D4—71no—
194C22×D4—71no—
195C23×C4—71no—
196C23×C4—71no—
197C23×C4—71no—
198C23×C4—71no—
199#123—144no—
200#123—144no—
201#121—144no—
202C4×D4—144no—
203C2×C4⋊C4—71no—
204C2×C4⋊C4—71no—
205C2×C4⋊C4—71no—
206C2×C4⋊C4—71no—
207C2×C4⋊C4C2×D711yescharacteristic
208C2×C4⋊C4C2×D711yescharacteristic
209C2×C4⋊C4C2×D711yescharacteristic
210C2×C4⋊C4C2×D711yescharacteristic
211C2×C22⋊C4—71no—
212C2×C22⋊C4—71no—
213C2×C22⋊C4—71no—
214C2×C22⋊C4—71no—
215C2×C22⋊C4—71no—
216C2×C22⋊C4—71no—
217C2×C22⋊C4—71no—
218C2×C22⋊C4—71no—
219C2×C22⋊C4—71no—
220C2×C22⋊C4—71no—
221C2×C22⋊C4—71no—
222C2×C42C2×D711yescharacteristic
223#95—71no—
224#95—71no—
225#95—71no—
226#95—71no—
227#95—71no—
228#95—71no—
229#95C2×D711yescharacteristic
230#95C2×D711yescharacteristic
231C22×C7C22×D411yescharacteristic
232C22×C7C22×D411yescharacteristic
233C22×C7C2×#4111yescharacteristic
234C22×C7C2×#4111yescharacteristic
235C22×C7C2×#4111yescharacteristic
236C22×C7H(5,𝔽2)11yescharacteristic
237C22×C72−1+411yescharacteristic
238C2×D7—22no—
239C2×D7—22no—
240C2×D7—22no—
241C2×D7—22no—
242C2×D7—22no—
243C2×D7—22no—
244C2×D7—22no—
245C2×D7—22no—
246C2×D7—22no—
247C2×D7—41no—
248C2×D7—41no—
249C2×D7—41no—
250C2×D7—41no—
251C2×D7—41no—
252C2×D7—41no—
253C2×D7—41no—
254C2×D7—41no—
255C2×D7—41no—
256C2×D7—41no—
257C2×D7—41no—
258C2×D7#12114yes—
259C4×C7—22no—
260C4×C7—22no—
261C4×C7—22no—
262C4×C7—22no—
263C4×C7—41no—
264C4×C7—41no—
265C4×C7—41no—
266Dic7—22no—
267Dic7—41no—
268Dic7—41no—
269Dic7—41no—
270Dic7—41no—
271C24—71no—
272C24—71no—
273C24—71no—
274C2×D4—74no—
275C2×D4—142no—
276C2×D4—142no—
277C2×D4—142no—
278C2×D4—142no—
279C2×D4—142no—
280C2×D4—142no—
281C2×D4—142no—
282C2×D4—281no—
283C22×C4—71no—
284C22×C4—71no—
285C22×C4—71no—
286C22×C4—71no—
287C22×C4—71no—
288C22×C4—142no—
289C22×C4—142no—
290C22×C4—142no—
291C22×C4—142no—
292C22×C4—142no—
293C22×C4—142no—
294C22×C4—142no—
295C22×C4—142no—
296C22×C4—142no—
297C22×C4—281no—
298C22×C4—281no—
299C22×C4—281no—
300C22×C4C22×D711yescharacteristic
301C22×C4C22×D711yescharacteristic
302C22×C4C22×D711yescharacteristic
303C22×C4C22×D711yescharacteristic
304C22×C4C22×D711yescharacteristic
305C22×C4C22×D711yescharacteristic
306C22×C4C22×D711yescharacteristic
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—142no—
312C4⋊C4—142no—
313C4⋊C4—142no—
314C4⋊C4—281no—
315C22⋊C4—74no—
316C22⋊C4—142no—
317C22⋊C4—142no—
318C22⋊C4—142no—
319C22⋊C4—142no—
320C22⋊C4—142no—
321C22⋊C4—142no—
322C22⋊C4—142no—
323C22⋊C4—142no—
324C22⋊C4—142no—
325C22⋊C4—281no—
326C42—22no—
327C2×C7C2×#12111yescharacteristic
328C2×C7#52911yescharacteristic
329C2×C7#53911yescharacteristic
330C2×C7#54611yescharacteristic
331C2×C7#54611yescharacteristic
332C2×C7#54711yescharacteristic
333C2×C7#54711yescharacteristic
334D7—24no—
335D7—42no—
336D7—81no—
337C23—72no—
338C23—72no—
339C23—72no—
340C23—72no—
341C23—141no—
342C23—141no—
343C23—141no—
344C23—141no—
345C23—141no—
346C23—141no—
347C23—141no—
348C23—141no—
349C23—141no—
350C23—141no—
351C23—141no—
352C23—141no—
353C23—141no—
354C23—141no—
355C23—141no—
356C23—141no—
357C23—141no—
358C23C23×D711yescharacteristic
359D4—148no—
360D4—284no—
361D4—284no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—72no—
380C2×C4—141no—
381C2×C4—141no—
382C2×C4—141no—
383C2×C4—141no—
384C2×C4—141no—
385C2×C4—141no—
386C2×C4—141no—
387C2×C4—141no—
388C2×C4—141no—
389C2×C4—141no—
390C2×C4—141no—
391C2×C4—141no—
392C2×C4—141no—
393C2×C4—141no—
394C2×C4—284no—
395C2×C4—284no—
396C2×C4—284no—
397C2×C4—284no—
398C2×C4#115712yes—
399C2×C4D4×D712yes—
400C2×C4D4×D712yes—
401C2×C4#116112yes—
402C7#242511yescharacteristic, complemented, Sylow
403C22—74no—
404C22—142no—
405C22—142no—
406C22—142no—
407C22—142no—
408C22—142no—
409C22—142no—
410C22—142no—
411C22—142no—
412C22—142no—
413C22—281no—
414C22—281no—
415C22—281no—
416C22—281no—
417C22—281no—
418C22—281no—
419C22—281no—
420C22—281no—
421C22—281no—
422C22—281no—
423C22—281no—
424C22C2×#115711yescharacteristic
425C22C2×D4×D711yescharacteristic
426C22C2×D4×D711yescharacteristic
427C22C2×#116111yescharacteristic
428C22#655111yescharacteristic
429C22D7×#4111yescharacteristic
430C22#655311yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—22no—
435C4—41no—
436C4—41no—
437C4—41no—
438C4—142no—
439C4—281no—
440C4—281no—
441C4—281no—
442C4—281no—
443C2—144no—
444C2—282no—
445C2—561no—
446C2#9019811yescharacteristic
447C2#9024111yescharacteristic
448C2#9027111yescharacteristic
449C2D7×#12111yescharacteristic
450C2#9032111yescharacteristic
451C2#9038311yescharacteristic
452C2#9041711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#389C2
prime radicalC2×#6458C2
hypercenter#389D7
hypocenterC7#2425
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
2. C2×#6457C2semidirect
3. C2×#6457C2semidirect
4. C2×C4×D28C2semidirect
5. C7×#389C2semidirect
6. #89766C2semidirect
7. #89765C2semidirect
8. #89763C2semidirect
9. #89761C2semidirect
10. #89756C2semidirect
11. #89712C2semidirect
12. #89439C2semidirect
13. #89437C2semidirect
14. #89437C2semidirect
15. D7×#95C2semidirect
107. (C2×D7)⋊C4D4semidirect
120. #389D7semidirect
402. C7#2425semidirect

Maximal subgroups

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

subgroupindex
C2×#64582
C2×#64572
C2×C4×D282
C7×#3892
#897662
#897652
#897632
#897612
#897562
#897122
#894392
#894372
D7×#952
#24257