Finite Group Databasebeta

← #11732267

Subgroups — #11732267

Order 896.

556 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6499C211yesmaximal, characteristic, complemented
2C2×#6448C211yesmaximal, characteristic, complemented
3C7×#386C211yesmaximal, characteristic, complemented
4#89995C211yesmaximal, characteristic
5#89985C211yesmaximal, characteristic, complemented
6#89982C211yesmaximal, characteristic, complemented
7#89731C211yesmaximal, characteristic, complemented
8#89731C211yesmaximal, characteristic, complemented
9#89724C211yesmaximal, characteristic, complemented
10#89719C211yesmaximal, characteristic, complemented
11#89717C211yesmaximal, characteristic, complemented
12#89445C211yesmaximal, characteristic, complemented
13#89441C211yesmaximal, characteristic, complemented
14#89418C211yesmaximal, characteristic, complemented
15#89418C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C23×C4×C7C2211yescharacteristic
18C22×C22⋊D7C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C22×Dic14C2211yescharacteristic
21C22×Dic14C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31#6499—24no—
32C2×(C2×D7)⋊C4C2211yescharacteristic
33C2×(C2×D7)⋊C4C2211yescharacteristic
34C2×(C2×D7)⋊C4C2211yescharacteristic
35C2×(C2×D7)⋊C4C2211yescharacteristic
36C2×(C2×D7)⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C2×Dic7⋊C4C2211yescharacteristic
40C2×C4×Dic7C2211yescharacteristic
41C2×C4×Dic7C2211yescharacteristic
42#6448—24no—
43C7×#95C2211yescharacteristic
44C7×#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53#2712—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C24×C7C2311yescharacteristic
56C23×D7C2311yescharacteristic
57C2×C7×D4—22no—
58C2×C7×D4—22no—
59C2×C7×D4—41no—
60C22×C4×C7—41no—
61C22×C4×C7—41no—
62C22×C4×C7—41no—
63C22×C4×C7C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C22×C4×C7C2311yescharacteristic
68C2×C22⋊D7—22no—
69C2×C22⋊D7—22no—
70C2×C22⋊D7—41no—
71C2×C22⋊D7—41no—
72C2×C22⋊D7—41no—
73C2×C22⋊D7—41no—
74C22×Dic7C2311yescharacteristic
75C22×Dic7C2311yescharacteristic
76C22×Dic7C2311yescharacteristic
77C22×Dic7C2311yescharacteristic
78C22×Dic7C2311yescharacteristic
79C22×Dic7C2311yescharacteristic
80C22×Dic7C2311yescharacteristic
81C2×Dic14—22no—
82C2×Dic14—22no—
83C2×Dic14—22no—
84C2×Dic14—22no—
85C2×Dic14—41no—
86C2×Dic14—41no—
87C7×C22⋊C4—22no—
88C7×C22⋊C4—41no—
89C7×C22⋊C4—41no—
90C22⋊Dic7—22no—
91C22⋊Dic7—22no—
92C22⋊Dic7—22no—
93C22⋊Dic7—22no—
94C22⋊Dic7—41no—
95C22⋊Dic7D414yescomplemented
96(C2×D7)⋊C4—22no—
97(C2×D7)⋊C4—22no—
98(C2×D7)⋊C4—22no—
99(C2×D7)⋊C4—41no—
100(C2×D7)⋊C4—41no—
101Dic7⋊C4—41no—
102Dic7⋊C4—41no—
103Dic7⋊C4—41no—
104C4×Dic7—22no—
105C4×Dic7—22no—
106C2×#124—71no—
107C2×#124—71no—
108#397—71no—
109#394—71no—
110#394—71no—
111#393—71no—
112#393—71no—
113#392—71no—
114#391—71no—
115#391—71no—
116#388—71no—
117#388—71no—
118#386—71no—
119#386D711yescharacteristic, complemented
120#385—71no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7—21no—
134C23×C7C2×D412yes—
135C23×C7C2411yescharacteristic
136C22×D7—21no—
137C22×D7—21no—
138C22×D7—21no—
139C22×D7—21no—
140C22×D7—21no—
141C22×D7—21no—
142C22×D7—21no—
143C7×D4—44no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
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—44no—
159C2×C4×C7#4112yes—
160C22⋊D7—44no—
161C22⋊D7—44no—
162C2×Dic7—21no—
163C2×Dic7—21no—
164C2×Dic7—21no—
165C2×Dic7—21no—
166C2×Dic7—21no—
167C2×Dic7—21no—
168C2×Dic7—21no—
169C2×Dic7—21no—
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×Dic7C2×D412yes—
181C2×Dic7C2×D412yes—
182C2×Dic7#4112yes—
183Dic14—44no—
184Dic14—44no—
185C22×Q8—71no—
186C22×Q8—71no—
187C22×D4—71no—
188C22×D4—71no—
189C22×D4C2×D711yescharacteristic
190C23×C4C2×D711yescharacteristic
191#124—144no—
192#124—144no—
193C2×C4⋊C4—71no—
194C2×C4⋊C4—71no—
195C2×C4⋊C4—71no—
196C2×C22⋊C4—71no—
197C2×C22⋊C4—71no—
198C2×C22⋊C4—71no—
199C2×C22⋊C4—71no—
200C2×C22⋊C4—71no—
201C2×C22⋊C4—71no—
202C2×C22⋊C4—71no—
203C2×C22⋊C4—71no—
204C2×C22⋊C4—71no—
205C2×C22⋊C4—71no—
206C2×C22⋊C4—71no—
207C2×C22⋊C4C2×D711yescharacteristic
208C2×C22⋊C4C2×D711yescharacteristic
209C2×C22⋊C4C2×D711yescharacteristic
210C2×C42—71no—
211C2×C42—71no—
212#95—71no—
213#95—71no—
214#95—71no—
215#95—71no—
216#95—71no—
217#95—71no—
218#95—71no—
219#95—71no—
220#95C2×D711yescharacteristic
221#95C2×D711yescharacteristic
222C22×C7—22no—
223C22×C7—22no—
224C22×C7—22no—
225C22×C7—41no—
226C22×C7—41no—
227C22×C7—41no—
228C22×C7—41no—
229C22×C7—41no—
230C22×C7—41no—
231C22×C7—41no—
232C22×C7—41no—
233C22×C7—41no—
234C22×C7—41no—
235C22×C7—41no—
236C22×C7C22×D411yescharacteristic
237C22×C7C2×#4111yescharacteristic
238C22×C7C2×#4111yescharacteristic
239C22×C7H(5,𝔽2)11yescharacteristic
240C22×C7H(5,𝔽2)11yescharacteristic
241C22×C7H(5,𝔽2)11yescharacteristic
242C22×C72−1+411yescharacteristic
243C2×D7—41no—
244C2×D7—41no—
245C2×D7—41no—
246C2×D7—41no—
247C2×D7—41no—
248C2×D7—41no—
249C2×D7—41no—
250C4×C7—22no—
251C4×C7—41no—
252C4×C7—41no—
253C4×C7—41no—
254C4×C7—41no—
255Dic7—22no—
256Dic7—22no—
257Dic7—22no—
258Dic7—41no—
259Dic7—41no—
260Dic7—41no—
261Dic7—41no—
262C24—71no—
263C24C22×D711yescharacteristic
264C24C22×D711yescharacteristic
265C2×Q8—142no—
266C2×Q8—142no—
267C2×Q8—142no—
268C2×Q8—142no—
269C2×Q8—281no—
270C2×Q8—281no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—41no—
274C2×D4—142no—
275C2×D4—142no—
276C2×D4—281no—
277C2×D4—281no—
278C2×D4—281no—
279C2×D4—281no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—71no—
284C22×C4—71no—
285C22×C4—71no—
286C22×C4—71no—
287C22×C4—71no—
288C22×C4—71no—
289C22×C4—71no—
290C22×C4C22×D711yescharacteristic
291C22×C4C22×D711yescharacteristic
292C22×C4C22×D711yescharacteristic
293C22×C4C22×D711yescharacteristic
294C22×C4C22×D711yescharacteristic
295C4⋊C4—281no—
296C4⋊C4—281no—
297C4⋊C4—281no—
298C22⋊C4—22no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—74no—
302C22⋊C4—142no—
303C22⋊C4—142no—
304C22⋊C4—142no—
305C22⋊C4—142no—
306C22⋊C4—142no—
307C22⋊C4—142no—
308C22⋊C4—142no—
309C22⋊C4—281no—
310C22⋊C4—281no—
311C22⋊C4—281no—
312C42—142no—
313C42—142no—
314C2×C7—42no—
315C2×C7—81no—
316C2×C7#53711yescharacteristic
317C2×C7#54211yescharacteristic
318C2×C7#54211yescharacteristic
319C2×C7#54611yescharacteristic
320C2×C7#54611yescharacteristic
321C2×C7#55911yescharacteristic
322C2×C7H(3,𝔽4)11yescharacteristic
323D7—81no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—141no—
338C23—141no—
339C23—141no—
340C23—141no—
341C23—141no—
342C23—141no—
343C23—141no—
344C23D4×D712yes—
345C23C23×D711yescharacteristic
346Q8—284no—
347Q8—284no—
348D4—44no—
349D4—284no—
350D4—284no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—44no—
366C2×C4—72no—
367C2×C4—72no—
368C2×C4—72no—
369C2×C4—141no—
370C2×C4—141no—
371C2×C4—141no—
372C2×C4—141no—
373C2×C4—141no—
374C2×C4—141no—
375C2×C4—141no—
376C2×C4—141no—
377C2×C4—141no—
378C2×C4—141no—
379C2×C4—141no—
380C2×C4—141no—
381C2×C4—141no—
382C2×C4—141no—
383C2×C4—141no—
384C2×C4—141no—
385C2×C4—141no—
386C2×C4—141no—
387C2×C4#115712yes—
388C7#271211yescharacteristic, complemented, Sylow
389C22—22no—
390C22—22no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—281no—
404C22—281no—
405C22—281no—
406C22—281no—
407C22—281no—
408C22—281no—
409C22—281no—
410C22C2×#115711yescharacteristic
411C22C2×D4×D711yescharacteristic
412C22#654811yescharacteristic
413C22#654811yescharacteristic
414C22#654811yescharacteristic
415C22D7×#4111yescharacteristic
416C22#655411yescharacteristic
417C4—22no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—142no—
423C4—142no—
424C4—142no—
425C4—281no—
426C4—281no—
427C4—281no—
428C4—281no—
429C2—42no—
430C2—81no—
431C2—561no—
432C2#9018111yescharacteristic
433C2#9025011yescharacteristic
434C2#9025211yescharacteristic
435C2#9028511yescharacteristic
436C2#9028811yescharacteristic
437C2#9030111yescharacteristic
438C2#9035311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#386C2
prime radical#89995C2
hypercenter#386D7
hypocenterC7#2712
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#6499C2semidirect
2. C2×#6448C2semidirect
3. C7×#386C2semidirect
5. #89985C2semidirect
6. #89982C2semidirect
7. #89731C2semidirect
8. #89731C2semidirect
9. #89724C2semidirect
10. #89719C2semidirect
11. #89717C2semidirect
12. #89445C2semidirect
13. #89441C2semidirect
14. #89418C2semidirect
15. #89418C2semidirect
95. C22⋊Dic7D4semidirect
119. #386D7semidirect
388. C7#2712semidirect

Maximal subgroups

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

subgroupindex
C2×#64992
C2×#64482
C7×#3862
#899952
#899852
#899822
#897312
#897242
#897192
#897172
#894452
#894412
#894182
#27127