Finite Group Databasebeta

← #11740323

Subgroups — #11740323

Order 896.

562 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×C7×#123C211yesmaximal, characteristic, complemented
2C2×#6460C211yesmaximal, characteristic, complemented
3#89995C211yesmaximal, characteristic
4#89993C211yesmaximal, characteristic, complemented
5#89985C211yesmaximal, characteristic, complemented
6#89984C211yesmaximal, characteristic, complemented
7#89762C211yesmaximal, characteristic, complemented
8#89758C211yesmaximal, characteristic, complemented
9#89753C211yesmaximal, characteristic, complemented
10#89733C211yesmaximal, characteristic, complemented
11#89728C211yesmaximal, characteristic, complemented
12#89728C211yesmaximal, characteristic, complemented
13#89725C211yesmaximal, characteristic, complemented
14#89719C211yesmaximal, characteristic, complemented
15#89719C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C23×C4×C7C2211yescharacteristic
18C22×C22⋊D7C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C22×C4×D7C2211yescharacteristic
21C22×Dic14C2211yescharacteristic
22C7×#123—24no—
23C2×C7×C4⋊C4C2211yescharacteristic
24C2×C7×C4⋊C4C2211yescharacteristic
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
34C2×(C2×D7)⋊C4C2211yescharacteristic
35C2×(C2×D7)⋊C4C2211yescharacteristic
36C2×(C2×D7)⋊C4C2211yescharacteristic
37C2×(C2×D7)⋊C4C2211yescharacteristic
38C2×C4⋊Dic7C2211yescharacteristic
39C2×C4⋊Dic7C2211yescharacteristic
40C2×C4⋊Dic7C2211yescharacteristic
41C2×C4×Dic7C2211yescharacteristic
42#6460—24no—
43C7⋊#95C2211yescharacteristic
44C7⋊#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53#2696—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C24×C7C2311yescharacteristic
56C23×D7C2311yescharacteristic
57C2×C7×D4—22no—
58C2×C7×D4—22no—
59C2×C7×D4—41no—
60C22×C4×C7—22no—
61C22×C4×C7—22no—
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—22no—
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×C4×D7—22no—
82C2×C4×D7—41no—
83C2×C4×D7—41no—
84C2×Dic14—22no—
85C2×Dic14—41no—
86C2×Dic14—41no—
87C7×C4⋊C4—22no—
88C7×C4⋊C4D414yescomplemented
89C7×C22⋊C4—22no—
90C7×C22⋊C4—22no—
91C7×C22⋊C4—22no—
92C22⋊Dic7—41no—
93C22⋊Dic7—41no—
94C22⋊Dic7—41no—
95C22⋊Dic7—41no—
96C22⋊Dic7—41no—
97C22⋊Dic7—41no—
98(C2×D7)⋊C4—22no—
99(C2×D7)⋊C4—22no—
100(C2×D7)⋊C4—22no—
101(C2×D7)⋊C4—41no—
102C4⋊Dic7—22no—
103C4⋊Dic7—22no—
104C4⋊Dic7—41no—
105C4×Dic7—41no—
106C2×#123D711yescharacteristic, complemented
107C2×C22⋊Q8—71no—
108#400—71no—
109#397—71no—
110#394—71no—
111#393—71no—
112#393—71no—
113#392—71no—
114#390—71no—
115#388—71no—
116#387—71no—
117#386—71no—
118#386—71no—
119#386—71no—
120#380—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×C7#4112yes—
135C23×C7C2411yescharacteristic
136C22×D7—21no—
137C22×D7—21no—
138C22×D7—21no—
139C22×D7—21no—
140C22×D7—21no—
141C22×D7—21no—
142C22×D7#4112yes—
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—44no—
157C2×C4×C7C2×D412yes—
158C2×C4×C7C2×D412yes—
159C2×C4×C7C2×D412yes—
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×Dic7—21no—
181C2×Dic7—21no—
182C2×Dic7—21no—
183C4×D7—44no—
184Dic14—44no—
185C22×Q8—71no—
186C22×D4—71no—
187C22×D4—71no—
188C22×D4C2×D711yescharacteristic
189C23×C4—71no—
190C23×C4C2×D711yescharacteristic
191#123—24no—
192C22⋊Q8—144no—
193C2×C4⋊C4—71no—
194C2×C4⋊C4—71no—
195C2×C4⋊C4—71no—
196C2×C4⋊C4C2×D711yescharacteristic
197C2×C4⋊C4C2×D711yescharacteristic
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⋊C4—71no—
208C2×C22⋊C4C2×D711yescharacteristic
209C2×C22⋊C4C2×D711yescharacteristic
210C2×C22⋊C4C2×D711yescharacteristic
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#95—71no—
221#95—71no—
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—22no—
244C2×D7—22no—
245C2×D7—22no—
246C2×D7—41no—
247C2×D7—41no—
248C2×D7—41no—
249C2×D7—41no—
250C4×C7—22no—
251C4×C7—22no—
252C4×C7—22no—
253C4×C7—41no—
254C4×C7—41no—
255Dic7—41no—
256Dic7—41no—
257Dic7—41no—
258Dic7—41no—
259Dic7—41no—
260Dic7—41no—
261Dic7—41no—
262C24—71no—
263C24C22×D711yescharacteristic
264C24C22×D711yescharacteristic
265C2×Q8—142no—
266C2×Q8—281no—
267C2×Q8—281no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—41no—
271C2×D4—142no—
272C2×D4—142no—
273C2×D4—142no—
274C2×D4—281no—
275C2×D4—281no—
276C2×D4—281no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—41no—
280C22×C4—71no—
281C22×C4—71no—
282C22×C4—71no—
283C22×C4—71no—
284C22×C4—71no—
285C22×C4—71no—
286C22×C4—71no—
287C22×C4—142no—
288C22×C4—281no—
289C22×C4—281no—
290C22×C4C22×D711yescharacteristic
291C22×C4C22×D711yescharacteristic
292C22×C4C22×D711yescharacteristic
293C22×C4C22×D711yescharacteristic
294C22×C4C22×D711yescharacteristic
295C4⋊C4—22no—
296C4⋊C4—142no—
297C4⋊C4—142no—
298C4⋊C4—281no—
299C4⋊C4C22⋊D714yescomplemented
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4—142no—
304C22⋊C4—142no—
305C22⋊C4—142no—
306C22⋊C4—281no—
307C22⋊C4—281no—
308C22⋊C4—281no—
309C22⋊C4—281no—
310C22⋊C4—281no—
311C22⋊C4—281no—
312C22⋊C4—281no—
313C42—281no—
314C2×C7—42no—
315C2×C7—81no—
316C2×C7#53511yescharacteristic
317C2×C7#53811yescharacteristic
318C2×C7#53911yescharacteristic
319C2×C7#54611yescharacteristic
320C2×C7#54811yescharacteristic
321C2×C7#55111yescharacteristic
322C2×C7#56211yescharacteristic
323D7—42no—
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—72no—
338C23—141no—
339C23—141no—
340C23—141no—
341C23—141no—
342C23—141no—
343C23—141no—
344C23#115912yes—
345C23C23×D711yescharacteristic
346Q8—284no—
347D4—44no—
348D4—284no—
349D4—284no—
350C2×C4—21no—
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—44no—
363C2×C4—141no—
364C2×C4—141no—
365C2×C4—141no—
366C2×C4—141no—
367C2×C4—141no—
368C2×C4—141no—
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—284no—
385C2×C4C2×C22⋊D712yes—
386C2×C4C2×C22⋊D712yes—
387C2×C4C2×C22⋊D712yes—
388C7#269611yescharacteristic, 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—142no—
404C22—142no—
405C22—142no—
406C22—281no—
407C22—281no—
408C22—281no—
409C22—281no—
410C22C2×#115911yescharacteristic
411C22#654811yescharacteristic
412C22D7×#4111yescharacteristic
413C22#655311yescharacteristic
414C22#655311yescharacteristic
415C22#655411yescharacteristic
416C22C22×C22⋊D711yescharacteristic
417C4—22no—
418C4—22no—
419C4—22no—
420C4—41no—
421C4—41no—
422C4—281no—
423C4—281no—
424C4—281no—
425C4—281no—
426C4—281no—
427C4—281no—
428C4—281no—
429C2—42no—
430C2—81no—
431C2—282no—
432C2#9034811yescharacteristic
433C2#9034911yescharacteristic
434C2#9035611yescharacteristic
435C2#9035811yescharacteristic
436C2#9049711yescharacteristic
437C2#9052111yescharacteristic
438C2#9052211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#123C2
prime radical#89995C2
hypercenterC2×#123D7
hypocenterC7#2696
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×C7×#123C2semidirect
2. C2×#6460C2semidirect
4. #89993C2semidirect
5. #89985C2semidirect
6. #89984C2semidirect
7. #89762C2semidirect
8. #89758C2semidirect
9. #89753C2semidirect
10. #89733C2semidirect
11. #89728C2semidirect
12. #89728C2semidirect
13. #89725C2semidirect
14. #89719C2semidirect
15. #89719C2semidirect
88. C7×C4⋊C4D4semidirect
106. C2×#123D7semidirect
299. C4⋊C4C22⋊D7semidirect
388. C7#2696semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1232
C2×#64602
#899952
#899932
#899852
#899842
#897622
#897582
#897532
#897332
#897282
#897252
#897192
#26967