Finite Group Databasebeta

← #11739724

Subgroups — #11739724

Order 896.

582 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#117C211yesmaximal, characteristic, complemented
2C2×#6457C211yesmaximal, characteristic, complemented
3#89985C211yesmaximal, characteristic, complemented
4#89985C211yesmaximal, characteristic, complemented
5#89765C211yesmaximal, characteristic, complemented
6#89764C211yesmaximal, characteristic, complemented
7#89753C211yesmaximal, characteristic, complemented
8#89732C211yesmaximal, characteristic
9#89729C211yesmaximal, characteristic, complemented
10#89728C211yesmaximal, characteristic, complemented
11#89720C211yesmaximal, characteristic, complemented
12#89712C211yesmaximal, characteristic, complemented
13#89711C211yesmaximal, characteristic, complemented
14C4×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
15#89705C211yesmaximal, characteristic, complemented
16C23×C4×C7C2211yescharacteristic
17C22×C22⋊D7C2211yescharacteristic
18C22×C22⋊D7C2211yescharacteristic
19C22×D28C2211yescharacteristic
20C22×C4×D7C2211yescharacteristic
21C7×#117—24no—
22C2×C7×C4⋊C4C2211yescharacteristic
23C2×C7×C4⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C42×C7C2211yescharacteristic
27C2×C42×C7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
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
43C2×C4×Dic7C2211yescharacteristic
44#6457—24no—
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53#2471—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C23×D7C2311yescharacteristic
56C23×D7C2311yescharacteristic
57C22×C4×C7—22no—
58C22×C4×C7—22no—
59C22×C4×C7—22no—
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C2×C22⋊D7—22no—
67C2×C22⋊D7—22no—
68C2×C22⋊D7—22no—
69C2×C22⋊D7—22no—
70C2×C22⋊D7—41no—
71C2×C22⋊D7—41no—
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C22×Dic7C2311yescharacteristic
75C22×Dic7C2311yescharacteristic
76C22×Dic7C2311yescharacteristic
77C22×Dic7C2311yescharacteristic
78C2×D28—22no—
79C2×D28—22no—
80C2×D28—41no—
81C2×C4×D7—22no—
82C2×C4×D7—22no—
83C2×C4×D7—41no—
84C7×C4⋊C4—22no—
85C7×C4⋊C4D414yescomplemented
86C7×C22⋊C4—22no—
87C7×C22⋊C4—22no—
88C42×C7—22no—
89C42×C7—22no—
90C22⋊Dic7—41no—
91C22⋊Dic7—41no—
92(C2×D7)⋊C4—22no—
93(C2×D7)⋊C4—22no—
94(C2×D7)⋊C4—22no—
95(C2×D7)⋊C4—22no—
96(C2×D7)⋊C4—22no—
97(C2×D7)⋊C4—22no—
98(C2×D7)⋊C4—41no—
99(C2×D7)⋊C4—41no—
100(C2×D7)⋊C4—41no—
101C4⋊Dic7—41no—
102Dic7⋊C4—22no—
103Dic7⋊C4—22no—
104Dic7⋊C4—41no—
105C4×Dic7—41no—
106C2×#123—71no—
107C2×#117D711yescharacteristic, complemented
108#399—71no—
109#397—71no—
110#396—71no—
111#392—71no—
112#390—71no—
113#390—71no—
114#388—71no—
115#387—71no—
116#387—71no—
117#386—71no—
118#386—71no—
119#386—71no—
120C4×C22⋊C4—71no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7#4112yes—
128C23×C7C2411yescharacteristic
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#4112yes—
143C2×C4×C7—28no—
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×C7C2×D412yes—
159C2×C4×C7C2×D412yes—
160C2×C4×C7C2×D412yes—
161C2×C4×C7#4112yes—
162C22⋊D7—44no—
163C22⋊D7—44no—
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#4112yes—
182D28—44no—
183C4×D7—44no—
184C22×D4—71no—
185C22×D4—71no—
186C22×D4—71no—
187C23×C4—71no—
188C23×C4C2×D711yescharacteristic
189#123—144no—
190#117—24no—
191C2×C4⋊C4—71no—
192C2×C4⋊C4—71no—
193C2×C4⋊C4—71no—
194C2×C4⋊C4—71no—
195C2×C4⋊C4C2×D711yescharacteristic
196C2×C4⋊C4C2×D711yescharacteristic
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⋊C4—71no—
208C2×C22⋊C4C2×D711yescharacteristic
209C2×C22⋊C4C2×D711yescharacteristic
210C2×C42—71no—
211C2×C42C2×D711yescharacteristic
212C2×C42C2×D711yescharacteristic
213#95—71no—
214#95—71no—
215#95—71no—
216#95—71no—
217#95—71no—
218#95—71no—
219#95—71no—
220#95—71no—
221C22×C7—22no—
222C22×C7—22no—
223C22×C7—22no—
224C22×C7—41no—
225C22×C7—41no—
226C22×C7—41no—
227C22×C7—41no—
228C22×C7C22×D411yescharacteristic
229C22×C7C2×#4111yescharacteristic
230C22×C7C2×#4111yescharacteristic
231C22×C7C2×#4111yescharacteristic
232C22×C7C2×#4111yescharacteristic
233C22×C7H(5,𝔽2)11yescharacteristic
234C22×C7H(5,𝔽2)11yescharacteristic
235C2×D7—22no—
236C2×D7—22no—
237C2×D7—22no—
238C2×D7—41no—
239C2×D7—41no—
240C2×D7—41no—
241C2×D7—41no—
242C2×D7—41no—
243C2×D7—41no—
244C2×D7—41no—
245C2×D7—41no—
246C2×D7—41no—
247C2×D7—41no—
248C2×D7—41no—
249C4×C7—22no—
250C4×C7—22no—
251C4×C7—22no—
252C4×C7—22no—
253C4×C7—41no—
254C4×C7—41no—
255Dic7—22no—
256Dic7—41no—
257Dic7—41no—
258Dic7—41no—
259Dic7—41no—
260Dic7—41no—
261C24—71no—
262C24—71no—
263C24C22×D711yescharacteristic
264C2×D4—142no—
265C2×D4—142no—
266C2×D4—142no—
267C2×D4—142no—
268C2×D4—142no—
269C2×D4—142no—
270C2×D4—281no—
271C2×D4—281no—
272C2×D4—281no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—71no—
277C22×C4—71no—
278C22×C4—71no—
279C22×C4—71no—
280C22×C4—71no—
281C22×C4—71no—
282C22×C4—142no—
283C22×C4—142no—
284C22×C4—281no—
285C22×C4C22×D711yescharacteristic
286C22×C4C22×D711yescharacteristic
287C22×C4C22×D711yescharacteristic
288C22×C4C22×D711yescharacteristic
289C22×C4C22×D711yescharacteristic
290C22×C4C22×D711yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—142no—
293C4⋊C4—142no—
294C4⋊C4—281no—
295C4⋊C4—281no—
296C4⋊C4C22⋊D714yescomplemented
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—142no—
300C22⋊C4—142no—
301C22⋊C4—142no—
302C22⋊C4—142no—
303C22⋊C4—142no—
304C22⋊C4—142no—
305C22⋊C4—281no—
306C22⋊C4—281no—
307C22⋊C4—281no—
308C22⋊C4—281no—
309C22⋊C4—281no—
310C42—22no—
311C42—22no—
312C42—281no—
313C2×C7—42no—
314C2×C7#52511yescharacteristic
315C2×C7#52911yescharacteristic
316C2×C7#54011yescharacteristic
317C2×C7#54611yescharacteristic
318C2×C7#55011yescharacteristic
319C2×C7#55111yescharacteristic
320C2×C7#55111yescharacteristic
321D7—42no—
322D7—81no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—72no—
330C23—141no—
331C23—141no—
332C23—141no—
333C23—141no—
334C23—141no—
335C23—141no—
336C23—141no—
337C23—141no—
338C23—141no—
339C23—141no—
340C23—141no—
341C23—141no—
342C23—141no—
343C23#115712yes—
344C23C23×D711yescharacteristic
345D4—284no—
346D4—284no—
347D4—284no—
348C2×C4—28no—
349C2×C4—21no—
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—21no—
363C2×C4—72no—
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—284no—
382C2×C4#115712yes—
383C2×C4C2×C22⋊D712yes—
384C2×C4C2×C22⋊D712yes—
385C2×C4C2×C22⋊D712yes—
386C7#247111yescharacteristic, complemented, Sylow
387C22—22no—
388C22—22no—
389C22—22no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—142no—
395C22—142no—
396C22—142no—
397C22—281no—
398C22—281no—
399C22—281no—
400C22—281no—
401C22—281no—
402C22—281no—
403C22—281no—
404C22—281no—
405C22—281no—
406C22—281no—
407C22—281no—
408C22C2×#115711yescharacteristic
409C22C2×#115711yescharacteristic
410C22D7×#4111yescharacteristic
411C22D7×#4111yescharacteristic
412C22#655311yescharacteristic
413C22#655311yescharacteristic
414C22C22×C22⋊D711yescharacteristic
415C4—22no—
416C4—22no—
417C4—22no—
418C4—22no—
419C4—41no—
420C4—41no—
421C4—142no—
422C4—281no—
423C4—281no—
424C4—281no—
425C4—281no—
426C4—281no—
427C2—42no—
428C2—282no—
429C2—561no—
430C2#9021811yescharacteristic
431C2#9021811yescharacteristic
432C2#9021911yescharacteristic
433C2#9022611yescharacteristic
434C2#9048211yescharacteristic
435C2#9052011yescharacteristic
436C2#9052511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#117C2
prime radical#89732C2
hypercenterC2×#117D7
hypocenterC7#2471
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×C7×#117C2semidirect
2. C2×#6457C2semidirect
3. #89985C2semidirect
4. #89985C2semidirect
5. #89765C2semidirect
6. #89764C2semidirect
7. #89753C2semidirect
9. #89729C2semidirect
10. #89728C2semidirect
11. #89720C2semidirect
12. #89712C2semidirect
13. #89711C2semidirect
14. C4×(C2×D7)⋊C4C2semidirect
15. #89705C2semidirect
85. C7×C4⋊C4D4semidirect
107. C2×#117D7semidirect
296. C4⋊C4C22⋊D7semidirect
386. C7#2471semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1172
C2×#64572
#899852
#897652
#897642
#897532
#897322
#897292
#897282
#897202
#897122
#897112
C4×(C2×D7)⋊C42
#897052
#24717