Finite Group Databasebeta

← #11740409

Subgroups — #11740409

Order 896.

646 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#124C211yesmaximal, characteristic, complemented
2C2×#6509C211yesmaximal, characteristic, complemented
3C2×#6498C211yesmaximal, characteristic, complemented
4C2×(C22⋊D7)⋊C4C211yesmaximal, characteristic, complemented
5#90001C211yesmaximal, characteristic, complemented
6#89995C211yesmaximal, characteristic
7#89731C211yesmaximal, characteristic, complemented
8#89729C211yesmaximal, characteristic, complemented
9#89728C211yesmaximal, characteristic, complemented
10#89723C211yesmaximal, characteristic, complemented
11#89720C211yesmaximal, characteristic, complemented
12#89718C211yesmaximal, characteristic, complemented
13#89716C211yesmaximal, characteristic, complemented
14#89712C211yesmaximal, characteristic, complemented
15C4×Dic7⋊C4C211yesmaximal, characteristic, complemented
16C22×C7×Q8C2211yescharacteristic
17C22×C7×D4C2211yescharacteristic
18C22×C22⋊D7C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C23×Dic7C2211yescharacteristic
21C22×C4×D7C2211yescharacteristic
22C7×#124—24no—
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C42×C7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31C2×C22⋊Dic7C2211yescharacteristic
32C2×C22⋊Dic7C2211yescharacteristic
33#6509—24no—
34#6498—24no—
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×Dic7⋊C4C2211yescharacteristic
44C2×Dic7⋊C4C2211yescharacteristic
45C2×Dic7⋊C4C2211yescharacteristic
46C2×C4×Dic7C2211yescharacteristic
47C2×C4×Dic7C2211yescharacteristic
48(C22⋊D7)⋊C4—24no—
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53C7⋊#95C2211yescharacteristic
54C7⋊#95C2211yescharacteristic
55#2461—71nomaximal, Sylow
56C24×C7C2311yescharacteristic
57C24×C7C2311yescharacteristic
58C23×D7C2311yescharacteristic
59C2×C7×Q8—22no—
60C2×C7×Q8—22no—
61C2×C7×Q8—41no—
62C2×C7×D4—22no—
63C2×C7×D4—22no—
64C2×C7×D4—41no—
65C22×C4×C7C2311yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C22×C4×C7C2311yescharacteristic
68C22×C4×C7C2311yescharacteristic
69C22×C4×C7C2311yescharacteristic
70C2×C22⋊D7—22no—
71C2×C22⋊D7—22no—
72C2×C22⋊D7—22no—
73C2×C22⋊D7—22no—
74C2×C22⋊D7—22no—
75C2×C22⋊D7—41no—
76C22×Dic7—22no—
77C22×Dic7—22no—
78C22×Dic7—41no—
79C22×Dic7C2311yescharacteristic
80C22×Dic7C2311yescharacteristic
81C22×Dic7C2311yescharacteristic
82C22×Dic7C2311yescharacteristic
83C22×Dic7C2311yescharacteristic
84C22×Dic7C2311yescharacteristic
85C22×Dic7C2311yescharacteristic
86C2×C4×D7—22no—
87C2×C4×D7—22no—
88C2×C4×D7—41no—
89C7×C22⋊C4—22no—
90C7×C22⋊C4—22no—
91C7×C22⋊C4—22no—
92C7×C22⋊C4D414yescomplemented
93C42×C7—22no—
94C22⋊Dic7—22no—
95C22⋊Dic7—22no—
96C22⋊Dic7—22no—
97C22⋊Dic7—22no—
98C22⋊Dic7—22no—
99(C2×D7)⋊C4—22no—
100(C2×D7)⋊C4—22no—
101(C2×D7)⋊C4—22no—
102(C2×D7)⋊C4—22no—
103C4⋊Dic7—22no—
104Dic7⋊C4—22no—
105Dic7⋊C4—22no—
106Dic7⋊C4—22no—
107Dic7⋊C4—22no—
108Dic7⋊C4—41no—
109Dic7⋊C4D414yescomplemented
110C4×Dic7—22no—
111C4×Dic7—22no—
112C2×#124D711yescharacteristic, complemented
113C2×#123—71no—
114C2×C22⋊Q8—71no—
115C2×C4×D4—71no—
116#396—71no—
117#395—71no—
118#394—71no—
119#392—71no—
120#390—71no—
121#388—71no—
122#388—71no—
123#388—71no—
124#386—71no—
125#385—71no—
126C4×C4⋊C4—71no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7—21no—
136C23×C7—21no—
137C23×C7—21no—
138C23×C7—21no—
139C23×C7—21no—
140C23×C7C2×D412yes—
141C23×C7C2411yescharacteristic
142C22×D7—21no—
143C22×D7—21no—
144C22×D7—21no—
145C22×D7—21no—
146C22×D7—21no—
147C22×D7—21no—
148C22×D7#4112yes—
149C7×Q8—44no—
150C7×D4—44no—
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×C7C2×D412yes—
164C2×C4×C7C2×D412yes—
165C2×C4×C7C2×D412yes—
166C22⋊D7—28no—
167C22⋊D7—44no—
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—
183C2×Dic7—21no—
184C2×Dic7—21no—
185C2×Dic7—44no—
186C2×Dic7C2×D412yes—
187C2×Dic7C2×D412yes—
188C2×Dic7#4112yes—
189C2×Dic7#4112yes—
190C4×D7—44no—
191C22×Q8C2×D711yescharacteristic
192C22×D4—71no—
193C22×D4—71no—
194C22×D4C2×D711yescharacteristic
195C23×C4—71no—
196C23×C4—71no—
197#124—24no—
198#123—144no—
199C22⋊Q8—144no—
200C4×D4—144no—
201C2×C4⋊C4—71no—
202C2×C4⋊C4—71no—
203C2×C4⋊C4—71no—
204C2×C4⋊C4—71no—
205C2×C4⋊C4—71no—
206C2×C4⋊C4—71no—
207C2×C4⋊C4—71no—
208C2×C22⋊C4—71no—
209C2×C22⋊C4—71no—
210C2×C22⋊C4—71no—
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⋊C4C2×D711yescharacteristic
218C2×C22⋊C4C2×D711yescharacteristic
219C2×C22⋊C4C2×D711yescharacteristic
220C2×C22⋊C4C2×D711yescharacteristic
221C2×C42—71no—
222C2×C42—71no—
223C2×C42C2×D711yescharacteristic
224#95—71no—
225#95—71no—
226#95—71no—
227#95—71no—
228#95—71no—
229#95—71no—
230C22×C7—22no—
231C22×C7—22no—
232C22×C7—22no—
233C22×C7—41no—
234C22×C7—41no—
235C22×C7—41no—
236C22×C7—41no—
237C22×C7—41no—
238C22×C7—41no—
239C22×C7—41no—
240C22×C7—41no—
241C22×C7—41no—
242C22×C7—41no—
243C22×C7—41no—
244C22×C7C22×D411yescharacteristic
245C22×C7C22×D411yescharacteristic
246C22×C7C2×#4111yescharacteristic
247C22×C7C2×#4111yescharacteristic
248C22×C7C2×#4111yescharacteristic
249C22×C7H(5,𝔽2)11yescharacteristic
250C22×C72−1+411yescharacteristic
251C2×D7—22no—
252C2×D7—22no—
253C2×D7—22no—
254C2×D7—41no—
255C2×D7—41no—
256C2×D7—41no—
257C2×D7—41no—
258C4×C7—22no—
259C4×C7—22no—
260C4×C7—22no—
261C4×C7—41no—
262C4×C7—41no—
263Dic7—22no—
264Dic7—22no—
265Dic7—22no—
266Dic7—22no—
267Dic7—41no—
268Dic7—41no—
269Dic7—41no—
270C24—71no—
271C24C22×D711yescharacteristic
272C24C22×D711yescharacteristic
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—41no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—41no—
279C2×D4—142no—
280C2×D4—142no—
281C2×D4—142no—
282C2×D4—142no—
283C2×D4—142no—
284C2×D4—281no—
285C22×C4—71no—
286C22×C4—71no—
287C22×C4—71no—
288C22×C4—71no—
289C22×C4—71no—
290C22×C4—71no—
291C22×C4—71no—
292C22×C4—142no—
293C22×C4—142no—
294C22×C4—142no—
295C22×C4—142no—
296C22×C4—281no—
297C22×C4—281no—
298C22×C4C22×D711yescharacteristic
299C22×C4C22×D711yescharacteristic
300C22×C4C22×D711yescharacteristic
301C22×C4C22×D711yescharacteristic
302C22×C4C22×D711yescharacteristic
303C4⋊C4—74no—
304C4⋊C4—142no—
305C4⋊C4—142no—
306C4⋊C4—142no—
307C4⋊C4—142no—
308C4⋊C4—142no—
309C4⋊C4—281no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—142no—
314C22⋊C4—142no—
315C22⋊C4—142no—
316C22⋊C4—142no—
317C22⋊C4—142no—
318C22⋊C4—142no—
319C22⋊C4—142no—
320C22⋊C4—142no—
321C22⋊C4—142no—
322C22⋊C4C22⋊D714yescomplemented
323C42—22no—
324C42—142no—
325C42—142no—
326C2×C7—42no—
327C2×C7—81no—
328C2×C7#52511yescharacteristic
329C2×C7#53211yescharacteristic
330C2×C7#54211yescharacteristic
331C2×C7D4211yescharacteristic
332C2×C7#54711yescharacteristic
333C2×C7#54811yescharacteristic
334C2×C7#55111yescharacteristic
335D7—42no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—72no—
350C23—141no—
351C23—141no—
352C23—141no—
353C23—141no—
354C23—141no—
355C23—141no—
356C23C2×C22⋊D712yes—
357C23C23×D711yescharacteristic
358Q8—44no—
359D4—44no—
360D4—148no—
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—72no—
375C2×C4—72no—
376C2×C4—72no—
377C2×C4—72no—
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—141no—
388C2×C4—141no—
389C2×C4—141no—
390C2×C4—141no—
391C2×C4—141no—
392C2×C4—141no—
393C2×C4—141no—
394C2×C4—141no—
395C2×C4—284no—
396C2×C4—284no—
397C2×C4D4×D712yes—
398C2×C4C2×C22⋊D712yes—
399C2×C4C2×C22⋊D712yes—
400C7#246111yescharacteristic, complemented, Sylow
401C22—22no—
402C22—22no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—142no—
416C22—142no—
417C22—142no—
418C22—281no—
419C22—281no—
420C22—281no—
421C22—281no—
422C22C2×D4×D711yescharacteristic
423C22#654811yescharacteristic
424C22D7×#4111yescharacteristic
425C22D7×#4111yescharacteristic
426C22D7×#4111yescharacteristic
427C22#655411yescharacteristic
428C22C22×C22⋊D711yescharacteristic
429C4—22no—
430C4—22no—
431C4—22no—
432C4—41no—
433C4—41no—
434C4—142no—
435C4—142no—
436C4—142no—
437C4—142no—
438C4—281no—
439C4—281no—
440C4—281no—
441C2—42no—
442C2—81no—
443C2—282no—
444C2#9035911yescharacteristic
445C2#9036011yescharacteristic
446C2#9036811yescharacteristic
447C2#9037011yescharacteristic
448C2D4×C22⋊D711yescharacteristic
449C2#9051511yescharacteristic
450C2#9051911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#124C2
prime radical#89995C2
hypercenterC2×#124D7
hypocenterC7#2461
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×C7×#124C2semidirect
2. C2×#6509C2semidirect
3. C2×#6498C2semidirect
4. C2×(C22⋊D7)⋊C4C2semidirect
5. #90001C2semidirect
7. #89731C2semidirect
8. #89729C2semidirect
9. #89728C2semidirect
10. #89723C2semidirect
11. #89720C2semidirect
12. #89718C2semidirect
13. #89716C2semidirect
14. #89712C2semidirect
15. C4×Dic7⋊C4C2semidirect
92. C7×C22⋊C4D4semidirect
109. Dic7⋊C4D4semidirect
112. C2×#124D7semidirect
322. C22⋊C4C22⋊D7semidirect
400. C7#2461semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1242
C2×#65092
C2×#64982
C2×(C22⋊D7)⋊C42
#900012
#899952
#897312
#897292
#897282
#897232
#897202
#897182
#897162
#897122
C4×Dic7⋊C42
#24617