Finite Group Databasebeta

← #13367595

Subgroups — #13367595

Order 1792.

950 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#495C211yesmaximal, characteristic
2Dic7×#124C212yesmaximal, complemented
3#11729819C212yesmaximal, complemented
4#11729802C212yesmaximal, complemented
5#11729801C212yesmaximal, complemented
6#11729791C212yesmaximal
7#11729479C212yesmaximal, complemented
8#11727517C211yesmaximal, characteristic, complemented
9#11727504C211yesmaximal, characteristic, complemented
10C22×C7×Q16C2211yescharacteristic
11C22×C7×SD16C2212yes—
12C22×C7×D8C2211yescharacteristic
13C2×C7×#124C2212yes—
14C2×Q8×Dic7C2212yes—
15C2×D4×Dic7C2212yes—
16C7×#495C418yescomplemented
17C2×C4×C7×C8C2211yescharacteristic
18#90002C2212yes—
19C2×#6409C2212yes—
20C2×#6409C2212yes—
21#89993C2212yes—
22C2×#6406C2212yes—
23C2×#6406C2212yes—
24#89723—22no—
25#89723—22no—
26C22⋊C4×Dic7—22no—
27C22⋊C4×Dic7—22no—
28#89708C2212yes—
29C42×Dic7C2211yescharacteristic
30C2×C4×C7⋊C8C2211yescharacteristic
31#89346C2212yes—
32#89346C2211yescharacteristic
33#89346C2211yescharacteristic
34#89345C2212yes—
35#89345C2212yes—
36#22271—71nomaximal, Sylow
37C22×C7×Q8C2312yes—
38C22×C7×D4C2312yes—
39C23×Dic7—22no—
40C2×C7×Q16—42no—
41C2×C7×Q16C2×C414yescomplemented
42C2×C7×SD16—42no—
43C2×C7×SD16—42no—
44C2×C7×SD16C2×C418yescomplemented
45C2×C7×D8—42no—
46C2×C7×D8C2×C414yes—
47C22×C7×C8C2311yescharacteristic
48C22×C7×C8C2311yescharacteristic
49C7×#124—24no—
50C7×#124C2×C418yescomplemented
51C2×C7×C22⋊C4—22no—
52C2×C7×C22⋊C4—22no—
53C2×C42×C7C2311yescharacteristic
54C2×C22⋊Dic7—22no—
55C2×C22⋊Dic7—22no—
56C2×C22⋊Dic7—22no—
57Q8×Dic7—42no—
58Q8×Dic7—42no—
59D4×Dic7—42no—
60D4×Dic7—42no—
61C2×C4⋊Dic7—22no—
62C2×C4⋊Dic7C2312yes—
63C2×C4⋊Dic7C2312yes—
64C2×C4×Dic7—22no—
65C2×C4×Dic7D412yes—
66C2×C4×Dic7D411yescharacteristic
67C2×C4×Dic7D411yescharacteristic
68C2×C4×Dic7C2312yes—
69C22×C7⋊C8C2312yes—
70C4×C7×C8C2×C414yescomplemented
71#6409—42no—
72#6409—42no—
73#6406—42no—
74#6406—42no—
75C7⋊#95—22no—
76C7⋊#95—22no—
77C7⋊#95—22no—
78C7⋊#95—22no—
79C4×C7⋊C8—22no—
80C2×#495D711yescharacteristic
81C4×#124—72no—
82#1958—72no—
83#1880—72no—
84#1879—72no—
85#1878—72no—
86#1877—72no—
87#1835—71no—
88#1744—71no—
89C24×C7—22no—
90C2×C7×Q8—22no—
91C2×C7×Q8—22no—
92C2×C7×Q8—22no—
93C2×C7×Q8—22no—
94C2×C7×Q8—22no—
95C2×C7×Q8C22×C414yes—
96C2×C7×D4—22no—
97C2×C7×D4—22no—
98C2×C7×D4—22no—
99C2×C7×D4—22no—
100C2×C7×D4—22no—
101C2×C7×D4C22×C414yes—
102C22×C4×C7—22no—
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7C2×D411yescharacteristic
105C22×C4×C7C2411yescharacteristic
106C22×Dic7—22no—
107C22×Dic7—22no—
108C22×Dic7—44no—
109C22×Dic7—44no—
110C22×Dic7—44no—
111C22×Dic7C2×D412yes—
112C22×Dic7C2×D412yes—
113C7×Q16—44no—
114C7×SD16—48no—
115C7×D8—44no—
116C2×C7×C8—22no—
117C2×C7×C8—22no—
118C2×C7×C8C22×C412yes—
119C2×C7×C8C22×C412yes—
120C7×C22⋊C4—28no—
121C7×C22⋊C4—28no—
122C42×C7C22×C412yes—
123C42×C7#4112yes—
124C22⋊Dic7—44no—
125C22⋊Dic7—44no—
126C22⋊Dic7—44no—
127C4⋊Dic7—22no—
128C4⋊Dic7—22no—
129C4⋊Dic7—22no—
130C4⋊Dic7—22no—
131C4⋊Dic7—44no—
132C4×Dic7—22no—
133C4×Dic7—22no—
134C4×Dic7—24no—
135C4×Dic7—41no—
136C4×Dic7—41no—
137C4×Dic7—44no—
138C2×C7⋊C8—22no—
139C2×C7⋊C8—22no—
140C2×C7⋊C8—22no—
141C22×Q16C2×D711yescharacteristic
142C22×SD16C2×D712yes—
143C22×D8C2×D711yescharacteristic
144C2×#124C2×D712yes—
145C2×C4×Q8—72no—
146C2×C4×D4—72no—
147#495Dic718yescomplemented
148C2×#103—72no—
149C2×#103—72no—
150C2×#102—72no—
151C2×#102—72no—
152C2×C4×C8—71no—
153C2×C4×C8C2×D711yescharacteristic
154#391—72no—
155#390—72no—
156#388—142no—
157#388—142no—
158#382—72no—
159C4×C22⋊C4—142no—
160C4×C22⋊C4—142no—
161C43—71no—
162#340—71no—
163#340—72no—
164#340—71no—
165#336—72no—
166#336—72no—
167C23×C7—24no—
168C23×C7—42no—
169C23×C7—42no—
170C23×C7—42no—
171C23×C7—42no—
172C23×C7—42no—
173C23×C7—42no—
174C23×C7C22×D411yescharacteristic
175C7×Q8—24no—
176C7×Q8—24no—
177C7×Q8—42no—
178C7×Q8—42no—
179C7×D4—24no—
180C7×D4—24no—
181C7×D4—42no—
182C7×D4—42no—
183C2×C4×C7—22no—
184C2×C4×C7—21no—
185C2×C4×C7—21no—
186C2×C4×C7—24no—
187C2×C4×C7—42no—
188C2×C4×C7—42no—
189C2×C4×C7C4×D412yes—
190C2×C4×C7C4×D412yes—
191C2×C4×C7C23×C411yescharacteristic
192C2×C4×C7C2×#4111yescharacteristic
193C2×C4×C7H(5,𝔽2)12yes—
194C2×C4×C7H(5,𝔽2)12yes—
195C2×Dic7—22no—
196C2×Dic7—22no—
197C2×Dic7—22no—
198C2×Dic7—22no—
199C2×Dic7—22no—
200C2×Dic7—42no—
201C2×Dic7—42no—
202C2×Dic7—42no—
203C2×Dic7—42no—
204C2×Dic7—42no—
205C2×Dic7—42no—
206C2×Dic7—88no—
207C2×Dic7#13514yes—
208C7×C8—22no—
209C7×C8—22no—
210C7⋊C8—42no—
211C22×Q8C22×D712yes—
212C22×D4C22×D712yes—
213C23×C4—142no—
214C2×Q16—42no—
215C2×Q16C2×Dic714yescomplemented
216C2×SD16—42no—
217C2×SD16—42no—
218C2×SD16C2×Dic718yescomplemented
219C2×D8—42no—
220C2×D8C2×Dic714yes—
221C22×C8—72no—
222C22×C8C22×D711yescharacteristic
223C22×C8C22×D711yescharacteristic
224#124—24no—
225#124C2×Dic718yescomplemented
226C4×Q8—282no—
227C4×Q8—282no—
228C4×D4—282no—
229C4×D4—282no—
230C2×C4⋊C4—72no—
231C2×C4⋊C4—72no—
232C2×C4⋊C4—142no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—142no—
236C2×C22⋊C4—142no—
237C2×C22⋊C4—142no—
238C2×C42—72no—
239C2×C42—72no—
240C2×C42—71no—
241C2×C42—71no—
242C2×C42—142no—
243C2×C42C22×D711yescharacteristic
244#103—282no—
245#103—282no—
246#102—282no—
247#102—282no—
248C4×C8—142no—
249C4×C8C2×Dic714yescomplemented
250#95—142no—
251#95—142no—
252#95—142no—
253#95—142no—
254C22×C7—44no—
255C22×C7—44no—
256C22×C7—44no—
257C22×C7—82no—
258C22×C7—82no—
259C22×C7—82no—
260C22×C7—82no—
261C22×C7C2×C4×D411yescharacteristic
262C22×C7#53512yes—
263C22×C7C2×#13512yes—
264C22×C7#57511yescharacteristic
265C22×C7#57511yescharacteristic
266C4×C7—22no—
267C4×C7—22no—
268C4×C7—44no—
269C4×C7#51811yescharacteristic
270C4×C7#51811yescharacteristic
271C4×C7#54012yes—
272Dic7—24no—
273Dic7—42no—
274Dic7—82no—
275Dic7—82no—
276C24—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8C22×Dic714yes—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4C22×Dic714yes—
289C22×C4—22no—
290C22×C4—72no—
291C22×C4—72no—
292C22×C4—142no—
293C22×C4—142no—
294C22×C4—284no—
295C22×C4—284no—
296C22×C4—284no—
297C22×C4D4×D711yescharacteristic
298C22×C4D4×D711yescharacteristic
299C22×C4C23×D711yescharacteristic
300Q16—44no—
301SD16—48no—
302D8—44no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—142no—
306C2×C8—142no—
307C2×C8—142no—
308C2×C8C22×Dic712yes—
309C2×C8C22×Dic712yes—
310C4⋊C4—142no—
311C4⋊C4—142no—
312C4⋊C4—142no—
313C4⋊C4—142no—
314C4⋊C4—284no—
315C22⋊C4—28no—
316C22⋊C4—28no—
317C22⋊C4—284no—
318C22⋊C4—284no—
319C22⋊C4—284no—
320C42—144no—
321C42—142no—
322C42—142no—
323C42—281no—
324C42—281no—
325C42—284no—
326C42#115912yes—
327C42C22×Dic712yes—
328C2×C7—84no—
329C2×C7#232811yescharacteristic
330C2×C7C4×#13511yescharacteristic
331C2×C7#294311yescharacteristic
332C2×C7#321112yes—
333C2×C7#321212yes—
334C23—24no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23C2×D4×D711yescharacteristic
342Q8—24no—
343Q8—24no—
344Q8—42no—
345Q8—42no—
346D4—24no—
347D4—24no—
348D4—42no—
349D4—42no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—24no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—74no—
357C2×C4—142no—
358C2×C4—142no—
359C2×C4—142no—
360C2×C4—142no—
361C2×C4—142no—
362C2×C4—282no—
363C2×C4—282no—
364C2×C4—282no—
365C2×C4—282no—
366C2×C4—282no—
367C2×C4—282no—
368C2×C4—568no—
369C2×C4D4×Dic712yes—
370C2×C4D4×Dic712yes—
371C2×C4C2×#115911yescharacteristic
372C2×C4#654812yes—
373C2×C4#654812yes—
374C2×C4C23×Dic711yescharacteristic
375C8—22no—
376C8—22no—
377C8—282no—
378C7#2227111yescharacteristic, complemented, Sylow
379C22—44no—
380C22—44no—
381C22—44no—
382C22—82no—
383C22—82no—
384C22—82no—
385C22—82no—
386C22#9040612yes—
387C22D7×#13512yes—
388C22#9045911yescharacteristic
389C22#9045911yescharacteristic
390C22C2×D4×Dic711yescharacteristic
391C4—22no—
392C4—22no—
393C4—44no—
394C4—144no—
395C4—282no—
396C4—562no—
397C4—562no—
398C4#9041012yes—
399C4#9048911yescharacteristic
400C4#9048911yescharacteristic
401C2—84no—
402C2#1173927012yes—
403C2#1173927112yes—
404C2#1174061711yescharacteristic
405C2Dic7×#13511yescharacteristic
406C2#1174134211yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#495C2
socleC23×C7C22×D4
prime radicalC22×C7×D8C22
cosocleC22×C4×C7C24
hypercenterC2×#495D7
hypocenterC7#22271
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#495

Complements

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

NH ≅ G/Ntype
2. Dic7×#124C2semidirect
3. #11729819C2semidirect
4. #11729802C2semidirect
5. #11729801C2semidirect
7. #11729479C2semidirect
8. #11727517C2semidirect
9. #11727504C2semidirect
16. C7×#495C4semidirect
41. C2×C7×Q16C2×C4semidirect
44. C2×C7×SD16C2×C4semidirect
50. C7×#124C2×C4semidirect
70. C4×C7×C8C2×C4semidirect
147. #495Dic7semidirect
215. C2×Q16C2×Dic7semidirect
218. C2×SD16C2×Dic7semidirect
225. #124C2×Dic7semidirect
249. C4×C8C2×Dic7semidirect
378. C7#22271semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4952
Dic7×#1242
#117298192
#117298022
#117298012
#117297912
#117294792
#117275172
#117275042
#222717