Finite Group Databasebeta

← #13339017

Subgroups — #13339017

Order 1792.

674 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740795C211yesmaximal, characteristic
2#11740768C211yesmaximal, characteristic, complemented
3#11736731C211yesmaximal, characteristic
4C2×#89613C211yesmaximal, characteristic
5C7×#1542C211yesmaximal, characteristic, complemented
6#11728532C211yesmaximal, characteristic, complemented
7#11728175C211yesmaximal, characteristic, complemented
8#11726250C214yesmaximal, complemented
9#11726198C214yesmaximal, complemented
10C7×#533C2211yescharacteristic
11C22×#1132—21no—
12C2×C4×D28C2211yescharacteristic
13C2×#6432C2211yescharacteristic
14C7×#431C2211yescharacteristic
15C2×C7×C4⋊C8C2211yescharacteristic
16C2×#6390—21no—
17#89859C2212yes—
18#89856C2212yes—
19#89853C2212yescomplemented
20#89851C2212yescomplemented
21#89838—22no—
22#89815—22no—
23#89792—22no—
24#89773—21no—
25#89767—21no—
26#89697C2211yescharacteristic
27#89696C2211yescharacteristic
28#89613—22no—
29#89613C2214yes—
30#89606C2214yes—
31D7×C22⋊C8—22no—
32C8×D28—22no—
33C7×#333C2212yescomplemented
34C7×#332C2212yescomplemented
35#89281C2214yescomplemented
36#89275C2214yescomplemented
37#19390—71nomaximal, Sylow
38C22×D28D411yescharacteristic
39C22×C4×D7—21no—
40C22×Dic14—21no—
41C22×C7×C8—21no—
42C7×C4⋊Q8C2311yescharacteristic
43C7×C4⋊Q8C2311yescharacteristic
44C7×#125—21no—
45C7×C22⋊Q8—21no—
46C7×C22⋊Q8—21no—
47C4×C7×Q8—21no—
48C4×C7×Q8—21no—
49C7×#117—21no—
50C2×C42×C7C2311yescharacteristic
51C2×(C2×D7)⋊C4—21no—
52C2×C4⋊Dic7—21no—
53C2×C4⋊Dic7—21no—
54C2×C4⋊Dic7D411yescharacteristic
55C2×#1136—21no—
56C2×#1132—24no—
57C2×#1132—44no—
58C2×C8×D7—24no—
59C4×D28—21no—
60C4×D28—21no—
61C4×D28D412yes—
62C4×D28C2312yes—
63#6432—21no—
64#6432—21no—
65#6432—21no—
66#6432C2312yes—
67C7×C4⋊C8—21no—
68C7×C4⋊C8C2312yes—
69C7×C4⋊C8C2312yes—
70C7×C22⋊C8—21no—
71C4×C7×C8—21no—
72#6409—24no—
73C22⋊(C7⋊C8)D412yes—
74(C2×D7)⋊C8—24no—
75#6390—24no—
76#6383—22no—
77#6383—22no—
78#6382—22no—
79#6382—24no—
80#6381—22no—
81C4⋊(C7⋊C8)—21no—
82C4⋊(C7⋊C8)D412yes—
83C4⋊(C7⋊C8)C2312yes—
84C4×C7⋊C8C2312yes—
85#3229—71no—
86#3077—71no—
87C2×#461—71no—
88#2972—71no—
89#1679—74no—
90#1671—74no—
91#1570—71no—
92#1546—71no—
93#1542D711yescharacteristic, complemented
94C23×D7—21no—
95C2×C7×Q8—21no—
96C2×C7×Q8—21no—
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C22×C4×C7#4111yescharacteristic
100C22×Dic7—21no—
101C22×Dic7—41no—
102C2×D28—21no—
103C2×D28—22no—
104C2×D28SD1614yes—
105C2×D28C2×D412yes—
106C2×C4×D7—22no—
107C2×C4×D7—22no—
108C2×C4×D7—82no—
109C2×Dic14—22no—
110C2×Dic14—42no—
111C2×Dic14—42no—
112C2×Dic14—41no—
113C2×C7×C8—22no—
114C2×C7×C8—22no—
115C2×C7×C8—42no—
116C7×C4⋊C4—21no—
117C7×C4⋊C4—21no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—41no—
121C7×C4⋊C4—41no—
122C7×C22⋊C4—41no—
123C42×C7—41no—
124C42×C7C2×D411yescharacteristic
125C42×C7C2×D411yescharacteristic
126C42×C7#4111yescharacteristic
127C42×C7C2411yescharacteristic
128(C2×D7)⋊C4—22no—
129(C2×D7)⋊C4—22no—
130C4⋊Dic7—21no—
131C4⋊Dic7—22no—
132C4⋊Dic7—22no—
133C4⋊Dic7—41no—
134C4⋊Dic7—41no—
135C4⋊Dic7C2×D412yes—
136#1136—42no—
137C2×C7⋊C8—22no—
138C2×C7⋊C8C2×D414yes—
139#1132—48no—
140C8×D7—84no—
141C22×SD16—141no—
142#533C2×D711yescharacteristic
143C2×C4⋊Q8—71no—
144C2×C4×D4—71no—
145#498—72no—
146#487—72no—
147#483—142no—
148#478—142no—
149#476—72no—
150#475—142no—
151#474—72no—
152#461—74no—
153#461—142no—
154C8×D4—74no—
155C8×D4—142no—
156#431—71no—
157#431C2×D711yescharacteristic
158#428—141no—
159#423—71no—
160C2×C4⋊C8C2×D711yescharacteristic
161#418—141no—
162C2×#103—141no—
163C2×C22⋊C8—142no—
164#334—74no—
165#333C2×D712yescomplemented
166#332—74no—
167#332C2×D712yescomplemented
168C23×C7#12111yescharacteristic
169C22×D7—22no—
170C22×D7—24no—
171C22×D7—81no—
172C22×D7—81no—
173C7×Q8—42no—
174C7×Q8—81no—
175C2×C4×C7—22no—
176C2×C4×C7—41no—
177C2×C4×C7—41no—
178C2×C4×C7—41no—
179C2×C4×C7#12111yescharacteristic
180C2×C4×C7#12111yescharacteristic
181C2×C4×C7#12111yescharacteristic
182C2×C4×C7C2×SD1612yes—
183C2×C4×C7#13512yes—
184C2×C4×C7C22×D411yescharacteristic
185C2×C4×C7C22×D411yescharacteristic
186C2×C4×C7C2×#4111yescharacteristic
187C2×Dic7—22no—
188C2×Dic7—42no—
189C2×Dic7—81no—
190C2×Dic7—82no—
191D28—22no—
192D28—24no—
193D28C2×SD1614yes—
194C4×D7—44no—
195C4×D7—82no—
196Dic14—42no—
197Dic14—42no—
198Dic14—82no—
199C7×C8—42no—
200C7×C8—42no—
201C7⋊C8—24no—
202C7⋊C8—42no—
203C22×Q8—141no—
204C22×D4—71no—
205C23×C4—141no—
206C2×SD16—144no—
207C2×SD16—284no—
208C2×M16—141no—
209C22×C8—21no—
210C22×C8—144no—
211C4⋊Q8—72no—
212C4⋊Q8—141no—
213C4⋊Q8—141no—
214C4⋊Q8—141no—
215C4⋊Q8C22×D711yescharacteristic
216C4⋊Q8C22×D711yescharacteristic
217#125—21no—
218C22⋊Q8—21no—
219C22⋊Q8—21no—
220C4×Q8—21no—
221C4×Q8—21no—
222C4×D4—72no—
223C4×D4—72no—
224C4×D4—141no—
225C4×D4—141no—
226#117—21no—
227C2×C4⋊C4—71no—
228C2×C4⋊C4—141no—
229C2×C4⋊C4—141no—
230C2×C22⋊C4—141no—
231C2×C42C22×D711yescharacteristic
232#107—142no—
233#106—142no—
234#106—144no—
235C4⋊C8—21no—
236C4⋊C8—72no—
237C4⋊C8—72no—
238C4⋊C8—141no—
239C4⋊C8C22×D712yes—
240C4⋊C8C22×D712yes—
241#103—144no—
242#103—144no—
243#102—142no—
244#102—142no—
245C22⋊C8—21no—
246C22⋊C8—72no—
247C22⋊C8—144no—
248C4×C8—21no—
249C4×C8—72no—
250C22×C7—21no—
251C22×C7—21no—
252C22×C7#46512yes—
253C22×C7C2×#12111yescharacteristic
254C2×D7—24no—
255C2×D7—42no—
256C2×D7—42no—
257C2×D7—84no—
258C4×C7—21no—
259C4×C7—21no—
260C4×C7—21no—
261C4×C7—81no—
262C4×C7—81no—
263C4×C7#46512yes—
264C4×C7C22×SD1611yescharacteristic
265C4×C7C2×#13511yescharacteristic
266C4×C7#57712yes—
267Dic7—42no—
268Dic7—82no—
269C24—141no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—142no—
273C2×Q8—281no—
274C2×Q8—282no—
275C2×Q8—282no—
276C2×D4—74no—
277C2×D4—72no—
278C2×D4—141no—
279C2×D4—142no—
280C22×C4—141no—
281C22×C4—142no—
282C22×C4—142no—
283C22×C4—281no—
284C22×C4—562no—
285C22×C4#115711yescharacteristic
286C22×C4D4×D711yescharacteristic
287C22×C4D4×D711yescharacteristic
288SD16—288no—
289M16—282no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—42no—
293C2×C8—74no—
294C2×C8—142no—
295C2×C8—564no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—72no—
303C4⋊C4—141no—
304C4⋊C4—142no—
305C4⋊C4—142no—
306C4⋊C4—281no—
307C4⋊C4—281no—
308C22⋊C4—41no—
309C22⋊C4—142no—
310C22⋊C4—142no—
311C42—41no—
312C42#115711yescharacteristic
313C42D4×D711yescharacteristic
314C42D4×D711yescharacteristic
315C42C23×D711yescharacteristic
316C2×C7—22no—
317C2×C7#304211yescharacteristic
318C2×C7C2×#46511yescharacteristic
319C2×C7#306911yescharacteristic
320D7—44no—
321C23—142no—
322C23—144no—
323C23—561no—
324C23—561no—
325C23#644711yescharacteristic
326Q8—42no—
327Q8—81no—
328Q8—282no—
329Q8—282no—
330Q8—562no—
331D4—74no—
332D4—142no—
333D4—144no—
334C2×C4—22no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—142no—
339C2×C4—284no—
340C2×C4—282no—
341C2×C4—562no—
342C2×C4—561no—
343C2×C4—562no—
344C2×C4#644711yescharacteristic
345C2×C4#644711yescharacteristic
346C2×C4#644711yescharacteristic
347C2×C4D7×SD1612yes—
348C2×C4#648212yes—
349C2×C4C2×#115711yescharacteristic
350C2×C4C2×D4×D711yescharacteristic
351C2×C4C2×D4×D711yescharacteristic
352C8—42no—
353C8—42no—
354C8—144no—
355C8—282no—
356C7#1939011yescharacteristic, complemented, Sylow
357C22—21no—
358C22—21no—
359C22—144no—
360C22—282no—
361C22—282no—
362C22—564no—
363C22#8957912yes—
364C22C2×#644711yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—81no—
369C4—81no—
370C4—282no—
371C4—562no—
372C4#8957912yes—
373C4C2×D7×SD1611yescharacteristic
374C4C2×#648211yescharacteristic
375C4#9046112yes—
376C2—22no—
377C2—284no—
378C2#1173616611yescharacteristic
379C2C2×#8957911yescharacteristic
380C2#1173630811yescharacteristic
subgroupG/H
centerC22C2×#6447
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#1542C2
socleC22×C7C2×#121
prime radicalC22×D28D4
hypercenter#1542D7
hypocenterC7#19390
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
2. #11740768C2semidirect
5. C7×#1542C2semidirect
6. #11728532C2semidirect
7. #11728175C2semidirect
8. #11726250C2semidirect
9. #11726198C2semidirect
19. #89853C22semidirect
20. #89851C22semidirect
33. C7×#333C22semidirect
34. C7×#332C22semidirect
35. #89281C22semidirect
36. #89275C22semidirect
93. #1542D7semidirect
165. #333C2×D7semidirect
167. #332C2×D7semidirect
356. C7#19390semidirect

Maximal subgroups

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

subgroupindex
#117407952
#117407682
#117367312
C2×#896132
C7×#15422
#117285322
#117281752
#117262502
#117261982
#193907