Finite Group Databasebeta

← #13362295

Subgroups — #13362295

Order 1792.

1126 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740672C211yesmaximal, characteristic, complemented
2C2×#89857C211yesmaximal, characteristic
3C2×#89589C212yesmaximal, complemented
4C2×#89560C212yesmaximal, complemented
5#11731598C211yesmaximal, characteristic
6C7×#1960C211yesmaximal, characteristic, complemented
7#11728014C212yesmaximal, complemented
8#11728011C212yesmaximal, complemented
9#11727577C211yesmaximal, characteristic
10#11727558C211yesmaximal, characteristic, complemented
11#11727303C211yesmaximal, characteristic, complemented
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×Q8×Dic7C2212yes—
14C22×#1143C2212yes—
15C22×#1143C2212yes—
16C2×#6456C2212yes—
17C2×#6437C2211yescharacteristic
18C2×C7×#103C2212yes—
19C2×C7×#103C2212yes—
20C2×C7×#97C2211yescharacteristic
21C7×#382C2211yescharacteristic
22#90002C2212yes—
23C2×#6395C2212yes—
24C2×#6395C2212yes—
25C2×#6389C2211yescharacteristic
26C2×#6389C2211yescharacteristic
27C2×C8×Dic7C2211yescharacteristic
28C2×C8×Dic7C2211yescharacteristic
29#89857C418yescomplemented
30#89760C2212yes—
31#89753—24no—
32C2×#6382C2212yes—
33C2×#6381C2212yes—
34#89708C2211yescharacteristic
35#89704C2211yescharacteristic
36C2×C4×C7⋊C8C2211yescharacteristic
37#89589—28no—
38#89560—28no—
39#89437—24no—
40#21695—71nomaximal, Sylow
41C22×C7×Q8C2312yes—
42C22×D28—22no—
43C22×D28C2312yes—
44C22×C4×D7—22no—
45C22×C7×C8C2311yescharacteristic
46C22×C7×C8C2311yescharacteristic
47C7×C4⋊Q8—22no—
48C7×C4⋊Q8C2×C414yescomplemented
49C2×C7×C4⋊C4—22no—
50C2×C7×C4⋊C4C2312yes—
51C2×C42×C7C2311yescharacteristic
52Q8×Dic7—24no—
53Q8×Dic7—24no—
54C2×#1143—24no—
55C2×#1143—24no—
56C2×#1143—24no—
57C2×#1143—24no—
58C2×#1143C2×C418yescomplemented
59C2×#1143C2×C418yescomplemented
60C2×(C2×D7)⋊C4—22no—
61C2×(C2×D7)⋊C4—24no—
62C2×C4⋊Dic7—22no—
63C2×C4⋊Dic7C2312yes—
64C2×C4×Dic7—22no—
65C2×C4×Dic7C2311yescharacteristic
66C2×C4×Dic7C2311yescharacteristic
67C22×C7⋊C8C2311yescharacteristic
68C22×C7⋊C8C2311yescharacteristic
69#6456—24no—
70#6456—24no—
71#6437—22no—
72#6437C2×C414yes—
73C7×#103—24no—
74C7×#103—24no—
75C7×#97—22no—
76C7×#95—22no—
77C7⋊#95—22no—
78C7⋊#95—24no—
79C7⋊#95D414yes—
80#6395—24no—
81#6395—24no—
82#6389—22no—
83#6389D414yescomplemented
84C8×Dic7—22no—
85C8×Dic7D414yescomplemented
86#6382—24no—
87#6381—24no—
88C4×C7⋊C8C2×C414yescomplemented
89C2×#492—71no—
90C2×#440—72no—
91C2×C4×SD16—72no—
92#2328—71no—
93#2327—71no—
94#1973—72no—
95#1968—72no—
96#1961—71no—
97#1960D711yescharacteristic, complemented
98#1846—71no—
99#1726—71no—
100C23×D7—22no—
101C2×C7×Q8—22no—
102C2×C7×Q8—22no—
103C2×C7×Q8—22no—
104C2×C7×Q8—22no—
105C2×C7×Q8—22no—
106C2×C7×Q8C22×C414yes—
107C22×C4×C7—22no—
108C22×C4×C7—22no—
109C22×C4×C7C2×D412yes—
110C22×C4×C7C2411yescharacteristic
111C22×Dic7—22no—
112C22×Dic7C2×D412yes—
113C22×Dic7C2×D412yes—
114C2×D28—22no—
115C2×D28—22no—
116C2×D28—22no—
117C2×D28—22no—
118C2×D28—22no—
119C2×D28—28no—
120C2×D28—84no—
121C2×D28C22×C414yes—
122C2×C4×D7—44no—
123C2×C4×D7—84no—
124C2×C7×C8—21no—
125C2×C7×C8—21no—
126C2×C7×C8—21no—
127C2×C7×C8—21no—
128C2×C7×C8C2×D412yes—
129C2×C7×C8C2×D412yes—
130C7×C4⋊C4—28no—
131C7×C4⋊C4—22no—
132C7×C4⋊C4—22no—
133C42×C7C22×C412yes—
134C42×C7#4112yes—
135#1143—216no—
136#1143—216no—
137(C2×D7)⋊C4—44no—
138(C2×D7)⋊C4—84no—
139C4⋊Dic7—22no—
140C4⋊Dic7—22no—
141C4⋊Dic7—44no—
142C4×Dic7—21no—
143C4×Dic7—21no—
144C4×Dic7—44no—
145C4×Dic7C2×D412yes—
146C4×Dic7C2×D412yes—
147C2×C7⋊C8C22×C412yes—
148C2×C7⋊C8C22×C412yes—
149C2×C7⋊C8C2×D412yes—
150C2×C7⋊C8C2×D412yes—
151C2×C7⋊C8#4112yes—
152C2×C7⋊C8#4112yes—
153C22×SD16—72no—
154C22×SD16—72no—
155C2×C4⋊Q8C2×D711yescharacteristic
156C2×#127—71no—
157C2×C4×Q8—72no—
158C2×C4×D4—72no—
159#492—78no—
160#440—148no—
161C4×SD16—148no—
162C2×#107—72no—
163C2×#106—72no—
164C2×#103C2×D712yes—
165C2×#103C2×D712yes—
166C2×#102—72no—
167C2×#102—72no—
168C2×#97—71no—
169C2×#97—71no—
170C2×#97C2×D711yescharacteristic
171C2×C4×C8—71no—
172C2×C4×C8—71no—
173C2×C4×C8—71no—
174#391—72no—
175#390—72no—
176#387—144no—
177#386—144no—
178#382—71no—
179#382C2×D711yescharacteristic
180#376—71no—
181C23×C7C22×D411yescharacteristic
182C22×D7—24no—
183C22×D7—42no—
184C22×D7—42no—
185C22×D7—44no—
186C22×D7—44no—
187C7×Q8—24no—
188C7×Q8—24no—
189C7×Q8—42no—
190C7×Q8—42no—
191C2×C4×C7—24no—
192C2×C4×C7—24no—
193C2×C4×C7—42no—
194C2×C4×C7—42no—
195C2×C4×C7—42no—
196C2×C4×C7—44no—
197C2×C4×C7C4×D414yes—
198C2×C4×C7C23×C411yescharacteristic
199C2×C4×C7C22×D411yescharacteristic
200C2×C4×C7C22×D411yescharacteristic
201C2×C4×C7C2×#4111yescharacteristic
202C2×C4×C7C2×#4111yescharacteristic
203C2×C4×C7C2×#4111yescharacteristic
204C2×Dic7—24no—
205C2×Dic7—22no—
206C2×Dic7—24no—
207C2×Dic7—42no—
208C2×Dic7—44no—
209C2×Dic7#12714yes—
210D28—24no—
211D28—24no—
212D28—42no—
213D28—42no—
214D28—88no—
215C4×D7—88no—
216C7×C8—22no—
217C7×C8—22no—
218C7⋊C8C4×D414yes—
219C7⋊C8C4×D414yes—
220C22×Q8C22×D712yes—
221C22×D4—72no—
222C22×D4—142no—
223C23×C4—142no—
224C2×SD16—78no—
225C2×SD16—78no—
226C2×SD16—144no—
227C2×SD16—144no—
228C2×SD16—144no—
229C2×SD16—144no—
230C22×C8—71no—
231C22×C8—71no—
232C22×C8C22×D711yescharacteristic
233C22×C8C22×D711yescharacteristic
234C4⋊Q8—22no—
235C4⋊Q8C4×D714yescomplemented
236#127—74no—
237#127—142no—
238C4×Q8—144no—
239C4×Q8—144no—
240C4×D4—144no—
241C4×D4—144no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—72no—
244C2×C4⋊C4—142no—
245C2×C4⋊C4C22×D712yes—
246C2×C22⋊C4—142no—
247C2×C22⋊C4—144no—
248C2×C42—71no—
249C2×C42—71no—
250C2×C42—142no—
251C2×C42C22×D711yescharacteristic
252#107—144no—
253#106—144no—
254#103—24no—
255#103—24no—
256#102—144no—
257#102—144no—
258#97—22no—
259#97—74no—
260#97—142no—
261C4×C8—74no—
262C4×C8—74no—
263C4×C8—142no—
264#95—22no—
265#95—74no—
266#95—144no—
267#95—142no—
268C22×C7C2×C4×D411yescharacteristic
269C22×C7C2×#12711yescharacteristic
270C22×C7#53211yescharacteristic
271C22×C7#57714yes—
272C2×D7—44no—
273C2×D7—48no—
274C2×D7—82no—
275C2×D7—82no—
276C2×D7—84no—
277C4×C7—24no—
278C4×C7—44no—
279C4×C7—82no—
280C4×C7C2×C4×D411yescharacteristic
281C4×C7C2×C4×D411yescharacteristic
282C4×C7#53211yescharacteristic
283C4×C7#53211yescharacteristic
284Dic7—24no—
285Dic7—42no—
286Dic7—82no—
287C24—142no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8C2×C4×D714yes—
294C2×D4—74no—
295C2×D4—148no—
296C2×D4—142no—
297C2×D4—142no—
298C2×D4—142no—
299C2×D4—142no—
300C2×D4—142no—
301C2×D4—564no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—72no—
305C22×C4—72no—
306C22×C4—142no—
307C22×C4—284no—
308C22×C4—564no—
309C22×C4D4×D712yes—
310C22×C4C23×D711yescharacteristic
311SD16—1416no—
312SD16—1416no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—72no—
318C2×C8—72no—
319C2×C8—72no—
320C2×C8—72no—
321C2×C8—72no—
322C2×C8—72no—
323C2×C8D4×D712yes—
324C2×C8D4×D712yes—
325C4⋊C4—28no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—142no—
329C4⋊C4—142no—
330C4⋊C4—284no—
331C22⋊C4—284no—
332C22⋊C4—564no—
333C42—72no—
334C42—72no—
335C42—141no—
336C42—141no—
337C42—284no—
338C42C2×C4×D712yes—
339C42#115912yes—
340C2×C7C4×#12711yescharacteristic
341C2×C7#294912yes—
342C2×C7#315412yes—
343C2×C7#317412yes—
344D7—84no—
345C23—144no—
346C23—282no—
347C23—284no—
348C23—284no—
349C23—282no—
350C23C2×D4×D711yescharacteristic
351Q8—24no—
352Q8—24no—
353Q8—42no—
354Q8—42no—
355D4—144no—
356D4—144no—
357D4—282no—
358D4—282no—
359D4—568no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—74no—
367C2×C4—144no—
368C2×C4—144no—
369C2×C4—142no—
370C2×C4—282no—
371C2×C4—284no—
372C2×C4—568no—
373C2×C4(C22⋊D7)⋊C414yes—
374C2×C4C22×C4×D711yescharacteristic
375C2×C4C2×D4×D711yescharacteristic
376C2×C4C2×D4×D711yescharacteristic
377C2×C4C2×#115911yescharacteristic
378C2×C4D7×#4111yescharacteristic
379C2×C4D7×#4111yescharacteristic
380C8—22no—
381C8—22no—
382C8—74no—
383C8—74no—
384C7#2169511yescharacteristic, complemented, Sylow
385C22—288no—
386C22—284no—
387C22—564no—
388C22—562no—
389C22—562no—
390C22C2×(C22⋊D7)⋊C411yescharacteristic
391C22#9035911yescharacteristic
392C22D7×#12711yescharacteristic
393C22#9046914yes—
394C4—24no—
395C4—44no—
396C4—82no—
397C4—144no—
398C4—282no—
399C4—562no—
400C4C4×D4×D711yescharacteristic
401C4C4×D4×D711yescharacteristic
402C4#9028711yescharacteristic
403C4#9028711yescharacteristic
404C2—564no—
405C2#1173373811yescharacteristic
406C2#1173639212yes—
407C2#1173917412yes—
408C2#1173931712yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1960C2
socleC23×C7C22×D4
prime radicalC2×#6437C22
cosocleC22×C4×C7C24
hypercenter#1960D7
hypocenterC7#21695
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
1. #11740672C2semidirect
3. C2×#89589C2semidirect
4. C2×#89560C2semidirect
6. C7×#1960C2semidirect
7. #11728014C2semidirect
8. #11728011C2semidirect
10. #11727558C2semidirect
11. #11727303C2semidirect
29. #89857C4semidirect
48. C7×C4⋊Q8C2×C4semidirect
58. C2×#1143C2×C4semidirect
59. C2×#1143C2×C4semidirect
83. #6389D4semidirect
85. C8×Dic7D4semidirect
88. C4×C7⋊C8C2×C4semidirect
97. #1960D7semidirect
235. C4⋊Q8C4×D7semidirect
384. C7#21695semidirect

Maximal subgroups

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

subgroupindex
#117406722
C2×#898572
C2×#895892
C2×#895602
#117315982
C7×#19602
#117280142
#117280112
#117275772
#117275582
#117273032
#216957