Finite Group Databasebeta

← #13510383

Subgroups — #13510383

Order 1792.

712 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89686C212yesmaximal, complemented
2#11739276C212yesmaximal, complemented
3C7×#2249C211yesmaximal, characteristic, complemented
4#11729257C212yesmaximal, complemented
5#11729242C212yesmaximal, complemented
6#11729221C212yesmaximal, complemented
7#11729077C212yesmaximal, complemented
8C4×D112C211yesmaximal, characteristic, complemented
9#11726809C211yesmaximal, characteristic, complemented
10C2×#6482C2212yes—
11C2×D7×D8C2212yescomplemented
12#90367—22no—
13C2×C7×Q32C2211yescharacteristic, complemented
14C2×C7×SD32C2212yescomplemented
15C2×C7×D16C2211yescharacteristic
16C7×#495C2212yescomplemented
17#89960C2212yescomplemented
18C2×#6402C2212yescomplemented
19#89944—22no—
20#89941—22no—
21#89929C2212yes—
22C2×#6400C2212yes—
23#89829—22no—
24#89686—28no—
25C2×D112C2211yescharacteristic
26C2×#6372C2212yescomplemented
27C4×D56C2212yescomplemented
28C8×D28C2211yescharacteristic
29C4×C7×C16C2211yescharacteristic, complemented
30#89359C2212yescomplemented
31#89357C2212yescomplemented
32#89304C2212yescomplemented
33(C2×D7)⋊C16C2212yescomplemented
34#89299C2211yescharacteristic, complemented
35C4⋊(C7⋊C16)C2211yescharacteristic, complemented
36#33839—71nomaximal, Sylow
37C2×#1161—22no—
38C2×D4×D7—22no—
39C2×C7×Q16C2312yes—
40C2×C7×SD16—22no—
41C2×C7×D8C2312yes—
42C7×#124—22no—
43#6509—22no—
44C2×#1143—22no—
45#6501—22no—
46C2×#1141—22no—
47#6482—24no—
48#6482—42no—
49D7×D8—24no—
50D7×D8—42no—
51C2×D56C2312yes—
52C2×C8×D7C2312yes—
53#6447—42no—
54#6446—42no—
55#6445—42no—
56D7×C22⋊C4—42no—
57C4×D28—22no—
58C4×D28D411yescharacteristic
59C7×Q32—41no—
60C7×SD32—24no—
61C7×D16—41no—
62C2×C7×C16C2311yescharacteristic
63C2×C7×C16C2311yescharacteristic
64C4×C7×C8C2311yescharacteristic
65#6409—22no—
66#6406—22no—
67#6402—24no—
68#6400—24no—
69#6395—22no—
70(C2×D7)⋊C8D412yes—
71#6392C2312yes—
72C2×C7⋊C16C2312yes—
73C4⋊(C7⋊C8)D411yescharacteristic
74D112D414yescomplemented
75#6372—24no—
76C2×#509—72no—
77#3280—72no—
78#2249D711yescharacteristic, complemented
79#2220—72no—
80#2210—72no—
81#2189—72no—
82#2184—72no—
83C4×D16—71no—
84#2169—71no—
85C23×D7—42no—
86C2×C7×Q8—22no—
87C2×C7×D4—22no—
88C2×C22⋊D7—42no—
89C2×C22⋊D7—42no—
90#1161—44no—
91#1161—82no—
92D4×D7—44no—
93D4×D7—82no—
94C2×D28—22no—
95C2×D28—42no—
96C2×D28D811yescharacteristic
97C2×C4×D7—42no—
98C2×C4×D7—42no—
99C2×C4×D7C2×D412yes—
100C7×Q16—22no—
101C7×Q16—22no—
102C7×SD16—82no—
103C7×D8—22no—
104C7×D8—22no—
105C2×C7×C8C2×D411yescharacteristic
106C2×C7×C8C2411yescharacteristic
107C7×C22⋊C4—42no—
108C7×C22⋊C4—42no—
109C42×C7C2×D411yescharacteristic
110C22⋊Dic7—42no—
111#1143—44no—
112#1141—44no—
113(C2×D7)⋊C4—42no—
114(C2×D7)⋊C4—42no—
115(C2×D7)⋊C4D812yes—
116C4⋊Dic7—22no—
117C4⋊Dic7D811yescharacteristic
118Dic7⋊C4—42no—
119C2×C7⋊C8C2×D412yes—
120D56—22no—
121D56C2×D414yes—
122C8×D7—22no—
123C8×D7—22no—
124C7×C16—21no—
125C7×C16C2×D412yes—
126C7⋊C16—22no—
127C2×#135—72no—
128C22×D8—72no—
129#546—142no—
130#509—148no—
131C2×Q32C2×D711yescharacteristic, complemented
132C2×SD32—72no—
133C2×SD32C2×D712yescomplemented
134C2×D16—72no—
135C2×D16—71no—
136C2×D16C2×D711yescharacteristic
137C2×M32—72no—
138#495C2×D712yescomplemented
139#467—72no—
140#466—72no—
141#463—142no—
142#450—142no—
143#449—142no—
144C4×D8—72no—
145C8×D4—71no—
146#366—71no—
147C4⋊C16—71no—
148#358—72no—
149#357—72no—
150#357—72no—
151C22⋊C16—72no—
152C4×C16C2×D711yescharacteristic, complemented
153C23×C7—42no—
154C22×D7—42no—
155C22×D7—84no—
156C22×D7—84no—
157C22×D7—162no—
158C22×D7C2×D812yes—
159C7×Q8—42no—
160C7×Q8—42no—
161C7×D4—42no—
162C7×D4—42no—
163C2×C4×C7—42no—
164C2×C4×C7C2×D811yescharacteristic
165C2×C4×C7C2×D811yescharacteristic
166C2×C4×C7C22×D411yescharacteristic
167C22⋊D7—84no—
168C22⋊D7—162no—
169C2×Dic7—42no—
170C2×Dic7C2×D812yes—
171D28—24no—
172D28—41no—
173D28—42no—
174D28—84no—
175C4×D7—22no—
176C4×D7—22no—
177C4×D7—84no—
178C4×D7—162no—
179C7×C8—21no—
180C7×C8C22×D411yescharacteristic
181C7×C8H(5,𝔽2)11yescharacteristic
182C7⋊C8—22no—
183C2×#41—142no—
184C22×D4—142no—
185#135—144no—
186#135—282no—
187C2×Q16C22×D712yes—
188C2×SD16—22no—
189C2×SD16—142no—
190C2×D8—72no—
191C2×D8—142no—
192C2×D8—144no—
193C2×D8—282no—
194C2×D8C22×D712yes—
195C22×C8—72no—
196#124—22no—
197#123—282no—
198C22⋊Q8—142no—
199#121—142no—
200#121—282no—
201#120—282no—
202C4×D4—71no—
203C4×D4—142no—
204C2×C22⋊C4—282no—
205Q32—41no—
206SD32—24no—
207SD32—144no—
208D16—41no—
209D16—74no—
210D16—144no—
211M32—144no—
212C2×C16—72no—
213C2×C16C22×D711yescharacteristic
214C2×C16C22×D711yescharacteristic
215#107—72no—
216C4⋊C8—71no—
217#103—142no—
218#102—142no—
219#102—142no—
220C22⋊C8—72no—
221C4×C8C22×D711yescharacteristic
222C22×C7—82no—
223C22×C7—82no—
224C22×C7—82no—
225C22×C7C22×D811yescharacteristic
226C2×D7—22no—
227C2×D7—22no—
228C2×D7—22no—
229C2×D7—44no—
230C2×D7—82no—
231C2×D7—82no—
232C2×D7—168no—
233C4×C7—21no—
234C4×C7—21no—
235C4×C7—82no—
236C4×C7D4211yescharacteristic
237C4×C7#57611yescharacteristic
238Dic7—22no—
239Dic7—82no—
240C24—282no—
241#41—284no—
242#41—562no—
243C2×Q8—22no—
244C2×D4—22no—
245C2×D4—71no—
246C2×D4—142no—
247C2×D4—282no—
248C2×D4—282no—
249C2×D4—284no—
250C2×D4—282no—
251C2×D4—562no—
252C22×C4—72no—
253C22×C4—282no—
254C22×C4—282no—
255Q16—22no—
256Q16—22no—
257SD16—82no—
258SD16—284no—
259D8—22no—
260D8—22no—
261D8—74no—
262D8—142no—
263D8—284no—
264C2×C8—72no—
265C2×C8—142no—
266C2×C8—142no—
267C2×C8D4×D711yescharacteristic
268C2×C8C23×D711yescharacteristic
269C4⋊C4—71no—
270C4⋊C4—142no—
271C4⋊C4—282no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—72no—
275C22⋊C4—282no—
276C22⋊C4—282no—
277C22⋊C4—282no—
278C42D4×D711yescharacteristic
279C16—21no—
280C16—142no—
281C16D4×D712yes—
282C2×C7—162no—
283C2×C7D4×D811yescharacteristic
284C2×C7#341411yescharacteristic
285C2×C7#341511yescharacteristic
286D7—42no—
287D7—84no—
288C23—42no—
289C23—72no—
290C23—282no—
291C23—564no—
292C23—564no—
293C23—1122no—
294Q8—42no—
295Q8—42no—
296D4—42no—
297D4—42no—
298D4—144no—
299D4—281no—
300D4—282no—
301D4—564no—
302D4—564no—
303D4—1122no—
304C2×C4—42no—
305C2×C4—72no—
306C2×C4—142no—
307C2×C4—142no—
308C2×C4—282no—
309C2×C4—564no—
310C2×C4—1122no—
311C2×C4D7×D811yescharacteristic
312C2×C4D7×D811yescharacteristic
313C2×C4C2×D4×D711yescharacteristic
314C8—21no—
315C8—142no—
316C8C2×D4×D711yescharacteristic
317C8#654811yescharacteristic
318C7#3383911yescharacteristic, complemented, Sylow
319C22—82no—
320C22—82no—
321C22—82no—
322C22—142no—
323C22—142no—
324C22—142no—
325C22—284no—
326C22—562no—
327C22—562no—
328C22—1128no—
329C22C2×D7×D811yescharacteristic
330C4—21no—
331C4—21no—
332C4—82no—
333C4—142no—
334C4—562no—
335C4#9040811yescharacteristic
336C4#9044811yescharacteristic
337C2—162no—
338C2—282no—
339C2—564no—
340C2#1173923411yescharacteristic
341C2#1173947511yescharacteristic
342C2#1173947611yescharacteristic
subgroupG/H
centerC22C2×D7×D8
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
FittingC7×#2249C2
socleC22×C7C22×D8
hypercenter#2249D7
hypocenterC7#33839
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2249

Complements

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

NH ≅ G/Ntype
1. C2×#89686C2semidirect
2. #11739276C2semidirect
3. C7×#2249C2semidirect
4. #11729257C2semidirect
5. #11729242C2semidirect
6. #11729221C2semidirect
7. #11729077C2semidirect
8. C4×D112C2semidirect
9. #11726809C2semidirect
11. C2×D7×D8C22semidirect
13. C2×C7×Q32C22semidirect
14. C2×C7×SD32C22semidirect
16. C7×#495C22semidirect
17. #89960C22semidirect
18. C2×#6402C22semidirect
26. C2×#6372C22semidirect
27. C4×D56C22semidirect
29. C4×C7×C16C22semidirect
30. #89359C22semidirect
31. #89357C22semidirect
32. #89304C22semidirect
33. (C2×D7)⋊C16C22semidirect
34. #89299C22semidirect
35. C4⋊(C7⋊C16)C22semidirect
74. D112D4semidirect
78. #2249D7semidirect
131. C2×Q32C2×D7semidirect
133. C2×SD32C2×D7semidirect
138. #495C2×D7semidirect
152. C4×C16C2×D7semidirect
318. C7#33839semidirect

Maximal subgroups

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

subgroupindex
C2×#896862
#117392762
C7×#22492
#117292572
#117292422
#117292212
#117290772
C4×D1122
#117268092
#338397