Finite Group Databasebeta

← #13410096

Subgroups — #13410096

Order 1792.

1370 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1370 conjugacy classes of subgroups — too many to draw. Only its 191 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
1#11742854C211yesmaximal, characteristic
2C7×#2542C211yesmaximal, characteristic, complemented
3#11740644C212yesmaximal, complemented
4C4×#6407C211yesmaximal, characteristic, complemented
5#11727416C214yesmaximal, complemented
6#11727347C214yesmaximal, complemented
7#11727246C212yesmaximal, complemented
8C2×#6548C2212yes—
9C2×C7×C4⋊Q8C2211yescharacteristic, complemented
10C2×C7×C22⋊Q8—22no—
11C2×C4×C7×D4C2211yescharacteristic
12C2×#6493—22no—
13#90246—22no—
14#90245—24no—
15D4×D28—22no—
16#90201C2212yes—
17#90184—22no—
18#90168C2212yes—
19C7×#399—22no—
20C7×#390C2212yes—
21C7×#384C2212yescomplemented
22C4×C7×C4⋊C4C2211yescharacteristic, complemented
23C2×#6407C2212yes—
24#89865—22no—
25#89861—24no—
26#89860—22no—
27#89738C2218yescomplemented
28C2×#6379C2218yescomplemented
29C4×#1136C2212yescomplemented
30#89325C2212yescomplemented
31#25870—71nomaximal, Sylow
32C22×C7×Q8—21no—
33C22×C7×D4C2311yescharacteristic
34C23×C4×C7D412yes—
35C22×C22⋊D7—24no—
36#6548—28no—
37#6548—44no—
38C2×#1159—24no—
39C2×D4×D7—24no—
40C2×#1157C2314yes—
41C22×D28—22no—
42C7×C4⋊Q8—22no—
43C7×C4⋊Q8—22no—
44C7×C22⋊Q8—48no—
45C4×C7×D4—22no—
46C4×C7×D4—22no—
47C2×C7×C4⋊C4—21no—
48C2×C7×C4⋊C4—24no—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4C2312yes—
51C2×C7×C4⋊C4C2311yescharacteristic
52C2×C7×C22⋊C4—24no—
53C2×C7×C22⋊C4D412yes—
54C2×C42×C7C2312yes—
55C2×C42×C7C2311yescharacteristic
56#6493—24no—
57#6493—24no—
58#6493—22no—
59#6493—22no—
60#6493—24no—
61C2×(C2×D7)⋊C4—24no—
62C2×C4⋊Dic7—22no—
63C2×#1136C2314yes—
64#6460—24no—
65#6458—24no—
66#6449—28no—
67#6445—28no—
68#6438D414yes—
69#6437D412yes—
70C4×D28—22no—
71C4×D28—22no—
72#6432D412yes—
73C7×#95—21no—
74#6407—24no—
75#6384—28no—
76#6383—28no—
77#6379—48no—
78#6377—22no—
79C4×C7⋊C8—22no—
80#3496—71no—
81#3141—72no—
82#2542D711yescharacteristic, complemented
83#2039—74no—
84#1980—72no—
85#1792—74no—
86C4×#100—71no—
87C24×C7C2×D412yes—
88C23×D7—24no—
89C2×C7×Q8—44no—
90C2×C7×Q8—41no—
91C2×C7×Q8—41no—
92C2×C7×D4—22no—
93C2×C7×D4—22no—
94C2×C7×D4#4114yes—
95C22×C4×C7—22no—
96C22×C4×C7—22no—
97C22×C4×C7—22no—
98C22×C4×C7—22no—
99C22×C4×C7—44no—
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7C2×D412yes—
102C22×C4×C7C2×D411yescharacteristic
103C22×C4×C7C2411yescharacteristic
104C2×C22⋊D7—28no—
105C2×C22⋊D7—24no—
106C2×C22⋊D7—48no—
107C2×C22⋊D7—48no—
108C22×Dic7—24no—
109#1159—416no—
110D4×D7—416no—
111#1157—28no—
112#1157—28no—
113C2×D28—22no—
114C2×D28—24no—
115C2×D28—22no—
116C2×D28—84no—
117C2×D28C2×D414yes—
118C2×C4×D7—24no—
119C2×Dic14C2×D414yes—
120C7×C4⋊C4—24no—
121C7×C4⋊C4—22no—
122C7×C4⋊C4—24no—
123C7×C4⋊C4—48no—
124C7×C4⋊C4—42no—
125C7×C22⋊C4—24no—
126C7×C22⋊C4—48no—
127C42×C7—24no—
128C42×C7C2×D412yes—
129C42×C7C2×D412yes—
130(C2×D7)⋊C4—28no—
131(C2×D7)⋊C4—24no—
132(C2×D7)⋊C4—24no—
133C4⋊Dic7—24no—
134C4⋊Dic7—22no—
135C4⋊Dic7—22no—
136#1136—24no—
137#1136—24no—
138C2×C7⋊C8—24no—
139C2×H(5,𝔽2)—72no—
140#547—142no—
141#546—144no—
142D42—142no—
143#537—72no—
144#534—142no—
145#532—72no—
146C2×C4⋊Q8C2×D711yescharacteristic, complemented
147C2×C22⋊Q8—22no—
148C2×#121—142no—
149C2×C4×D4C2×D711yescharacteristic
150#489—144no—
151#487—142no—
152#486—142no—
153C2×#104—78no—
154#417—78no—
155C2×#100—72no—
156C4×M16—72no—
157#399—22no—
158#390C2×D712yes—
159#384C2×D712yescomplemented
160C4×C4⋊C4C2×D711yescharacteristic, complemented
161#343—72no—
162C23×C7—22no—
163C23×C7—22no—
164C23×C7—42no—
165C23×C7—42no—
166C23×C7#12414yes—
167C23×C7C22×D411yescharacteristic
168C22×D7—24no—
169C22×D7—48no—
170C22×D7—84no—
171C22×D7—84no—
172C7×Q8—82no—
173C7×Q8—82no—
174C7×D4—28no—
175C2×C4×C7—22no—
176C2×C4×C7—24no—
177C2×C4×C7—22no—
178C2×C4×C7—41no—
179C2×C4×C7—42no—
180C2×C4×C7—42no—
181C2×C4×C7—44no—
182C2×C4×C7—42no—
183C2×C4×C7—44no—
184C2×C4×C7—42no—
185C2×C4×C7—82no—
186C2×C4×C7#12012yes—
187C2×C4×C7#12011yescharacteristic
188C2×C4×C7#12011yescharacteristic
189C2×C4×C7C22×D412yes—
190C2×C4×C7C2×#4112yes—
191C2×C4×C7H(5,𝔽2)12yes—
192C22⋊D7—416no—
193C22⋊D7—44no—
194C22⋊D7—44no—
195C2×Dic7—24no—
196C2×Dic7—48no—
197D28—28no—
198D28—44no—
199D28—82no—
200D28—84no—
201C4×D7—48no—
202Dic14—28no—
203C7⋊C8—44no—
204H(5,𝔽2)—148no—
205H(5,𝔽2)—284no—
206C2×#41—74no—
207C2×#41—144no—
208C22×Q8—21no—
209C22×D4—142no—
210C22×D4—144no—
211C22×D4—144no—
212C22×D4C22×D711yescharacteristic
213C23×C4C22⋊D712yes—
214C2×M16—74no—
215C4⋊Q8—22no—
216C4⋊Q8—22no—
217C4⋊Q8—72no—
218#127—72no—
219#124—74no—
220#123—148no—
221C22⋊Q8—48no—
222C22⋊Q8—144no—
223#121—144no—
224#121—144no—
225#121—144no—
226#121—142no—
227#121—144no—
228#121—142no—
229#120—148no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—142no—
233C4×D4—142no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—142no—
238C2×C4⋊C4C22×D712yes—
239C2×C4⋊C4C22×D711yescharacteristic
240C2×C22⋊C4—24no—
241C2×C22⋊C4—144no—
242C2×C22⋊C4C22⋊D712yes—
243C2×C42C22×D712yes—
244C2×C42C22×D711yescharacteristic
245#104—288no—
246#103—148no—
247#102—148no—
248#100—144no—
249#97—142no—
250C4×C8—142no—
251#95—21no—
252C22×C7—21no—
253C22×C7—21no—
254C22×C7—24no—
255C22×C7—44no—
256C22×C7—42no—
257C22×C7—42no—
258C22×C7—82no—
259C22×C7C2×#12011yescharacteristic
260C22×C7C2×#12411yescharacteristic
261C22×C7#53511yescharacteristic
262C2×D7—44no—
263C2×D7—44no—
264C2×D7—44no—
265C2×D7—168no—
266C4×C7—21no—
267C4×C7—22no—
268C4×C7—44no—
269C4×C7—42no—
270C4×C7—82no—
271C4×C7#45312yes—
272C4×C7#54614yes—
273Dic7—44no—
274C24—144no—
275C24C2×C22⋊D712yes—
276#41—148no—
277#41—148no—
278#41—2816no—
279C2×Q8—44no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—74no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—74no—
286C2×D4—144no—
287C2×D4—148no—
288C2×D4—144no—
289C2×D4—142no—
290C2×D4—142no—
291C2×D4—2816no—
292C2×D4—288no—
293C2×D4—288no—
294C2×D4—564no—
295C2×D4#116114yes—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—44no—
301C22×C4—144no—
302C22×C4—144no—
303C22×C4C2×C22⋊D711yescharacteristic
304C22×C4C2×C22⋊D712yes—
305C22×C4C2×C22⋊D711yescharacteristic
306C22×C4C23×D711yescharacteristic
307M16—144no—
308M16—144no—
309C2×C8—144no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—24no—
313C4⋊C4—48no—
314C4⋊C4—42no—
315C4⋊C4—144no—
316C4⋊C4—142no—
317C4⋊C4—142no—
318C22⋊C4—24no—
319C22⋊C4—48no—
320C22⋊C4—148no—
321C22⋊C4—144no—
322C22⋊C4—144no—
323C42—24no—
324C42D4×D712yes—
325C42D4×D712yes—
326C2×C7—21no—
327C2×C7—21no—
328C2×C7—44no—
329C2×C7#243511yescharacteristic
330C2×C7C2×#45311yescharacteristic
331C2×C7#302211yescharacteristic
332D7—84no—
333C23—22no—
334C23—22no—
335C23—42no—
336C23—42no—
337C23—144no—
338C23—288no—
339C23—564no—
340C23—564no—
341C23#651014yes—
342C23C22×C22⋊D711yescharacteristic
343Q8—82no—
344Q8—82no—
345Q8—148no—
346D4—28no—
347D4—148no—
348D4—2816no—
349D4—284no—
350D4—284no—
351D4—284no—
352D4—562no—
353D4—564no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—82no—
365C2×C4—144no—
366C2×C4—288no—
367C2×C4—288no—
368C2×C4#650012yes—
369C2×C4#650011yescharacteristic
370C2×C4#650011yescharacteristic
371C2×C4C2×D4×D712yes—
372C2×C4C2×#116112yes—
373C2×C4#655312yes—
374C8—284no—
375C7#2587011yescharacteristic, complemented, Sylow
376C22—21no—
377C22—21no—
378C22—24no—
379C22—44no—
380C22—42no—
381C22—42no—
382C22—82no—
383C22—284no—
384C22—284no—
385C22—284no—
386C22—1128no—
387C22C2×#650011yescharacteristic
388C22C2×#651011yescharacteristic
389C22#9052111yescharacteristic
390C4—22no—
391C4—21no—
392C4—44no—
393C4—42no—
394C4—82no—
395C4—284no—
396C4#8984912yes—
397C4#9032114yes—
398C2—21no—
399C2—21no—
400C2—44no—
401C2—564no—
402C2#1174020611yescharacteristic
403C2C2×#8984911yescharacteristic
404C2#1174086111yescharacteristic
subgroupG/H
centerC22C2×#6500
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2542C2
socleC22×C7C2×#120
prime radical#11742854C2
hypercenter#2542D7
hypocenterC7#25870
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2542

Complements

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

NH ≅ G/Ntype
2. C7×#2542C2semidirect
3. #11740644C2semidirect
4. C4×#6407C2semidirect
5. #11727416C2semidirect
6. #11727347C2semidirect
7. #11727246C2semidirect
9. C2×C7×C4⋊Q8C22semidirect
21. C7×#384C22semidirect
22. C4×C7×C4⋊C4C22semidirect
27. #89738C22semidirect
28. C2×#6379C22semidirect
29. C4×#1136C22semidirect
30. #89325C22semidirect
82. #2542D7semidirect
146. C2×C4⋊Q8C2×D7semidirect
159. #384C2×D7semidirect
160. C4×C4⋊C4C2×D7semidirect
375. C7#25870semidirect

Maximal subgroups

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

subgroupindex
#117428542
C7×#25422
#117406442
C4×#64072
#117274162
#117273472
#117272462
#258707