Finite Group Databasebeta

← #13328587

Subgroups — #13328587

Order 1792.

1290 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1290 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#11740661C212yesmaximal, complemented
4#11729355C211yesmaximal, characteristic, complemented
5#11727468C218yesmaximal, complemented
6#11727223C212yesmaximal, complemented
7C2×#6548—21no—
8C2×C7×C4⋊Q8C2211yescharacteristic, complemented
9C2×C7×C22⋊Q8—22no—
10C2×C4×C7×D4C2211yescharacteristic
11C2×#6493—22no—
12#90246—22no—
13#90245—24no—
14D4×D28—22no—
15#90201—21no—
16#90184C2214yes—
17#90168C2212yes—
18C7×#399C2214yescomplemented
19C7×#390—21no—
20C7×#384—21no—
21C4×C7×C4⋊C4C2211yescharacteristic, complemented
22#89860—22no—
23#89856—24no—
24#89851—22no—
25#89740C2218yescomplemented
26#89697C2212yescomplemented
27#89320C2214yescomplemented
28#89246C2218yescomplemented
29#18395—71nomaximal, Sylow
30C22×C7×Q8—21no—
31C22×C7×D4D411yescharacteristic
32C23×C4×C7D412yes—
33C22×C22⋊D7—24no—
34#6548—44no—
35#6548—44no—
36C2×#1159—24no—
37C2×D4×D7—24no—
38C2×#1157—22no—
39C22×D28—22no—
40C7×C4⋊Q8—22no—
41C7×C4⋊Q8—22no—
42C7×C22⋊Q8—48no—
43C4×C7×D4—22no—
44C4×C7×D4—22no—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4—24no—
47C2×C7×C4⋊C4—21no—
48C2×C7×C4⋊C4D411yescharacteristic
49C2×C7×C4⋊C4C2312yes—
50C2×C7×C22⋊C4—24no—
51C2×C7×C22⋊C4C2312yes—
52C2×C42×C7—21no—
53C2×C42×C7C2311yescharacteristic
54#6493—24no—
55#6493—24no—
56#6493—22no—
57#6493—24no—
58#6493C2314yes—
59C2×(C2×D7)⋊C4—24no—
60C2×C4⋊Dic7—22no—
61#6460—24no—
62#6458—24no—
63#6449—28no—
64#6445—28no—
65#6438—22no—
66#6437D412yes—
67C4×D28—22no—
68C4×D28D414yes—
69#6432D412yes—
70C7×#95C2312yes—
71C22⋊(C7⋊C8)C2314yes—
72#6383—28no—
73#6382—28no—
74C4⋊(C7⋊C8)—22no—
75C4×C7⋊C8—22no—
76#3496—71no—
77#3227—72no—
78#2542D711yescharacteristic, complemented
79#1613—78no—
80#1502—72no—
81#1460—71no—
82C24×C7C2×D412yes—
83C23×D7—24no—
84C2×C7×Q8—44no—
85C2×C7×Q8—41no—
86C2×C7×Q8—41no—
87C2×C7×D4—22no—
88C2×C7×D4—22no—
89C2×C7×D4SD1614yes—
90C22×C4×C7—22no—
91C22×C4×C7—22no—
92C22×C4×C7—22no—
93C22×C4×C7—22no—
94C22×C4×C7—44no—
95C22×C4×C7C2×D411yescharacteristic
96C22×C4×C7C2×D412yes—
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2411yescharacteristic
99C2×C22⋊D7—28no—
100C2×C22⋊D7—24no—
101C2×C22⋊D7—48no—
102C2×C22⋊D7—48no—
103C22×Dic7—24no—
104#1159—416no—
105D4×D7—416no—
106#1157—44no—
107#1157—44no—
108C2×D28—22no—
109C2×D28—22no—
110C2×D28—24no—
111C2×D28—84no—
112C2×D28C2×D414yes—
113C2×C4×D7—24no—
114C2×Dic14—22no—
115C7×C4⋊C4—24no—
116C7×C4⋊C4—22no—
117C7×C4⋊C4—24no—
118C7×C4⋊C4—48no—
119C7×C4⋊C4—42no—
120C7×C22⋊C4—24no—
121C7×C22⋊C4—48no—
122C42×C7—42no—
123C42×C7C2×D412yes—
124C42×C7C2×D412yes—
125(C2×D7)⋊C4—28no—
126(C2×D7)⋊C4—24no—
127(C2×D7)⋊C4—24no—
128C4⋊Dic7—24no—
129C4⋊Dic7—22no—
130C4⋊Dic7C2×D414yes—
131C2×C7⋊C8—24no—
132C2×H(5,𝔽2)—141no—
133#547—142no—
134#546—144no—
135D42—142no—
136#537—141no—
137#534—74no—
138#532—72no—
139C2×C4⋊Q8C2×D711yescharacteristic, complemented
140C2×C22⋊Q8—22no—
141C2×#121—142no—
142C2×C4×D4C2×D711yescharacteristic
143#498—142no—
144#486—142no—
145#481—78no—
146#476—144no—
147#431—72no—
148#399C2×D714yescomplemented
149#390—21no—
150#384—21no—
151C4×C4⋊C4C2×D711yescharacteristic, complemented
152#327—78no—
153#323—74no—
154C23×C7—22no—
155C23×C7—22no—
156C23×C7—42no—
157C23×C7—42no—
158C23×C7C2×SD1614yes—
159C23×C7C22×D411yescharacteristic
160C22×D7—24no—
161C22×D7—48no—
162C22×D7—84no—
163C22×D7—84no—
164C7×Q8—82no—
165C7×Q8—82no—
166C7×D4—28no—
167C2×C4×C7—22no—
168C2×C4×C7—24no—
169C2×C4×C7—22no—
170C2×C4×C7—41no—
171C2×C4×C7—42no—
172C2×C4×C7—42no—
173C2×C4×C7—44no—
174C2×C4×C7—42no—
175C2×C4×C7—44no—
176C2×C4×C7—42no—
177C2×C4×C7—82no—
178C2×C4×C7#12011yescharacteristic
179C2×C4×C7#12012yes—
180C2×C4×C7#12011yescharacteristic
181C2×C4×C7C2×SD1612yes—
182C2×C4×C7AGL(1,ℤ/8)12yes—
183C2×C4×C7C22×D412yes—
184C22⋊D7—416no—
185C22⋊D7—44no—
186C22⋊D7—44no—
187C2×Dic7—24no—
188C2×Dic7—48no—
189D28—44no—
190D28—44no—
191D28—44no—
192D28—84no—
193C4×D7—48no—
194Dic14—44no—
195C7⋊C8—44no—
196H(5,𝔽2)—284no—
197H(5,𝔽2)—284no—
198C2×#41—144no—
199C2×#41—142no—
200C22×Q8—21no—
201C22×D4—142no—
202C22×D4—144no—
203C22×D4—144no—
204C22×D4C22⋊D711yescharacteristic
205C23×C4C22⋊D712yes—
206C4⋊Q8—22no—
207C4⋊Q8—22no—
208C4⋊Q8—72no—
209#127—72no—
210#124—142no—
211#123—148no—
212C22⋊Q8—48no—
213C22⋊Q8—144no—
214#121—74no—
215#121—144no—
216#121—144no—
217#121—144no—
218#121—142no—
219#121—144no—
220#120—148no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—74no—
224C4×D4—142no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—142no—
229C2×C4⋊C4C22⋊D711yescharacteristic
230C2×C4⋊C4C22×D712yes—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—144no—
233C2×C22⋊C4C22×D712yes—
234C2×C42—21no—
235C2×C42C22×D711yescharacteristic
236#106—148no—
237C4⋊C8—142no—
238#102—148no—
239C22⋊C8—74no—
240C4×C8—142no—
241#95C22×D712yes—
242C22×C7—21no—
243C22×C7—21no—
244C22×C7—24no—
245C22×C7—44no—
246C22×C7—42no—
247C22×C7—42no—
248C22×C7—82no—
249C22×C7C2×#12011yescharacteristic
250C22×C7C22×SD1611yescharacteristic
251C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
252C2×D7—44no—
253C2×D7—44no—
254C2×D7—44no—
255C2×D7—168no—
256C4×C7—21no—
257C4×C7—22no—
258C4×C7—44no—
259C4×C7—42no—
260C4×C7—82no—
261C4×C7#45014yes—
262C4×C7#45312yes—
263Dic7—44no—
264C24—144no—
265C24C2×C22⋊D712yes—
266#41—2816no—
267#41—284no—
268#41—284no—
269C2×Q8—44no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—142no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—74no—
276C2×D4—144no—
277C2×D4—148no—
278C2×D4—144no—
279C2×D4—142no—
280C2×D4—142no—
281C2×D4—2816no—
282C2×D4—288no—
283C2×D4—288no—
284C2×D4—564no—
285C2×D4#114314yes—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—44no—
291C22×C4—144no—
292C22×C4—144no—
293C22×C4C2×C22⋊D711yescharacteristic
294C22×C4C2×C22⋊D712yes—
295C22×C4C2×C22⋊D711yescharacteristic
296C22×C4C23×D711yescharacteristic
297C2×C8—144no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—48no—
302C4⋊C4—42no—
303C4⋊C4—74no—
304C4⋊C4—144no—
305C4⋊C4—142no—
306C22⋊C4—24no—
307C22⋊C4—48no—
308C22⋊C4—148no—
309C22⋊C4—144no—
310C22⋊C4—144no—
311C42—42no—
312C42D4×D712yes—
313C42D4×D712yes—
314C2×C7—21no—
315C2×C7—21no—
316C2×C7—44no—
317C2×C7C2×#45011yescharacteristic
318C2×C7C2×#45311yescharacteristic
319C2×C7#302811yescharacteristic
320D7—84no—
321C23—22no—
322C23—22no—
323C23—42no—
324C23—42no—
325C23—144no—
326C23—288no—
327C23—564no—
328C23—564no—
329C23C2×#114314yes—
330C23C22×C22⋊D711yescharacteristic
331Q8—82no—
332Q8—82no—
333Q8—284no—
334D4—28no—
335D4—284no—
336D4—2816no—
337D4—284no—
338D4—284no—
339D4—284no—
340D4—284no—
341D4—564no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—82no—
353C2×C4—144no—
354C2×C4—288no—
355C2×C4—288no—
356C2×C4#650011yescharacteristic
357C2×C4#650012yes—
358C2×C4#650011yescharacteristic
359C2×C4C2×#114312yes—
360C2×C4#651212yes—
361C2×C4C2×D4×D712yes—
362C8—284no—
363C7#1839511yescharacteristic, complemented, Sylow
364C22—21no—
365C22—21no—
366C22—24no—
367C22—44no—
368C22—42no—
369C22—42no—
370C22—82no—
371C22—284no—
372C22—284no—
373C22—284no—
374C22—1128no—
375C22C2×#650011yescharacteristic
376C22C22×#114311yescharacteristic
377C22C2×#651211yescharacteristic
378C4—22no—
379C4—21no—
380C4—44no—
381C4—42no—
382C4—82no—
383C4—284no—
384C4#8981814yes—
385C4#8984912yes—
386C2—21no—
387C2—21no—
388C2—44no—
389C2—564no—
390C2C2×#8981811yescharacteristic
391C2C2×#8984911yescharacteristic
392C2#1174086211yescharacteristic
subgroupG/H
centerC22C2×#6500
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2542C2
socleC22×C7C2×#120
prime radical#11742854C2
hypercenter#2542D7
hypocenterC7#18395
LCSG ▷ C22×C4×C7 ▷ C22×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. #11740661C2semidirect
4. #11729355C2semidirect
5. #11727468C2semidirect
6. #11727223C2semidirect
8. C2×C7×C4⋊Q8C22semidirect
18. C7×#399C22semidirect
21. C4×C7×C4⋊C4C22semidirect
25. #89740C22semidirect
26. #89697C22semidirect
27. #89320C22semidirect
28. #89246C22semidirect
78. #2542D7semidirect
139. C2×C4⋊Q8C2×D7semidirect
148. #399C2×D7semidirect
151. C4×C4⋊C4C2×D7semidirect
363. C7#18395semidirect

Maximal subgroups

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

subgroupindex
#117428542
C7×#25422
#117406612
#117293552
#117274682
#117272232
#183957