Finite Group Databasebeta

← #14220547

Subgroups — #14220547

Order 1792.

5128 conjugacy classes of subgroups; only the 807 normal subgroups are shown.

normal subgroups only

This group has 5128 conjugacy classes of subgroups — too many to draw. Only its 807 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
1C7×H(7,𝔽2)C211yesmaximal, characteristic, complemented
2#11744352C213yesmaximal, complemented
3#11741571C214yesmaximal, complemented
4#11741570C214yesmaximal, complemented
5#11741555C218yesmaximal, complemented
6C2×#90016C218yesmaximal, complemented
7#11741552C213yesmaximal, complemented
8C7×#585—212no—
9C7×#585C2214yescomplemented
10C2×C7×H(5,𝔽2)—212no—
11C2×C7×H(5,𝔽2)C2213yes—
12C2×C7×H(5,𝔽2)C2218yescomplemented
13C22×D4×D7C2211yescharacteristic, complemented
14#90521C2216yescomplemented
15#90520—212no—
16#90519C2216yescomplemented
17#90495C22112yescomplemented
18D4×C22⋊D7—212no—
19C2×#6500C22112yescomplemented
20#90489C2213yescomplemented
21#90016C22116yescomplemented
22#90016C22132yescomplemented
23#90015C22116yescomplemented
24#90012C22112yescomplemented
25C2×#6408C22124yescomplemented
26#60399—71nomaximal, Sylow
27C24×D7C2312yescomplemented
28C7×2−1+4—216no—
29C7×2−1+4—212no—
30C7×H(5,𝔽2)—248no—
31C7×H(5,𝔽2)—224no—
32C7×H(5,𝔽2)—212no—
33C7×H(5,𝔽2)—248no—
34C7×H(5,𝔽2)C23116yes—
35C2×C7×#41—248no—
36C2×C7×#41—224no—
37C2×C7×#41—224no—
38C2×C7×#41D4112yescomplemented
39C2×C7×#41C2316yes—
40C22×C7×D4—216no—
41C22×C7×D4—224no—
42C22×C7×D4D4112yescomplemented
43C22×C7×D4C23112yes—
44C22×C7×D4C2311yescharacteristic
45C22×C22⋊D7—212no—
46C22×C22⋊D7C2312yescomplemented
47C2×D4×D7—212no—
48C22×D28C2311yescharacteristic, complemented
49C22×C4×D7C2311yescharacteristic, complemented
50#6516—212no—
51C2×C22⋊Dic7C23112yescomplemented
52#6510—26no—
53#6509—26no—
54#6503—26no—
55#6502—224no—
56#6501—26no—
57#6500—224no—
58#6500C23112yescomplemented
59#6500C23124yescomplemented
60#6498C23112yescomplemented
61D4×Dic7—212no—
62#6493—212no—
63#6492—212no—
64C4×C22⋊D7—212no—
65C2×(C2×D7)⋊C4—212no—
66#6489C2316yescomplemented
67#6408C23148yescomplemented
68H(7,𝔽2)D711yescharacteristic, complemented
69#3484—73no—
70#3027—74no—
71#3026—78no—
72#3025—74no—
73C2×B(4,𝔽2)—78no—
74#2883—73no—
75C24×C7—28no—
76C24×C7C2×D4112yescomplemented
77C24×C7C2412yes—
78C23×D7—212no—
79C23×D7—84no—
80C23×D7C2411yescharacteristic
81C23×D7C2412yes—
82C7×#41—296no—
83C7×#41—248no—
84C7×#41—216no—
85C7×#41—248no—
86C7×#41—248no—
87C7×#41—224no—
88C2×C7×Q8—224no—
89C2×C7×Q8—28no—
90C2×C7×Q8C2×D416yescomplemented
91C2×C7×D4—296no—
92C2×C7×D4—224no—
93C2×C7×D4—224no—
94C2×C7×D4—248no—
95C2×C7×D4—248no—
96C2×C7×D4—248no—
97C2×C7×D4C2×D4112yes—
98C2×C7×D4C2×D4124yescomplemented
99C2×C7×D4C2×D416yescomplemented
100C2×C7×D4C24112yes—
101C22×C4×C7—224no—
102C22×C4×C7—216no—
103C22×C4×C7C2×D4112yescomplemented
104C22×C4×C7C2×D4112yescomplemented
105C22×C4×C7C2411yescharacteristic
106C2×C22⋊D7—212no—
107C2×C22⋊D7—212no—
108C2×C22⋊D7—224no—
109C2×C22⋊D7—824no—
110C22×Dic7—212no—
111C22×Dic7C2411yescharacteristic
112D4×D7—416no—
113C2×D28—26no—
114C2×C4×D7—26no—
115C22⋊Dic7—212no—
116C22⋊Dic7—212no—
117C22⋊Dic7C24112yes—
118(C2×D7)⋊C4—212no—
119(C2×D7)⋊C4—224no—
120C4⋊Dic7—26no—
121Dic7⋊C4—212no—
122C4×Dic7—26no—
123#585—212no—
124#585C2×D714yescomplemented
125C2×H(5,𝔽2)—212no—
126C2×H(5,𝔽2)C2×D713yes—
127C2×H(5,𝔽2)C2×D718yescomplemented
128C23×D4—71no—
129#546—1412no—
130D42—1412no—
131#535—76no—
132#534—712no—
133#525—76no—
134C2×#120—712no—
135#518—73no—
136#458—716no—
137B(4,𝔽2)—716no—
138B(4,𝔽2)—732no—
139#410—712no—
140C2×#99—724no—
141C23×C7—216no—
142C23×C7—224no—
143C23×C7—82no—
144C23×C7—812no—
145C23×C7—168no—
146C23×C7C22×D4112yes—
147C23×C7C22×D4112yes—
148C23×C7C2511yescharacteristic
149C22×D7—23no—
150C22×D7—26no—
151C22×D7—42no—
152C22×D7—424no—
153C22×D7—48no—
154C22×D7—82no—
155C22×D7—812no—
156C7×Q8—216no—
157C7×Q8—212no—
158C7×D4—248no—
159C7×D4—248no—
160C7×D4—224no—
161C7×D4—212no—
162C7×D4#120116yes—
163C2×C4×C7—248no—
164C2×C4×C7—224no—
165C2×C4×C7—224no—
166C2×C4×C7C22×D4112yes—
167C2×C4×C7C22×D416yes—
168C22⋊D7—424no—
169C22⋊D7—48no—
170C22⋊D7—824no—
171C2×Dic7—212no—
172C2×Dic7—23no—
173C2×Dic7—812no—
174D28—44no—
175C4×D7—44no—
176C25—72no—
1772−1+4—216no—
1782−1+4—212no—
179H(5,𝔽2)—248no—
180H(5,𝔽2)—224no—
181H(5,𝔽2)—212no—
182H(5,𝔽2)—248no—
183H(5,𝔽2)C22×D7116yes—
184C2×#41—248no—
185C2×#41—224no—
186C2×#41—224no—
187C2×#41C22⋊D7112yescomplemented
188C2×#41C22×D716yes—
189C22×D4—216no—
190C22×D4—224no—
191C22×D4—72no—
192C22×D4—71no—
193C22×D4—1412no—
194C22×D4—1412no—
195C22×D4C22⋊D7112yescomplemented
196C22×D4C22×D7112yes—
197C22×D4C22×D711yescharacteristic
198C23×C4—71no—
199#127—146no—
200#124—146no—
201#123—712no—
202#123—1412no—
203C22⋊Q8—146no—
204#121—146no—
205#121—1424no—
206#121—1412no—
207#120—712no—
208#120—724no—
209#120—1412no—
210#120—1424no—
211C4×D4—1412no—
212C4×D4—1412no—
213#117—76no—
214C2×C22⋊C4—712no—
215C2×C22⋊C4—1412no—
216#99—748no—
217C22×C7—212no—
218C22×C7—412no—
219C22×C7—41no—
220C22×C7—412no—
221C22×C7—824no—
222C22×C7—816no—
223C22×C7C2×#12018yes—
224C22×C7C23×D413yes—
225C2×D7—26no—
226C2×D7—43no—
227C2×D7—41no—
228C2×D7—412no—
229C2×D7—42no—
230C2×D7—816no—
231C4×C7—212no—
232C4×C7C2×#12014yes—
233Dic7—412no—
234Dic7—41no—
235C24—28no—
236C24—71no—
237C24—72no—
238C24—1412no—
239C24—564no—
240C24C2×C22⋊D7112yescomplemented
241C24C23×D712yes—
242#41—296no—
243#41—248no—
244#41—216no—
245#41—248no—
246#41—248no—
247#41—224no—
248C2×Q8—224no—
249C2×Q8—28no—
250C2×Q8C2×C22⋊D716yescomplemented
251C2×D4—296no—
252C2×D4—224no—
253C2×D4—224no—
254C2×D4—248no—
255C2×D4—248no—
256C2×D4—248no—
257C2×D4—1412no—
258C2×D4—1412no—
259C2×D4—1424no—
260C2×D4—146no—
261C2×D4—2816no—
262C2×D4—5624no—
263C2×D4C2×C22⋊D7112yes—
264C2×D4C2×C22⋊D7124yescomplemented
265C2×D4C2×C22⋊D716yescomplemented
266C2×D4C23×D7112yes—
267C22×C4—224no—
268C22×C4—216no—
269C22×C4—71no—
270C22×C4—1412no—
271C22×C4—146no—
272C22×C4C2×C22⋊D7112yescomplemented
273C22×C4C2×C22⋊D7112yescomplemented
274C22×C4C23×D711yescharacteristic
275C4⋊C4—1412no—
276C4⋊C4—146no—
277C22⋊C4—712no—
278C22⋊C4—1412no—
279C22⋊C4—1412no—
280C22⋊C4—1412no—
281C22⋊C4—1424no—
282C42—146no—
283C2×C7—23no—
284C2×C7—28no—
285C2×C7—412no—
286C2×C7C22×#12011yescharacteristic
287D7—42no—
288D7—82no—
289C23—216no—
290C23—224no—
291C23—82no—
292C23—812no—
293C23—143no—
294C23—146no—
295C23—168no—
296C23—282no—
297C23—2824no—
298C23—288no—
299C23—562no—
300C23—5612no—
301C23C22×C22⋊D7112yes—
302C23C22×C22⋊D7112yes—
303C23C24×D711yescharacteristic
304Q8—216no—
305Q8—212no—
306D4—248no—
307D4—248no—
308D4—224no—
309D4—212no—
310D4—2824no—
311D4—288no—
312D4—284no—
313D4—5624no—
314D4#6516116yes—
315C2×C4—248no—
316C2×C4—224no—
317C2×C4—224no—
318C2×C4—1412no—
319C2×C4—143no—
320C2×C4—284no—
321C2×C4—5612no—
322C2×C4C22×C22⋊D7112yes—
323C2×C4C22×C22⋊D716yes—
324C7#6039911yescharacteristic, complemented, Sylow
325C22—212no—
326C22—412no—
327C22—41no—
328C22—412no—
329C22—824no—
330C22—816no—
331C22—146no—
332C22—283no—
333C22—2812no—
334C22—281no—
335C22—282no—
336C22—5616no—
337C22C2×#651618yes—
338C22C23×C22⋊D713yes—
339C4—212no—
340C4—2812no—
341C4—281no—
342C4C2×#651614yes—
343C2—23no—
344C2—28no—
345C2—412no—
346C2—282no—
347C2—562no—
348C2C22×#651611yescharacteristic
subgroupG/H
centerC2C22×#6516
commutator, cosocleC23×C7C25
FrattiniC23C24×D7
FittingC7×H(7,𝔽2)C2
socleC2×C7C22×#120
hypercenterH(7,𝔽2)D7
hypocenterC7#60399
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C22×D4 ◁ H(7,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C7×H(7,𝔽2)C2semidirect
2. #11744352C2semidirect
3. #11741571C2semidirect
4. #11741570C2semidirect
5. #11741555C2semidirect
6. C2×#90016C2semidirect
7. #11741552C2semidirect
9. C7×#585C22semidirect
12. C2×C7×H(5,𝔽2)C22semidirect
13. C22×D4×D7C22semidirect
14. #90521C22semidirect
16. #90519C22semidirect
17. #90495C22semidirect
19. C2×#6500C22semidirect
20. #90489C22semidirect
21. #90016C22semidirect
22. #90016C22semidirect
23. #90015C22semidirect
24. #90012C22semidirect
25. C2×#6408C22semidirect
27. C24×D7C23semidirect
38. C2×C7×#41D4semidirect
42. C22×C7×D4D4semidirect
46. C22×C22⋊D7C23semidirect
48. C22×D28C23semidirect
49. C22×C4×D7C23semidirect
51. C2×C22⋊Dic7C23semidirect
58. #6500C23semidirect
59. #6500C23semidirect
60. #6498C23semidirect
66. #6489C23semidirect
67. #6408C23semidirect
68. H(7,𝔽2)D7semidirect
76. C24×C7C2×D4semidirect
90. C2×C7×Q8C2×D4semidirect
98. C2×C7×D4C2×D4semidirect
99. C2×C7×D4C2×D4semidirect
103. C22×C4×C7C2×D4semidirect
104. C22×C4×C7C2×D4semidirect
124. #585C2×D7semidirect
127. C2×H(5,𝔽2)C2×D7semidirect
187. C2×#41C22⋊D7semidirect
195. C22×D4C22⋊D7semidirect
240. C24C2×C22⋊D7semidirect
250. C2×Q8C2×C22⋊D7semidirect
264. C2×D4C2×C22⋊D7semidirect
265. C2×D4C2×C22⋊D7semidirect
272. C22×C4C2×C22⋊D7semidirect
273. C22×C4C2×C22⋊D7semidirect
324. C7#60399semidirect

Maximal subgroups

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

subgroupindex
C7×H(7,𝔽2)2
#117443522
#117415712
#117415702
#117415552
C2×#900162
#117415522
#603997