Finite Group Databasebeta

← AGL(1,𝔽7)×#3773 #72217854

Subgroups — AGL(1,𝔽7)×#3773 #72217854

Order 6048.

648 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 99 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
1D7×#3773C311yesmaximal, characteristic, complemented
2#72217606C212yesmaximal, complemented
3#72217594C211yesmaximal, characteristic, complemented
4ΓL(1,𝔽8)×#3773C211yesmaximal, characteristic, complemented
5AGL(1,𝔽7)×#624C212yesmaximal
6C2×#157×AGL(1,𝔽7)C211yesmaximal, characteristic
7C2×C3⋊D3×AGL(1,𝔽7)C2211yescharacteristic
8C2×ΓL(1,𝔽8)⋊#157C2211yescharacteristic
9C2×ΓL(1,𝔽8)×#157C2211yescharacteristic
10#157×AGL(1,𝔽7)C414yescomplemented
11#92693⋊C3C2212yes—
12#92687⋊C3—22no—
13ΓL(1,𝔽8)×#624C2212yes—
14C3×Dic3×AGL(1,𝔽7)—22no—
15C2×D7×#157C2×C311yescharacteristic
16D7×#624C2×C312yes—
17#11759408C2×C311yescharacteristic, complemented
18#11759402C2×C312yescomplemented
19C7×#3773C2×C311yescharacteristic, complemented
20C4×D3×AGL(1,𝔽7)—62no—
21C2×C3×#3773—71nomaximal, Hall
22C2×C3⋊(C3⋊AGL(1,𝔽7))D411yescharacteristic
23C2×C3⋊D3×ΓL(1,𝔽8)C2311yescharacteristic
24C2×C32×AGL(1,𝔽7)D411yescharacteristic
25C3⋊D3×AGL(1,𝔽7)C2×C412yes—
26C3⋊D3×AGL(1,𝔽7)Q812yes—
27ΓL(1,𝔽8)⋊#157C2×C412yescomplemented
28ΓL(1,𝔽8)×#157C2×C412yescomplemented
29C3×C3⋊(ΓL(1,𝔽8)⋊C4)—22no—
30C3×Dic3×ΓL(1,𝔽8)—22no—
31C4⋊C4×AGL(1,𝔽7)—91nomaximal
32C2×D7×C3⋊D3C22×C311yescharacteristic
33C2×C7⋊#157C22×C311yescharacteristic
34C2×C7×#157C22×C311yescharacteristic
35D7×#157C3×C414yescomplemented
36C2×D3×AGL(1,𝔽7)—62no—
37#92693C22×C312yes—
38#92687—22no—
39C7×#624C22×C312yes—
40C3×Dic3×D7—22no—
41C4×C3⋊AGL(1,𝔽7)—62no—
42C4×D3×ΓL(1,𝔽8)—62no—
43C3×C4×AGL(1,𝔽7)—62no—
44(C3⋊AGL(1,𝔽7))⋊C4—62no—
45D3×ΓL(1,𝔽8)⋊C4—62no—
46Dic3×AGL(1,𝔽7)—22no—
47Dic3×AGL(1,𝔽7)—42no—
48C22×C3×#157—71no—
49C2×C3×#624—72no—
50C3×#3773—72no—
51C3×#3773—71no—
52C3×#3773—71no—
53C2×C32×ΓL(1,𝔽8)C2×D411yescharacteristic
54C3⋊(C3⋊AGL(1,𝔽7))C4⋊C412yescomplemented
55C3⋊D3×ΓL(1,𝔽8)C22×C411yescharacteristic
56C3⋊D3×ΓL(1,𝔽8)C2×Q811yescharacteristic
57C32×AGL(1,𝔽7)C4⋊C412yescomplemented
58C4×D3×D7—62no—
59C2×C4×AGL(1,𝔽7)—92no—
60C2×C4×AGL(1,𝔽7)—91no—
61C4⋊C4×ΓL(1,𝔽8)—91no—
62C4⋊(ΓL(1,𝔽8)⋊C4)—91no—
63(ΓL(1,𝔽8)⋊C4)⋊C4—92no—
64C2×#3773—71no—
65C2×C3⋊D21C3×D411yescharacteristic
66C2×C7×C3⋊D3C23×C311yescharacteristic
67C2×C32×D7C3×D411yescharacteristic
68D7×C3⋊D3C2×C3×C412yes—
69D7×C3⋊D3C3×Q812yes—
70C7⋊#157C2×C3×C412yescomplemented
71C7×#157C2×C3×C412yescomplemented
72C2×C3⋊AGL(1,𝔽7)—62no—
73C2×D3×ΓL(1,𝔽8)—62no—
74C2×C3×AGL(1,𝔽7)—22no—
75C2×C3×AGL(1,𝔽7)—42no—
76D3×AGL(1,𝔽7)—64no—
77D3×AGL(1,𝔽7)—64no—
78C3×Dic21—22no—
79C3×C7×Dic3—22no—
80C3×C4×ΓL(1,𝔽8)—62no—
81C3⋊(ΓL(1,𝔽8)⋊C4)—22no—
82C3⋊(ΓL(1,𝔽8)⋊C4)—42no—
83Dic3×ΓL(1,𝔽8)—22no—
84Dic3×ΓL(1,𝔽8)—42no—
85C3×ΓL(1,𝔽8)⋊C4—62no—
86D7×C4⋊C4—91noHall
87C22×C3×C3⋊D3—71no—
88C2×C3×#157—71no—
89C2×C3×#157—74no—
90C2×C3×#157—71no—
91C2×C32×Dic3—142no—
92C3×#624—72no—
93C3×#624—72no—
94C3×#624—142no—
95C32×ΓL(1,𝔽8)C2×C4⋊C411yescharacteristic, complemented, Hall
96C2×D3×D7—62no—
97C22×AGL(1,𝔽7)—91no—
98C4×D21—62no—
99C4×D3×C7—62no—
100C3×C4×D7—62no—
101C2×C4×ΓL(1,𝔽8)—92no—
102C2×C4×ΓL(1,𝔽8)—91no—
103D21⋊C4—62no—
104D3×Dic7—62no—
105Dic3×D7—22no—
106C2×ΓL(1,𝔽8)⋊C4—92no—
107C2×ΓL(1,𝔽8)⋊C4—91no—
108C4×AGL(1,𝔽7)—62no—
109C4×AGL(1,𝔽7)—94no—
110C4×AGL(1,𝔽7)—122no—
111C4×AGL(1,𝔽7)—182no—
112C22×#157—71no—
113C2×C3×C4×D3—422no—
114C2×#624—72no—
115#3773—71no—
116#3773—72no—
117#3773AGL(1,𝔽7)11yescharacteristic, direct product factor
118C2×C32×C7C2×C3×D411yescharacteristic
119C3⋊D21C3×C4⋊C412yescomplemented
120C7×C3⋊D3C22×C3×C411yescharacteristic
121C7×C3⋊D3C2×C3×Q811yescharacteristic
122C32×D7C3×C4⋊C412yescomplemented
123C2×C3×ΓL(1,𝔽8)—22no—
124C2×C3×ΓL(1,𝔽8)—42no—
125C3⋊AGL(1,𝔽7)—64no—
126D3×ΓL(1,𝔽8)—62no—
127D3×ΓL(1,𝔽8)—62no—
128C3×AGL(1,𝔽7)—24no—
129C3×AGL(1,𝔽7)—44no—
130C2×C4×D7—92no—
131C2×C4×D7—91no—
132C7×C4⋊C4—91no—
133C4⋊Dic7—91no—
134Dic7⋊C4—92no—
135C22×C33—71no—
136C2×C3×C3⋊D3—71no—
137C2×C3×C3⋊D3—72no—
138C2×C3×C3⋊D3—72no—
139C2×C3×C3⋊D3—71no—
140C3×#157—72no—
141C3×#157—72no—
142C32×Dic3—142no—
143C32×Dic3—142no—
144C2×C3×C4⋊C4—631no—
145C2×D21—62no—
146C2×D3×C7—62no—
147C2×C3×D7—22no—
148C22×ΓL(1,𝔽8)—91no—
149D3×D7—64no—
150D3×D7—64no—
151C2×AGL(1,𝔽7)—42no—
152C2×AGL(1,𝔽7)—91no—
153C2×AGL(1,𝔽7)—92no—
154C2×AGL(1,𝔽7)—92no—
155C2×AGL(1,𝔽7)GO+(4,𝔽2)11yescharacteristic
156C3×C4×C7—62no—
157Dic21—22no—
158C3×Dic7—62no—
159C7×Dic3—22no—
160C4×ΓL(1,𝔽8)—62no—
161C4×ΓL(1,𝔽8)—92no—
162C4×ΓL(1,𝔽8)—122no—
163ΓL(1,𝔽8)⋊C4—62no—
164ΓL(1,𝔽8)⋊C4—92no—
165ΓL(1,𝔽8)⋊C4—122no—
166C22×C3⋊D3—71no—
167C22×C3×D3—422no—
168C2×#157—71no—
169C2×#157—74no—
170C2×#157C2×AGL(1,𝔽7)11yescharacteristic
171C2×C32×C4—422no—
172C2×C3×Dic3—142no—
173C2×C3×Dic3—142no—
174C2×C3×Dic3—282no—
175C3×C4×D3—422no—
176C3×C4×D3—422no—
177C3×C4×D3—422no—
178C3×C4×D3—422no—
179#624—72no—
180#624—142no—
181#624C2×AGL(1,𝔽7)12yes—
182C32×C7C2×C3×C4⋊C411yescharacteristic, complemented
183C3×ΓL(1,𝔽8)—22no—
184C3×ΓL(1,𝔽8)—42no—
185C22×D7—91no—
186C2×C4×C7—92no—
187C2×C4×C7—91no—
188C2×Dic7—92no—
189C2×Dic7—91no—
190C4×D7—62no—
191C4×D7—94no—
192C4×D7—182no—
193C2×C33—72no—
194C2×C33—71no—
195C3×C3⋊D3—72no—
196C3×C3⋊D3—71no—
197C3×C3⋊D3—71no—
198C22×C3×C4—632no—
199C22×C3×C4—631no—
200C2×C4×D3—422no—
201C3×C4⋊C4—631no—
202C3×C4⋊C4—632no—
203C3×C4⋊C4—631no—
204C2×C3×C7—22no—
205D21—64no—
206C3×D7—24no—
207D3×C7—62no—
208D3×C7—62no—
209C2×ΓL(1,𝔽8)—42no—
210C2×ΓL(1,𝔽8)—91no—
211C2×ΓL(1,𝔽8)—91no—
212C2×ΓL(1,𝔽8)C2×GO+(4,𝔽2)11yescharacteristic
213AGL(1,𝔽7)—44no—
214AGL(1,𝔽7)—92no—
215AGL(1,𝔽7)#377312yesdirect product factor
216C22×C32—71no—
217C22×C32—142no—
218C22×C32—282no—
219C2×C3⋊D3—71no—
220C2×C3⋊D3—72no—
221C2×C3⋊D3—72no—
222C2×C3⋊D3C22×AGL(1,𝔽7)11yescharacteristic
223C2×C3×D3—422no—
224C2×C3×D3—424no—
225C2×C3×D3—424no—
226C2×C3×D3—422no—
227#157—72no—
228#157C4×AGL(1,𝔽7)12yescomplemented
229C32×C4—422no—
230C32×C4—422no—
231C3×Dic3—22no—
232C3×Dic3—142no—
233C3×Dic3—142no—
234C3×Dic3—142no—
235C3×Dic3—282no—
236C3×Dic3—282no—
237C2×C4⋊C4—631noSylow
238C22×C7—91no—
239C2×D7—91no—
240C2×D7—92no—
241C2×D7—92no—
242C2×D7C3×GO+(4,𝔽2)11yescharacteristic
243C4×C7—62no—
244C4×C7—92no—
245Dic7—62no—
246Dic7—92no—
247C33—71noSylow
248C23×C3—631no—
249C22×D3—422no—
250C2×C3×C4—422no—
251C2×C3×C4—422no—
252C2×C3×C4—632no—
253C2×C3×C4—631no—
254C2×C3×C4—632no—
255C2×C3×C4—634no—
256C2×C3×C4—631no—
257C2×C3×C4—842no—
258C2×C3×C4—1262no—
259C2×Dic3—142no—
260C4×D3—62no—
261C4×D3—422no—
262C4×D3—422no—
263C4×D3—422no—
264C3×C7—22no—
265ΓL(1,𝔽8)—42no—
266ΓL(1,𝔽8)C2×#377311yescharacteristic, complemented
267C2×C32—72no—
268C2×C32—142no—
269C2×C32—144no—
270C2×C32—282no—
271C2×C32—284no—
272C2×C32D4×AGL(1,𝔽7)11yescharacteristic
273C3⋊D3—72no—
274C3⋊D3C2×C4×AGL(1,𝔽7)11yescharacteristic
275C3⋊D3Q8×AGL(1,𝔽7)11yescharacteristic
276C3×D3—422no—
277C3×D3—422no—
278C3×D3—424no—
279C22×C4—632no—
280C22×C4—631no—
281C4⋊C4—91no—
282C4⋊C4—632no—
283C4⋊C4—631no—
284C2×C7—91no—
285C2×C7—91no—
286C2×C7C2×C3×GO+(4,𝔽2)11yescharacteristic
287D7—92no—
288D7C3×#377312yescomplemented
289C22×C3—71no—
290C22×C3—142no—
291C22×C3—282no—
292C22×C3—631no—
293C22×C3—632no—
294C22×C3—631no—
295C22×C3—632no—
296C2×D3—62no—
297C2×D3—424no—
298C2×D3—424no—
299C2×D3—422no—
300C3×C4—62no—
301C3×C4—422no—
302C3×C4—422no—
303C3×C4—422no—
304C3×C4—632no—
305C3×C4—632no—
306C3×C4—842no—
307C3×C4—842no—
308Dic3—22no—
309Dic3—142no—
310C32—142no—
311C32—282no—
312C32C4⋊C4×AGL(1,𝔽7)11yescharacteristic, complemented
313C23—631no—
314C2×C4—92no—
315C2×C4—91no—
316C2×C4—422no—
317C2×C4—634no—
318C2×C4—632no—
319C2×C4—631no—
320C2×C4—1262no—
321C7C2×C3×#377311yescharacteristic, complemented, Sylow
322C2×C3—22no—
323C2×C3—72no—
324C2×C3—71no—
325C2×C3—144no—
326C2×C3—284no—
327C2×C3—282no—
328C2×C3—631no—
329C2×C3—631no—
330C2×C3—632no—
331D3—62no—
332D3—62no—
333D3—424no—
334C22—71no—
335C22—91no—
336C22—632no—
337C22—632no—
338C22—631no—
339C4—62no—
340C4—92no—
341C4—422no—
342C4—632no—
343C3—22no—
344C3—71no—
345C3—282no—
346C2—72no—
347C2—91no—
348C2—91no—
349C2—632no—
350C2AGL(1,𝔽7)×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2AGL(1,𝔽7)×GO+(4,𝔽2)
commutatorC7×C3⋊D3C22×C3×C4
Fitting, socleC2×C32×C7C2×C3×D4
prime radicalC2×C3⋊D3×AGL(1,𝔽7)C22
cosocleC2×C7×C3⋊D3C23×C3
hypocenterC32×C7C2×C3×C4⋊C4
LCSG ▷ C7×C3⋊D3 ▷ C32×C7
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. D7×#3773C3semidirect
2. #72217606C2semidirect
3. #72217594C2semidirect
4. ΓL(1,𝔽8)×#3773C2semidirect
10. #157×AGL(1,𝔽7)C4semidirect
17. #11759408C2×C3semidirect
18. #11759402C2×C3semidirect
19. C7×#3773C2×C3semidirect
27. ΓL(1,𝔽8)⋊#157C2×C4semidirect
28. ΓL(1,𝔽8)×#157C2×C4semidirect
35. D7×#157C3×C4semidirect
54. C3⋊(C3⋊AGL(1,𝔽7))C4⋊C4semidirect
57. C32×AGL(1,𝔽7)C4⋊C4semidirect
70. C7⋊#157C2×C3×C4semidirect
71. C7×#157C2×C3×C4semidirect
95. C32×ΓL(1,𝔽8)C2×C4⋊C4semidirect
117. #3773AGL(1,𝔽7)direct (N × H)
119. C3⋊D21C3×C4⋊C4semidirect
122. C32×D7C3×C4⋊C4semidirect
182. C32×C7C2×C3×C4⋊C4semidirect
215. AGL(1,𝔽7)#3773direct (N × H)
228. #157C4×AGL(1,𝔽7)semidirect
266. ΓL(1,𝔽8)C2×#3773semidirect
288. D7C3×#3773semidirect
312. C32C4⋊C4×AGL(1,𝔽7)semidirect
321. C7C2×C3×#3773semidirect

Maximal subgroups

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

subgroupindex
D7×#37733
#722176062
#722175942
ΓL(1,𝔽8)×#37732
AGL(1,𝔽7)×#6242
C2×#157×AGL(1,𝔽7)2
C2×C3×#37737
C4⋊C4×AGL(1,𝔽7)9