Finite Group Databasebeta

← AGL(1,𝔽7)×#627 #72246912

Subgroups — AGL(1,𝔽7)×#627 #72246912

Order 3024.

420 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 94 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#92696⋊C3C211yesmaximal, characteristic, complemented
2#92691⋊C3C211yesmaximal, characteristic, complemented
3#92691⋊C3C211yesmaximal, characteristic, complemented
4ΓL(1,𝔽8)×#627C211yesmaximal, characteristic, complemented
5C3⋊Dic3×AGL(1,𝔽7)C211yesmaximal, characteristic
6C3×Dic3×AGL(1,𝔽7)C211yesmaximal, characteristic
7C3×Dic3×AGL(1,𝔽7)C211yesmaximal, characteristic
8D7×#627C311yesmaximal, characteristic, complemented
9Dic6×AGL(1,𝔽7)—31nomaximal
10Dic6×AGL(1,𝔽7)—31nomaximal
11C2×C32×AGL(1,𝔽7)C2211yescharacteristic
12C3⋊(C3⋊(ΓL(1,𝔽8)⋊C4))C2211yescharacteristic
13ΓL(1,𝔽8)×C3⋊Dic3C2211yescharacteristic
14C3×C3⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic
15C3×C3⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic
16C3×Dic3×ΓL(1,𝔽8)C2211yescharacteristic
17C3×Dic3×ΓL(1,𝔽8)C2211yescharacteristic
18#92696C2×C311yescharacteristic, complemented
19#92691C2×C311yescharacteristic, complemented
20#92691C2×C311yescharacteristic, complemented
21C7×#627C2×C311yescharacteristic, complemented
22D7×C3⋊Dic3C2×C311yescharacteristic
23C3×Dic3×D7C2×C311yescharacteristic
24C3×Dic3×D7C2×C311yescharacteristic
25Dic42⋊C3—31no—
26Dic42⋊C3—31no—
27ΓL(1,𝔽8)×Dic6—31no—
28ΓL(1,𝔽8)×Dic6—31no—
29C3×C4×AGL(1,𝔽7)—31no—
30C3×C4×AGL(1,𝔽7)—31no—
31#4248⋊C3—31no—
32#4248⋊C3—31no—
33#4248⋊C3—31no—
34#4248⋊C3—31no—
35Dic3×AGL(1,𝔽7)—21no—
36Dic3×AGL(1,𝔽7)—21no—
37Dic3×AGL(1,𝔽7)—31no—
38Dic3×AGL(1,𝔽7)—31no—
39Dic3×AGL(1,𝔽7)—61no—
40Dic3×AGL(1,𝔽7)D311yescharacteristic
41Dic3×AGL(1,𝔽7)D311yescharacteristic
42C2×C3×#627—71nomaximal, Hall
43C2×C32×ΓL(1,𝔽8)C2311yescharacteristic
44C32×AGL(1,𝔽7)Q812yescomplemented
45D7×Dic6—31no—
46D7×Dic6—31no—
47Q8×AGL(1,𝔽7)—91no—
48C2×C32×D7C22×C311yescharacteristic
49C2×C3×AGL(1,𝔽7)—21no—
50C2×C3×AGL(1,𝔽7)—21no—
51C2×C3×AGL(1,𝔽7)—21no—
52C2×C3×AGL(1,𝔽7)—41no—
53C2×C3×AGL(1,𝔽7)C2×D311yescharacteristic
54C2×C3×AGL(1,𝔽7)C2×D311yescharacteristic
55C3⋊Dic21C22×C311yescharacteristic
56C7×C3⋊Dic3C22×C311yescharacteristic
57C3×Dic21C22×C311yescharacteristic
58C3×Dic21C22×C311yescharacteristic
59C3×C7×Dic3C22×C311yescharacteristic
60C3×C7×Dic3C22×C311yescharacteristic
61C3×C4×ΓL(1,𝔽8)—31no—
62C3×C4×ΓL(1,𝔽8)—31no—
63C3⋊(ΓL(1,𝔽8)⋊C4)—21no—
64C3⋊(ΓL(1,𝔽8)⋊C4)—21no—
65C3⋊(ΓL(1,𝔽8)⋊C4)—31no—
66C3⋊(ΓL(1,𝔽8)⋊C4)—31no—
67C3⋊(ΓL(1,𝔽8)⋊C4)—61no—
68C3⋊(ΓL(1,𝔽8)⋊C4)C2×D311yescharacteristic
69C3⋊(ΓL(1,𝔽8)⋊C4)C2×D311yescharacteristic
70Dic3×ΓL(1,𝔽8)—21no—
71Dic3×ΓL(1,𝔽8)—21no—
72Dic3×ΓL(1,𝔽8)—31no—
73Dic3×ΓL(1,𝔽8)—31no—
74Dic3×ΓL(1,𝔽8)—61no—
75Dic3×ΓL(1,𝔽8)C2×D311yescharacteristic
76Dic3×ΓL(1,𝔽8)C2×D311yescharacteristic
77C3×ΓL(1,𝔽8)⋊C4—31no—
78C3×ΓL(1,𝔽8)⋊C4—31no—
79C2×C3×C3⋊Dic3—71no—
80C2×C32×Dic3—71no—
81C2×C32×Dic3—71no—
82C3×#627—71no—
83C3×#627—71no—
84C3×#627—71no—
85C3×#627—141no—
86C32×ΓL(1,𝔽8)C2×Q811yescharacteristic, complemented, Hall
87Dic42—31no—
88Dic42—31no—
89C7×Dic6—31no—
90C7×Dic6—31no—
91C3×C4×D7—31no—
92C3×C4×D7—31no—
93Q8×ΓL(1,𝔽8)—91no—
94#4248—31no—
95#4248—31no—
96#4248—31no—
97#4248—31no—
98Dic3×D7—31no—
99Dic3×D7—31no—
100Dic3×D7—61no—
101Dic3×D7C3×D311yescharacteristic
102Dic3×D7C3×D311yescharacteristic
103C4×AGL(1,𝔽7)—31no—
104C4×AGL(1,𝔽7)—31no—
105C4×AGL(1,𝔽7)—61no—
106C4×AGL(1,𝔽7)—61no—
107C4×AGL(1,𝔽7)—91no—
108ΓL(1,𝔽8)⋊Q8—91no—
109ΓL(1,𝔽8)⋊Q8—91no—
110ΓL(1,𝔽8)⋊Q8—91no—
111C2×C3×Dic6—211no—
112C2×C3×Dic6—211no—
113C2×#627—71no—
114C2×C32×C7C23×C311yescharacteristic
115C32×D7C3×Q812yescomplemented
116C2×C3×ΓL(1,𝔽8)—21no—
117C2×C3×ΓL(1,𝔽8)—21no—
118C2×C3×ΓL(1,𝔽8)—21no—
119C2×C3×ΓL(1,𝔽8)—41no—
120C2×C3×ΓL(1,𝔽8)C22×D311yescharacteristic
121C2×C3×ΓL(1,𝔽8)C22×D311yescharacteristic
122C3×AGL(1,𝔽7)—22no—
123C3×AGL(1,𝔽7)—22no—
124C3×AGL(1,𝔽7)—22no—
125C3×AGL(1,𝔽7)—42no—
126C3×AGL(1,𝔽7)Dic612yescomplemented
127C3×AGL(1,𝔽7)Dic612yescomplemented
128Q8×D7—91noHall
129C22×C33—71no—
130C3×C3⋊Dic3—71no—
131C3×C3⋊Dic3—141no—
132C32×Dic3—71no—
133C32×Dic3—71no—
134C32×Dic3—141no—
135C32×Dic3—141no—
136C2×C3×D7—21no—
137C2×C3×D7C2×C3×D311yescharacteristic
138C2×C3×D7C2×C3×D311yescharacteristic
139C2×AGL(1,𝔽7)—21no—
140C2×AGL(1,𝔽7)—21no—
141C2×AGL(1,𝔽7)—41no—
142C2×AGL(1,𝔽7)D3211yescharacteristic
143C3×C4×C7—31no—
144C3×C4×C7—31no—
145Dic21—31no—
146Dic21—31no—
147Dic21—61no—
148Dic21C2×C3×D311yescharacteristic
149Dic21C2×C3×D311yescharacteristic
150C3×Dic7—31no—
151C3×Dic7—31no—
152C7×Dic3—31no—
153C7×Dic3—31no—
154C7×Dic3—61no—
155C7×Dic3C2×C3×D311yescharacteristic
156C7×Dic3C2×C3×D311yescharacteristic
157C4×ΓL(1,𝔽8)—31no—
158C4×ΓL(1,𝔽8)—31no—
159C4×ΓL(1,𝔽8)—61no—
160C4×ΓL(1,𝔽8)—61no—
161C4×ΓL(1,𝔽8)—91no—
162ΓL(1,𝔽8)⋊C4—31no—
163ΓL(1,𝔽8)⋊C4—31no—
164ΓL(1,𝔽8)⋊C4—61no—
165ΓL(1,𝔽8)⋊C4—61no—
166ΓL(1,𝔽8)⋊C4—91no—
167C2×C32×C4—211no—
168C2×C32×C4—211no—
169C2×C3⋊Dic3—71no—
170C2×C3×Dic3—71no—
171C2×C3×Dic3—71no—
172C2×C3×Dic3—71no—
173C2×C3×Dic3—71no—
174C2×C3×Dic3—141no—
175C2×C3×Dic3—141no—
176C2×C3×Dic3—211no—
177C2×C3×Dic3—211no—
178C2×C3×Dic3—421no—
179C3×Dic6—211no—
180C3×Dic6—211no—
181C3×Dic6—211no—
182C3×Dic6—211no—
183C3×Dic6—211no—
184C3×Dic6—211no—
185C3×Dic6—421no—
186C3×Dic6—421no—
187#627—71no—
188#627—71no—
189#627—71no—
190#627AGL(1,𝔽7)11yescharacteristic, direct product factor
191C32×C7C2×C3×Q811yescharacteristic, complemented
192C3×ΓL(1,𝔽8)—21no—
193C3×ΓL(1,𝔽8)—21no—
194C3×ΓL(1,𝔽8)—21no—
195C3×ΓL(1,𝔽8)—41no—
196C3×ΓL(1,𝔽8)C2×Dic611yescharacteristic, complemented
197C3×ΓL(1,𝔽8)C2×Dic611yescharacteristic, complemented
198C7×Q8—91no—
199C4×D7—31no—
200C4×D7—31no—
201C4×D7—91no—
202Dic14—91no—
203Dic14—91no—
204Dic14—91no—
205C2×C33—72no—
206C2×C33—141no—
207C2×C3×Q8—631no—
208C2×Dic6—211no—
209C2×Dic6—211no—
210C2×C3×C7—21no—
211C2×C3×C7C22×C3×D311yescharacteristic
212C2×C3×C7C22×C3×D311yescharacteristic
213C3×D7—22no—
214C3×D7C3×Dic612yescomplemented
215C3×D7C3×Dic612yescomplemented
216C2×ΓL(1,𝔽8)—21no—
217C2×ΓL(1,𝔽8)—21no—
218C2×ΓL(1,𝔽8)—41no—
219C2×ΓL(1,𝔽8)C2×D3211yescharacteristic
220AGL(1,𝔽7)—22no—
221AGL(1,𝔽7)—22no—
222AGL(1,𝔽7)—42no—
223AGL(1,𝔽7)#62712yesdirect product factor
224C22×C32—71no—
225C22×C32—71no—
226C22×C32—71no—
227C22×C32—141no—
228C22×C32—141no—
229C22×C32—141no—
230C22×C32—281no—
231C32×C4—211no—
232C32×C4—211no—
233C32×C4—421no—
234C32×C4—421no—
235C3⋊Dic3—71no—
236C3⋊Dic3C2×AGL(1,𝔽7)11yescharacteristic
237C3×Dic3—71no—
238C3×Dic3—71no—
239C3×Dic3—71no—
240C3×Dic3—71no—
241C3×Dic3—141no—
242C3×Dic3—141no—
243C3×Dic3—141no—
244C3×Dic3—141no—
245C3×Dic3—211no—
246C3×Dic3—211no—
247C3×Dic3—281no—
248C3×Dic3—281no—
249C3×Dic3—421no—
250C3×Dic3—421no—
251C3×Dic3—421no—
252C3×Dic3—841no—
253C3×Dic3C2×AGL(1,𝔽7)11yescharacteristic
254C3×Dic3C2×AGL(1,𝔽7)11yescharacteristic
255C2×D7C3×D3211yescharacteristic
256C4×C7—31no—
257C4×C7—31no—
258C4×C7—91no—
259Dic7—31no—
260Dic7—31no—
261Dic7—91no—
262C33—141noSylow
263C3×Q8—631no—
264C3×Q8—631no—
265C3×Q8—631no—
266C3×Q8—1261no—
267C2×C3×C4—211no—
268C2×C3×C4—211no—
269C2×C3×C4—211no—
270C2×C3×C4—211no—
271C2×C3×C4—421no—
272C2×C3×C4—421no—
273C2×C3×C4—631no—
274C2×Dic3—71no—
275C2×Dic3—71no—
276C2×Dic3—211no—
277C2×Dic3—211no—
278C2×Dic3—421no—
279Dic6—31no—
280Dic6—31no—
281Dic6—211no—
282Dic6—211no—
283Dic6—211no—
284Dic6—211no—
285Dic6—211no—
286Dic6—211no—
287C3×C7—21no—
288C3×C7C2×C3×Dic611yescharacteristic, complemented
289C3×C7C2×C3×Dic611yescharacteristic, complemented
290ΓL(1,𝔽8)—21no—
291ΓL(1,𝔽8)—21no—
292ΓL(1,𝔽8)—41no—
293ΓL(1,𝔽8)C2×#62711yescharacteristic, complemented
294C2×C32—72no—
295C2×C32—72no—
296C2×C32—72no—
297C2×C32—142no—
298C2×C32—142no—
299C2×C32—142no—
300C2×C32—141no—
301C2×C32—141no—
302C2×C32—281no—
303C2×C32—282no—
304C2×C32—281no—
305C2×C32—281no—
306C2×C32—561no—
307C2×C32C22×AGL(1,𝔽7)11yescharacteristic
308C2×Q8—631noSylow
309C2×C7C2×C3×D3211yescharacteristic
310D7C3×#62712yescomplemented
311C22×C3—71no—
312C22×C3—71no—
313C22×C3—71no—
314C22×C3—141no—
315C22×C3—141no—
316C22×C3—141no—
317C22×C3—281no—
318C3×C4—31no—
319C3×C4—31no—
320C3×C4—211no—
321C3×C4—211no—
322C3×C4—211no—
323C3×C4—211no—
324C3×C4—421no—
325C3×C4—421no—
326C3×C4—421no—
327C3×C4—421no—
328C3×C4—421no—
329C3×C4—631no—
330C3×C4—841no—
331C3×C4—1261no—
332Dic3—31no—
333Dic3—31no—
334Dic3—61no—
335Dic3—71no—
336Dic3—71no—
337Dic3—211no—
338Dic3—211no—
339Dic3—421no—
340Dic3D3×AGL(1,𝔽7)11yescharacteristic
341Dic3D3×AGL(1,𝔽7)11yescharacteristic
342C32—141no—
343C32—141no—
344C32—281no—
345C32—281no—
346C32—281no—
347C32—561no—
348C32Q8×AGL(1,𝔽7)11yescharacteristic, complemented
349Q8—91no—
350Q8—631no—
351Q8—631no—
352Q8—631no—
353C2×C4—211no—
354C2×C4—211no—
355C2×C4—631no—
356C7C2×C3×#62711yescharacteristic, complemented, Sylow
357C2×C3—21no—
358C2×C3—72no—
359C2×C3—72no—
360C2×C3—72no—
361C2×C3—142no—
362C2×C3—142no—
363C2×C3—142no—
364C2×C3—141no—
365C2×C3—141no—
366C2×C3—282no—
367C2×C3—281no—
368C2×C3—561no—
369C2×C3C2×D3×AGL(1,𝔽7)11yescharacteristic
370C2×C3C2×D3×AGL(1,𝔽7)11yescharacteristic
371C22—71no—
372C4—31no—
373C4—31no—
374C4—91no—
375C4—211no—
376C4—211no—
377C4—631no—
378C3—21no—
379C3—141no—
380C3—141no—
381C3—281no—
382C3—561no—
383C3Dic6×AGL(1,𝔽7)11yescharacteristic, complemented
384C3Dic6×AGL(1,𝔽7)11yescharacteristic, complemented
385C2—72no—
386C2D32×AGL(1,𝔽7)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D32×AGL(1,𝔽7)
commutator, Fitting, socle, cosocleC2×C32×C7C23×C3
prime radicalC2×C32×AGL(1,𝔽7)C22
hypocenterC32×C7C2×C3×Q8
LCSG ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. #92696⋊C3C2semidirect
2. #92691⋊C3C2semidirect
3. #92691⋊C3C2semidirect
4. ΓL(1,𝔽8)×#627C2semidirect
8. D7×#627C3semidirect
18. #92696C2×C3semidirect
19. #92691C2×C3semidirect
20. #92691C2×C3semidirect
21. C7×#627C2×C3semidirect
44. C32×AGL(1,𝔽7)Q8semidirect
86. C32×ΓL(1,𝔽8)C2×Q8semidirect
115. C32×D7C3×Q8semidirect
126. C3×AGL(1,𝔽7)Dic6semidirect
127. C3×AGL(1,𝔽7)Dic6semidirect
190. #627AGL(1,𝔽7)direct (N × H)
191. C32×C7C2×C3×Q8semidirect
196. C3×ΓL(1,𝔽8)C2×Dic6semidirect
197. C3×ΓL(1,𝔽8)C2×Dic6semidirect
214. C3×D7C3×Dic6semidirect
215. C3×D7C3×Dic6semidirect
223. AGL(1,𝔽7)#627direct (N × H)
288. C3×C7C2×C3×Dic6semidirect
289. C3×C7C2×C3×Dic6semidirect
293. ΓL(1,𝔽8)C2×#627semidirect
310. D7C3×#627semidirect
348. C32Q8×AGL(1,𝔽7)semidirect
356. C7C2×C3×#627semidirect
383. C3Dic6×AGL(1,𝔽7)semidirect
384. C3Dic6×AGL(1,𝔽7)semidirect

Maximal subgroups

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

subgroupindex
#92696⋊C32
#92691⋊C32
ΓL(1,𝔽8)×#6272
C3⋊Dic3×AGL(1,𝔽7)2
C3×Dic3×AGL(1,𝔽7)2
D7×#6273
Dic6×AGL(1,𝔽7)3
C2×C3×#6277