Finite Group Databasebeta

← ΓL(1,𝔽8)×#1790 #72277254

Subgroups — ΓL(1,𝔽8)×#1790 #72277254

Order 2688.

1340 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 1340 conjugacy classes of subgroups — too many to draw. Only its 204 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
1C2×ΓL(1,𝔽8)×H(5,𝔽2)C211yesmaximal, characteristic
2C2×ΓL(1,𝔽8)×#99C213yesmaximal, complemented
3ΓL(1,𝔽8)×#386C213yesmaximal, complemented
4C7×#1790C311yesmaximal, characteristic, complemented, Hall
5ΓL(1,𝔽8)×H(5,𝔽2)—26no—
6ΓL(1,𝔽8)×H(5,𝔽2)C414yescomplemented
7C2×#41×ΓL(1,𝔽8)—23no—
8C22×D4×ΓL(1,𝔽8)—23no—
9C22×D4×ΓL(1,𝔽8)C2213yes—
10C23×C4×ΓL(1,𝔽8)C2211yescharacteristic, complemented
11C2×C22⋊C4×ΓL(1,𝔽8)—23no—
12C2×C22⋊C4×ΓL(1,𝔽8)C2213yescomplemented
13ΓL(1,𝔽8)×#99—26no—
14ΓL(1,𝔽8)×#95—23no—
15C2×C7×H(5,𝔽2)C2×C311yescharacteristic
16C2×C7×#99C2×C313yescomplemented
17C7×#386C2×C313yescomplemented
18C3×#1790—71nomaximal, Hall
19C24×ΓL(1,𝔽8)—21no—
20C24×ΓL(1,𝔽8)D413yes—
21C24×ΓL(1,𝔽8)C2311yescharacteristic
22#41×ΓL(1,𝔽8)—212no—
23#41×ΓL(1,𝔽8)—212no—
24C2×Q8×ΓL(1,𝔽8)—21no—
25C2×D4×ΓL(1,𝔽8)—23no—
26C2×D4×ΓL(1,𝔽8)—212no—
27C2×D4×ΓL(1,𝔽8)—212no—
28C2×D4×ΓL(1,𝔽8)—26no—
29C2×D4×ΓL(1,𝔽8)—26no—
30C2×D4×ΓL(1,𝔽8)C2×C416yes—
31C2×D4×ΓL(1,𝔽8)D416yes—
32C22×C4×ΓL(1,𝔽8)—23no—
33C22×C4×ΓL(1,𝔽8)—23no—
34C22×C4×ΓL(1,𝔽8)—23no—
35C22×C4×ΓL(1,𝔽8)—42no—
36C22×C4×ΓL(1,𝔽8)D413yes—
37C22⋊C4×ΓL(1,𝔽8)—26no—
38C22⋊C4×ΓL(1,𝔽8)—46no—
39C7×H(5,𝔽2)—26no—
40C7×H(5,𝔽2)C3×C414yescomplemented
41C2×C7×#41—23no—
42C22×C7×D4—23no—
43C22×C7×D4C22×C313yes—
44C23×C4×C7C22×C311yescharacteristic, complemented
45C2×C7×C22⋊C4—23no—
46C2×C7×C22⋊C4C22×C313yescomplemented
47C7×#99—26no—
48C7×#95—23no—
49C2×C3×H(5,𝔽2)—71no—
50C2×C3×#99—73no—
51C3×#386—73no—
52C23×ΓL(1,𝔽8)—24no—
53C23×ΓL(1,𝔽8)—23no—
54C23×ΓL(1,𝔽8)—23no—
55C23×ΓL(1,𝔽8)—41no—
56C23×ΓL(1,𝔽8)—41no—
57C23×ΓL(1,𝔽8)—43no—
58C23×ΓL(1,𝔽8)—43no—
59C23×ΓL(1,𝔽8)—81no—
60C23×ΓL(1,𝔽8)—81no—
61C23×ΓL(1,𝔽8)C22⋊C416yescomplemented
62C23×ΓL(1,𝔽8)C22×C411yescharacteristic
63C23×ΓL(1,𝔽8)C2×D413yes—
64C23×ΓL(1,𝔽8)C2×D413yes—
65Q8×ΓL(1,𝔽8)—24no—
66D4×ΓL(1,𝔽8)—212no—
67D4×ΓL(1,𝔽8)—212no—
68D4×ΓL(1,𝔽8)—212no—
69C2×C4×ΓL(1,𝔽8)—212no—
70C2×C4×ΓL(1,𝔽8)—23no—
71C2×C4×ΓL(1,𝔽8)—23no—
72C2×C4×ΓL(1,𝔽8)—26no—
73C2×C4×ΓL(1,𝔽8)—43no—
74C2×C4×ΓL(1,𝔽8)—41no—
75C2×C4×ΓL(1,𝔽8)—43no—
76C2×C4×ΓL(1,𝔽8)—43no—
77C2×C4×ΓL(1,𝔽8)—43no—
78C2×C4×ΓL(1,𝔽8)C22⋊C416yescomplemented
79#1790ΓL(1,𝔽8)11yescharacteristic, direct product factor, Sylow
80C24×C7—21no—
81C24×C7C3×D413yes—
82C24×C7C23×C311yescharacteristic
83C7×#41—212no—
84C7×#41—212no—
85C2×C7×Q8—21no—
86C2×C7×D4—23no—
87C2×C7×D4—212no—
88C2×C7×D4—212no—
89C2×C7×D4—26no—
90C2×C7×D4—26no—
91C2×C7×D4C2×C3×C416yes—
92C2×C7×D4C3×D416yes—
93C22×C4×C7—23no—
94C22×C4×C7—23no—
95C22×C4×C7—23no—
96C22×C4×C7—42no—
97C22×C4×C7C3×D413yes—
98C7×C22⋊C4—26no—
99C7×C22⋊C4—46no—
100C3×H(5,𝔽2)—74no—
101C3×H(5,𝔽2)—146no—
102C2×C3×#41—143no—
103C22×C3×D4—73no—
104C22×C3×D4—143no—
105C23×C3×C4—71no—
106C2×C3×C22⋊C4—73no—
107C2×C3×C22⋊C4—143no—
108C3×#99—146no—
109C3×#95—143no—
110C22×ΓL(1,𝔽8)—23no—
111C22×ΓL(1,𝔽8)—23no—
112C22×ΓL(1,𝔽8)—26no—
113C22×ΓL(1,𝔽8)—41no—
114C22×ΓL(1,𝔽8)—43no—
115C22×ΓL(1,𝔽8)—46no—
116C22×ΓL(1,𝔽8)—43no—
117C22×ΓL(1,𝔽8)—43no—
118C22×ΓL(1,𝔽8)—83no—
119C22×ΓL(1,𝔽8)—84no—
120C22×ΓL(1,𝔽8)C2×C22⋊C413yes—
121C22×ΓL(1,𝔽8)#12013yes—
122C22×ΓL(1,𝔽8)#12011yescharacteristic
123C4×ΓL(1,𝔽8)—26no—
124C4×ΓL(1,𝔽8)—42no—
125C4×ΓL(1,𝔽8)—83no—
126C2×H(5,𝔽2)C2×ΓL(1,𝔽8)11yescharacteristic
127C2×#99C2×ΓL(1,𝔽8)13yescomplemented
128#386C2×ΓL(1,𝔽8)13yescomplemented
129C23×C7—24no—
130C23×C7—23no—
131C23×C7—23no—
132C23×C7—41no—
133C23×C7—41no—
134C23×C7—43no—
135C23×C7—43no—
136C23×C7—81no—
137C23×C7—81no—
138C23×C7C3×C22⋊C416yescomplemented
139C23×C7C22×C3×C411yescharacteristic
140C23×C7C2×C3×D413yes—
141C23×C7C2×C3×D413yes—
142C7×Q8—24no—
143C7×D4—212no—
144C7×D4—212no—
145C7×D4—212no—
146C2×C4×C7—212no—
147C2×C4×C7—23no—
148C2×C4×C7—23no—
149C2×C4×C7—26no—
150C2×C4×C7—43no—
151C2×C4×C7—41no—
152C2×C4×C7—43no—
153C2×C4×C7—43no—
154C2×C4×C7—43no—
155C2×C4×C7C3×C22⋊C416yescomplemented
156C24×C3—73no—
157C24×C3—71no—
158C24×C3—141no—
159C3×#41—1412no—
160C3×#41—1412no—
161C2×C3×Q8—141no—
162C2×C3×D4—76no—
163C2×C3×D4—76no—
164C2×C3×D4—146no—
165C2×C3×D4—146no—
166C2×C3×D4—1412no—
167C2×C3×D4—143no—
168C2×C3×D4—1412no—
169C22×C3×C4—73no—
170C22×C3×C4—143no—
171C22×C3×C4—143no—
172C22×C3×C4—143no—
173C22×C3×C4—282no—
174C3×C22⋊C4—146no—
175C3×C22⋊C4—286no—
176C2×ΓL(1,𝔽8)—23no—
177C2×ΓL(1,𝔽8)—23no—
178C2×ΓL(1,𝔽8)—46no—
179C2×ΓL(1,𝔽8)#37911yescharacteristic
180C2×ΓL(1,𝔽8)B(4,𝔽2)11yescharacteristic
181C2×ΓL(1,𝔽8)#45811yescharacteristic
182H(5,𝔽2)—26no—
183H(5,𝔽2)C4×ΓL(1,𝔽8)14yescomplemented
184C2×#41—23no—
185C22×D4—23no—
186C22×D4C22×ΓL(1,𝔽8)13yes—
187C23×C4C22×ΓL(1,𝔽8)11yescharacteristic, complemented
188C2×C22⋊C4—23no—
189C2×C22⋊C4C22×ΓL(1,𝔽8)13yescomplemented
190#99—26no—
191#95—23no—
192C22×C7—23no—
193C22×C7—23no—
194C22×C7—26no—
195C22×C7—41no—
196C22×C7—43no—
197C22×C7—46no—
198C22×C7—43no—
199C22×C7—43no—
200C22×C7—83no—
201C22×C7—84no—
202C22×C7C2×C3×C22⋊C413yes—
203C22×C7C3×#12013yes—
204C22×C7C3×#12011yescharacteristic
205C4×C7—26no—
206C4×C7—42no—
207C4×C7—83no—
208C23×C3—76no—
209C23×C3—71no—
210C23×C3—73no—
211C23×C3—73no—
212C23×C3—143no—
213C23×C3—143no—
214C23×C3—144no—
215C23×C3—281no—
216C23×C3—281no—
217C23×C3—283no—
218C23×C3—283no—
219C23×C3—561no—
220C23×C3—561no—
221C3×Q8—144no—
222C3×D4—1412no—
223C3×D4—1412no—
224C3×D4—1412no—
225C2×C3×C4—76no—
226C2×C3×C4—143no—
227C2×C3×C4—143no—
228C2×C3×C4—1412no—
229C2×C3×C4—146no—
230C2×C3×C4—281no—
231C2×C3×C4—283no—
232C2×C3×C4—283no—
233C2×C3×C4—283no—
234C2×C3×C4—283no—
235ΓL(1,𝔽8)#179011yescharacteristic, direct product factor, Hall
236C24—21no—
237C24D4×ΓL(1,𝔽8)13yes—
238C24C23×ΓL(1,𝔽8)11yescharacteristic
239#41—212no—
240#41—212no—
241C2×Q8—21no—
242C2×D4—23no—
243C2×D4—212no—
244C2×D4—212no—
245C2×D4—26no—
246C2×D4—26no—
247C2×D4C2×C4×ΓL(1,𝔽8)16yes—
248C2×D4D4×ΓL(1,𝔽8)16yes—
249C22×C4—23no—
250C22×C4—23no—
251C22×C4—23no—
252C22×C4—42no—
253C22×C4D4×ΓL(1,𝔽8)13yes—
254C22⋊C4—26no—
255C22⋊C4—46no—
256C2×C7—23no—
257C2×C7—23no—
258C2×C7—46no—
259C2×C7C3×#37911yescharacteristic
260C2×C7C3×B(4,𝔽2)11yescharacteristic
261C2×C7C3×#45811yescharacteristic
262C22×C3—73no—
263C22×C3—73no—
264C22×C3—71no—
265C22×C3—143no—
266C22×C3—143no—
267C22×C3—146no—
268C22×C3—283no—
269C22×C3—283no—
270C22×C3—281no—
271C22×C3—283no—
272C22×C3—286no—
273C22×C3—563no—
274C22×C3—564no—
275C3×C4—146no—
276C3×C4—282no—
277C3×C4—563no—
278C23—24no—
279C23—23no—
280C23—23no—
281C23—41no—
282C23—41no—
283C23—43no—
284C23—43no—
285C23—81no—
286C23—81no—
287C23C22⋊C4×ΓL(1,𝔽8)16yescomplemented
288C23C22×C4×ΓL(1,𝔽8)11yescharacteristic
289C23C2×D4×ΓL(1,𝔽8)13yes—
290C23C2×D4×ΓL(1,𝔽8)13yes—
291Q8—24no—
292D4—212no—
293D4—212no—
294D4—212no—
295C2×C4—212no—
296C2×C4—23no—
297C2×C4—23no—
298C2×C4—26no—
299C2×C4—43no—
300C2×C4—41no—
301C2×C4—43no—
302C2×C4—43no—
303C2×C4—43no—
304C2×C4C22⋊C4×ΓL(1,𝔽8)16yescomplemented
305C7C3×#179011yescharacteristic, complemented, Sylow
306C2×C3—71no—
307C2×C3—71no—
308C2×C3—71no—
309C2×C3—143no—
310C2×C3—143no—
311C2×C3—286no—
312C22—23no—
313C22—23no—
314C22—26no—
315C22—41no—
316C22—43no—
317C22—46no—
318C22—43no—
319C22—43no—
320C22—83no—
321C22—84no—
322C22C2×C22⋊C4×ΓL(1,𝔽8)13yes—
323C22ΓL(1,𝔽8)×#12013yes—
324C22ΓL(1,𝔽8)×#12011yescharacteristic
325C4—26no—
326C4—42no—
327C4—83no—
328C3—71noSylow
329C2—23no—
330C2—23no—
331C2—46no—
332C2ΓL(1,𝔽8)×#37911yescharacteristic
333C2ΓL(1,𝔽8)×B(4,𝔽2)11yescharacteristic
334C2ΓL(1,𝔽8)×#45811yescharacteristic
subgroupG/H
centerC22ΓL(1,𝔽8)×#120
commutatorC23×C7C22×C3×C4
FrattiniC24C23×ΓL(1,𝔽8)
FittingC7×#1790C3
socleC22×C7C3×#120
prime radicalC2×ΓL(1,𝔽8)×H(5,𝔽2)C2
cosocleC24×C7C23×C3
hypercenter#1790ΓL(1,𝔽8)
hypocenterC7C3×#1790
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C24 ◁ #1790

Complements

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

NH ≅ G/Ntype
2. C2×ΓL(1,𝔽8)×#99C2semidirect
3. ΓL(1,𝔽8)×#386C2semidirect
4. C7×#1790C3semidirect
6. ΓL(1,𝔽8)×H(5,𝔽2)C4semidirect
10. C23×C4×ΓL(1,𝔽8)C22semidirect
12. C2×C22⋊C4×ΓL(1,𝔽8)C22semidirect
16. C2×C7×#99C2×C3semidirect
17. C7×#386C2×C3semidirect
40. C7×H(5,𝔽2)C3×C4semidirect
44. C23×C4×C7C22×C3semidirect
46. C2×C7×C22⋊C4C22×C3semidirect
61. C23×ΓL(1,𝔽8)C22⋊C4semidirect
78. C2×C4×ΓL(1,𝔽8)C22⋊C4semidirect
79. #1790ΓL(1,𝔽8)direct (N × H)
127. C2×#99C2×ΓL(1,𝔽8)semidirect
128. #386C2×ΓL(1,𝔽8)semidirect
138. C23×C7C3×C22⋊C4semidirect
155. C2×C4×C7C3×C22⋊C4semidirect
183. H(5,𝔽2)C4×ΓL(1,𝔽8)semidirect
187. C23×C4C22×ΓL(1,𝔽8)semidirect
189. C2×C22⋊C4C22×ΓL(1,𝔽8)semidirect
235. ΓL(1,𝔽8)#1790direct (N × H)
287. C23C22⋊C4×ΓL(1,𝔽8)semidirect
304. C2×C4C22⋊C4×ΓL(1,𝔽8)semidirect
305. C7C3×#1790semidirect

Maximal subgroups

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

subgroupindex
C2×ΓL(1,𝔽8)×H(5,𝔽2)2
C2×ΓL(1,𝔽8)×#992
ΓL(1,𝔽8)×#3862
C7×#17903
C3×#17907