Finite Group Databasebeta

← ΓL(1,𝔽8)×#15947 #72277227

Subgroups — ΓL(1,𝔽8)×#15947 #72277227

Order 5376.

12292 conjugacy classes of subgroups; only the 741 normal subgroups are shown.

normal subgroups only

This group has 12292 conjugacy classes of subgroups — too many to draw. Only its 741 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
1ΓL(1,𝔽8)×#1438C219yesmaximal, complemented
2C22×ΓL(1,𝔽8)×#120C216yesmaximal, complemented
3C7×#15947C311yesmaximal, characteristic, complemented, Hall
4C26×ΓL(1,𝔽8)C2212yescomplemented
5C23×D4×ΓL(1,𝔽8)C2219yes—
6C2×ΓL(1,𝔽8)×#120—272no—
7C22×C22⋊C4×ΓL(1,𝔽8)C22118yescomplemented
8C2×ΓL(1,𝔽8)×#95C2216yescomplemented
9C22×C7×#120C2×C316yescomplemented
10C7×#1438C2×C319yescomplemented
11C3×#15947—71nomaximal, Hall
12C25×ΓL(1,𝔽8)—236no—
13C25×ΓL(1,𝔽8)—412no—
14C25×ΓL(1,𝔽8)C2316yes—
15C22×D4×ΓL(1,𝔽8)—272no—
16C22×D4×ΓL(1,𝔽8)—418no—
17C22×D4×ΓL(1,𝔽8)D4136yescomplemented
18C23×C4×ΓL(1,𝔽8)C2319yes—
19ΓL(1,𝔽8)×#120—2192no—
20C2×C22⋊C4×ΓL(1,𝔽8)—236no—
21C2×C22⋊C4×ΓL(1,𝔽8)—272no—
22ΓL(1,𝔽8)×#95—46no—
23C26×C7C22×C312yescomplemented
24C23×C7×D4C22×C319yes—
25C2×C7×#120—272no—
26C22×C7×C22⋊C4C22×C3118yescomplemented
27C2×C7×#95C22×C316yescomplemented
28C22×C3×#120—76no—
29C3×#1438—79no—
30C24×ΓL(1,𝔽8)—236no—
31C24×ΓL(1,𝔽8)—236no—
32C24×ΓL(1,𝔽8)—248no—
33C24×ΓL(1,𝔽8)—4144no—
34C24×ΓL(1,𝔽8)—472no—
35C24×ΓL(1,𝔽8)—416no—
36C24×ΓL(1,𝔽8)—424no—
37C24×ΓL(1,𝔽8)C2×D4136yes—
38C24×ΓL(1,𝔽8)C2411yescharacteristic
39C2×D4×ΓL(1,𝔽8)—2144no—
40C2×D4×ΓL(1,𝔽8)—436no—
41C2×D4×ΓL(1,𝔽8)—4144no—
42C22×C4×ΓL(1,𝔽8)—236no—
43C22×C4×ΓL(1,𝔽8)—218no—
44C22×C4×ΓL(1,𝔽8)C2×D4118yes—
45C22⋊C4×ΓL(1,𝔽8)—2144no—
46#15947ΓL(1,𝔽8)11yescharacteristic, direct product factor, Sylow
47C25×C7—236no—
48C25×C7—412no—
49C25×C7C23×C316yes—
50C22×C7×D4—272no—
51C22×C7×D4—418no—
52C22×C7×D4C3×D4136yescomplemented
53C23×C4×C7C23×C319yes—
54C7×#120—2192no—
55C2×C7×C22⋊C4—236no—
56C2×C7×C22⋊C4—272no—
57C7×#95—46no—
58C26×C3—72no—
59C23×C3×D4—79no—
60C2×C3×#120—1472no—
61C22×C3×C22⋊C4—718no—
62C2×C3×#95—76no—
63C23×ΓL(1,𝔽8)—236no—
64C23×ΓL(1,𝔽8)—236no—
65C23×ΓL(1,𝔽8)—2144no—
66C23×ΓL(1,𝔽8)—4288no—
67C23×ΓL(1,𝔽8)—4192no—
68C23×ΓL(1,𝔽8)—412no—
69C23×ΓL(1,𝔽8)—436no—
70C23×ΓL(1,𝔽8)—436no—
71C23×ΓL(1,𝔽8)—412no—
72C23×ΓL(1,𝔽8)#120124yescomplemented
73C23×ΓL(1,𝔽8)C22×D419yes—
74C23×ΓL(1,𝔽8)H(5,𝔽2)16yes—
75D4×ΓL(1,𝔽8)—4144no—
76C2×C4×ΓL(1,𝔽8)—236no—
77C2×C4×ΓL(1,𝔽8)—218no—
78C2×C4×ΓL(1,𝔽8)—436no—
79C22×#120C2×ΓL(1,𝔽8)16yescomplemented
80#1438C2×ΓL(1,𝔽8)19yescomplemented
81C24×C7—236no—
82C24×C7—236no—
83C24×C7—248no—
84C24×C7—4144no—
85C24×C7—472no—
86C24×C7—416no—
87C24×C7—424no—
88C24×C7C2×C3×D4136yes—
89C24×C7C24×C311yescharacteristic
90C2×C7×D4—2144no—
91C2×C7×D4—436no—
92C2×C7×D4—4144no—
93C22×C4×C7—236no—
94C22×C4×C7—218no—
95C22×C4×C7C2×C3×D4118yes—
96C7×C22⋊C4—2144no—
97C25×C3—76no—
98C25×C3—1436no—
99C25×C3—2812no—
100C22×C3×D4—736no—
101C22×C3×D4—1472no—
102C22×C3×D4—2818no—
103C23×C3×C4—79no—
104C3×#120—14192no—
105C2×C3×C22⋊C4—1472no—
106C2×C3×C22⋊C4—1436no—
107C3×#95—286no—
108C22×ΓL(1,𝔽8)—272no—
109C22×ΓL(1,𝔽8)—4128no—
110C22×ΓL(1,𝔽8)—472no—
111C22×ΓL(1,𝔽8)—472no—
112C22×ΓL(1,𝔽8)C2×#12016yes—
113C22×ΓL(1,𝔽8)#53419yes—
114C22×ΓL(1,𝔽8)D42118yes—
115C22×ΓL(1,𝔽8)H(3,𝔽4)12yes—
116C4×ΓL(1,𝔽8)—418no—
117C26C22×ΓL(1,𝔽8)12yescomplemented
118C23×D4C22×ΓL(1,𝔽8)19yes—
119C2×#120—272no—
120C22×C22⋊C4C22×ΓL(1,𝔽8)118yescomplemented
121C2×#95C22×ΓL(1,𝔽8)16yescomplemented
122C23×C7—236no—
123C23×C7—236no—
124C23×C7—2144no—
125C23×C7—4288no—
126C23×C7—4192no—
127C23×C7—412no—
128C23×C7—412no—
129C23×C7—436no—
130C23×C7—436no—
131C23×C7C3×#120124yescomplemented
132C23×C7C22×C3×D419yes—
133C23×C7C3×H(5,𝔽2)16yes—
134C7×D4—4144no—
135C2×C4×C7—236no—
136C2×C4×C7—218no—
137C2×C4×C7—436no—
138C24×C3—736no—
139C24×C3—71no—
140C24×C3—1436no—
141C24×C3—1436no—
142C24×C3—1448no—
143C24×C3—2824no—
144C24×C3—28144no—
145C24×C3—2872no—
146C24×C3—2816no—
147C2×C3×D4—14144no—
148C2×C3×D4—2836no—
149C2×C3×D4—28144no—
150C22×C3×C4—718no—
151C22×C3×C4—1418no—
152C22×C3×C4—1436no—
153C3×C22⋊C4—14144no—
154C2×ΓL(1,𝔽8)—424no—
155C2×ΓL(1,𝔽8)#240319yes—
156C2×ΓL(1,𝔽8)#267916yes—
157C25—236no—
158C25—412no—
159C25C23×ΓL(1,𝔽8)16yes—
160C22×D4—272no—
161C22×D4—418no—
162C22×D4D4×ΓL(1,𝔽8)136yescomplemented
163C23×C4C23×ΓL(1,𝔽8)19yes—
164#120—2192no—
165C2×C22⋊C4—236no—
166C2×C22⋊C4—272no—
167#95—46no—
168C22×C7—272no—
169C22×C7—4128no—
170C22×C7—472no—
171C22×C7—472no—
172C22×C7C2×C3×#12016yes—
173C22×C7C3×#53419yes—
174C22×C7C3×D42118yes—
175C22×C7C3×H(3,𝔽4)12yes—
176C4×C7—418no—
177C23×C3—79no—
178C23×C3—76no—
179C23×C3—724no—
180C23×C3—1436no—
181C23×C3—1436no—
182C23×C3—14144no—
183C23×C3—2812no—
184C23×C3—2836no—
185C23×C3—2836no—
186C23×C3—2812no—
187C23×C3—28288no—
188C23×C3—28192no—
189C3×D4—28144no—
190C2×C3×C4—1436no—
191C2×C3×C4—1418no—
192C2×C3×C4—2836no—
193ΓL(1,𝔽8)#1594711yescharacteristic, direct product factor, Hall
194C24—236no—
195C24—236no—
196C24—248no—
197C24—4144no—
198C24—472no—
199C24—416no—
200C24—424no—
201C24C2×D4×ΓL(1,𝔽8)136yes—
202C24C24×ΓL(1,𝔽8)11yescharacteristic
203C2×D4—2144no—
204C2×D4—436no—
205C2×D4—4144no—
206C22×C4—236no—
207C22×C4—218no—
208C22×C4C2×D4×ΓL(1,𝔽8)118yes—
209C22⋊C4—2144no—
210C2×C7—424no—
211C2×C7C3×#240319yes—
212C2×C7C3×#267916yes—
213C22×C3—76no—
214C22×C3—79no—
215C22×C3—718no—
216C22×C3—72no—
217C22×C3—1472no—
218C22×C3—2872no—
219C22×C3—2872no—
220C22×C3—28128no—
221C3×C4—2818no—
222C23—236no—
223C23—236no—
224C23—2144no—
225C23—4288no—
226C23—4192no—
227C23—412no—
228C23—436no—
229C23—436no—
230C23—412no—
231C23ΓL(1,𝔽8)×#120124yescomplemented
232C23C22×D4×ΓL(1,𝔽8)19yes—
233C23ΓL(1,𝔽8)×H(5,𝔽2)16yes—
234D4—4144no—
235C2×C4—236no—
236C2×C4—218no—
237C2×C4—436no—
238C7C3×#1594711yescharacteristic, complemented, Sylow
239C2×C3—76no—
240C2×C3—79no—
241C2×C3—2824no—
242C22—272no—
243C22—4128no—
244C22—472no—
245C22—472no—
246C22C2×ΓL(1,𝔽8)×#12016yes—
247C22ΓL(1,𝔽8)×#53419yes—
248C22D42×ΓL(1,𝔽8)118yes—
249C22ΓL(1,𝔽8)×H(3,𝔽4)12yes—
250C4—418no—
251C3—71noSylow
252C2—424no—
253C2ΓL(1,𝔽8)×#240319yes—
254C2ΓL(1,𝔽8)×#267916yes—
subgroupG/H
center, FrattiniC24C24×ΓL(1,𝔽8)
commutator, socle, cosocleC24×C7C24×C3
FittingC7×#15947C3
hypercenter#15947ΓL(1,𝔽8)
hypocenterC7C3×#15947
LCSG ▷ C24×C7 ▷ C7
UCSC1 ◁ C24 ◁ #15947

Complements

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

NH ≅ G/Ntype
1. ΓL(1,𝔽8)×#1438C2semidirect
2. C22×ΓL(1,𝔽8)×#120C2semidirect
3. C7×#15947C3semidirect
4. C26×ΓL(1,𝔽8)C22semidirect
7. C22×C22⋊C4×ΓL(1,𝔽8)C22semidirect
8. C2×ΓL(1,𝔽8)×#95C22semidirect
9. C22×C7×#120C2×C3semidirect
10. C7×#1438C2×C3semidirect
17. C22×D4×ΓL(1,𝔽8)D4semidirect
23. C26×C7C22×C3semidirect
26. C22×C7×C22⋊C4C22×C3semidirect
27. C2×C7×#95C22×C3semidirect
46. #15947ΓL(1,𝔽8)direct (N × H)
52. C22×C7×D4C3×D4semidirect
72. C23×ΓL(1,𝔽8)#120semidirect
79. C22×#120C2×ΓL(1,𝔽8)semidirect
80. #1438C2×ΓL(1,𝔽8)semidirect
117. C26C22×ΓL(1,𝔽8)semidirect
120. C22×C22⋊C4C22×ΓL(1,𝔽8)semidirect
121. C2×#95C22×ΓL(1,𝔽8)semidirect
131. C23×C7C3×#120semidirect
162. C22×D4D4×ΓL(1,𝔽8)semidirect
193. ΓL(1,𝔽8)#15947direct (N × H)
231. C23ΓL(1,𝔽8)×#120semidirect
238. C7C3×#15947semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×#14382
C22×ΓL(1,𝔽8)×#1202
C7×#159473
C3×#159477