Finite Group Databasebeta

← #89444⋊C3 #13051574

Subgroups — #89444⋊C3 #13051574

Order 1344.

476 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C22×C4⋊AGL(1,𝔽7)C211yesmaximal, characteristic
2C2×(C2×AGL(1,𝔽7))⋊C4C212yesmaximal, complemented
3C2×(C2×AGL(1,𝔽7))⋊C4C212yesmaximal, complemented
4C2×(ΓL(1,𝔽8)⋊C4)⋊C4C211yesmaximal, characteristic, complemented
5ΓL(1,𝔽8)×#95C211yesmaximal, characteristic, complemented
6#89444C311yesmaximal, characteristic, complemented, Hall
7C23×AGL(1,𝔽7)C2212yes—
8C22×C4×ΓL(1,𝔽8)C2212yes—
9C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
10C22×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
11C22×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
12C2×C4⋊AGL(1,𝔽7)—24no—
13C2×C4⋊AGL(1,𝔽7)—22no—
14(C2×AGL(1,𝔽7))⋊C4—24no—
15(C2×AGL(1,𝔽7))⋊C4—24no—
16(ΓL(1,𝔽8)⋊C4)⋊C4—22no—
17C22×D28C2×C311yescharacteristic
18C2×(C2×D7)⋊C4C2×C312yescomplemented
19C2×(C2×D7)⋊C4C2×C312yescomplemented
20C2×Dic7⋊C4C2×C311yescharacteristic, complemented
21C7×#95C2×C311yescharacteristic, complemented
22C3×#394—71nomaximal, Hall
23C23×ΓL(1,𝔽8)C2311yescharacteristic
24C22×AGL(1,𝔽7)—22no—
25C22×AGL(1,𝔽7)—22no—
26C22×AGL(1,𝔽7)—22no—
27C22×AGL(1,𝔽7)—22no—
28C22×AGL(1,𝔽7)—22no—
29C22×AGL(1,𝔽7)—22no—
30C22×AGL(1,𝔽7)D414yes—
31C2×C4×ΓL(1,𝔽8)—22no—
32C2×C4×ΓL(1,𝔽8)—22no—
33C2×C4×ΓL(1,𝔽8)—22no—
34C2×C4×ΓL(1,𝔽8)—22no—
35C2×C4×ΓL(1,𝔽8)D412yes—
36C2×ΓL(1,𝔽8)⋊C4—21no—
37C2×ΓL(1,𝔽8)⋊C4—21no—
38C2×ΓL(1,𝔽8)⋊C4—21no—
39C2×ΓL(1,𝔽8)⋊C4—22no—
40C2×ΓL(1,𝔽8)⋊C4D412yescomplemented
41C4⋊AGL(1,𝔽7)—28no—
42C23×D7C22×C312yes—
43C22×C4×C7C22×C312yes—
44C22×C4×C7C22×C311yescharacteristic
45C22×Dic7C22×C311yescharacteristic
46C22×Dic7C22×C311yescharacteristic
47C2×D28—24no—
48C2×D28—22no—
49(C2×D7)⋊C4—24no—
50(C2×D7)⋊C4—24no—
51Dic7⋊C4—22no—
52C22×C3×D4—71no—
53C2×C3×C4⋊C4—71no—
54C2×C3×C22⋊C4—72no—
55C2×C3×C22⋊C4—72no—
56C3×#95—71no—
57C22×ΓL(1,𝔽8)C2×D412yes—
58C22×ΓL(1,𝔽8)C2×D411yescharacteristic
59C22×ΓL(1,𝔽8)C2×D411yescharacteristic
60C22×ΓL(1,𝔽8)#4111yescharacteristic
61C22×ΓL(1,𝔽8)#4112yes—
62C2×AGL(1,𝔽7)—24no—
63C2×AGL(1,𝔽7)—24no—
64C2×AGL(1,𝔽7)—24no—
65C2×AGL(1,𝔽7)—42no—
66C2×AGL(1,𝔽7)—42no—
67C2×AGL(1,𝔽7)—42no—
68C2×AGL(1,𝔽7)—42no—
69C4×ΓL(1,𝔽8)—22no—
70C4×ΓL(1,𝔽8)—42no—
71ΓL(1,𝔽8)⋊C4—22no—
72ΓL(1,𝔽8)⋊C4—41no—
73#394—71noSylow
74C23×C7C23×C311yescharacteristic
75C22×D7—22no—
76C22×D7—22no—
77C22×D7—22no—
78C22×D7—22no—
79C22×D7—22no—
80C22×D7—22no—
81C22×D7C3×D414yes—
82C2×C4×C7—22no—
83C2×C4×C7—22no—
84C2×C4×C7—22no—
85C2×C4×C7—22no—
86C2×C4×C7C3×D412yes—
87C2×Dic7—21no—
88C2×Dic7—21no—
89C2×Dic7—21no—
90C2×Dic7—22no—
91C2×Dic7C3×D412yescomplemented
92D28—28no—
93C24×C3—72no—
94C2×C3×D4—142no—
95C2×C3×D4—144no—
96C22×C3×C4—71no—
97C22×C3×C4—71no—
98C22×C3×C4—71no—
99C22×C3×C4—72no—
100C3×C4⋊C4—142no—
101C3×C22⋊C4—144no—
102C3×C22⋊C4—144no—
103C2×ΓL(1,𝔽8)#12011yescharacteristic
104C2×ΓL(1,𝔽8)#12111yescharacteristic
105C2×ΓL(1,𝔽8)#12112yes—
106C2×ΓL(1,𝔽8)#12312yes—
107C2×ΓL(1,𝔽8)#12411yescharacteristic
108AGL(1,𝔽7)—44no—
109C22×D4—71no—
110C2×C4⋊C4—71no—
111C2×C22⋊C4—72no—
112C2×C22⋊C4—72no—
113#95AGL(1,𝔽7)11yescharacteristic, complemented
114C22×C7C2×C3×D411yescharacteristic
115C22×C7C2×C3×D412yes—
116C22×C7C2×C3×D411yescharacteristic
117C22×C7C3×#4112yes—
118C22×C7C3×#4111yescharacteristic
119C2×D7—24no—
120C2×D7—24no—
121C2×D7—24no—
122C2×D7—42no—
123C2×D7—42no—
124C2×D7—42no—
125C2×D7—42no—
126C4×C7—22no—
127C4×C7—42no—
128Dic7—22no—
129Dic7—41no—
130C23×C3—74no—
131C23×C3—71no—
132C23×C3—142no—
133C23×C3—142no—
134C23×C3—142no—
135C23×C3—142no—
136C23×C3—142no—
137C23×C3—142no—
138C3×D4—148no—
139C2×C3×C4—72no—
140C2×C3×C4—72no—
141C2×C3×C4—142no—
142C2×C3×C4—142no—
143C2×C3×C4—142no—
144C2×C3×C4—142no—
145C2×C3×C4—141no—
146C2×C3×C4—141no—
147C2×C3×C4—142no—
148C2×C3×C4—141no—
149ΓL(1,𝔽8)#39411yescharacteristic, complemented, Hall
150C24—72no—
151C2×D4—144no—
152C2×D4—142no—
153C22×C4—71no—
154C22×C4—71no—
155C22×C4C2×AGL(1,𝔽7)12yes—
156C22×C4C2×AGL(1,𝔽7)11yescharacteristic
157C4⋊C4—142no—
158C22⋊C4—144no—
159C22⋊C4—144no—
160C2×C7C3×#12011yescharacteristic
161C2×C7C3×#12112yes—
162C2×C7C3×#12111yescharacteristic
163C2×C7C3×#12312yes—
164C2×C7C3×#12411yescharacteristic
165D7—44no—
166C22×C3—71no—
167C22×C3—72no—
168C22×C3—71no—
169C22×C3—72no—
170C22×C3—71no—
171C22×C3—144no—
172C22×C3—144no—
173C22×C3—144no—
174C22×C3—282no—
175C22×C3—282no—
176C22×C3—282no—
177C22×C3—282no—
178C3×C4—142no—
179C3×C4—142no—
180C3×C4—282no—
181C3×C4—281no—
182C23—74no—
183C23—142no—
184C23—142no—
185C23—142no—
186C23—142no—
187C23—142no—
188C23—142no—
189C23C22×AGL(1,𝔽7)11yescharacteristic
190D4—148no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—72no—
196C2×C4—141no—
197C2×C4—141no—
198C2×C4—141no—
199C2×C4—142no—
200C2×C4C4⋊AGL(1,𝔽7)12yes—
201C7C3×#39411yescharacteristic, complemented, Sylow
202C2×C3—71no—
203C2×C3—71no—
204C2×C3—72no—
205C2×C3—72no—
206C2×C3—71no—
207C2×C3—284no—
208C22—144no—
209C22—144no—
210C22—144no—
211C22—282no—
212C22—282no—
213C22—282no—
214C22—282no—
215C22C2×C4⋊AGL(1,𝔽7)11yescharacteristic
216C22#1157⋊C312yes—
217C22D4×AGL(1,𝔽7)12yes—
218C22D4×AGL(1,𝔽7)11yescharacteristic
219C22#6665911yescharacteristic
220C4—22no—
221C4—42no—
222C4—142no—
223C4—281no—
224C3—71noSylow
225C2—284no—
226C2#6438⋊C311yescharacteristic
227C2#6445⋊C311yescharacteristic
228C2#6447⋊C312yes—
229C2#6457⋊C312yes—
230C2#6458⋊C311yescharacteristic
subgroupG/H
center, FrattiniC23C22×AGL(1,𝔽7)
commutator, socle, cosocleC23×C7C23×C3
FittingC7×#95C2×C3
prime radicalC22×C4⋊AGL(1,𝔽7)C2
hypercenter#95AGL(1,𝔽7)
hypocenterC7C3×#394
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
2. C2×(C2×AGL(1,𝔽7))⋊C4C2semidirect
3. C2×(C2×AGL(1,𝔽7))⋊C4C2semidirect
4. C2×(ΓL(1,𝔽8)⋊C4)⋊C4C2semidirect
5. ΓL(1,𝔽8)×#95C2semidirect
6. #89444C3semidirect
18. C2×(C2×D7)⋊C4C2×C3semidirect
19. C2×(C2×D7)⋊C4C2×C3semidirect
20. C2×Dic7⋊C4C2×C3semidirect
21. C7×#95C2×C3semidirect
40. C2×ΓL(1,𝔽8)⋊C4D4semidirect
91. C2×Dic7C3×D4semidirect
113. #95AGL(1,𝔽7)semidirect
149. ΓL(1,𝔽8)#394semidirect
201. C7C3×#394semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊AGL(1,𝔽7)2
C2×(C2×AGL(1,𝔽7))⋊C42
C2×(ΓL(1,𝔽8)⋊C4)⋊C42
ΓL(1,𝔽8)×#952
#894443
C3×#3947