Finite Group Databasebeta

← #89995⋊C3 #13052123

Subgroups — #89995⋊C3 #13052123

Order 1344.

644 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 112 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×D4×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2C22×(C22⋊D7)⋊C3C212yesmaximal, complemented
3C2×(C22⋊Dic7)⋊C3C212yesmaximal, complemented
4C2×(C2×AGL(1,𝔽7))⋊C4C211yesmaximal, characteristic, complemented
5ΓL(1,𝔽8)⋊#95C211yesmaximal, characteristic, complemented
6#89995C311yesmaximal, characteristic, complemented, Hall
7C24×ΓL(1,𝔽8)C2212yes—
8C23×AGL(1,𝔽7)C2211yescharacteristic
9C2×D4×ΓL(1,𝔽8)—24no—
10C2×D4×ΓL(1,𝔽8)—22no—
11C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
12C2×(C22⋊D7)⋊C3—24no—
13C2×(C22⋊D7)⋊C3—24no—
14C2×(C22⋊D7)⋊C3—24no—
15C22×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
16C22×ΓL(1,𝔽8)⋊C4C2212yes—
17(C22⋊Dic7)⋊C3—24no—
18(C2×AGL(1,𝔽7))⋊C4—22no—
19C22×C7×D4C2×C311yescharacteristic, complemented
20C22×C22⋊D7C2×C312yescomplemented
21C2×C22⋊Dic7C2×C312yescomplemented
22C2×(C2×D7)⋊C4C2×C311yescharacteristic, complemented
23C7⋊#95C2×C311yescharacteristic, complemented
24C3×#392—71nomaximal, Hall
25C23×ΓL(1,𝔽8)—22no—
26C23×ΓL(1,𝔽8)—22no—
27C23×ΓL(1,𝔽8)—22no—
28C23×ΓL(1,𝔽8)—22no—
29C23×ΓL(1,𝔽8)—22no—
30C23×ΓL(1,𝔽8)—22no—
31C23×ΓL(1,𝔽8)D414yescomplemented
32C23×ΓL(1,𝔽8)C2311yescharacteristic
33C22×AGL(1,𝔽7)—22no—
34C22×AGL(1,𝔽7)—22no—
35C22×AGL(1,𝔽7)—21no—
36C22×AGL(1,𝔽7)—21no—
37C22×AGL(1,𝔽7)D412yescomplemented
38D4×ΓL(1,𝔽8)—28no—
39C2×C4×ΓL(1,𝔽8)—22no—
40C2×C4×ΓL(1,𝔽8)D412yescomplemented
41(C22⋊D7)⋊C3—216no—
42C2×ΓL(1,𝔽8)⋊C4—22no—
43C2×ΓL(1,𝔽8)⋊C4—22no—
44C2×ΓL(1,𝔽8)⋊C4—21no—
45C2×ΓL(1,𝔽8)⋊C4—22no—
46C2×ΓL(1,𝔽8)⋊C4D414yescomplemented
47C24×C7C22×C312yes—
48C23×D7C22×C311yescharacteristic
49C2×C7×D4—24no—
50C2×C7×D4—22no—
51C22×C4×C7C22×C311yescharacteristic
52C2×C22⋊D7—24no—
53C2×C22⋊D7—24no—
54C2×C22⋊D7—24no—
55C22×Dic7C22×C311yescharacteristic
56C22×Dic7C22×C312yes—
57C22⋊Dic7—24no—
58(C2×D7)⋊C4—22no—
59C22×C3×D4—72no—
60C22×C3×D4—71no—
61C2×C3×C22⋊C4—71no—
62C2×C3×C22⋊C4—72no—
63C3×#95—71no—
64C22×ΓL(1,𝔽8)—24no—
65C22×ΓL(1,𝔽8)—24no—
66C22×ΓL(1,𝔽8)—24no—
67C22×ΓL(1,𝔽8)—42no—
68C22×ΓL(1,𝔽8)—42no—
69C22×ΓL(1,𝔽8)—42no—
70C22×ΓL(1,𝔽8)—42no—
71C22×ΓL(1,𝔽8)C2×D411yescharacteristic
72C22×ΓL(1,𝔽8)C2×D412yes—
73C22×ΓL(1,𝔽8)C2×D412yes—
74C22×ΓL(1,𝔽8)C2×D411yescharacteristic
75C22×ΓL(1,𝔽8)#4111yescharacteristic
76C2×AGL(1,𝔽7)—24no—
77C2×AGL(1,𝔽7)—22no—
78C2×AGL(1,𝔽7)—41no—
79C2×AGL(1,𝔽7)—42no—
80C2×AGL(1,𝔽7)—41no—
81C4×ΓL(1,𝔽8)—22no—
82ΓL(1,𝔽8)⋊C4—24no—
83ΓL(1,𝔽8)⋊C4—41no—
84#392—71noSylow
85C23×C7—22no—
86C23×C7—22no—
87C23×C7—22no—
88C23×C7—22no—
89C23×C7—22no—
90C23×C7—22no—
91C23×C7C3×D414yescomplemented
92C23×C7C23×C311yescharacteristic
93C22×D7—21no—
94C22×D7—22no—
95C22×D7—22no—
96C22×D7—21no—
97C22×D7C3×D412yescomplemented
98C7×D4—28no—
99C2×C4×C7—22no—
100C2×C4×C7C3×D412yescomplemented
101C22⋊D7—216no—
102C2×Dic7—22no—
103C2×Dic7—22no—
104C2×Dic7—21no—
105C2×Dic7—22no—
106C2×Dic7C3×D414yescomplemented
107C24×C3—71no—
108C24×C3—72no—
109C2×C3×D4—142no—
110C2×C3×D4—144no—
111C2×C3×D4—144no—
112C2×C3×D4—144no—
113C2×C3×D4—144no—
114C22×C3×C4—72no—
115C22×C3×C4—71no—
116C22×C3×C4—71no—
117C3×C22⋊C4—142no—
118C3×C22⋊C4—144no—
119C2×ΓL(1,𝔽8)—44no—
120C2×ΓL(1,𝔽8)#12012yes—
121C2×ΓL(1,𝔽8)#12011yescharacteristic
122C2×ΓL(1,𝔽8)#12111yescharacteristic
123C2×ΓL(1,𝔽8)#12112yes—
124C2×ΓL(1,𝔽8)#12711yescharacteristic
125AGL(1,𝔽7)—42no—
126C22×D4—72no—
127C22×D4AGL(1,𝔽7)11yescharacteristic, complemented
128C2×C22⋊C4—72no—
129C2×C22⋊C4—71no—
130#95—71no—
131C22×C7—24no—
132C22×C7—24no—
133C22×C7—24no—
134C22×C7—42no—
135C22×C7—42no—
136C22×C7—42no—
137C22×C7—42no—
138C22×C7C2×C3×D411yescharacteristic
139C22×C7C2×C3×D412yes—
140C22×C7C2×C3×D412yes—
141C22×C7C2×C3×D411yescharacteristic
142C22×C7C3×#4111yescharacteristic
143C2×D7—24no—
144C2×D7—22no—
145C2×D7—41no—
146C2×D7—42no—
147C2×D7—41no—
148C4×C7—22no—
149Dic7—24no—
150Dic7—41no—
151C23×C3—72no—
152C23×C3—74no—
153C23×C3—71no—
154C23×C3—142no—
155C23×C3—142no—
156C23×C3—142no—
157C23×C3—142no—
158C23×C3—142no—
159C23×C3—142no—
160C23×C3—141no—
161C23×C3—142no—
162C23×C3—142no—
163C23×C3—141no—
164C3×D4—148no—
165C3×D4—1416no—
166C2×C3×C4—74no—
167C2×C3×C4—72no—
168C2×C3×C4—142no—
169C2×C3×C4—142no—
170C2×C3×C4—142no—
171C2×C3×C4—142no—
172C2×C3×C4—141no—
173ΓL(1,𝔽8)#39211yescharacteristic, complemented, Hall
174C24—71no—
175C24C2×AGL(1,𝔽7)12yes—
176C2×D4—24no—
177C2×D4—22no—
178C2×D4—144no—
179C2×D4—144no—
180C2×D4—144no—
181C22×C4—72no—
182C22×C4—71no—
183C22×C4C2×AGL(1,𝔽7)11yescharacteristic
184C22⋊C4—144no—
185C22⋊C4—142no—
186C2×C7—44no—
187C2×C7C3×#12012yes—
188C2×C7C3×#12011yescharacteristic
189C2×C7C3×#12111yescharacteristic
190C2×C7C3×#12112yes—
191C2×C7C3×#12711yescharacteristic
192D7—42no—
193C22×C3—72no—
194C22×C3—71no—
195C22×C3—71no—
196C22×C3—71no—
197C22×C3—72no—
198C22×C3—144no—
199C22×C3—144no—
200C22×C3—144no—
201C22×C3—144no—
202C22×C3—142no—
203C22×C3—282no—
204C22×C3—282no—
205C22×C3—282no—
206C22×C3—282no—
207C22×C3—282no—
208C22×C3—281no—
209C22×C3—281no—
210C3×C4—142no—
211C3×C4—144no—
212C3×C4—281no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—72no—
220C23—142no—
221C23—141no—
222C23—141no—
223C23—142no—
224C23(C22⋊D7)⋊C314yescomplemented
225C23C22×AGL(1,𝔽7)11yescharacteristic
226D4—28no—
227D4—1416no—
228C2×C4—22no—
229C2×C4—74no—
230C2×C4—142no—
231C2×C4—142no—
232C2×C4—142no—
233C2×C4—141no—
234C2×C4(C22⋊D7)⋊C312yescomplemented
235C7C3×#39211yescharacteristic, complemented, Sylow
236C2×C3—72no—
237C2×C3—71no—
238C2×C3—72no—
239C2×C3—71no—
240C2×C3—71no—
241C2×C3—284no—
242C2×C3—282no—
243C22—24no—
244C22—24no—
245C22—24no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—144no—
251C22—142no—
252C22—282no—
253C22—281no—
254C22—281no—
255C22D4×AGL(1,𝔽7)11yescharacteristic
256C22D4×AGL(1,𝔽7)12yes—
257C22#1159⋊C311yescharacteristic
258C22C2×(C22⋊D7)⋊C312yes—
259C22C2×(C22⋊D7)⋊C311yescharacteristic
260C4—22no—
261C4—144no—
262C4—281no—
263C3—71noSylow
264C2—44no—
265C2—282no—
266C2#6500⋊C312yes—
267C2#6501⋊C311yescharacteristic
268C2#6502⋊C312yes—
269C2#6503⋊C311yescharacteristic
270C2#1062078911yescharacteristic
subgroupG/H
center, FrattiniC23C22×AGL(1,𝔽7)
commutator, socle, cosocleC23×C7C23×C3
FittingC22×C7×D4C2×C3
hypercenterC22×D4AGL(1,𝔽7)
hypocenterC7C3×#392
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×D4×ΓL(1,𝔽8)C2semidirect
2. C22×(C22⋊D7)⋊C3C2semidirect
3. C2×(C22⋊Dic7)⋊C3C2semidirect
4. C2×(C2×AGL(1,𝔽7))⋊C4C2semidirect
5. ΓL(1,𝔽8)⋊#95C2semidirect
6. #89995C3semidirect
19. C22×C7×D4C2×C3semidirect
20. C22×C22⋊D7C2×C3semidirect
21. C2×C22⋊Dic7C2×C3semidirect
22. C2×(C2×D7)⋊C4C2×C3semidirect
23. C7⋊#95C2×C3semidirect
31. C23×ΓL(1,𝔽8)D4semidirect
37. C22×AGL(1,𝔽7)D4semidirect
40. C2×C4×ΓL(1,𝔽8)D4semidirect
46. C2×ΓL(1,𝔽8)⋊C4D4semidirect
91. C23×C7C3×D4semidirect
97. C22×D7C3×D4semidirect
100. C2×C4×C7C3×D4semidirect
106. C2×Dic7C3×D4semidirect
127. C22×D4AGL(1,𝔽7)semidirect
173. ΓL(1,𝔽8)#392semidirect
224. C23(C22⋊D7)⋊C3semidirect
234. C2×C4(C22⋊D7)⋊C3semidirect
235. C7C3×#392semidirect

Maximal subgroups

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

subgroupindex
C22×D4×ΓL(1,𝔽8)2
C22×(C22⋊D7)⋊C32
C2×(C22⋊Dic7)⋊C32
C2×(C2×AGL(1,𝔽7))⋊C42
ΓL(1,𝔽8)⋊#952
#899953
C3×#3927