Finite Group Databasebeta

← #89447⋊C3 #13051576

Subgroups — #89447⋊C3 #13051576

Order 1344.

340 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 90 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×(C2×AGL(1,𝔽7))⋊C4C211yesmaximal, characteristic
2C2×(C2×AGL(1,𝔽7))⋊C4C211yesmaximal, characteristic
3C2×(C2×AGL(1,𝔽7))⋊C4C211yesmaximal, characteristic
4C2×(ΓL(1,𝔽8)⋊C4)⋊C4C211yesmaximal, characteristic, complemented
5ΓL(1,𝔽8)×#95C211yesmaximal, characteristic, complemented
6ΓL(1,𝔽8)⋊#95C211yesmaximal, characteristic, complemented
7ΓL(1,𝔽8)⋊#95C211yesmaximal, characteristic, complemented
8#89447C311yesmaximal, characteristic, complemented, Hall
9C23×AGL(1,𝔽7)C2211yescharacteristic
10C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
11C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
12C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
13C22×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
14C22×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
15C22×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
16(C2×AGL(1,𝔽7))⋊C4—22no—
17(C2×AGL(1,𝔽7))⋊C4—22no—
18(C2×AGL(1,𝔽7))⋊C4—22no—
19(ΓL(1,𝔽8)⋊C4)⋊C4—22no—
20C2×(C2×D7)⋊C4C2×C311yescharacteristic
21C2×(C2×D7)⋊C4C2×C311yescharacteristic
22C2×(C2×D7)⋊C4C2×C311yescharacteristic
23C2×Dic7⋊C4C2×C311yescharacteristic, complemented
24C7×#95C2×C311yescharacteristic, complemented
25C7⋊#95C2×C311yescharacteristic, complemented
26C7⋊#95C2×C311yescharacteristic, complemented
27C3×#397—71nomaximal, Hall
28C23×ΓL(1,𝔽8)C2311yescharacteristic
29C22×AGL(1,𝔽7)—21no—
30C22×AGL(1,𝔽7)—21no—
31C22×AGL(1,𝔽7)—21no—
32C22×AGL(1,𝔽7)—21no—
33C22×AGL(1,𝔽7)—21no—
34C22×AGL(1,𝔽7)—21no—
35C22×AGL(1,𝔽7)D412yes—
36C2×C4×ΓL(1,𝔽8)—21no—
37C2×C4×ΓL(1,𝔽8)—21no—
38C2×C4×ΓL(1,𝔽8)—21no—
39C2×C4×ΓL(1,𝔽8)—21no—
40C2×C4×ΓL(1,𝔽8)—21no—
41C2×C4×ΓL(1,𝔽8)—21no—
42C2×C4×ΓL(1,𝔽8)—21no—
43C2×C4×ΓL(1,𝔽8)—21no—
44C2×C4×ΓL(1,𝔽8)—21no—
45C2×ΓL(1,𝔽8)⋊C4—21no—
46C2×ΓL(1,𝔽8)⋊C4—21no—
47C2×ΓL(1,𝔽8)⋊C4—21no—
48C2×ΓL(1,𝔽8)⋊C4—21no—
49C2×ΓL(1,𝔽8)⋊C4—21no—
50C2×ΓL(1,𝔽8)⋊C4—21no—
51C2×ΓL(1,𝔽8)⋊C4—21no—
52C2×ΓL(1,𝔽8)⋊C4—21no—
53C2×ΓL(1,𝔽8)⋊C4D412yes—
54C23×D7C22×C311yescharacteristic
55C22×C4×C7C22×C311yescharacteristic
56C22×C4×C7C22×C311yescharacteristic
57C22×C4×C7C22×C311yescharacteristic
58C22×Dic7C22×C311yescharacteristic
59C22×Dic7C22×C311yescharacteristic
60C22×Dic7C22×C311yescharacteristic
61(C2×D7)⋊C4—22no—
62(C2×D7)⋊C4—22no—
63(C2×D7)⋊C4—22no—
64Dic7⋊C4—22no—
65C2×C3×C4⋊C4—71no—
66C2×C3×C22⋊C4—71no—
67C2×C3×C22⋊C4—71no—
68C2×C3×C22⋊C4—71no—
69C3×#95—71no—
70C3×#95—71no—
71C3×#95—71no—
72C22×ΓL(1,𝔽8)C2×D411yescharacteristic
73C22×ΓL(1,𝔽8)C2×D411yescharacteristic
74C22×ΓL(1,𝔽8)#4111yescharacteristic
75C22×ΓL(1,𝔽8)#4111yescharacteristic
76C22×ΓL(1,𝔽8)#4111yescharacteristic
77C22×ΓL(1,𝔽8)#4111yescharacteristic
78C22×ΓL(1,𝔽8)#4111yescharacteristic
79C2×AGL(1,𝔽7)—22no—
80C2×AGL(1,𝔽7)—22no—
81C2×AGL(1,𝔽7)—22no—
82C2×AGL(1,𝔽7)—41no—
83C2×AGL(1,𝔽7)—41no—
84C2×AGL(1,𝔽7)—41no—
85C2×AGL(1,𝔽7)—41no—
86C4×ΓL(1,𝔽8)—41no—
87C4×ΓL(1,𝔽8)—41no—
88C4×ΓL(1,𝔽8)—41no—
89ΓL(1,𝔽8)⋊C4—22no—
90ΓL(1,𝔽8)⋊C4—41no—
91ΓL(1,𝔽8)⋊C4—41no—
92#397—71noSylow
93C23×C7C23×C311yescharacteristic
94C22×D7—21no—
95C22×D7—21no—
96C22×D7—21no—
97C22×D7—21no—
98C22×D7—21no—
99C22×D7—21no—
100C22×D7C3×D412yes—
101C2×C4×C7—21no—
102C2×C4×C7—21no—
103C2×C4×C7—21no—
104C2×C4×C7—21no—
105C2×C4×C7—21no—
106C2×C4×C7—21no—
107C2×C4×C7—21no—
108C2×C4×C7—21no—
109C2×C4×C7—21no—
110C2×Dic7—21no—
111C2×Dic7—21no—
112C2×Dic7—21no—
113C2×Dic7—21no—
114C2×Dic7—21no—
115C2×Dic7—21no—
116C2×Dic7—21no—
117C2×Dic7—21no—
118C2×Dic7C3×D412yes—
119C24×C3—71no—
120C22×C3×C4—71no—
121C22×C3×C4—71no—
122C22×C3×C4—71no—
123C22×C3×C4—71no—
124C22×C3×C4—71no—
125C22×C3×C4—71no—
126C3×C4⋊C4—142no—
127C3×C22⋊C4—142no—
128C3×C22⋊C4—142no—
129C3×C22⋊C4—142no—
130C2×ΓL(1,𝔽8)#12111yescharacteristic
131C2×ΓL(1,𝔽8)#12311yescharacteristic
132C2×ΓL(1,𝔽8)#12311yescharacteristic
133C2×ΓL(1,𝔽8)#12311yescharacteristic
134C2×ΓL(1,𝔽8)#12411yescharacteristic
135C2×ΓL(1,𝔽8)#12611yescharacteristic
136C2×ΓL(1,𝔽8)#12611yescharacteristic
137AGL(1,𝔽7)—42no—
138C2×C4⋊C4—71no—
139C2×C22⋊C4—71no—
140C2×C22⋊C4—71no—
141C2×C22⋊C4—71no—
142#95—71no—
143#95—71no—
144#95AGL(1,𝔽7)11yescharacteristic, complemented
145C22×C7C2×C3×D411yescharacteristic
146C22×C7C2×C3×D411yescharacteristic
147C22×C7C3×#4111yescharacteristic
148C22×C7C3×#4111yescharacteristic
149C22×C7C3×#4111yescharacteristic
150C22×C7C3×#4111yescharacteristic
151C22×C7C3×#4111yescharacteristic
152C2×D7—22no—
153C2×D7—22no—
154C2×D7—22no—
155C2×D7—41no—
156C2×D7—41no—
157C2×D7—41no—
158C2×D7—41no—
159C4×C7—41no—
160C4×C7—41no—
161C4×C7—41no—
162Dic7—22no—
163Dic7—41no—
164Dic7—41no—
165C23×C3—72no—
166C23×C3—71no—
167C23×C3—141no—
168C23×C3—141no—
169C23×C3—141no—
170C23×C3—141no—
171C23×C3—141no—
172C23×C3—141no—
173C2×C3×C4—72no—
174C2×C3×C4—141no—
175C2×C3×C4—141no—
176C2×C3×C4—141no—
177C2×C3×C4—141no—
178C2×C3×C4—141no—
179C2×C3×C4—141no—
180C2×C3×C4—141no—
181C2×C3×C4—141no—
182C2×C3×C4—141no—
183C2×C3×C4—141no—
184C2×C3×C4—141no—
185C2×C3×C4—141no—
186C2×C3×C4—141no—
187C2×C3×C4—141no—
188C2×C3×C4—141no—
189C2×C3×C4—141no—
190C2×C3×C4—141no—
191ΓL(1,𝔽8)#39711yescharacteristic, complemented, Hall
192C24—71no—
193C22×C4—71no—
194C22×C4—71no—
195C22×C4—71no—
196C22×C4C2×AGL(1,𝔽7)11yescharacteristic
197C22×C4C2×AGL(1,𝔽7)11yescharacteristic
198C22×C4C2×AGL(1,𝔽7)11yescharacteristic
199C4⋊C4—142no—
200C22⋊C4—142no—
201C22⋊C4—142no—
202C22⋊C4—142no—
203C2×C7C3×#12111yescharacteristic
204C2×C7C3×#12311yescharacteristic
205C2×C7C3×#12311yescharacteristic
206C2×C7C3×#12311yescharacteristic
207C2×C7C3×#12411yescharacteristic
208C2×C7C3×#12611yescharacteristic
209C2×C7C3×#12611yescharacteristic
210D7—42no—
211C22×C3—71no—
212C22×C3—71no—
213C22×C3—71no—
214C22×C3—71no—
215C22×C3—71no—
216C22×C3—71no—
217C22×C3—71no—
218C22×C3—142no—
219C22×C3—142no—
220C22×C3—142no—
221C22×C3—281no—
222C22×C3—281no—
223C22×C3—281no—
224C22×C3—281no—
225C3×C4—142no—
226C3×C4—281no—
227C3×C4—281no—
228C3×C4—281no—
229C3×C4—281no—
230C3×C4—281no—
231C23—72no—
232C23—141no—
233C23—141no—
234C23—141no—
235C23—141no—
236C23—141no—
237C23—141no—
238C23C22×AGL(1,𝔽7)11yescharacteristic
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—72no—
249C2×C4—141no—
250C2×C4—141no—
251C2×C4—141no—
252C2×C4—141no—
253C2×C4—141no—
254C2×C4—141no—
255C2×C4—141no—
256C2×C4—141no—
257C7C3×#39711yescharacteristic, complemented, Sylow
258C2×C3—71no—
259C2×C3—71no—
260C2×C3—71no—
261C2×C3—71no—
262C2×C3—71no—
263C2×C3—71no—
264C2×C3—71no—
265C2×C3—282no—
266C22—142no—
267C22—142no—
268C22—142no—
269C22—281no—
270C22—281no—
271C22—281no—
272C22—281no—
273C22#1157⋊C311yescharacteristic
274C22#1157⋊C311yescharacteristic
275C22#1157⋊C311yescharacteristic
276C22D4×AGL(1,𝔽7)11yescharacteristic
277C22D4×AGL(1,𝔽7)11yescharacteristic
278C22#1159⋊C311yescharacteristic
279C22#6665911yescharacteristic
280C4—41no—
281C4—41no—
282C4—41no—
283C4—142no—
284C4—281no—
285C4—281no—
286C3—71noSylow
287C2—282no—
288C2#1062071211yescharacteristic
289C2#6446⋊C311yescharacteristic
290C2#6447⋊C311yescharacteristic
291C2#6448⋊C311yescharacteristic
292C2#6457⋊C311yescharacteristic
293C2#6457⋊C311yescharacteristic
294C2#6461⋊C311yescharacteristic
subgroupG/H
center, FrattiniC23C22×AGL(1,𝔽7)
commutator, socle, cosocleC23×C7C23×C3
FittingC7×#95C2×C3
prime radicalC23×AGL(1,𝔽7)C22
hypercenter#95AGL(1,𝔽7)
hypocenterC7C3×#397
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
4. C2×(ΓL(1,𝔽8)⋊C4)⋊C4C2semidirect
5. ΓL(1,𝔽8)×#95C2semidirect
6. ΓL(1,𝔽8)⋊#95C2semidirect
7. ΓL(1,𝔽8)⋊#95C2semidirect
8. #89447C3semidirect
23. C2×Dic7⋊C4C2×C3semidirect
24. C7×#95C2×C3semidirect
25. C7⋊#95C2×C3semidirect
26. C7⋊#95C2×C3semidirect
144. #95AGL(1,𝔽7)semidirect
191. ΓL(1,𝔽8)#397semidirect
257. C7C3×#397semidirect

Maximal subgroups

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

subgroupindex
C2×(C2×AGL(1,𝔽7))⋊C42
C2×(ΓL(1,𝔽8)⋊C4)⋊C42
ΓL(1,𝔽8)×#952
ΓL(1,𝔽8)⋊#952
#894473
C3×#3977