Finite Group Databasebeta

← #72269385

Subgroups — #72269385

Order 4704.

434 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 56 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
1C7×#66560C211yesmaximal, characteristic, complemented
2C3×#11716033C211yesmaximal, characteristic, complemented
3#72269288C211yesmaximal, characteristic, complemented
4C7×#66567C211yesmaximal, characteristic, complemented
5#72254257C211yesmaximal, characteristic, complemented
6#72254206C211yesmaximal, characteristic, complemented
7#72245541C211yesmaximal, characteristic, complemented
8#13096375—31nomaximal, Hall
9C2×D3×C7⋊D7—21no—
10C4×C3⋊(C7⋊D7)C2211yescharacteristic
11C7×Dic42C2211yescharacteristic, complemented
12C72×D12C2211yescharacteristic
13C3×C4⋊(C7⋊D7)C2211yescharacteristic
14C3×C7×D28C2211yescharacteristic, complemented
15#11924249—21no—
16D7×Dic21—21no—
17#11924240—21no—
18C7×C3⋊(C7⋊C8)C2211yescharacteristic, complemented
19C3×C7×C7⋊C8C2211yescharacteristic, complemented
20D4×C7⋊D7—31no—
21#11716100—31no—
22C7×#1142—31no—
23C7×#1141—31no—
24#11716036—31no—
25#11716033D311yescharacteristic, complemented
26#11716031—31no—
27#10621044—71nomaximal
28#10620820—71nomaximal
29C2×C3⋊(C7⋊D7)D411yescharacteristic
30C2×D3×C72—21no—
31C2×C3×C7⋊D7—21no—
32C2×C3×C7×D7—21no—
33D3×C7⋊D7—42no—
34C3×C4×C72C2311yescharacteristic
35C3⋊(C7⋊Dic7)D411yescharacteristic
36C7×Dic21—21no—
37C22×C7⋊D7—61no—
38C72×D4—31no—
39C22⋊(C7⋊D7)—61no—
40C4⋊(C7⋊D7)C2×D311yescharacteristic
41C4×C7⋊D7—31no—
42C7×D28C2×D311yescharacteristic, complemented
43C7×Dic14—31no—
44#87641—61no—
45D7×Dic7—61no—
46C7×C7⋊C8—31no—
47C7×C7⋊C8C2×D311yescharacteristic, complemented
48#66730—71no—
49#66681—71no—
50#66681—71no—
51#66681—143no—
52#66670—71no—
53#66622—71no—
54C7×#213—71no—
55C7×#204—71no—
56C3×#1141—71no—
57C3×D56—71no—
58#66567D711yescharacteristic, complemented
59#66565—71no—
60#66563—71no—
61#66560D711yescharacteristic, complemented
62#66559—71no—
63C2×C3×C72C2×D411yescharacteristic
64C3⋊(C7⋊D7)—21no—
65D3×C72—41no—
66C3×C7⋊D7—41no—
67C3×C7×D7—41no—
68#6477—211no—
69#6474—211no—
70C22×C72—61no—
71C2×C7⋊D7—21no—
72C2×C7⋊D7—31no—
73C2×C7⋊D7—122no—
74C2×C7×D7—21no—
75C4×C72C22×D311yescharacteristic
76C7⋊Dic7—31no—
77C7×Dic7—61no—
78C2×D3×D7—141no—
79C2×D3×D7—141no—
80C2×D3×D7—146no—
81C3×C7×D4—71no—
82#4268—141no—
83C2×Dic21—141no—
84C4×D21—71no—
85C4×D21—71no—
86C4×D21—143no—
87Dic42C2×D711yescharacteristic, complemented
88C7×D12—23no—
89C7×D12C2×D711yescharacteristic
90C7×D12C2×D711yescharacteristic
91C7×Dic6—71no—
92C3×D28—71no—
93C3×D28—71no—
94C3×D28—143no—
95C3×D28C2×D711yescharacteristic, complemented
96#4246—141no—
97#4245—141no—
98#4245—141no—
99#4245—146no—
100Dic3×D7—141no—
101C3×C7×C8—71no—
102C3⋊(C7⋊C8)C2×D711yescharacteristic, complemented
103C3×C7⋊C8C2×D711yescharacteristic, complemented
104C7×C3⋊C8—71no—
105C3×C72AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
106#1161—211no—
107#1159—211no—
108D4×D7—211no—
109D4×D7—211no—
110D4×D7—423no—
111C7×SD16—211no—
112C7×D8—211no—
113#1143—211no—
114#1142—31no—
115#1141—31no—
116#1141—71no—
117D56—71no—
118#1132—211no—
119#1131—211no—
120#1131—211no—
121C2×C72—121no—
122C2×C72D3×D411yescharacteristic
123C7⋊D7—41no—
124C7⋊D7—61no—
125C7×D7—41no—
126#911—491noHall
127C22×C3×C7—141no—
128C2×D21—71no—
129C2×D21—71no—
130C2×D21—143no—
131C2×D3×C7—26no—
132C2×D3×C7—21no—
133C2×D3×C7—21no—
134C2×C3×D7—21no—
135C2×C3×D7—146no—
136C2×C3×D7—141no—
137C2×C3×D7—141no—
138D3×D7—2812no—
139D3×D7—282no—
140D3×D7—282no—
141C3×C4×C7—23no—
142C3×C4×C7C22×D711yescharacteristic
143C3×C4×C7C22×D711yescharacteristic
144Dic21—21no—
145Dic21—71no—
146Dic21—71no—
147Dic21—143no—
148C7×Dic3—141no—
149C22×D7—421no—
150C22×D7—421no—
151C22×D7—426no—
152C7×Q8—211no—
153C7×D4—31no—
154C7×D4—31no—
155C7×D4—63no—
156C7×D4—71no—
157C22⋊D7—421no—
158C22⋊D7—421no—
159C22⋊D7—421no—
160C22⋊D7—426no—
161C2×Dic7—421no—
162D28—71no—
163D28—71no—
164D28—143no—
165D28—421no—
166D28D3×D711yescharacteristic, complemented
167C4×D7—211no—
168C4×D7—211no—
169C4×D7—421no—
170C4×D7—423no—
171Dic14—31no—
172C7×C8—71no—
173C7×C8—211no—
174C7⋊C8—31no—
175C7⋊C8D3×D711yescharacteristic, complemented
176C72#91111yescharacteristic, complemented, Sylow
177#237—491no—
178D3×D4—491no—
179C3×D8—491no—
180#214—491no—
181#213—71no—
182#204—71no—
183ΓL(1,𝔽25)—491no—
184C2×C3×C7—23no—
185C2×C3×C7—281no—
186C2×C3×C7D4×D711yescharacteristic
187C2×C3×C7D4×D711yescharacteristic
188D21—146no—
189D21—141no—
190D21—141no—
191C3×D7—41no—
192C3×D7—286no—
193C3×D7—281no—
194C3×D7—281no—
195D3×C7—46no—
196D3×C7—41no—
197D3×C7—41no—
198AGL(1,ℤ/8)—1471noSylow
199C22×C7—66no—
200C22×C7—61no—
201C22×C7—61no—
202C22×C7—141no—
203C2×D7—21no—
204C2×D7—146no—
205C2×D7—141no—
206C2×D7—141no—
207C2×D7—211no—
208C2×D7—211no—
209C2×D7—423no—
210C2×D7—8412no—
211C2×D7—842no—
212C2×D7—842no—
213C4×C7—23no—
214C4×C7—421no—
215C4×C7C2×D3×D711yescharacteristic
216C4×C7C2×D3×D711yescharacteristic
217Dic7—61no—
218Dic7—211no—
219Dic7—211no—
220Dic7—423no—
221C22×D3—981no—
222C3×D4—71no—
223C3×D4—491no—
224#67—981no—
225#67—981no—
226C2×Dic3—981no—
227D12D7211yescharacteristic
228C4×D3—491no—
229Dic6—71no—
230C3×C8—71no—
231C3⋊C8—71no—
232C3×C7—23no—
233C3×C7#647411yescharacteristic, complemented
234C3×C7#647711yescharacteristic, complemented
235#41—1471no—
236C2×D4—1471no—
237SD16—211no—
238SD16—1471no—
239D8—211no—
240D8—491no—
241M16—1471no—
242C2×C7—23no—
243C2×C7—126no—
244C2×C7—121no—
245C2×C7—121no—
246C2×C7—281no—
247C2×C7#6668111yescharacteristic
248C2×C7#6669411yescharacteristic
249D7—41no—
250D7—286no—
251D7—281no—
252D7—281no—
253D7—426no—
254D7—421no—
255D7—421no—
256C22×C3—141no—
257C22×C3—981no—
258C2×D3—21no—
259C2×D3—491no—
260C2×D3—1962no—
261C3×C4C2×D7211yescharacteristic
262Dic3—141no—
263Dic3—491no—
264C23—2941no—
265Q8—211no—
266D4—31no—
267D4—71no—
268D4—2941no—
269D4—2941no—
270D4—491no—
271C2×C4—2941no—
272C2×C4—1471no—
273C8—71no—
274C8—211no—
275C7—23no—
276C7#1062082011yescharacteristic, complemented
277C7#1062104411yescharacteristic, complemented
278C2×C3—281no—
279C2×C3—1961no—
280C2×C3#1171611511yescharacteristic
281D3—41no—
282D3—981no—
283C22—61no—
284C22—141no—
285C22—5882no—
286C22—981no—
287C22—1471no—
288C4—421no—
289C4—1471no—
290C4D3×D7211yescharacteristic
291C3#1309637511yescharacteristic, complemented, Sylow
292C2—121no—
293C2—281no—
294C2—2941no—
295C2—1961no—
296C2#7222210911yescharacteristic
subgroupG/H
centerC2#72222109
commutator, Fitting, cosocleC3×C4×C72C23
Frattini, hypercenterC4D3×D72
socleC2×C3×C72C2×D4
hypocenterC3×C72AGL(1,ℤ/8)
LCSG ▷ C3×C4×C72 ▷ C2×C3×C72 ▷ C3×C72
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C7×#66560C2semidirect
2. C3×#11716033C2semidirect
3. #72269288C2semidirect
4. C7×#66567C2semidirect
5. #72254257C2semidirect
6. #72254206C2semidirect
7. #72245541C2semidirect
11. C7×Dic42C22semidirect
14. C3×C7×D28C22semidirect
18. C7×C3⋊(C7⋊C8)C22semidirect
19. C3×C7×C7⋊C8C22semidirect
25. #11716033D3semidirect
42. C7×D28C2×D3semidirect
47. C7×C7⋊C8C2×D3semidirect
58. #66567D7semidirect
61. #66560D7semidirect
87. Dic42C2×D7semidirect
95. C3×D28C2×D7semidirect
102. C3⋊(C7⋊C8)C2×D7semidirect
103. C3×C7⋊C8C2×D7semidirect
105. C3×C72AGL(1,ℤ/8)semidirect
166. D28D3×D7semidirect
175. C7⋊C8D3×D7semidirect
176. C72#911semidirect
233. C3×C7#6474semidirect
234. C3×C7#6477semidirect
276. C7#10620820semidirect
277. C7#10621044semidirect
291. C3#13096375semidirect

Maximal subgroups

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

subgroupindex
C7×#665602
C3×#117160332
#722692882
C7×#665672
#722542572
#722542062
#722455412
#130963753
#106210447
#106208207