Finite Group Databasebeta

← #72269387

Subgroups — #72269387

Order 4704.

434 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 62 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
1#72269322C211yesmaximal, characteristic, complemented
2C3×#11716033C211yesmaximal, characteristic, complemented
3#72269288C211yesmaximal, characteristic, complemented
4#72269280C211yesmaximal, characteristic, complemented
5C7×#66669C211yesmaximal, characteristic, complemented
6C7×#66558C211yesmaximal, characteristic, complemented
7D3×C4⋊(C7⋊D7)C211yesmaximal, characteristic, complemented
8#13096291—31nomaximal, Hall
9C2×D3×C7⋊D7—21no—
10C4⋊(C3⋊(C7⋊D7))C2211yescharacteristic
11C7×Dic42C2211yescharacteristic, complemented
12C4×D3×C72C2211yescharacteristic
13C3×C4⋊(C7⋊D7)C2211yescharacteristic
14C3×C7×D28C2211yescharacteristic, complemented
15#11924241—21no—
16C7×#4245—21no—
17C7×Dic3×D7—21no—
18C7×C3⋊(C7⋊C8)C2211yescharacteristic, complemented
19C3×C7×C7⋊C8C2211yescharacteristic, complemented
20C2×C4⋊(C7⋊D7)—31no—
21C7×#1157—31no—
22C7×#1136—31no—
23#11716036—31no—
24#11716036—31no—
25#11716033—31no—
26#11716033D311yescharacteristic, complemented
27#10621042—71nomaximal
28#10620816—71nomaximal
29C2×C3⋊(C7⋊D7)—21no—
30C2×D3×C72D411yescharacteristic
31C2×C3×C7⋊D7—21no—
32C2×C3×C7×D7—21no—
33D3×C7⋊D7—42no—
34C3×C4×C72C2311yescharacteristic
35C7×Dic21—21no—
36C72×Dic3D411yescharacteristic
37C22×C7⋊D7—61no—
38C2×C4×C72—31no—
39C4⋊(C7⋊D7)—31no—
40C4⋊(C7⋊D7)—61no—
41C4⋊(C7⋊D7)C2×D311yescharacteristic
42C7×C22⋊D7—61no—
43C7×D28C2×D311yescharacteristic, complemented
44C4×C7×D7—61no—
45C7×Dic14—31no—
46C7×C7⋊C8—31no—
47C7×C7⋊C8C2×D311yescharacteristic, complemented
48C7×#237—71no—
49D3×D28—71no—
50D3×D28—71no—
51D3×D28—143no—
52#66669D711yescharacteristic, complemented
53#66632—71no—
54#66623—71no—
55C7×ΓL(1,𝔽25)—71no—
56C3×#1141—71no—
57C3×D56—71no—
58#66568—71no—
59#66565—71no—
60#66563—71no—
61#66561—71no—
62#66558D711yescharacteristic, complemented
63C2×C3×C72C2×D411yescharacteristic
64C3⋊(C7⋊D7)—41no—
65D3×C72—21no—
66C3×C7⋊D7—41no—
67C3×C7×D7—41no—
68#6512—211no—
69#6471—211no—
70C22×C72—31no—
71C2×C7⋊D7—21no—
72C2×C7⋊D7—61no—
73C2×C7⋊D7—122no—
74C2×C7×D7—21no—
75C4×C72—31no—
76C4×C72C22×D311yescharacteristic
77C7×Dic7—61no—
78C2×D3×D7—146no—
79C2×D3×D7—141no—
80C2×D3×D7—141no—
81C3×C7×D4—71no—
82D84—71no—
83D84—71no—
84D84—143no—
85Dic42C2×D711yescharacteristic, complemented
86C7×#67—141no—
87C2×C7×Dic3—141no—
88C4×D3×C7—23no—
89C4×D3×C7C2×D711yescharacteristic
90C4×D3×C7C2×D711yescharacteristic
91C7×Dic6—71no—
92C3×D28—71no—
93C3×D28—71no—
94C3×D28—143no—
95C3×D28C2×D711yescharacteristic, complemented
96#4246—146no—
97#4246—141no—
98#4246—141no—
99#4245—21no—
100Dic3×D7—21no—
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
106C7×#41—211no—
107#1157—31no—
108C2×D28—211no—
109C2×D28—211no—
110C2×D28—423no—
111C7×M16—211no—
112#1143—211no—
113#1143—211no—
114#1141—71no—
115#1141—211no—
116#1136—31no—
117D56—71no—
118D56—211no—
119#1132—211no—
120#1132—211no—
121C2×C72—61no—
122C2×C72D3×D411yescharacteristic
123C7⋊D7—41no—
124C7⋊D7—121no—
125C7×D7—41no—
126#911—491noHall
127C22×C3×C7—141no—
128C2×D21—146no—
129C2×D21—141no—
130C2×D21—141no—
131C2×D3×C7—23no—
132C2×D3×C7D2811yescharacteristic
133C2×D3×C7C22⋊D711yescharacteristic
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—
145C7×Dic3—23no—
146C7×Dic3—141no—
147C7×Dic3D2811yescharacteristic
148C7×Dic3C22⋊D711yescharacteristic
149C22×D7—421no—
150C22×D7—421no—
151C22×D7—426no—
152C7×Q8—211no—
153C7×D4—71no—
154C7×D4—421no—
155C2×C4×C7—31no—
156C2×C4×C7—31no—
157C2×C4×C7—63no—
158C2×C4×C7—421no—
159C22⋊D7—61no—
160D28—71no—
161D28—71no—
162D28—143no—
163D28—211no—
164D28—211no—
165D28—421no—
166D28—421no—
167D28—426no—
168D28—423no—
169D28D3×D711yescharacteristic, complemented
170C4×D7—61no—
171Dic14—31no—
172C7×C8—71no—
173C7×C8—211no—
174C7⋊C8—31no—
175C7⋊C8D3×D711yescharacteristic, complemented
176C72#91111yescharacteristic, complemented, Sylow
177#237—71no—
178D3×D4—491no—
179C3×D8—491no—
180#214—491no—
181#213—491no—
182#204—491no—
183ΓL(1,𝔽25)—71no—
184C2×C3×C7—23no—
185C2×C3×C7—281no—
186C2×C3×C7C2×D2811yescharacteristic
187C2×C3×C7C2×C22⋊D711yescharacteristic
188D21—286no—
189D21—281no—
190D21—281no—
191C3×D7—41no—
192C3×D7—286no—
193C3×D7—281no—
194C3×D7—281no—
195D3×C7—26no—
196D3×C7—21no—
197D3×C7—21no—
198AGL(1,ℤ/8)—1471noSylow
199C22×C7—31no—
200C22×C7—31no—
201C22×C7—63no—
202C22×C7—141no—
203C2×D7—21no—
204C2×D7—146no—
205C2×D7—141no—
206C2×D7—141no—
207C2×D7—426no—
208C2×D7—421no—
209C2×D7—421no—
210C2×D7—8412no—
211C2×D7—842no—
212C2×D7—842no—
213C4×C7—23no—
214C4×C7—31no—
215C4×C7—31no—
216C4×C7—63no—
217C4×C7—421no—
218C4×C7C2×D3×D711yescharacteristic
219C4×C7C2×D3×D711yescharacteristic
220Dic7—61no—
221C22×D3—981no—
222C3×D4—71no—
223C3×D4—491no—
224#67—141no—
225#67—981no—
226C2×Dic3—141no—
227D12—491no—
228C4×D3D7211yescharacteristic
229Dic6—71no—
230C3×C8—71no—
231C3⋊C8—71no—
232C3×C7—23no—
233C3×C7#647111yescharacteristic, complemented
234C3×C7#651211yescharacteristic, complemented
235#41—211no—
236C2×D4—1471no—
237SD16—1471no—
238SD16—1471no—
239D8—491no—
240D8—1471no—
241M16—211no—
242C2×C7—23no—
243C2×C7—66no—
244C2×C7—61no—
245C2×C7—61no—
246C2×C7—281no—
247C2×C7D3×D2811yescharacteristic
248C2×C7D3×C22⋊D711yescharacteristic
249D7—41no—
250D7—286no—
251D7—281no—
252D7—281no—
253D7—846no—
254D7—841no—
255D7—841no—
256C22×C3—141no—
257C22×C3—981no—
258C2×D3—981no—
259C2×D3—1962no—
260C2×D3#8764111yescharacteristic
261C3×C4C2×D7211yescharacteristic
262Dic3—141no—
263Dic3#8764111yescharacteristic
264C23—2941no—
265Q8—211no—
266D4—71no—
267D4—2941no—
268D4—421no—
269D4—491no—
270D4—1471no—
271C2×C4—31no—
272C2×C4—421no—
273C8—71no—
274C8—211no—
275C7—23no—
276C7#1062081611yescharacteristic, complemented
277C7#1062104211yescharacteristic, complemented
278C2×C3—281no—
279C2×C3—1961no—
280C2×C3C2×#8764111yescharacteristic
281D3—21no—
282D3—1961no—
283C22—31no—
284C22—141no—
285C22—2941no—
286C22—5882no—
287C22—981no—
288C4—31no—
289C4—421no—
290C4D3×D7211yescharacteristic
291C3#1309629111yescharacteristic, complemented, Sylow
292C2—61no—
293C2—281no—
294C2—5881no—
295C2—1961no—
296C2D3×#8764111yescharacteristic
subgroupG/H
centerC2D3×#87641
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. #72269322C2semidirect
2. C3×#11716033C2semidirect
3. #72269288C2semidirect
4. #72269280C2semidirect
5. C7×#66669C2semidirect
6. C7×#66558C2semidirect
7. D3×C4⋊(C7⋊D7)C2semidirect
11. C7×Dic42C22semidirect
14. C3×C7×D28C22semidirect
18. C7×C3⋊(C7⋊C8)C22semidirect
19. C3×C7×C7⋊C8C22semidirect
26. #11716033D3semidirect
43. C7×D28C2×D3semidirect
47. C7×C7⋊C8C2×D3semidirect
52. #66669D7semidirect
62. #66558D7semidirect
85. Dic42C2×D7semidirect
95. C3×D28C2×D7semidirect
102. C3⋊(C7⋊C8)C2×D7semidirect
103. C3×C7⋊C8C2×D7semidirect
105. C3×C72AGL(1,ℤ/8)semidirect
169. D28D3×D7semidirect
175. C7⋊C8D3×D7semidirect
176. C72#911semidirect
233. C3×C7#6471semidirect
234. C3×C7#6512semidirect
276. C7#10620816semidirect
277. C7#10621042semidirect
291. C3#13096291semidirect

Maximal subgroups

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

subgroupindex
#722693222
C3×#117160332
#722692882
#722692802
C7×#666692
C7×#665582
D3×C4⋊(C7⋊D7)2
#130962913
#106210427
#106208167