Finite Group Databasebeta

← #32003469

Subgroups — #32003469

Order 18176.

1306 conjugacy classes of subgroups; only the 279 normal subgroups are shown.

normal subgroups only

This group has 1306 conjugacy classes of subgroups — too many to draw. Only its 279 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#22597732C211yesmaximal, characteristic
2#22597041C213yesmaximal, complemented
3#22588758C213yesmaximal, complemented
4#22588626C213yesmaximal, complemented
5#22588291C211yesmaximal, characteristic, complemented
6#22588140C213yesmaximal, complemented
7C71×#3573C211yesmaximal, characteristic, complemented
8C22×#11765848C2211yescharacteristic, complemented
9#17986506C2213yes—
10#17986189C414yes—
11#17986186C414yes—
12#17986075—26no—
13#17986070—26no—
14#17986027C2213yescomplemented
15#17985983C2213yes—
16#17985973C2213yes—
17#17985966C2213yes—
18#17985962C2213yescomplemented
19#17985954C2213yescomplemented
20#17985770C2216yescomplemented
21#585×C71C2211yescharacteristic
22#568×C71—26no—
23#568×C71C2213yes—
24C2×#131×C71—26no—
25C2×#131×C71C2213yes—
26C22×C4×D71C2311yescharacteristic
27C2×#11765849C2313yes—
28C2×#11765848—23no—
29C2×#11765848—23no—
30C22×C71⋊C8C2311yescharacteristic
31#14556109C2×C416yes—
32#14556104—23no—
33#14556099—23no—
34#14556096C2×C416yes—
35C4×C22⋊D71—26no—
36#14556087C2313yes—
37Dic71⋊C8—26no—
38#14556047—23no—
39#14556045—23no—
40#14556043C2×C4112yes—
41C22⋊(C71⋊C8)—26no—
42(C2×D71)⋊C8—26no—
43#14556038—23no—
44C8×Dic71—23no—
452−1+4×C71—23no—
46H(5,𝔽2)×C71—23no—
47H(5,𝔽2)×C71C2×C414yes—
48C2×#41×C71—26no—
49C2×#41×C71C2313yes—
50#131×C71—24no—
51#131×C71—212no—
52#131×C71—26no—
53#131×C71—212no—
54#131×C71—26no—
55C2×M16×C71—212no—
56C2×M16×C71—26no—
57C2×M16×C71D416yescomplemented
58C2×M16×C71C2313yes—
59C22×C8×C71—24no—
60C22×C8×C71D416yescomplemented
61C22×C8×C71C2311yescharacteristic
62C23×D71C22×C411yescharacteristic
63C2×C22⋊D71—26no—
64C22×Dic71C22×C411yescharacteristic
65C2×C4×D71—23no—
66C2×C4×D71—23no—
67#11765849—26no—
68#11765848—41no—
69#11765848—43no—
70C2×C71⋊C8—23no—
71C2×C71⋊C8—23no—
72C4⋊Dic71—23no—
73Dic71⋊C4—26no—
74C22⋊Dic71—26no—
75C22⋊Dic71C22×C416yes—
76(C2×D71)⋊C4—26no—
77C4×Dic71—23no—
78#41×C71—24no—
79#41×C71—212no—
80#41×C71—212no—
81#41×C71—26no—
82#41×C71—26no—
83C2×Q8×C71—26no—
84C2×Q8×C71#4113yes—
85C2×D4×C71—26no—
86C2×D4×C71—212no—
87C2×D4×C71C22×C416yes—
88C2×D4×C71#4113yes—
89C22×C4×C71—24no—
90C22×C4×C71C2×D416yes—
91C22×C4×C71C2411yescharacteristic
92M16×C71—212no—
93M16×C71—26no—
94M16×C71—212no—
95C2×C8×C71—212no—
96C2×C8×C71—26no—
97C2×C8×C71—26no—
98C2×C8×C71C2×D416yes—
99C2×C8×C71C2×D413yes—
100C2×C8×C71C2×D413yes—
101C23×C71—24no—
102C23×C71C4×D416yes—
103C23×C71C23×C411yescharacteristic
104C22×D71—23no—
105C22×D71—81no—
106Q8×C71—24no—
107Q8×C71—26no—
108D4×C71—212no—
109D4×C71—212no—
110D4×C71—26no—
111C2×C4×C71—212no—
112C2×C4×C71—26no—
113C2×C4×C71—26no—
114C2×C4×C71C4×D416yes—
115C2×C4×C71C22×D413yes—
116C2×C4×C71C2×#4113yes—
117C22⋊D71—86no—
118C2×Dic71—26no—
119C2×Dic71—23no—
120C4×D71—41no—
121C4×D71—43no—
122C8×C71—26no—
123C8×C71#12011yescharacteristic
124C8×C71#12013yes—
125C71⋊C8—41no—
126C71⋊C8—43no—
127C22×C71—26no—
128C22×C71—41no—
129C22×C71—46no—
130C22×C71—84no—
131C22×C71C2×C4×D413yes—
132C2×D71—43no—
133C2×D71—43no—
134C2×D71—41no—
135C4×C71—26no—
136C4×C71C2×#12011yescharacteristic
137C4×C71#52513yes—
138Dic71—46no—
139Dic71—41no—
140#21484—711nomaximal, Sylow
141C2×C71—23no—
142C2×C71—46no—
143C2×C71C4×#12011yescharacteristic
144D71—81no—
145#3573D7111yescharacteristic, complemented
146#3025—711no—
147#2930—713no—
148#1891—713no—
149#1868—713no—
150#1791—711no—
151#1757—713no—
152C71#2148411yescharacteristic, complemented, Sylow
153#585C2×D7111yescharacteristic
154#568—26no—
155#568C2×D7113yes—
156C2×#131—26no—
157C2×#131C2×D7113yes—
158C22×M16—711no—
159#525—713no—
160#458—714no—
161B(4,𝔽2)—714no—
162#436—1426no—
163#435—1426no—
164#424—713no—
165#413—713no—
166#410—713no—
167#408—713no—
168#407—713no—
169#405—713no—
170#344—716no—
1712−1+4—23no—
172H(5,𝔽2)—23no—
173H(5,𝔽2)C4×D7114yes—
174C2×#41—26no—
175C2×#41C22×D7113yes—
176C23×C4—711no—
177#131—24no—
178#131—212no—
179#131—26no—
180#131—212no—
181#131—26no—
182C2×M16—212no—
183C2×M16—26no—
184C2×M16—713no—
185C2×M16—1423no—
186C2×M16—1423no—
187C2×M16C22⋊D7116yescomplemented
188C2×M16C22×D7113yes—
189C22×C8—24no—
190C22×C8—711no—
191C22×C8C22⋊D7116yescomplemented
192C22×C8C22×D7111yescharacteristic
193#123—716no—
194C22⋊Q8—1423no—
195#121—1423no—
196#120—716no—
197C4×D4—1426no—
198#117—713no—
199C4⋊C8—1426no—
200#101—1423no—
201#100—1423no—
202#99—7112no—
203C22⋊C8—1426no—
204C22⋊C8—1426no—
205#97—1423no—
206C4×C8—1423no—
207C24—711no—
208#41—24no—
209#41—212no—
210#41—26no—
211#41—26no—
212#41—212no—
213C2×Q8—26no—
214C2×Q8#1176586513yes—
215C2×D4—26no—
216C2×D4—212no—
217C2×D4—1426no—
218C2×D4C2×C4×D7116yes—
219C2×D4#1176586513yes—
220C22×C4—24no—
221C22×C4—711no—
222C22×C4—1423no—
223C22×C4—1423no—
224C22×C4C2×C22⋊D7116yes—
225C22×C4C23×D7111yescharacteristic
226M16—212no—
227M16—26no—
228M16—212no—
229M16—2841no—
230M16—2843no—
231M16—1426no—
232C2×C8—212no—
233C2×C8—26no—
234C2×C8—26no—
235C2×C8—1423no—
236C2×C8—1423no—
237C2×C8C2×C22⋊D7116yes—
238C2×C8C2×C22⋊D7113yes—
239C2×C8C2×C22⋊D7113yes—
240C4⋊C4—1423no—
241C4⋊C4—1426no—
242C22⋊C4—716no—
243C22⋊C4—1426no—
244C22⋊C4—1426no—
245C42—1423no—
246C23—24no—
247C23—5681no—
248C23—1423no—
249C23C4×C22⋊D7116yes—
250C23C22×C4×D7111yescharacteristic
251Q8—24no—
252Q8—26no—
253D4—212no—
254D4—212no—
255D4—26no—
256D4—5686no—
257C2×C4—212no—
258C2×C4—26no—
259C2×C4—26no—
260C2×C4—2841no—
261C2×C4—2843no—
262C2×C4—1426no—
263C2×C4—1423no—
264C2×C4C4×C22⋊D7116yes—
265C2×C4C22×C22⋊D7113yes—
266C2×C4C2×#1176586513yes—
267C8—26no—
268C8—2841no—
269C8—2843no—
270C8#1455609711yescharacteristic
271C8#1455609713yes—
272C22—26no—
273C22—41no—
274C22—46no—
275C22—84no—
276C22—2843no—
277C22—2843no—
278C22—2841no—
279C22C2×C4×C22⋊D7113yes—
280C4—26no—
281C4—2846no—
282C4—2841no—
283C4C2×#1455609711yescharacteristic
284C4#1798651013yes—
285C2—23no—
286C2—46no—
287C2—5681no—
288C2C4×#1455609711yescharacteristic
subgroupG/H
centerC4C2×#14556097
commutatorC23×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#3573C2
socleC2×C71C4×#120
prime radical#22597732C2
cosocleC22×C4×C71C24
hypercenter#3573D71
hypocenterC71#21484
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ C22×C8 ◁ #3573

Complements

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

NH ≅ G/Ntype
2. #22597041C2semidirect
3. #22588758C2semidirect
4. #22588626C2semidirect
5. #22588291C2semidirect
6. #22588140C2semidirect
7. C71×#3573C2semidirect
8. C22×#11765848C22semidirect
14. #17986027C22semidirect
18. #17985962C22semidirect
19. #17985954C22semidirect
20. #17985770C22semidirect
57. C2×M16×C71D4semidirect
60. C22×C8×C71D4semidirect
145. #3573D71semidirect
152. C71#21484semidirect
187. C2×M16C22⋊D71semidirect
191. C22×C8C22⋊D71semidirect

Maximal subgroups

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

subgroupindex
#225977322
#225970412
#225887582
#225886262
#225882912
#225881402
C71×#35732
#2148471