Finite Group Databasebeta

← #31891854

Subgroups — #31891854

Order 18176.

392 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 95 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#22597411C211yesmaximal, characteristic
2#22587105C212yesmaximal, complemented
3#22586932C211yesmaximal, characteristic
4#22586233C212yesmaximal, complemented
5C71×#1963C211yesmaximal, characteristic
6#17986071—22no—
7#17986071—22no—
8#17986056C2212yes—
9#17985966—21no—
10#17985966—21no—
11C4×#11765849C2212yes—
12#17985751—22no—
13#17985749—22no—
14#17985739C411yescharacteristic, complemented
15#17985739C412yescomplemented
16#17985739C411yescharacteristic, complemented
17#532×C71C2211yescharacteristic
18#423×C71C2211yescharacteristic
19C2×#104×C71—21no—
20C2×#104×C71—21no—
21#376×C71C2211yescharacteristic
22C2×#11765849—21no—
23C2×#11765849—21no—
24C2×#11765849—22no—
25C22×C71⋊C8—21no—
26C22×C71⋊C8—21no—
27C4⋊(C71⋊C8)D412yes—
28C4⋊(C71⋊C8)D412yes—
29C22⋊(C71⋊C8)—22no—
30C22⋊(C71⋊C8)—22no—
31C22⋊(C71⋊C8)—22no—
32#14556039—22no—
33#14556039C2×C412yes—
34#14556039C2×C412yes—
35C4×C71⋊C8—22no—
36C2×#41×C71D411yescharacteristic
37C2×#41×C71D411yescharacteristic
38C2×M16×C71—21no—
39C4⋊Q8×C71Q811yescharacteristic
40#127×C71Q811yescharacteristic
41#124×C71C2×C411yescharacteristic
42#124×C71C2×C411yescharacteristic
43#121×C71—22no—
44C4×D4×C71—22no—
45C2×C42×C71—21no—
46C2×C42×C71C2311yescharacteristic
47C4⋊C8×C71—22no—
48#104×C71—42no—
49#104×C71—42no—
50#95×C71—22no—
51#11765849—44no—
52#11765849—81no—
53#11765849—81no—
54C2×C71⋊C8—22no—
55C2×C71⋊C8—22no—
56C2×C71⋊C8—22no—
57C2×C71⋊C8—22no—
58C2×C71⋊C8—81no—
59C2×C71⋊C8—81no—
60#41×C71—41no—
61#41×C71—41no—
62#41×C71—41no—
63#41×C71—41no—
64C2×Q8×C71C4⋊C411yescharacteristic
65C2×Q8×C71C4⋊C411yescharacteristic
66C2×D4×C71—21no—
67C2×D4×C71—21no—
68C2×D4×C71—22no—
69C2×D4×C71C4⋊C411yescharacteristic
70C2×D4×C71C4⋊C411yescharacteristic
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—41no—
74C22×C4×C71C2×D411yescharacteristic
75C22×C4×C71#4112yes—
76M16×C71—41no—
77M16×C71—41no—
78C2×C8×C71—41no—
79C4⋊C4×C71—22no—
80C22⋊C4×C71—22no—
81C22⋊C4×C71—22no—
82C42×C71—22no—
83C42×C71—41no—
84C42×C71C22×C411yescharacteristic
85C42×C71C2×D412yes—
86C42×C71C2×Q811yescharacteristic
87C23×C71—21no—
88C23×C71—21no—
89C23×C71#12311yescharacteristic
90Q8×C71—41no—
91Q8×C71—41no—
92D4×C71—41no—
93D4×C71—41no—
94D4×C71—82no—
95D4×C71—81no—
96D4×C71—81no—
97C2×C4×C71—21no—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—21no—
101C2×C4×C71—42no—
102C2×C4×C71—41no—
103C2×C4×C71—41no—
104C2×C4×C71—41no—
105C2×C4×C71—41no—
106C2×C4×C71—42no—
107C2×C4×C71—82no—
108C2×C4×C71C2×C4⋊C411yescharacteristic
109C2×C4×C71C4×D412yes—
110C2×C4×C71#12011yescharacteristic
111C2×C4×C71C22⋊Q812yes—
112C8×C71—81no—
113C71⋊C8—24no—
114C71⋊C8—42no—
115C71⋊C8—42no—
116C71⋊C8—42no—
117C22×C71—21no—
118C22×C71—22no—
119C22×C71—41no—
120C22×C71—41no—
121C22×C71—42no—
122C22×C71—42no—
123C22×C71#38511yescharacteristic
124C4×C71—21no—
125C4×C71—22no—
126C4×C71—22no—
127C4×C71—41no—
128C4×C71—41no—
129C4×C71—82no—
130C4×C71#45311yescharacteristic
131C4×C71#45511yescharacteristic
132#8918—711nomaximal, Sylow
133C2×C71—41no—
134C2×C71—81no—
135C2×C71—81no—
136C2×C71#185611yescharacteristic
137C2×C71#186112yes—
138#2980—711no—
139#1963D7111yescharacteristic
140#1569—712no—
141#1525—711no—
142#1300—712no—
143C71#891811yescharacteristic, complemented, Sylow
144#532C2×D7111yescharacteristic
145#435—1422no—
146#435—1422no—
147#432—712no—
148#423C2×D7111yescharacteristic
149C2×#104—21no—
150C2×#104—21no—
151#408—1421no—
152#408—1421no—
153C4×M16—712no—
154#376C2×D7111yescharacteristic
155#335—1422no—
156#334—1422no—
157#329—711no—
158#329—712no—
159#329—711no—
160C2×#41C22⋊D7111yescharacteristic
161C2×#41C22⋊D7111yescharacteristic
162C2×M16—21no—
163C2×M16—1422no—
164C2×M16—1421no—
165C2×M16—1421no—
166C22×C8—1421no—
167C22×C8—1421no—
168C4⋊Q8Dic14211yescharacteristic
169#127Dic14211yescharacteristic
170#124C4×D7111yescharacteristic
171#124C4×D7111yescharacteristic
172#121—22no—
173C4×D4—22no—
174C2×C42—21no—
175C2×C42C22×D7111yescharacteristic
176C4⋊C8—22no—
177C4⋊C8—712no—
178C4⋊C8—712no—
179#104—42no—
180#104—42no—
181C22⋊C8—1422no—
182C22⋊C8—1422no—
183C22⋊C8—1422no—
184#97—712no—
185#97—712no—
186#97—1422no—
187C4×C8—1422no—
188#95—22no—
189#41—41no—
190#41—41no—
191#41—41no—
192#41—41no—
193C2×Q8Dic71⋊C411yescharacteristic
194C2×Q8Dic71⋊C411yescharacteristic
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—22no—
198C2×D4Dic71⋊C411yescharacteristic
199C2×D4Dic71⋊C411yescharacteristic
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—41no—
203C22×C4C2×C22⋊D7111yescharacteristic
204C22×C4#1176586412yes—
205M16—41no—
206M16—41no—
207M16—2844no—
208M16—5681no—
209M16—5681no—
210C2×C8—41no—
211C2×C8—5681no—
212C2×C8—5681no—
213C2×C8—1422no—
214C2×C8—1422no—
215C2×C8—1422no—
216C2×C8—1422no—
217C4⋊C4—22no—
218C22⋊C4—22no—
219C22⋊C4—22no—
220C42—22no—
221C42—41no—
222C42C2×C4×D7111yescharacteristic
223C42D4×D7112yes—
224C42C2×Dic14211yescharacteristic
225C23—21no—
226C23—21no—
227C23#1455610911yescharacteristic
228Q8—41no—
229Q8—41no—
230D4—41no—
231D4—41no—
232D4—82no—
233D4—81no—
234D4—81no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—82no—
246C2×C4C2×Dic71⋊C411yescharacteristic
247C2×C4(C22⋊D71)⋊C412yes—
248C2×C4#1455609611yescharacteristic
249C2×C4#1455610512yes—
250C8—81no—
251C8—2842no—
252C8—2842no—
253C8—2842no—
254C8—1424no—
255C22—21no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—42no—
260C22—42no—
261C22#1798587111yescharacteristic
262C4—21no—
263C4—22no—
264C4—22no—
265C4—41no—
266C4—41no—
267C4—82no—
268C4#1798616611yescharacteristic
269C4#1798617611yescharacteristic
270C2—41no—
271C2—81no—
272C2—81no—
273C2#2258854911yescharacteristic
274C2#2258857812yes—
subgroupG/H
centerC2×C4#14556096
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1963C2
socleC22×C71#385
prime radical#532×C71C22
cosocleC2×C42×C71C23
hypercenter#1963D71
hypocenterC71#8918
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1963

Complements

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

NH ≅ G/Ntype
2. #22587105C2semidirect
4. #22586233C2semidirect
14. #17985739C4semidirect
15. #17985739C4semidirect
16. #17985739C4semidirect
143. C71#8918semidirect

Maximal subgroups

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

subgroupindex
#225974112
#225871052
#225869322
#225862332
C71×#19632
#891871