Finite Group Databasebeta

← #31889615

Subgroups — #31889615

Order 18176.

658 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 143 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#22589281C211yesmaximal, characteristic
2#22587040C212yesmaximal
3#22586955C212yesmaximal
4#22586684C211yesmaximal, characteristic
5C71×#3502C211yesmaximal, characteristic
6#17986056C2212yes—
7C2×C4⋊(C71⋊C8)C2211yescharacteristic
8C2×#14556039C2211yescharacteristic
9#17985754—22no—
10#17985745—22no—
11#17985745—22no—
12#17985741—24no—
13#17985725—22no—
14Q82×C71—22no—
15#557×C71—22no—
16#557×C71—24no—
17#544×C71—22no—
18#544×C71C412yes—
19#533×C71—22no—
20#533×C71C2212yes—
21#519×C71C412yes—
22C2×C4×Q8×C71C2211yescharacteristic
23C22×C71⋊C8—21no—
24C22×C71⋊C8—21no—
25C4⋊(C71⋊C8)—22no—
26C4⋊(C71⋊C8)—22no—
27C22⋊(C71⋊C8)—22no—
28#14556039—22no—
29#14556039—22no—
30C22×Q8×C71C2×C411yescharacteristic
31C4⋊Q8×C71—22no—
32C4⋊Q8×C71—22no—
33C4⋊Q8×C71—28no—
34C4⋊Q8×C71—24no—
35C4⋊Q8×C71D412yes—
36C4⋊Q8×C71D412yes—
37#125×C71—24no—
38#125×C71—28no—
39#125×C71—24no—
40#125×C71D414yes—
41C22⋊Q8×C71—24no—
42C22⋊Q8×C71—22no—
43C22⋊Q8×C71—22no—
44C4×Q8×C71—24no—
45C4×Q8×C71—24no—
46C4×Q8×C71—24no—
47C4×Q8×C71—24no—
48C4×Q8×C71—22no—
49C4×Q8×C71—22no—
50C4×Q8×C71—22no—
51C4×Q8×C71—22no—
52#117×C71—24no—
53#117×C71C2×C414yes—
54C2×C4⋊C4×C71C2×C411yescharacteristic
55C2×C4⋊C4×C71D412yes—
56C2×C42×C71D412yes—
57C2×C42×C71C2311yescharacteristic
58C2×C71⋊C8—22no—
59C2×C71⋊C8—41no—
60C2×C71⋊C8—41no—
61C2×C71⋊C8—42no—
62C2×C71⋊C8—42no—
63C2×C71⋊C8—41no—
64C2×Q8×C71—24no—
65C2×Q8×C71—24no—
66C2×Q8×C71—22no—
67C2×Q8×C71—22no—
68C2×Q8×C71—22no—
69C22×C4×C71C22⋊C412yes—
70C22×C4×C71C22⋊C412yes—
71C22×C4×C71C22×C411yescharacteristic
72C22×C4×C71C2×D411yescharacteristic
73C22×C4×C71C2×D411yescharacteristic
74C4⋊C4×C71—24no—
75C4⋊C4×C71—24no—
76C4⋊C4×C71—24no—
77C4⋊C4×C71—28no—
78C4⋊C4×C71—22no—
79C4⋊C4×C71—24no—
80C4⋊C4×C71—24no—
81C4⋊C4×C71—22no—
82C4⋊C4×C71—22no—
83C4⋊C4×C71—24no—
84C4⋊C4×C71C22⋊C414yes—
85C4⋊C4×C71C22⋊C414yes—
86C22⋊C4×C71—24no—
87C42×C71—28no—
88C42×C71—24no—
89C42×C71—22no—
90C42×C71—22no—
91C42×C71C2×D412yes—
92C42×C71C2×D412yes—
93C23×C71C2×C22⋊C411yescharacteristic
94Q8×C71—28no—
95Q8×C71—42no—
96Q8×C71—42no—
97Q8×C71—82no—
98Q8×C71—84no—
99C2×C4×C71—22no—
100C2×C4×C71—22no—
101C2×C4×C71—22no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—24no—
105C2×C4×C71—24no—
106C2×C4×C71—22no—
107C2×C4×C71—41no—
108C2×C4×C71—42no—
109C2×C4×C71#10114yes—
110C2×C4×C71C2×C22⋊C412yes—
111C2×C4×C71#12011yescharacteristic
112C2×C4×C71#12012yes—
113C2×C4×C71#12011yescharacteristic
114C71⋊C8—42no—
115C71⋊C8—82no—
116C71⋊C8—81no—
117C22×C71—21no—
118C22×C71—21no—
119C22×C71#37911yescharacteristic
120C22×C71C2×#10112yes—
121C4×C71—21no—
122C4×C71—22no—
123C4×C71—24no—
124C4×C71—22no—
125C4×C71—44no—
126C4×C71—42no—
127C4×C71—44no—
128C4×C71#45411yescharacteristic
129C4×C71#45611yescharacteristic
130#8460—711nomaximal, Sylow
131C2×C71—22no—
132C2×C71#178411yescharacteristic
133C2×C71#179311yescharacteristic
134C2×C71#179411yescharacteristic
135#3502D7111yescharacteristic
136#1985—711no—
137#1552—712no—
138#1531—712no—
139#1467—711no—
140C71#846011yescharacteristic, complemented, Sylow
141Q82—22no—
142#557—22no—
143#557—24no—
144#544—22no—
145#544Dic7112yes—
146#533—22no—
147#533C2×D7112yes—
148#519Dic7112yes—
149C2×C4×Q8C2×D7111yescharacteristic
150#432—712no—
151C2×C4⋊C8—711no—
152C2×#97—711no—
153#336—1422no—
154#332—1422no—
155#332—1422no—
156#330—1424no—
157#324—1422no—
158C22×Q8C2×Dic7111yescharacteristic
159C22×C8—1421no—
160C22×C8—1421no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—28no—
164C4⋊Q8—24no—
165C4⋊Q8C22⋊D7112yes—
166C4⋊Q8C22⋊D7112yes—
167#125—24no—
168#125—28no—
169#125—24no—
170#125C22⋊D7114yes—
171C22⋊Q8—24no—
172C22⋊Q8—22no—
173C22⋊Q8—22no—
174C4×Q8—24no—
175C4×Q8—24no—
176C4×Q8—24no—
177C4×Q8—24no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×Q8—22no—
182#117—24no—
183#117C2×Dic7114yes—
184C2×C4⋊C4C2×Dic7111yescharacteristic
185C2×C4⋊C4C22⋊D7112yes—
186C2×C42C22⋊D7112yes—
187C2×C42C22×D7111yescharacteristic
188C4⋊C8—1422no—
189C4⋊C8—1422no—
190C22⋊C8—1422no—
191#97—1422no—
192#97—1422no—
193C2×Q8—24no—
194C2×Q8—24no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—22no—
198C22×C4C22⋊Dic7112yes—
199C22×C4C22⋊Dic7112yes—
200C22×C4C22×Dic7111yescharacteristic
201C22×C4C2×C22⋊D7111yescharacteristic
202C22×C4C2×C22⋊D7111yescharacteristic
203C2×C8—2841no—
204C2×C8—2841no—
205C2×C8—2842no—
206C2×C8—2842no—
207C2×C8—2841no—
208C2×C8—1422no—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—28no—
213C4⋊C4—22no—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—24no—
219C4⋊C4C22⋊Dic7114yes—
220C4⋊C4C22⋊Dic7114yes—
221C22⋊C4—24no—
222C42—28no—
223C42—24no—
224C42—22no—
225C42—22no—
226C42C2×C22⋊D7112yes—
227C42C2×C22⋊D7112yes—
228C23C2×C22⋊Dic7111yescharacteristic
229Q8—28no—
230Q8—42no—
231Q8—42no—
232Q8—82no—
233Q8—84no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4#1455604714yes—
245C2×C4C2×C22⋊Dic7112yes—
246C2×C4#1455609711yescharacteristic
247C2×C4#1455609712yes—
248C2×C4#1455609711yescharacteristic
249C8—2842no—
250C8—5682no—
251C8—5681no—
252C22—21no—
253C22—21no—
254C22#1798585511yescharacteristic
255C22C2×#1455604712yes—
256C4—24no—
257C4—22no—
258C4—21no—
259C4—22no—
260C4—44no—
261C4—42no—
262C4—44no—
263C4#1798617211yescharacteristic
264C4#1798618211yescharacteristic
265C2—22no—
266C2#2258826411yescharacteristic
267C2#2258829911yescharacteristic
268C2#2258830411yescharacteristic
subgroupG/H
centerC22#17985855
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#3502C2
socleC22×C71#379
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenter#3502D71
hypocenterC71#8460
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3502

Complements

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

NH ≅ G/Ntype
140. C71#8460semidirect

Maximal subgroups

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

subgroupindex
#225892812
#225870402
#225869552
#225866842
C71×#35022
#846071