Finite Group Databasebeta

← #31885421

Subgroups — #31885421

Order 18176.

528 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 159 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#22588086C212yesmaximal, complemented
2#22586166C212yesmaximal, complemented
3#22586161C212yesmaximal
4C4×#124×C71C211yesmaximal, characteristic
5C2×C4×C71⋊C8—22no—
6#17985754—22no—
7#17985754C2214yes—
8C2×#124×C71C411yescharacteristic
9C2×C4×Q8×C71C411yescharacteristic
10C2×C4×D4×C71C411yescharacteristic
11#391×C71C2211yescharacteristic
12#390×C71C2211yescharacteristic
13#388×C71—22no—
14#382×C71C411yescharacteristic
15C4×C22⋊C4×C71—22no—
16C43×C71C2211yescharacteristic
17C22×C71⋊C8—24no—
18C4×C71⋊C8—28no—
19C22×Q8×C71C2×C411yescharacteristic
20C22×D4×C71C2×C411yescharacteristic
21C23×C4×C71—21no—
22#124×C71—22no—
23#124×C71C814yescomplemented
24C4×Q8×C71—42no—
25C4×D4×C71—42no—
26C2×C4⋊C4×C71—21no—
27C2×C4⋊C4×C71C2×C411yescharacteristic
28C2×C4⋊C4×C71C2×C411yescharacteristic
29C2×C22⋊C4×C71—22no—
30C2×C22⋊C4×C71—21no—
31C2×C22⋊C4×C71—22no—
32C2×C42×C71—21no—
33C2×C42×C71C2×C411yescharacteristic
34C2×C42×C71C2×C411yescharacteristic
35C2×C42×C71D412yes—
36C2×C42×C71D412yes—
37C2×C42×C71C2311yescharacteristic
38#95×C71—22no—
39#95×C71—22no—
40C2×C71⋊C8—28no—
41C2×C71⋊C8—44no—
42C2×C71⋊C8—44no—
43C24×C71—21no—
44C2×Q8×C71—21no—
45C2×Q8×C71—41no—
46C2×Q8×C71—41no—
47C2×Q8×C71C2×C812yes—
48C2×D4×C71—21no—
49C2×D4×C71—41no—
50C2×D4×C71—41no—
51C2×D4×C71C2×C812yes—
52C22×C4×C71—21no—
53C22×C4×C71—21no—
54C22×C4×C71—21no—
55C22×C4×C71—42no—
56C22×C4×C71—44no—
57C22×C4×C71C22⋊C411yescharacteristic
58C22×C4×C71C22⋊C412yes—
59C22×C4×C71C22⋊C411yescharacteristic
60C22×C4×C71C22×C411yescharacteristic
61C22×C4×C71C2×D412yes—
62C4⋊C4×C71—41no—
63C4⋊C4×C71—41no—
64C4⋊C4×C71—42no—
65C22⋊C4×C71—28no—
66C22⋊C4×C71—42no—
67C22⋊C4×C71—44no—
68C42×C71—22no—
69C42×C71—24no—
70C42×C71—24no—
71C42×C71—41no—
72C42×C71—42no—
73C42×C71C2×C812yes—
74C42×C71M1612yes—
75C23×C71—22no—
76C23×C71—41no—
77C23×C71—41no—
78C23×C71—42no—
79C23×C71—42no—
80C23×C71C2×C22⋊C411yescharacteristic
81Q8×C71—42no—
82Q8×C71—42no—
83D4×C71—42no—
84D4×C71—42no—
85C2×C4×C71—21no—
86C2×C4×C71—22no—
87C2×C4×C71—21no—
88C2×C4×C71—22no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—21no—
93C2×C4×C71—22no—
94C2×C4×C71—41no—
95C2×C4×C71—41no—
96C2×C4×C71—41no—
97C2×C4×C71—41no—
98C2×C4×C71—42no—
99C2×C4×C71—42no—
100C2×C4×C71—84no—
101C2×C4×C71C22⋊C814yes—
102C2×C4×C71#10012yes—
103C2×C4×C71#10414yes—
104C2×C4×C71C22×C811yescharacteristic
105C2×C4×C71C2×M1611yescharacteristic
106C2×C4×C71#13514yes—
107C71⋊C8—48no—
108C22×C71—42no—
109C22×C71—44no—
110C22×C71—81no—
111C22×C71—82no—
112C22×C71—81no—
113C22×C71C2×C22⋊C811yescharacteristic
114C22×C71#41011yescharacteristic
115C22×C71C2×#10011yescharacteristic
116C22×C71#41612yes—
117C22×C71C2×#10412yes—
118C4×C71—21no—
119C4×C71—21no—
120C4×C71—24no—
121C4×C71—24no—
122C4×C71—41no—
123C4×C71—42no—
124C4×C71—81no—
125C4×C71—81no—
126C4×C71#32914yes—
127#7567—711nomaximal, Sylow
128C2×C71—82no—
129C2×C71#146011yescharacteristic
130C2×C71#147612yes—
131C2×C71#150012yes—
132C2×C71C2×#32911yescharacteristic
133C2×C71#154311yescharacteristic
134C4×#124D7111yescharacteristic
135#1744—712no—
136#1272—712no—
137#1270—712no—
138C71#756711yescharacteristic, complemented, Sylow
139C2×#124Dic7111yescharacteristic
140C2×C4×Q8Dic7111yescharacteristic
141C2×C4×D4Dic7111yescharacteristic
142C2×C4×C8—1422no—
143#391C2×D7111yescharacteristic
144#390C2×D7111yescharacteristic
145#388—22no—
146#382Dic7111yescharacteristic
147C4×C22⋊C4—22no—
148C43C2×D7111yescharacteristic
149#336—714no—
150#336—1422no—
151C22×Q8C2×Dic7111yescharacteristic
152C22×D4C2×Dic7111yescharacteristic
153C23×C4—21no—
154C22×C8—1424no—
155#124—22no—
156#124C71⋊C814yescomplemented
157C4×Q8—42no—
158C4×D4—42no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4C2×Dic7111yescharacteristic
161C2×C4⋊C4C2×Dic7111yescharacteristic
162C2×C22⋊C4—22no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—22no—
165C2×C42—21no—
166C2×C42C2×Dic7111yescharacteristic
167C2×C42C2×Dic7111yescharacteristic
168C2×C42C22⋊D7112yes—
169C2×C42C22⋊D7112yes—
170C2×C42C22×D7111yescharacteristic
171C4×C8—1428no—
172#95—22no—
173#95—22no—
174C24—21no—
175C2×Q8—21no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8C2×C71⋊C812yes—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4C2×C71⋊C812yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—42no—
187C22×C4—44no—
188C22×C4C22⋊Dic7111yescharacteristic
189C22×C4C22⋊Dic7112yes—
190C22×C4C22⋊Dic7111yescharacteristic
191C22×C4C22×Dic7111yescharacteristic
192C22×C4C2×C22⋊D7112yes—
193C2×C8—2844no—
194C2×C8—2844no—
195C2×C8—1428no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—42no—
199C22⋊C4—28no—
200C22⋊C4—42no—
201C22⋊C4—44no—
202C42—22no—
203C42—24no—
204C42—24no—
205C42—41no—
206C42—42no—
207C42C2×C71⋊C812yes—
208C42#1176584912yes—
209C23—22no—
210C23—41no—
211C23—41no—
212C23—42no—
213C23—42no—
214C23C2×C22⋊Dic7111yescharacteristic
215Q8—42no—
216Q8—42no—
217D4—42no—
218D4—42no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—84no—
235C2×C4C22⋊(C71⋊C8)14yes—
236C2×C4#1455604512yes—
237C2×C4#1455605514yes—
238C2×C4C22×C71⋊C811yescharacteristic
239C2×C4C2×#1176584911yescharacteristic
240C2×C4#1455614814yes—
241C8—2848no—
242C22—42no—
243C22—44no—
244C22—81no—
245C22—81no—
246C22—82no—
247C22C2×C22⋊(C71⋊C8)11yescharacteristic
248C22#1798597311yescharacteristic
249C22C2×#1455604511yescharacteristic
250C22#1798599512yes—
251C22C2×#1455605512yes—
252C4—21no—
253C4—21no—
254C4—24no—
255C4—24no—
256C4—42no—
257C4—41no—
258C4—81no—
259C4—81no—
260C4#1798573914yes—
261C2—82no—
262C2#2258666111yescharacteristic
263C2#2258672212yes—
264C2#2258682612yes—
265C2C2×#1798573911yescharacteristic
266C2#2258700211yescharacteristic
subgroupG/H
centerC22×C4C22⋊Dic71
commutatorC2×C4×C71C22×C8
FrattiniC2×C42C22×D71
FittingC4×#124×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC22×D4×C71C2×C4
cosocleC2×C42×C71C23
hypercenterC4×#124D71
hypocenterC71#7567
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ C43 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
1. #22588086C2semidirect
2. #22586166C2semidirect
23. #124×C71C8semidirect
138. C71#7567semidirect
156. #124C71⋊C8semidirect

Maximal subgroups

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

subgroupindex
#225880862
#225861662
#225861612
C4×#124×C712
#756771