Finite Group Databasebeta

← #31885687

Subgroups — #31885687

Order 18176.

658 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 139 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#22588102C212yesmaximal
2#22586166C214yesmaximal, complemented
3C2×#391×C71C211yesmaximal, characteristic
4C2×C22⋊(C71⋊C8)—22no—
5C2×#14556039—22no—
6#17985754C2214yes—
7C23×Q8×C71C411yescharacteristic
8C22×C4⋊C4×C71C411yescharacteristic
9C22×C42×C71C2211yescharacteristic
10#391×C71—22no—
11#391×C71C412yes—
12#391×C71C2212yes—
13C2×#95×C71—22no—
14C22×C71⋊C8—24no—
15C22⋊(C71⋊C8)—28no—
16#14556039—44no—
17C22×Q8×C71—24no—
18C22×Q8×C71—21no—
19C22×Q8×C71—42no—
20C22×Q8×C71—42no—
21C22×Q8×C71C2×C412yes—
22C23×C4×C71—21no—
23C23×C4×C71—21no—
24C23×C4×C71C2×C411yescharacteristic
25C23×C4×C71D412yes—
26C2×C4⋊C4×C71—21no—
27C2×C4⋊C4×C71—42no—
28C2×C4⋊C4×C71C2×C412yes—
29C2×C42×C71—42no—
30C2×C42×C71C2×C411yescharacteristic
31C2×C42×C71D412yes—
32C2×C42×C71C2311yescharacteristic
33#95×C71—24no—
34#95×C71—24no—
35C2×C71⋊C8—28no—
36C2×C71⋊C8—44no—
37C2×C71⋊C8—44no—
38C24×C71C22⋊C411yescharacteristic
39C2×Q8×C71—24no—
40C2×Q8×C71—22no—
41C2×Q8×C71—416no—
42C2×Q8×C71—42no—
43C2×Q8×C71—42no—
44C2×Q8×C71—42no—
45C2×Q8×C71—42no—
46C2×Q8×C71M1614yes—
47C22×C4×C71—22no—
48C22×C4×C71—22no—
49C22×C4×C71—22no—
50C22×C4×C71—24no—
51C22×C4×C71—24no—
52C22×C4×C71—22no—
53C22×C4×C71—24no—
54C22×C4×C71—22no—
55C22×C4×C71—42no—
56C22×C4×C71—42no—
57C22×C4×C71—42no—
58C22×C4×C71—42no—
59C22×C4×C71—44no—
60C22×C4×C71C22⋊C412yes—
61C22×C4×C71C22⋊C411yescharacteristic
62C22×C4×C71C22×C411yescharacteristic
63C22×C4×C71C2×D412yes—
64C4⋊C4×C71—42no—
65C4⋊C4×C71—42no—
66C42×C71—24no—
67C42×C71—41no—
68C42×C71—41no—
69C23×C71—21no—
70C23×C71—21no—
71C23×C71—22no—
72C23×C71#10012yes—
73C23×C71#10414yes—
74C23×C71C2×C22⋊C411yescharacteristic
75Q8×C71—44no—
76Q8×C71—48no—
77Q8×C71—44no—
78C2×C4×C71—28no—
79C2×C4×C71—21no—
80C2×C4×C71—22no—
81C2×C4×C71—21no—
82C2×C4×C71—21no—
83C2×C4×C71—21no—
84C2×C4×C71—22no—
85C2×C4×C71—24no—
86C2×C4×C71—22no—
87C2×C4×C71—22no—
88C2×C4×C71—22no—
89C2×C4×C71—48no—
90C2×C4×C71—48no—
91C2×C4×C71—42no—
92C2×C4×C71—42no—
93C2×C4×C71—42no—
94C2×C4×C71—44no—
95C2×C4×C71—84no—
96C2×C4×C71#9912yes—
97C2×C4×C71C2×M1612yes—
98C2×C4×C71#13714yes—
99C71⋊C8—48no—
100C22×C71—22no—
101C22×C71—22no—
102C22×C71—24no—
103C22×C71—24no—
104C22×C71#32914yes—
105C22×C71#40711yescharacteristic
106C22×C71C2×#9911yescharacteristic
107C22×C71C2×#10011yescharacteristic
108C22×C71#41912yes—
109C22×C71C2×#10412yes—
110C4×C71—22no—
111C4×C71—22no—
112C4×C71—24no—
113C4×C71—44no—
114C4×C71—42no—
115C4×C71—82no—
116#7629—711nomaximal, Sylow
117C2×C71—24no—
118C2×C71#146211yescharacteristic
119C2×C71#149112yes—
120C2×C71#150812yes—
121C2×C71C2×#32911yescharacteristic
122C2×C71#155811yescharacteristic
123C2×#391D7111yescharacteristic
124#1749—712no—
125#1272—714no—
126C71#762911yescharacteristic, complemented, Sylow
127C23×Q8Dic7111yescharacteristic
128C22×C4⋊C4Dic7111yescharacteristic
129C22×C42C2×D7111yescharacteristic
130C2×C22⋊C8—1422no—
131C2×#97—1422no—
132#391—22no—
133#391Dic7112yes—
134#391C2×D7112yes—
135C2×#95—22no—
136#336—714no—
137C22×Q8—24no—
138C22×Q8—21no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8C2×Dic7112yes—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4C2×Dic7111yescharacteristic
145C23×C4C22⋊D7112yes—
146C22×C8—1424no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4C2×Dic7112yes—
150C2×C42—42no—
151C2×C42C2×Dic7111yescharacteristic
152C2×C42C22⋊D7112yes—
153C2×C42C22×D7111yescharacteristic
154C22⋊C8—1428no—
155#97—2844no—
156#95—24no—
157#95—24no—
158C24C22⋊Dic7111yescharacteristic
159C2×Q8—24no—
160C2×Q8—22no—
161C2×Q8—416no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8#1176584914yes—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—44no—
180C22×C4C22⋊Dic7112yes—
181C22×C4C22⋊Dic7111yescharacteristic
182C22×C4C22×Dic7111yescharacteristic
183C22×C4C2×C22⋊D7112yes—
184C2×C8—2844no—
185C2×C8—2844no—
186C2×C8—1428no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C42—24no—
190C42—41no—
191C42—41no—
192C23—21no—
193C23—21no—
194C23—22no—
195C23#1455604512yes—
196C23#1455605514yes—
197C23C2×C22⋊Dic7111yescharacteristic
198Q8—44no—
199Q8—48no—
200Q8—44no—
201C2×C4—28no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—48no—
213C2×C4—48no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—44no—
218C2×C4—84no—
219C2×C4#1455604312yes—
220C2×C4C2×#1176584912yes—
221C2×C4#1455615914yes—
222C8—2848no—
223C22—22no—
224C22—24no—
225C22—22no—
226C22—24no—
227C22#1798573914yes—
228C22#1798596311yescharacteristic
229C22C2×#1455604311yescharacteristic
230C22C2×#1455604511yescharacteristic
231C22#1798600812yes—
232C22C2×#1455605512yes—
233C4—22no—
234C4—22no—
235C4—24no—
236C4—44no—
237C4—42no—
238C4—82no—
239C2—24no—
240C2#2258666711yescharacteristic
241C2#2258678712yes—
242C2#2258686012yes—
243C2C2×#1798573911yescharacteristic
244C2#2258706611yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC2×#391×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenterC2×#391D71
hypocenterC71#7629
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
2. #22586166C2semidirect
126. C71#7629semidirect

Maximal subgroups

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

subgroupindex
#225881022
#225861662
C2×#391×C712
#762971