Finite Group Databasebeta

← #31885567

Subgroups — #31885567

Order 18176.

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

normal subgroups only

This group has 492 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#22588457C212yesmaximal, 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)—44no—
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—44no—
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—44no—
56C22×C4×C71—42no—
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—42no—
68C22⋊C4×C71—42no—
69C42×C71—22no—
70C42×C71—22no—
71C42×C71—22no—
72C42×C71—42no—
73C42×C71—41no—
74C42×C71—42no—
75C42×C71C2×C812yes—
76C42×C71M1612yes—
77C23×C71—22no—
78C23×C71—41no—
79C23×C71—41no—
80C23×C71—42no—
81C23×C71—42no—
82C23×C71C2×C22⋊C411yescharacteristic
83Q8×C71—42no—
84Q8×C71—42no—
85D4×C71—42no—
86D4×C71—42no—
87C2×C4×C71—21no—
88C2×C4×C71—22no—
89C2×C4×C71—21no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71—22no—
94C2×C4×C71—22no—
95C2×C4×C71—21no—
96C2×C4×C71—22no—
97C2×C4×C71—41no—
98C2×C4×C71—41no—
99C2×C4×C71—41no—
100C2×C4×C71—41no—
101C2×C4×C71—42no—
102C2×C4×C71—42no—
103C2×C4×C71—84no—
104C2×C4×C71C22⋊C814yes—
105C2×C4×C71#10012yes—
106C2×C4×C71C22×C811yescharacteristic
107C2×C4×C71C2×M1611yescharacteristic
108C2×C4×C71AGL(1,ℤ/8)12yes—
109C2×C4×C71#13712yes—
110C71⋊C8—84no—
111C22×C71—42no—
112C22×C71—44no—
113C22×C71—81no—
114C22×C71—82no—
115C22×C71—81no—
116C22×C71C2×C22⋊C811yescharacteristic
117C22×C71#41011yescharacteristic
118C22×C71C2×#10011yescharacteristic
119C22×C71#41712yes—
120C22×C71#42112yes—
121C4×C71—21no—
122C4×C71—21no—
123C4×C71—24no—
124C4×C71—22no—
125C4×C71—22no—
126C4×C71—22no—
127C4×C71—41no—
128C4×C71—42no—
129C4×C71—81no—
130C4×C71—81no—
131#7601—711nomaximal, Sylow
132C2×C71—82no—
133C2×C71#146011yescharacteristic
134C2×C71#147712yes—
135C2×C71#151212yes—
136C2×C71#153311yescharacteristic
137C2×C71#156011yescharacteristic
138C4×#124D7111yescharacteristic
139#1831—712no—
140#1272—712no—
141#1270—712no—
142C71#760111yescharacteristic, complemented, Sylow
143C2×#124Dic7111yescharacteristic
144C2×C4×Q8Dic7111yescharacteristic
145C2×C4×D4Dic7111yescharacteristic
146C2×C4⋊C8—1422no—
147#391C2×D7111yescharacteristic
148#390C2×D7111yescharacteristic
149#388—22no—
150#382Dic7111yescharacteristic
151C4×C22⋊C4—22no—
152C43C2×D7111yescharacteristic
153#336—714no—
154#336—1422no—
155C22×Q8C2×Dic7111yescharacteristic
156C22×D4C2×Dic7111yescharacteristic
157C23×C4—21no—
158C22×C8—1424no—
159#124—22no—
160#124C71⋊C814yescomplemented
161C4×Q8—42no—
162C4×D4—42no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4C2×Dic7111yescharacteristic
165C2×C4⋊C4C2×Dic7111yescharacteristic
166C2×C22⋊C4—22no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—22no—
169C2×C42—21no—
170C2×C42C2×Dic7111yescharacteristic
171C2×C42C2×Dic7111yescharacteristic
172C2×C42C22⋊D7112yes—
173C2×C42C22⋊D7112yes—
174C2×C42C22×D7111yescharacteristic
175C4⋊C8—2844no—
176#95—22no—
177#95—22no—
178C24—21no—
179C2×Q8—21no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8C2×C71⋊C812yes—
183C2×D4—21no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4C2×C71⋊C812yes—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4C22⋊Dic7111yescharacteristic
193C22×C4C22⋊Dic7112yes—
194C22×C4C22⋊Dic7111yescharacteristic
195C22×C4C22×Dic7111yescharacteristic
196C22×C4C2×C22⋊D7112yes—
197C2×C8—2844no—
198C2×C8—2844no—
199C2×C8—2844no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—42no—
203C22⋊C4—28no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C42—22no—
208C42—22no—
209C42—22no—
210C42—42no—
211C42—41no—
212C42—42no—
213C42C2×C71⋊C812yes—
214C42#1176584912yes—
215C23—22no—
216C23—41no—
217C23—41no—
218C23—42no—
219C23—42no—
220C23C2×C22⋊Dic7111yescharacteristic
221Q8—42no—
222Q8—42no—
223D4—42no—
224D4—42no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—84no—
242C2×C4C22⋊(C71⋊C8)14yes—
243C2×C4#1455604512yes—
244C2×C4C22×C71⋊C811yescharacteristic
245C2×C4C2×#1176584911yescharacteristic
246C2×C4#1455615312yes—
247C2×C4#1455615812yes—
248C8—5684no—
249C22—42no—
250C22—44no—
251C22—81no—
252C22—81no—
253C22—82no—
254C22C2×C22⋊(C71⋊C8)11yescharacteristic
255C22#1798597311yescharacteristic
256C22C2×#1455604511yescharacteristic
257C22#1798600012yes—
258C22#1798601612yes—
259C4—21no—
260C4—21no—
261C4—22no—
262C4—24no—
263C4—22no—
264C4—22no—
265C4—42no—
266C4—41no—
267C4—81no—
268C4—81no—
269C2—82no—
270C2#2258666111yescharacteristic
271C2#2258672712yes—
272C2#2258687812yes—
273C2#2258696411yescharacteristic
274C2#2258707511yescharacteristic
subgroupG/H
centerC23C2×C22⋊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#7601
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C43 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
1. #22588457C2semidirect
2. #22586166C2semidirect
23. #124×C71C8semidirect
142. C71#7601semidirect
160. #124C71⋊C8semidirect

Maximal subgroups

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

subgroupindex
#225884572
#225861662
#225861612
C4×#124×C712
#760171