Finite Group Databasebeta

← #31892484

Subgroups — #31892484

Order 18176.

572 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#17985938C211yesmaximal, characteristic
2#22590943C211yesmaximal, characteristic
3#22588431C212yesmaximal
4#22586220C212yesmaximal
5C2×#324×C71C211yesmaximal, characteristic
6C22×C4⋊Dic71C2211yescharacteristic
7#17985938—22no—
8#17985938C414yescomplemented
9#17985885—24no—
10#17985862—21no—
11#17985859C2212yes—
12C4×C22⋊Dic71—21no—
13C2×C71⋊#95—21no—
14C2×C71⋊#95—21no—
15C2×C71⋊#95C2211yescharacteristic
16#17985762—24no—
17C22×C4⋊C4×C71C2211yescharacteristic
18C2×C22⋊C8×C71C2212yes—
19#324×C71—22no—
20C23×Dic71—21no—
21C23×Dic71—21no—
22C2×C4⋊Dic71—22no—
23C2×C4⋊Dic71—21no—
24C2×C4⋊Dic71—22no—
25C2×C4⋊Dic71C2×C412yes—
26C2×C4⋊Dic71D414yes—
27C2×C22⋊Dic71—22no—
28C2×C4×Dic71—22no—
29C71⋊#95—22no—
30C71⋊#95—24no—
31C71⋊#95—24no—
32C71⋊#95—24no—
33C71⋊#95—21no—
34C71⋊#95—22no—
35C71⋊#95C2×C412yes—
36C23×C4×C71D411yescharacteristic
37C23×C4×C71D411yescharacteristic
38C23×C4×C71C2311yescharacteristic
39C22×C8×C71—22no—
40C2×C4⋊C4×C71—22no—
41C2×C4⋊C4×C71—22no—
42C2×C4⋊C4×C71C2×C412yes—
43C2×C4⋊C4×C71D412yes—
44C22⋊C8×C71—22no—
45C22⋊C8×C71—22no—
46C22×Dic71—22no—
47C22×Dic71—22no—
48C22×Dic71—24no—
49C22×Dic71—42no—
50C22×Dic71—42no—
51C22×Dic71—44no—
52C22×Dic71—41no—
53C22×Dic71—41no—
54C22×Dic71—42no—
55C4⋊Dic71—24no—
56C4⋊Dic71—42no—
57C4⋊Dic71—44no—
58C22⋊Dic71—42no—
59C22⋊Dic71—42no—
60C4×Dic71—82no—
61C24×C71C2×D411yescharacteristic
62C22×C4×C71—21no—
63C22×C4×C71—21no—
64C22×C4×C71—21no—
65C22×C4×C71—22no—
66C22×C4×C71—22no—
67C22×C4×C71—21no—
68C22×C4×C71—42no—
69C22×C4×C71—42no—
70C22×C4×C71C22⋊C412yes—
71C22×C4×C71C22⋊C412yes—
72C22×C4×C71C22×C411yescharacteristic
73C22×C4×C71C2×D412yes—
74C22×C4×C71C2×D411yescharacteristic
75C22×C4×C71#4112yes—
76C2×C8×C71—44no—
77C2×C8×C71—42no—
78C4⋊C4×C71—24no—
79C4⋊C4×C71—24no—
80C23×C71—22no—
81C23×C71—22no—
82C23×C71C2×C22⋊C411yescharacteristic
83C23×C71#12011yescharacteristic
84C23×C71#12311yescharacteristic
85C23×C71#13512yes—
86C23×C71#13712yes—
87C2×C4×C71—24no—
88C2×C4×C71—22no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—41no—
93C2×C4×C71—41no—
94C2×C4×C71—42no—
95C2×C4×C71—42no—
96C2×C4×C71—42no—
97C2×C4×C71—42no—
98C2×C4×C71—42no—
99C2×C4×C71—81no—
100C2×C4×C71C4×D412yes—
101C2×C4×C71#12112yes—
102C2×Dic71—42no—
103C2×Dic71—42no—
104C2×Dic71—44no—
105C2×Dic71—42no—
106C2×Dic71—42no—
107C2×Dic71—44no—
108C2×Dic71—44no—
109C2×Dic71—44no—
110C2×Dic71—84no—
111C2×Dic71—82no—
112C2×Dic71—82no—
113C8×C71—82no—
114C22×C71—24no—
115C22×C71—24no—
116C22×C71—21no—
117C22×C71—21no—
118C22×C71—21no—
119C22×C71—21no—
120C22×C71#38611yescharacteristic
121C22×C71#41611yescharacteristic
122C22×C71#41911yescharacteristic
123C22×C71#45212yes—
124C22×C71#45412yes—
125C22×C71#45812yes—
126C22×C71#48512yes—
127C4×C71—22no—
128C4×C71—22no—
129C4×C71—42no—
130C4×C71—42no—
131Dic71—82no—
132Dic71—84no—
133Dic71—82no—
134#9022—711nomaximal, Sylow
135C2×C71—21no—
136C2×C71—21no—
137C2×C71—22no—
138C2×C71#161514yes—
139C2×C71#187311yescharacteristic
140C2×C71#188811yescharacteristic
141C2×C71#189011yescharacteristic
142C2×#400—711no—
143#2313—711no—
144#1824—712no—
145C2×#324D7111yescharacteristic
146#1294—712no—
147C71#902211yescharacteristic, complemented, Sylow
148C22×C4⋊C4—711no—
149C22×C4⋊C4C2×D7111yescharacteristic
150C2×C22⋊C8C2×D7112yes—
151#400—714no—
152#400—1422no—
153#387—1424no—
154#381—1421no—
155#380—712no—
156C4×C22⋊C4—1421no—
157C2×#95—711no—
158C2×#95—1421no—
159C2×#95—1421no—
160#340—1424no—
161#324—22no—
162C23×C4—1421no—
163C23×C4—1421no—
164C23×C4D28411yescharacteristic
165C23×C4D28411yescharacteristic
166C23×C4C22×D7111yescharacteristic
167C22×C8—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—714no—
171C2×C4⋊C4—712no—
172C2×C4⋊C4—1421no—
173C2×C4⋊C4—1422no—
174C2×C4⋊C4—1422no—
175C2×C4⋊C4C4×D7112yes—
176C2×C4⋊C4C22⋊D7112yes—
177C2×C22⋊C4—1422no—
178C2×C42—1422no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181#95—712no—
182#95—1422no—
183#95—1424no—
184#95—1424no—
185#95—1424no—
186#95—1422no—
187#95—1421no—
188C24C2×D28411yescharacteristic
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—2842no—
198C22×C4—2842no—
199C22×C4—2844no—
200C22×C4—2841no—
201C22×C4—2841no—
202C22×C4—2842no—
203C22×C4—1422no—
204C22×C4—1422no—
205C22×C4—1424no—
206C22×C4(C2×D71)⋊C412yes—
207C22×C4(C2×D71)⋊C412yes—
208C22×C4C2×C4×D7111yescharacteristic
209C22×C4D4×D7112yes—
210C22×C4C2×C22⋊D7111yescharacteristic
211C22×C4#1176586412yes—
212C2×C8—44no—
213C2×C8—42no—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—2844no—
217C4⋊C4—2842no—
218C4⋊C4—1424no—
219C22⋊C4—2842no—
220C22⋊C4—2842no—
221C42—5682no—
222C23—22no—
223C23—22no—
224C23C2×(C2×D71)⋊C411yescharacteristic
225C23#1455609811yescharacteristic
226C23#1455611211yescharacteristic
227C23#1455615012yes—
228C23#1455616112yes—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—81no—
242C2×C4—2842no—
243C2×C4—2842no—
244C2×C4—2844no—
245C2×C4—2842no—
246C2×C4—2844no—
247C2×C4—2844no—
248C2×C4—2844no—
249C2×C4—2842no—
250C2×C4—5684no—
251C2×C4—5682no—
252C2×C4—5682no—
253C2×C4(C22⋊D71)⋊C412yes—
254C2×C4#1455610012yes—
255C8—82no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—24no—
261C22—24no—
262C22#1798587611yescharacteristic
263C22#1798599611yescharacteristic
264C22#1798600911yescharacteristic
265C22#1798616312yes—
266C22#1798617412yes—
267C22#1798619012yes—
268C22#1798634112yes—
269C4—22no—
270C4—22no—
271C4—42no—
272C4—42no—
273C4—5682no—
274C4—5682no—
275C4—5684no—
276C2—21no—
277C2—22no—
278C2—21no—
279C2#2258737314yes—
280C2#2258865011yescharacteristic
281C2#2258874111yescharacteristic
282C2#2258875111yescharacteristic
subgroupG/H
centerC23#14556098
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC2×#324×C71C2
socleC23×C71#120
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenterC2×#324D71
hypocenterC71#9022
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
8. #17985938C4semidirect
147. C71#9022semidirect

Maximal subgroups

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

subgroupindex
C2×#179859382
#225909432
#225884312
#225862202
C2×#324×C712
#902271