Finite Group Databasebeta

← #31892487

Subgroups — #31892487

Order 18176.

572 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 135 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#22590943C211yesmaximal, characteristic
2#22588430C212yesmaximal
3C2×#17985725C211yesmaximal, characteristic
4#22586219C212yesmaximal
5C2×#400×C71C211yesmaximal, characteristic
6C2×C22⋊(C71⋊C8)C2212yes—
7#17985885—24no—
8#17985862—21no—
9#17985859C2212yes—
10C4×C22⋊Dic71—21no—
11#17985761—24no—
12#17985725—22no—
13C22×C4⋊C4×C71C2211yescharacteristic
14C22×C4⋊C4×C71C2211yescharacteristic
15#400×C71—22no—
16#400×C71C414yescomplemented
17C2×#95×C71—21no—
18C2×#95×C71—21no—
19C2×#95×C71C2211yescharacteristic
20C22×C71⋊C8—22no—
21C2×C4⋊Dic71D414yes—
22C2×C22⋊Dic71—22no—
23C2×C4×Dic71—22no—
24C22⋊(C71⋊C8)—22no—
25C22⋊(C71⋊C8)—22no—
26C71⋊#95—22no—
27C71⋊#95—24no—
28C71⋊#95—22no—
29C23×C4×C71—21no—
30C23×C4×C71—21no—
31C23×C4×C71D411yescharacteristic
32C23×C4×C71D411yescharacteristic
33C23×C4×C71C2311yescharacteristic
34C2×C4⋊C4×C71—22no—
35C2×C4⋊C4×C71—22no—
36C2×C4⋊C4×C71—22no—
37C2×C4⋊C4×C71—22no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71C2×C412yes—
40C2×C4⋊C4×C71C2×C412yes—
41C2×C4⋊C4×C71D412yes—
42#95×C71—24no—
43#95×C71—24no—
44#95×C71—21no—
45#95×C71C2×C412yes—
46C22×Dic71—24no—
47C2×C71⋊C8—44no—
48C2×C71⋊C8—42no—
49C4⋊Dic71—44no—
50C22⋊Dic71—42no—
51C22⋊Dic71—42no—
52C4×Dic71—82no—
53C24×C71C2×D411yescharacteristic
54C22×C4×C71—21no—
55C22×C4×C71—21no—
56C22×C4×C71—21no—
57C22×C4×C71—22no—
58C22×C4×C71—22no—
59C22×C4×C71—21no—
60C22×C4×C71—22no—
61C22×C4×C71—22no—
62C22×C4×C71—42no—
63C22×C4×C71—42no—
64C22×C4×C71—42no—
65C22×C4×C71—42no—
66C22×C4×C71—44no—
67C22×C4×C71—41no—
68C22×C4×C71—41no—
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—
76C4⋊C4×C71—24no—
77C4⋊C4×C71—24no—
78C4⋊C4×C71—24no—
79C4⋊C4×C71—42no—
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—42no—
100C2×C4×C71—42no—
101C2×C4×C71—44no—
102C2×C4×C71—42no—
103C2×C4×C71—42no—
104C2×C4×C71—81no—
105C2×C4×C71—84no—
106C2×C4×C71—82no—
107C2×C4×C71—82no—
108C2×C4×C71C4×D412yes—
109C2×C4×C71#12112yes—
110C2×Dic71—44no—
111C2×Dic71—44no—
112C2×Dic71—44no—
113C71⋊C8—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—
131C4×C71—82no—
132C4×C71—82no—
133Dic71—84no—
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×#400D7111yescharacteristic
143#2313—711no—
144#1824—712no—
145C2×#324—711no—
146#1294—712no—
147C71#902211yescharacteristic, complemented, Sylow
148C22×C4⋊C4C2×D7111yescharacteristic
149C22×C4⋊C4C2×D7111yescharacteristic
150C2×C22⋊C8—712no—
151#400—22no—
152#400Dic7114yescomplemented
153#387—1424no—
154#381—1421no—
155#380—712no—
156C4×C22⋊C4—1421no—
157C2×#95—21no—
158C2×#95—21no—
159C2×#95C2×D7111yescharacteristic
160#340—1424no—
161#324—1422no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4C22⋊D7111yescharacteristic
165C23×C4C22⋊D7111yescharacteristic
166C23×C4C22×D7111yescharacteristic
167C22×C8—1422no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—714no—
174C2×C4⋊C4C2×Dic7112yes—
175C2×C4⋊C4C2×Dic7112yes—
176C2×C4⋊C4C22⋊D7112yes—
177C2×C22⋊C4—1422no—
178C2×C42—1422no—
179C22⋊C8—1422no—
180C22⋊C8—1422no—
181#95—24no—
182#95—24no—
183#95—21no—
184#95—1422no—
185#95—1424no—
186#95—1422no—
187#95C2×Dic7112yes—
188C24C2×C22⋊D7111yescharacteristic
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—1424no—
206C22×C4C22⋊Dic7112yes—
207C22×C4C22⋊Dic7112yes—
208C22×C4C22×Dic7111yescharacteristic
209C22×C4D4×D7112yes—
210C22×C4C2×C22⋊D7111yescharacteristic
211C22×C4#1176586412yes—
212C2×C8—2844no—
213C2×C8—2842no—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—42no—
218C4⋊C4—2844no—
219C22⋊C4—2842no—
220C22⋊C4—2842no—
221C42—5682no—
222C23—22no—
223C23—22no—
224C23C2×C22⋊Dic7111yescharacteristic
225C23#1455609611yescharacteristic
226C23#1455610911yescharacteristic
227C23#1455614812yes—
228C23#1455615912yes—
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—42no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—81no—
247C2×C4—84no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—2844no—
251C2×C4—2844no—
252C2×C4—2844no—
253C2×C4D4×Dic7112yes—
254C2×C4#1455610012yes—
255C8—5682no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—24no—
261C22—24no—
262C22#1798587911yescharacteristic
263C22#1798599511yescharacteristic
264C22#1798600811yescharacteristic
265C22#1798615912yes—
266C22#1798617112yes—
267C22#1798618812yes—
268C22#1798633712yes—
269C4—22no—
270C4—22no—
271C4—42no—
272C4—42no—
273C4—82no—
274C4—82no—
275C4—5684no—
276C2—21no—
277C2—22no—
278C2—21no—
279C2#2258736914yes—
280C2#2258865311yescharacteristic
281C2#2258874411yescharacteristic
282C2#2258875411yescharacteristic
subgroupG/H
centerC23#14556096
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC2×#400×C71C2
socleC23×C71#120
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenterC2×#400D71
hypocenterC71#9022
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
16. #400×C71C4semidirect
147. C71#9022semidirect
152. #400Dic71semidirect

Maximal subgroups

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

subgroupindex
#225909432
#225884302
C2×#179857252
#225862192
C2×#400×C712
#902271