Finite Group Databasebeta

← #31900965

Subgroups — #31900965

Order 18176.

630 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 133 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#22588496C211yesmaximal, characteristic, complemented
2#22588340C212yesmaximal, complemented
3C2×D4⋊(C71⋊C8)C212yesmaximal, complemented
4#22586223C211yesmaximal, characteristic, complemented
5C71×#2537C211yesmaximal, characteristic
6C2×#14556059—22no—
7C2×C4⋊(C71⋊C8)C2212yes—
8C2×#14556049—22no—
9C2×C4×C71⋊C8C2211yescharacteristic
10#17985869C2211yescharacteristic
11#17985761—22no—
12D4⋊(C71⋊C8)—24no—
13C2×#121×C71—22no—
14C2×C4×D4×C71C2212yes—
15#394×C71—22no—
16#384×C71C2211yescharacteristic
17#380×C71—22no—
18C22×C71⋊C8—21no—
19C22×C71⋊C8—22no—
20C2×C4⋊Dic71—22no—
21C2×C4⋊Dic71—21no—
22#14556059—44no—
23C4⋊(C71⋊C8)—22no—
24C4⋊(C71⋊C8)—22no—
25#14556049—44no—
26C4×C71⋊C8—21no—
27C4×C71⋊C8D412yescomplemented
28C22×D4×C71—22no—
29C22×D4×C71D412yes—
30C23×C4×C71—22no—
31#121×C71—44no—
32#121×C71—44no—
33C4×D4×C71—22no—
34C4×D4×C71—22no—
35C4×D4×C71—22no—
36C4×D4×C71—22no—
37C2×C4⋊C4×C71—22no—
38C2×C4⋊C4×C71D412yes—
39C2×C4⋊C4×C71D412yes—
40C2×C22⋊C4×C71—24no—
41C2×C22⋊C4×C71—22no—
42C2×C42×C71C2311yescharacteristic
43#95×C71—22no—
44C22×Dic71—41no—
45C2×C71⋊C8—22no—
46C2×C71⋊C8—22no—
47C2×C71⋊C8—22no—
48C2×C71⋊C8—42no—
49C2×C71⋊C8—42no—
50C2×C71⋊C8—42no—
51C4⋊Dic71—42no—
52C4⋊Dic71—42no—
53C4⋊Dic71—42no—
54C24×C71—22no—
55C2×D4×C71—22no—
56C2×D4×C71—22no—
57C2×D4×C71—22no—
58C2×D4×C71—22no—
59C2×D4×C71—22no—
60C2×D4×C71—48no—
61C2×D4×C71—82no—
62C2×D4×C71SD1614yes—
63C22×C4×C71—22no—
64C22×C4×C71—22no—
65C22×C4×C71—42no—
66C22×C4×C71—42no—
67C22×C4×C71—42no—
68C22×C4×C71—42no—
69C22×C4×C71—42no—
70C22×C4×C71—42no—
71C22×C4×C71C2×D411yescharacteristic
72C22×C4×C71C2×D412yes—
73C4⋊C4×C71—22no—
74C4⋊C4×C71—22no—
75C4⋊C4×C71—42no—
76C4⋊C4×C71—42no—
77C4⋊C4×C71—42no—
78C22⋊C4×C71—48no—
79C22⋊C4×C71—42no—
80C22⋊C4×C71—42no—
81C42×C71C2×D411yescharacteristic
82C42×C71#4111yescharacteristic
83C42×C71#4112yes—
84C23×C71—24no—
85C23×C71—42no—
86C23×C71—42no—
87C23×C71—42no—
88C23×C71—42no—
89C23×C71—42no—
90C23×C71—42no—
91C23×C71#12011yescharacteristic
92D4×C71—24no—
93D4×C71—24no—
94D4×C71—42no—
95D4×C71—42no—
96D4×C71—88no—
97C2×C4×C71—21no—
98C2×C4×C71—21no—
99C2×C4×C71—42no—
100C2×C4×C71—42no—
101C2×C4×C71—42no—
102C2×C4×C71—44no—
103C2×C4×C71—42no—
104C2×C4×C71—82no—
105C2×C4×C71—82no—
106C2×C4×C71—82no—
107C2×C4×C71—82no—
108C2×C4×C71#12111yescharacteristic
109C2×C4×C71#12112yes—
110C2×C4×C71#12312yes—
111C2×C4×C71#12411yescharacteristic
112C2×C4×C71C2×SD1612yes—
113C2×C4×C71#13512yes—
114C2×C4×C71AGL(1,ℤ/8)12yes—
115C2×C4×C71AGL(1,ℤ/8)12yes—
116C2×Dic71—82no—
117C2×Dic71—81no—
118C71⋊C8—24no—
119C71⋊C8—82no—
120C22×C71—44no—
121C22×C71—44no—
122C22×C71—44no—
123C22×C71—82no—
124C22×C71—82no—
125C22×C71—82no—
126C22×C71—82no—
127C22×C71#39411yescharacteristic
128C22×C71#44912yes—
129C22×C71#45012yes—
130C22×C71#45311yescharacteristic
131C22×C71#45611yescharacteristic
132C4×C71—21no—
133C4×C71—21no—
134C4×C71—82no—
135C4×C71—82no—
136C4×C71#46512yes—
137C4×C71#46912yes—
138C4×C71#48112yes—
139C4×C71#48212yes—
140Dic71—161no—
141#10432—711nomaximal, Sylow
142C2×C71—84no—
143C2×C71#162411yescharacteristic
144C2×C71#162911yescharacteristic
145C2×C71#165412yes—
146C2×C71#203412yes—
147C2×C71#203911yescharacteristic
148#2537D7111yescharacteristic
149#1844—711no—
150#1802—712no—
151C2×D4⋊C8—712no—
152#1296—711no—
153C71#1043211yescharacteristic, complemented, Sylow
154C2×#121—22no—
155C2×C4×D4C2×D7112yes—
156C2×#106—1422no—
157C2×C4⋊C8—712no—
158C2×#102—1422no—
159C2×C4×C8—711no—
160#394—22no—
161#384—711no—
162#384C2×D7111yescharacteristic
163#380—22no—
164#340—1422no—
165D4⋊C8—1424no—
166C22×D4—22no—
167C22×D4C22⋊D7112yes—
168C23×C4—22no—
169C22×C8—1422no—
170C22×C8—1421no—
171#121—44no—
172#121—44no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—1422no—
179C2×C4⋊C4—1421no—
180C2×C4⋊C4C22⋊D7112yes—
181C2×C4⋊C4C22⋊D7112yes—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—22no—
184C2×C42C22×D7111yescharacteristic
185#106—2844no—
186C4⋊C8—1422no—
187C4⋊C8—1422no—
188#102—2844no—
189C4×C8—712no—
190C4×C8—1421no—
191#95—22no—
192C24—22no—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—48no—
199C2×D4—82no—
200C2×D4#1176585314yes—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—2841no—
210C22×C4C2×C22⋊D7111yescharacteristic
211C22×C4C2×C22⋊D7112yes—
212C2×C8—2842no—
213C2×C8—2842no—
214C2×C8—2842no—
215C2×C8—1422no—
216C2×C8—1422no—
217C2×C8—1422no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—2842no—
224C4⋊C4—2842no—
225C4⋊C4—2842no—
226C22⋊C4—48no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C42D4×D7111yescharacteristic
230C42#1176586411yescharacteristic
231C42#1176586412yes—
232C23—24no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23#1455609711yescharacteristic
240D4—24no—
241D4—24no—
242D4—42no—
243D4—42no—
244D4—88no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—5682no—
257C2×C4—5681no—
258C2×C4#1455609911yescharacteristic
259C2×C4#1455610012yes—
260C2×C4#1455610912yes—
261C2×C4#1455611511yescharacteristic
262C2×C4C2×#1176585312yes—
263C2×C4#1455614812yes—
264C2×C4#1455615312yes—
265C2×C4#1455615412yes—
266C8—5682no—
267C8—1424no—
268C22—44no—
269C22—44no—
270C22—44no—
271C22—82no—
272C22—82no—
273C22—82no—
274C22—82no—
275C22#1798591311yescharacteristic
276C22#1798614412yes—
277C22#1798615012yes—
278C22#1798616711yescharacteristic
279C22#1798618211yescharacteristic
280C4—21no—
281C4—21no—
282C4—82no—
283C4—82no—
284C4—11361no—
285C4#1798622612yes—
286C4#1798625012yes—
287C4#1798631512yes—
288C4#1798632012yes—
289C2—84no—
290C2#2258742311yescharacteristic
291C2#2258745611yescharacteristic
292C2#2258762412yes—
293C2#2258952612yes—
294C2#2258956111yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
Fitting, prime radicalC71×#2537C2
socleC23×C71#120
hypercenter#2537D71
hypocenterC71#10432
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2537

Complements

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

NH ≅ G/Ntype
1. #22588496C2semidirect
2. #22588340C2semidirect
3. C2×D4⋊(C71⋊C8)C2semidirect
4. #22586223C2semidirect
27. C4×C71⋊C8D4semidirect
153. C71#10432semidirect

Maximal subgroups

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

subgroupindex
#225884962
#225883402
C2×D4⋊(C71⋊C8)2
#225862232
C71×#25372
#1043271