Finite Group Databasebeta

← #31892446

Subgroups — #31892446

Order 18176.

756 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 131 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×#17985919C211yesmaximal, characteristic
2#22588274C212yesmaximal
3C2×#17985725C211yesmaximal, characteristic
4#22586218C212yesmaximal
5C71×#2313C211yesmaximal, characteristic
6C23×Dic142C2211yescharacteristic
7C22×Dic71⋊C4—21no—
8C2×#14556051—24no—
9C2×C22⋊(C71⋊C8)C2212yes—
10#17985919—22no—
11#17985919C414yescomplemented
12C2×C71⋊#95—21no—
13C2×C71⋊#95C2211yescharacteristic
14#17985725—22no—
15C22×C4⋊C4×C71C2211yescharacteristic
16#387×C71—24no—
17#381×C71—21no—
18#380×C71C2212yes—
19C4×C22⋊C4×C71—21no—
20C22×Dic142—21no—
21C22×Dic142—22no—
22C22×Dic142—22no—
23C22×Dic142—44no—
24C22×Dic142C2×C412yes—
25C23×Dic71—21no—
26C23×Dic71—21no—
27C22×C71⋊C8—22no—
28C2×Dic71⋊C4—24no—
29C2×Dic71⋊C4—44no—
30#14556051—48no—
31C22⋊(C71⋊C8)—22no—
32C22⋊(C71⋊C8)—22no—
33C71⋊#95—24no—
34C71⋊#95—21no—
35C71⋊#95C2×C412yes—
36C23×C4×C71D411yescharacteristic
37C23×C4×C71D411yescharacteristic
38C23×C4×C71C2311yescharacteristic
39C2×C4⋊C4×C71—22no—
40C2×C4⋊C4×C71—22no—
41C2×C4⋊C4×C71C2×C412yes—
42C2×C4⋊C4×C71D412yes—
43C2×C4⋊C4×C71D414yes—
44C2×C22⋊C4×C71—22no—
45C2×C42×C71—22no—
46#95×C71—22no—
47#95×C71—24no—
48#95×C71—22no—
49C2×Dic142—28no—
50C2×Dic142—24no—
51C2×Dic142—44no—
52C2×Dic142—48no—
53C2×Dic142—44no—
54C2×Dic142—44no—
55C2×Dic142—42no—
56C22×Dic71—24no—
57C22×Dic71—24no—
58C22×Dic71—22no—
59C22×Dic71—22no—
60C22×Dic71—42no—
61C22×Dic71—42no—
62C22×Dic71—41no—
63C22×Dic71—41no—
64C22×Dic71—42no—
65C2×C71⋊C8—44no—
66C2×C71⋊C8—42no—
67Dic71⋊C4—48no—
68C24×C71C2×D411yescharacteristic
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—22no—
73C22×C4×C71—22no—
74C22×C4×C71—21no—
75C22×C4×C71—24no—
76C22×C4×C71—42no—
77C22×C4×C71—42no—
78C22×C4×C71C22⋊C412yes—
79C22×C4×C71C22⋊C412yes—
80C22×C4×C71C22×C411yescharacteristic
81C22×C4×C71C2×D412yes—
82C22×C4×C71C2×D411yescharacteristic
83C22×C4×C71#4112yes—
84C4⋊C4×C71—24no—
85C4⋊C4×C71—24no—
86C4⋊C4×C71—44no—
87C22⋊C4×C71—42no—
88C22⋊C4×C71—42no—
89C42×C71—82no—
90C23×C71—22no—
91C23×C71—22no—
92C23×C71C2×C22⋊C411yescharacteristic
93C23×C71#12011yescharacteristic
94C23×C71#12311yescharacteristic
95C23×C71#13512yes—
96C23×C71#13712yes—
97C2×C4×C71—24no—
98C2×C4×C71—22no—
99C2×C4×C71—22no—
100C2×C4×C71—22no—
101C2×C4×C71—22no—
102C2×C4×C71—41no—
103C2×C4×C71—41no—
104C2×C4×C71—42no—
105C2×C4×C71—42no—
106C2×C4×C71—42no—
107C2×C4×C71—42no—
108C2×C4×C71—42no—
109C2×C4×C71—44no—
110C2×C4×C71—44no—
111C2×C4×C71—44no—
112C2×C4×C71—81no—
113C2×C4×C71C4×D412yes—
114C2×C4×C71#12112yes—
115C2×Dic71—24no—
116C2×Dic71—24no—
117C2×Dic71—42no—
118C2×Dic71—48no—
119C2×Dic71—42no—
120C2×Dic71—44no—
121C2×Dic71—42no—
122C2×Dic71—42no—
123C2×Dic71—44no—
124Dic142—28no—
125Dic142—48no—
126Dic142—44no—
127C71⋊C8—82no—
128C22×C71—24no—
129C22×C71—24no—
130C22×C71—21no—
131C22×C71—21no—
132C22×C71—21no—
133C22×C71—21no—
134C22×C71#38611yescharacteristic
135C22×C71#41611yescharacteristic
136C22×C71#41911yescharacteristic
137C22×C71#45212yes—
138C22×C71#45612yes—
139C22×C71B(4,𝔽2)12yes—
140C22×C71#48512yes—
141C4×C71—22no—
142C4×C71—22no—
143C4×C71—42no—
144C4×C71—42no—
145C4×C71—84no—
146Dic71—44no—
147Dic71—44no—
148#9014—711nomaximal, Sylow
149C2×C71—21no—
150C2×C71—21no—
151C2×C71—22no—
152C2×C71#160314yes—
153C2×C71#187311yescharacteristic
154C2×C71#188611yescharacteristic
155C2×C71#188911yescharacteristic
156C2×#395—711no—
157#2313D7111yescharacteristic
158#1787—712no—
159C2×#324—711no—
160#1294—712no—
161C71#901411yescharacteristic, complemented, Sylow
162C23×Q8—711no—
163C22×C4⋊C4—1421no—
164C22×C4⋊C4C2×D7111yescharacteristic
165C2×#103—1424no—
166C2×C22⋊C8—712no—
167#395—714no—
168#395—1422no—
169#387—24no—
170#381—21no—
171#380C2×D7112yes—
172C4×C22⋊C4—21no—
173C2×#95—711no—
174C2×#95—1421no—
175#324—1422no—
176C22×Q8—2844no—
177C22×Q8—712no—
178C22×Q8—1422no—
179C22×Q8—1422no—
180C22×Q8—1421no—
181C23×C4—1421no—
182C23×C4—1421no—
183C23×C4C22⋊D7111yescharacteristic
184C23×C4C22⋊D7111yescharacteristic
185C23×C4C22×D7111yescharacteristic
186C22×C8—1422no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—2844no—
190C2×C4⋊C4—1424no—
191C2×C4⋊C4C4×D7112yes—
192C2×C4⋊C4D28412yes—
193C2×C4⋊C4C22⋊D7114yes—
194C2×C22⋊C4—22no—
195C2×C42—22no—
196#103—2848no—
197C22⋊C8—1422no—
198C22⋊C8—1422no—
199#95—22no—
200#95—24no—
201#95—22no—
202#95—712no—
203#95—1424no—
204#95—1421no—
205C24C2×C22⋊D7111yescharacteristic
206C2×Q8—2844no—
207C2×Q8—2848no—
208C2×Q8—2844no—
209C2×Q8—2844no—
210C2×Q8—2842no—
211C2×Q8—1428no—
212C2×Q8—1424no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—2842no—
223C22×C4—2842no—
224C22×C4—2841no—
225C22×C4—2841no—
226C22×C4—2842no—
227C22×C4—1424no—
228C22×C4—1424no—
229C22×C4—1422no—
230C22×C4—1422no—
231C22×C4(C2×D71)⋊C412yes—
232C22×C4(C2×D71)⋊C412yes—
233C22×C4C2×C4×D7111yescharacteristic
234C22×C4C2×C22⋊D7112yes—
235C22×C4C2×D28411yescharacteristic
236C22×C4#1176586512yes—
237C2×C8—2844no—
238C2×C8—2842no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—44no—
242C4⋊C4—2848no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C42—82no—
246C23—22no—
247C23—22no—
248C23C2×(C2×D71)⋊C411yescharacteristic
249C23#1455609711yescharacteristic
250C23#1455611111yescharacteristic
251C23#1455614812yes—
252C23#1455615812yes—
253Q8—2848no—
254Q8—2844no—
255Q8—1428no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—81no—
272C2×C4—2842no—
273C2×C4—2842no—
274C2×C4—2844no—
275C2×C4—2848no—
276C2×C4—2842no—
277C2×C4—2844no—
278C2×C4—2842no—
279C2×C4—1424no—
280C2×C4—1424no—
281C2×C4C4×C22⋊D7112yes—
282C2×C4#1455610212yes—
283C8—5682no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—24no—
289C22—24no—
290C22#1798587811yescharacteristic
291C22#1798599311yescharacteristic
292C22#1798600711yescharacteristic
293C22#1798616112yes—
294C22#1798618212yes—
295C22#1798618612yes—
296C22#1798634012yes—
297C4—22no—
298C4—22no—
299C4—42no—
300C4—42no—
301C4—84no—
302C4—2844no—
303C4—2844no—
304C2—21no—
305C2—22no—
306C2—21no—
307C2#2258730314yes—
308C2#2258865211yescharacteristic
309C2#2258873311yescharacteristic
310C2#2258874811yescharacteristic
subgroupG/H
centerC23#14556097
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2313C2
socleC23×C71#120
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenter#2313D71
hypocenterC71#9014
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
11. #17985919C4semidirect
161. C71#9014semidirect

Maximal subgroups

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

subgroupindex
C2×#179859192
#225882742
C2×#179857252
#225862182
C71×#23132
#901471