Finite Group Databasebeta

← #31893217

Subgroups — #31893217

Order 18176.

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

normal subgroups only

This group has 586 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
1#22596679C211yesmaximal, characteristic
2#22588408C211yesmaximal, characteristic
3C2×#17985735C212yesmaximal
4#22586219C212yesmaximal
5C2×#383×C71C211yesmaximal, characteristic
6C2×#14556107C2211yescharacteristic
7C2×C22⋊(C71⋊C8)C2212yes—
8#17985905C412yescomplemented
9#17985897C412yescomplemented
10#17985883—22no—
11#17985872—21no—
12#17985862—21no—
13#17985859C2211yescharacteristic
14#17985754—22no—
15#17985735—24no—
16C22×C42×C71C2211yescharacteristic
17#383×C71—24no—
18C2×#95×C71—21no—
19C2×#95×C71—21no—
20C2×#95×C71C2212yes—
21C22×Dic142C2×C411yescharacteristic
22C22×C71⋊C8—22no—
23#14556107—22no—
24#14556107—22no—
25C2×C4⋊Dic71C2×C411yescharacteristic
26C2×C4⋊Dic71C2×C412yes—
27C2×Dic71⋊C4—22no—
28C2×Dic71⋊C4—21no—
29C2×C22⋊Dic71—21no—
30C2×C22⋊Dic71—21no—
31C22⋊(C71⋊C8)—22no—
32C22⋊(C71⋊C8)—22no—
33C71⋊#95—22no—
34C71⋊#95—22no—
35C71⋊#95—22no—
36C71⋊#95—21no—
37C23×C4×C71—22no—
38C23×C4×C71D411yescharacteristic
39C23×C4×C71D411yescharacteristic
40C23×C4×C71C2311yescharacteristic
41C2×C42×C71—22no—
42C2×C42×C71—22no—
43C2×C42×C71C2×C412yes—
44C2×C42×C71D412yes—
45#95×C71—24no—
46#95×C71—24no—
47#95×C71—22no—
48#95×C71D414yes—
49C2×Dic142—22no—
50C2×Dic142—42no—
51C22×Dic71—21no—
52C22×Dic71—22no—
53C22×Dic71—21no—
54C2×C71⋊C8—44no—
55C2×C71⋊C8—42no—
56C4⋊Dic71—22no—
57C4⋊Dic71—42no—
58Dic71⋊C4—44no—
59Dic71⋊C4—42no—
60C22⋊Dic71—41no—
61C22⋊Dic71—41no—
62C22⋊Dic71—42no—
63C24×C71C2×D411yescharacteristic
64C22×C4×C71—21no—
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71—22no—
69C22×C4×C71—22no—
70C22×C4×C71—24no—
71C22×C4×C71—42no—
72C22×C4×C71—42no—
73C22×C4×C71—44no—
74C22×C4×C71—44no—
75C22×C4×C71—42no—
76C22×C4×C71—42no—
77C22×C4×C71C22⋊C412yes—
78C22×C4×C71C22⋊C412yes—
79C22×C4×C71C22×C411yescharacteristic
80C22×C4×C71C2×D412yes—
81C22×C4×C71C2×D411yescharacteristic
82C22×C4×C71#4112yes—
83C42×C71—24no—
84C42×C71—24no—
85C23×C71—22no—
86C23×C71—22no—
87C23×C71C2×C22⋊C411yescharacteristic
88C23×C71#12011yescharacteristic
89C23×C71#12311yescharacteristic
90C23×C71#13512yes—
91C23×C71#13712yes—
92C2×C4×C71—22no—
93C2×C4×C71—22no—
94C2×C4×C71—22no—
95C2×C4×C71—22no—
96C2×C4×C71—22no—
97C2×C4×C71—22no—
98C2×C4×C71—22no—
99C2×C4×C71—22no—
100C2×C4×C71—42no—
101C2×C4×C71—42no—
102C2×C4×C71—42no—
103C2×C4×C71—42no—
104C2×C4×C71—42no—
105C2×C4×C71—42no—
106C2×C4×C71—44no—
107C2×C4×C71—44no—
108C2×C4×C71—44no—
109C2×C4×C71—84no—
110C2×C4×C71—84no—
111C2×C4×C71C4×D412yes—
112C2×C4×C71#12112yes—
113C2×Dic71—41no—
114C2×Dic71—41no—
115C2×Dic71—42no—
116C2×Dic71—42no—
117C2×Dic71—42no—
118C2×Dic71—42no—
119C2×Dic71—42no—
120Dic142—82no—
121C71⋊C8—82no—
122C22×C71—21no—
123C22×C71—21no—
124C22×C71—21no—
125C22×C71—21no—
126C22×C71—24no—
127C22×C71—24no—
128C22×C71#38611yescharacteristic
129C22×C71#41012yes—
130C22×C71#41611yescharacteristic
131C22×C71#41911yescharacteristic
132C22×C71#42112yes—
133C22×C71#45212yes—
134C22×C71#48512yes—
135C4×C71—22no—
136C4×C71—22no—
137C4×C71—42no—
138C4×C71—42no—
139C4×C71—84no—
140Dic71—81no—
141Dic71—81no—
142Dic71—82no—
143#9153—711nomaximal, Sylow
144C2×C71—22no—
145C2×C71—22no—
146C2×C71#151512yes—
147C2×C71#152112yes—
148C2×C71#187311yescharacteristic
149C2×C71#189711yescharacteristic
150C2×C71#189911yescharacteristic
151#2855—711no—
152C2×#383D7111yescharacteristic
153#1817—711no—
154C2×#328—712no—
155#1294—712no—
156C71#915311yescharacteristic, complemented, Sylow
157C2×C22⋊Q8—711no—
158C22×C42C2×D7111yescharacteristic
159C2×C22⋊C8—712no—
160#391—712no—
161#389—712no—
162#387—1422no—
163#385—1421no—
164#383—24no—
165#381—1421no—
166#380—711no—
167C2×#95—21no—
168C2×#95—21no—
169C2×#95C2×D7112yes—
170#336—1422no—
171#328—1424no—
172C22×Q8—711no—
173C23×C4—22no—
174C23×C4C22⋊D7111yescharacteristic
175C23×C4C22⋊D7111yescharacteristic
176C23×C4C22×D7111yescharacteristic
177C22×C8—1422no—
178C22⋊Q8—1422no—
179C22⋊Q8—1422no—
180C2×C4⋊C4—712no—
181C2×C4⋊C4—711no—
182C2×C4⋊C4—1422no—
183C2×C4⋊C4—1421no—
184C2×C22⋊C4—1421no—
185C2×C22⋊C4—1421no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42C4×D7112yes—
189C2×C42D28412yes—
190C22⋊C8—1422no—
191C22⋊C8—1422no—
192#95—24no—
193#95—24no—
194#95—22no—
195#95—1422no—
196#95—1422no—
197#95—1422no—
198#95—1421no—
199#95C22⋊D7114yes—
200C24C2×C22⋊D7111yescharacteristic
201C2×Q8—2842no—
202C2×Q8—1422no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—1422no—
217C22×C4—1421no—
218C22×C4—1421no—
219C22×C4(C2×D71)⋊C412yes—
220C22×C4(C2×D71)⋊C412yes—
221C22×C4C2×C4×D7111yescharacteristic
222C22×C4C2×C22⋊D7112yes—
223C22×C4C2×D28411yescharacteristic
224C22×C4#1176586512yes—
225C2×C8—2844no—
226C2×C8—2842no—
227C4⋊C4—2842no—
228C4⋊C4—2844no—
229C4⋊C4—2842no—
230C4⋊C4—1422no—
231C22⋊C4—2841no—
232C22⋊C4—2841no—
233C22⋊C4—2842no—
234C42—24no—
235C42—24no—
236C23—22no—
237C23—22no—
238C23C2×(C2×D71)⋊C411yescharacteristic
239C23#1455609711yescharacteristic
240C23#1455611111yescharacteristic
241C23#1455614812yes—
242C23#1455615812yes—
243Q8—5682no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—84no—
262C2×C4—84no—
263C2×C4—2841no—
264C2×C4—2841no—
265C2×C4—2842no—
266C2×C4—2842no—
267C2×C4—2842no—
268C2×C4—2842no—
269C2×C4—2842no—
270C2×C4C4×C22⋊D7112yes—
271C2×C4#1455610212yes—
272C8—5682no—
273C22—24no—
274C22—24no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22#1798587811yescharacteristic
280C22#1798597212yes—
281C22#1798599311yescharacteristic
282C22#1798600711yescharacteristic
283C22#1798601712yes—
284C22#1798616112yes—
285C22#1798634012yes—
286C4—22no—
287C4—22no—
288C4—42no—
289C4—42no—
290C4—84no—
291C4—5682no—
292C4—5681no—
293C4—5681no—
294C2—22no—
295C2—22no—
296C2#2258689012yes—
297C2#2258691512yes—
298C2#2258865211yescharacteristic
299C2#2258878811yescharacteristic
300C2#2258879811yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC2×#383×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenterC2×#383D71
hypocenterC71#9153
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#383

Complements

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

NH ≅ G/Ntype
8. #17985905C4semidirect
9. #17985897C4semidirect
156. C71#9153semidirect

Maximal subgroups

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

subgroupindex
#225966792
#225884082
C2×#179857352
#225862192
C2×#383×C712
#915371