Finite Group Databasebeta

← #31895141

Subgroups — #31895141

Order 18176.

498 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 107 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#22593513C211yesmaximal, characteristic
2#22587056C212yesmaximal, complemented
3#22586681C211yesmaximal, characteristic
4#22586197C212yesmaximal, complemented
5C71×#2973C211yesmaximal, characteristic
6C2×#14556128C2211yescharacteristic
7#17986056C2212yes—
8#17985911—21no—
9#17985911—21no—
10#17985902C412yes—
11#17985864C412yes—
12#17985859—21no—
13#17985859—21no—
14C4×C4⋊Dic71C2211yescharacteristic
15#17985744—24no—
16#17985723—21no—
17#17985723—21no—
18C2×C4×D4×C71C2211yescharacteristic
19#435×C71—24no—
20#432×C71C2212yes—
21#407×C71—21no—
22C2×C22⋊C8×C71—21no—
23C22×Dic142—21no—
24#14556128—22no—
25#14556128—22no—
26C2×C4⋊Dic71—21no—
27C2×C4⋊Dic71—21no—
28C2×C4⋊Dic71C2×C412yes—
29C2×C22⋊Dic71—22no—
30C2×C22⋊Dic71—22no—
31C2×C4×Dic71C2×C412yes—
32C4⋊(C71⋊C8)—22no—
33C22⋊(C71⋊C8)—22no—
34#14556039—22no—
35C71⋊#95—22no—
36C71⋊#95—22no—
37C71⋊#95—21no—
38C22×D4×C71C2×C411yescharacteristic
39C23×C4×C71D411yescharacteristic
40C23×C4×C71D411yescharacteristic
41C2×M16×C71—22no—
42C22×C8×C71—22no—
43C4×D4×C71—22no—
44C4×D4×C71—22no—
45C2×C4⋊C4×C71C2×C411yescharacteristic
46C2×C22⋊C4×C71D411yescharacteristic
47C2×C22⋊C4×C71D411yescharacteristic
48C2×C42×C71C2311yescharacteristic
49C4⋊C8×C71D414yes—
50C22⋊C8×C71—22no—
51C22⋊C8×C71—22no—
52C22⋊C8×C71—22no—
53#97×C71—22no—
54C2×Dic142—42no—
55C2×Dic142—82no—
56C22×Dic71—22no—
57C22×Dic71—22no—
58C2×C71⋊C8—42no—
59C4⋊Dic71—24no—
60C4⋊Dic71—42no—
61C4⋊Dic71—41no—
62C4⋊Dic71—41no—
63C22⋊Dic71—82no—
64C22⋊Dic71—82no—
65C4×Dic71—42no—
66C24×C71C22⋊C411yescharacteristic
67C24×C71C22⋊C411yescharacteristic
68C2×D4×C71—22no—
69C2×D4×C71—22no—
70C2×D4×C71—41no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—42no—
76C22×C4×C71—42no—
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22×C411yescharacteristic
80C22×C4×C71C2×D411yescharacteristic
81C22×C4×C71C2×D411yescharacteristic
82M16×C71—82no—
83C2×C8×C71—24no—
84C2×C8×C71—82no—
85C4⋊C4×C71—22no—
86C22⋊C4×C71—22no—
87C22⋊C4×C71—22no—
88C42×C71C2×D412yes—
89C42×C71#4112yes—
90C23×C71—21no—
91C23×C71—21no—
92C23×C71—21no—
93C23×C71—21no—
94C23×C71—21no—
95C23×C71—41no—
96C23×C71—41no—
97C23×C71—41no—
98C23×C71—41no—
99C23×C71#10112yes—
100C23×C71C2×C22⋊C411yescharacteristic
101D4×C71—44no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—22no—
105C2×C4×C71—21no—
106C2×C4×C71—21no—
107C2×C4×C71—41no—
108C2×C4×C71—42no—
109C2×C4×C71—42no—
110C2×C4×C71—41no—
111C2×C4×C71—41no—
112C2×C4×C71—81no—
113C2×C4×C71—81no—
114C2×C4×C71C4×D412yes—
115C2×C4×C71#12011yescharacteristic
116C2×C4×C71#12112yes—
117C2×C4×C71#12311yescharacteristic
118C2×Dic71—42no—
119C2×Dic71—42no—
120C2×Dic71—42no—
121C2×Dic71—42no—
122C2×Dic71—42no—
123C2×Dic71—42no—
124Dic142—84no—
125C8×C71—44no—
126C71⋊C8—82no—
127C22×C71—21no—
128C22×C71—21no—
129C22×C71—22no—
130C22×C71—42no—
131C22×C71—42no—
132C22×C71—41no—
133C22×C71—41no—
134C22×C71—41no—
135C22×C71—41no—
136C22×C71—41no—
137C22×C71—81no—
138C22×C71—81no—
139C22×C71#38611yescharacteristic
140C22×C71C2×#10111yescharacteristic
141C22×C71#41311yescharacteristic
142C4×C71—21no—
143C4×C71—22no—
144C4×C71—22no—
145C4×C71—41no—
146C4×C71—41no—
147C4×C71#45512yes—
148Dic71—82no—
149Dic71—82no—
150#9453—711nomaximal, Sylow
151C2×C71—21no—
152C2×C71—21no—
153C2×C71—42no—
154C2×C71—81no—
155C2×C71#188511yescharacteristic
156C2×C71#190111yescharacteristic
157C2×C71#191011yescharacteristic
158#2973D7111yescharacteristic
159#2563—711no—
160#1556—712no—
161#1466—711no—
162#1285—712no—
163C71#945311yescharacteristic, complemented, Sylow
164C2×C4⋊Q8—711no—
165C2×C4×D4C2×D7111yescharacteristic
166#435—24no—
167#432—712no—
168#432C2×D7112yes—
169#407—21no—
170C2×C22⋊C8—21no—
171#393—1421no—
172#393—1421no—
173#390—712no—
174#382—712no—
175#380—1421no—
176#380—1421no—
177C4×C4⋊C4—711no—
178#332—1424no—
179#323—1421no—
180#323—1421no—
181C22×Q8—1421no—
182C22×D4C4×D7111yescharacteristic
183C23×C4C22⋊D7111yescharacteristic
184C23×C4C22⋊D7111yescharacteristic
185C2×M16—22no—
186C22×C8—22no—
187C4⋊Q8—1422no—
188C4⋊Q8—1422no—
189C4×D4—22no—
190C4×D4—22no—
191C2×C4⋊C4—712no—
192C2×C4⋊C4—1421no—
193C2×C4⋊C4—1421no—
194C2×C4⋊C4C4×D7111yescharacteristic
195C2×C22⋊C4—1422no—
196C2×C22⋊C4—1422no—
197C2×C22⋊C4D28411yescharacteristic
198C2×C22⋊C4D28411yescharacteristic
199C2×C42—712no—
200C2×C42C22×D7111yescharacteristic
201C4⋊C8—1422no—
202C4⋊C8C22⋊D7114yes—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—1422no—
207#97—22no—
208#97—1422no—
209#95—1422no—
210#95—1422no—
211#95—1421no—
212C24(C2×D71)⋊C411yescharacteristic
213C24(C2×D71)⋊C411yescharacteristic
214C2×Q8—2842no—
215C2×Q8—5682no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—41no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—1422no—
226C22×C4—1422no—
227C22×C4(C2×D71)⋊C411yescharacteristic
228C22×C4(C2×D71)⋊C411yescharacteristic
229C22×C4C2×C4×D7111yescharacteristic
230C22×C4C2×C22⋊D7111yescharacteristic
231C22×C4C2×D28411yescharacteristic
232M16—82no—
233C2×C8—24no—
234C2×C8—82no—
235C2×C8—2842no—
236C4⋊C4—22no—
237C4⋊C4—2842no—
238C4⋊C4—2841no—
239C4⋊C4—2841no—
240C4⋊C4—1424no—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—5682no—
244C22⋊C4—5682no—
245C42—2842no—
246C42C2×C22⋊D7112yes—
247C42#1176586512yes—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23#1455604612yes—
258C23C2×(C2×D71)⋊C411yescharacteristic
259Q8—5684no—
260D4—44no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—2842no—
274C2×C4—2842no—
275C2×C4—2842no—
276C2×C4—2842no—
277C2×C4—2842no—
278C2×C4—2842no—
279C2×C4C4×C22⋊D7112yes—
280C2×C4#1455609711yescharacteristic
281C2×C4#1455610212yes—
282C2×C4#1455611111yescharacteristic
283C8—44no—
284C8—5682no—
285C22—21no—
286C22—21no—
287C22—22no—
288C22—42no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22#1798587811yescharacteristic
298C22C2×#1455604611yescharacteristic
299C22#1798598111yescharacteristic
300C4—21no—
301C4—22no—
302C4—22no—
303C4—41no—
304C4—41no—
305C4—5682no—
306C4—5682no—
307C4#1798617712yes—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—81no—
312C2#2258872811yescharacteristic
313C2#2258880811yescharacteristic
314C2#2258886711yescharacteristic
subgroupG/H
centerC22#17985878
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2973C2
socleC22×C71#386
prime radicalC22×D4×C71C2×C4
cosocleC2×C42×C71C23
hypercenter#2973D71
hypocenterC71#9453
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #22587056C2semidirect
4. #22586197C2semidirect
163. C71#9453semidirect

Maximal subgroups

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

subgroupindex
#225935132
#225870562
#225866812
#225861972
C71×#29732
#945371