Finite Group Databasebeta

← #31895187

Subgroups — #31895187

Order 18176.

700 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 115 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#22590926C211yesmaximal, characteristic
2#22588264C211yesmaximal, characteristic
3#22586873C212yesmaximal
4#22586193C212yesmaximal
5C71×#2434C211yesmaximal, characteristic
6C2×#14556087—21no—
7C2×#14556047—22no—
8#17985963C2212yes—
9#17985906—24no—
10#17985862—21no—
11#17985862C2212yes—
12#17985736—24no—
13C23×Q8×C71C2211yescharacteristic
14C2×C4⋊Q8×C71C412yescomplemented
15C2×C22⋊Q8×C71C2211yescharacteristic
16#395×C71—22no—
17#391×C71C412yescomplemented
18#385×C71—21no—
19#385×C71—21no—
20C4×C22⋊C4×C71C2211yescharacteristic
21C2×#11765849—22no—
22#14556087—44no—
23C2×C4⋊Dic71—22no—
24C2×C22⋊Dic71—22no—
25C2×C4×Dic71—22no—
26#14556047—44no—
27C22⋊(C71⋊C8)—22no—
28C22⋊(C71⋊C8)—22no—
29C71⋊#95—24no—
30C71⋊#95—22no—
31C71⋊#95D414yes—
32C22×Q8×C71—22no—
33C22×Q8×C71—22no—
34C22×Q8×C71—44no—
35C22×Q8×C71C2×C411yescharacteristic
36C22×Q8×C71C2×C412yes—
37C22×Q8×C71D412yes—
38C23×C4×C71D411yescharacteristic
39C23×C4×C71D411yescharacteristic
40C23×C4×C71C2311yescharacteristic
41C4⋊Q8×C71—44no—
42C22⋊Q8×C71—22no—
43C22⋊Q8×C71—22no—
44C2×C4⋊C4×C71—22no—
45C2×C4⋊C4×C71—22no—
46C2×C4⋊C4×C71—21no—
47C2×C4⋊C4×C71C2×C411yescharacteristic
48C2×C22⋊C4×C71—21no—
49C2×C22⋊C4×C71—21no—
50C2×C42×C71C2×C412yes—
51#95×C71—22no—
52#95×C71—22no—
53#95×C71—21no—
54C22×Dic71—24no—
55#11765849—44no—
56C2×C71⋊C8—42no—
57C4⋊Dic71—44no—
58C22⋊Dic71—44no—
59C4×Dic71—44no—
60C24×C71C2×D411yescharacteristic
61C2×Q8×C71—24no—
62C2×Q8×C71—24no—
63C2×Q8×C71—24no—
64C2×Q8×C71—24no—
65C2×Q8×C71—22no—
66C2×Q8×C71—42no—
67C2×Q8×C71—42no—
68C2×Q8×C71—48no—
69C2×Q8×C71—44no—
70C2×Q8×C71—44no—
71C2×Q8×C71—42no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—24no—
75C22×C4×C71—22no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71—42no—
81C22×C4×C71—42no—
82C22×C4×C71C22⋊C412yes—
83C22×C4×C71C22⋊C412yes—
84C22×C4×C71C22×C411yescharacteristic
85C22×C4×C71C2×D412yes—
86C22×C4×C71C2×D411yescharacteristic
87C22×C4×C71#4112yes—
88C4⋊C4×C71—22no—
89C4⋊C4×C71—44no—
90C4⋊C4×C71—44no—
91C4⋊C4×C71—42no—
92C22⋊C4×C71—24no—
93C22⋊C4×C71—41no—
94C22⋊C4×C71—41no—
95C42×C71—42no—
96C23×C71—22no—
97C23×C71—22no—
98C23×C71—41no—
99C23×C71C2×C22⋊C411yescharacteristic
100C23×C71#12011yescharacteristic
101C23×C71#12311yescharacteristic
102Q8×C71—48no—
103Q8×C71—48no—
104Q8×C71—82no—
105C2×C4×C71—24no—
106C2×C4×C71—22no—
107C2×C4×C71—22no—
108C2×C4×C71—22no—
109C2×C4×C71—22no—
110C2×C4×C71—22no—
111C2×C4×C71—22no—
112C2×C4×C71—24no—
113C2×C4×C71—22no—
114C2×C4×C71—22no—
115C2×C4×C71—22no—
116C2×C4×C71—42no—
117C2×C4×C71—42no—
118C2×C4×C71—42no—
119C2×C4×C71—42no—
120C2×C4×C71—42no—
121C2×C4×C71—42no—
122C2×C4×C71—42no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—42no—
126C2×C4×C71#9914yes—
127C2×C4×C71C4×D412yes—
128C2×C4×C71#12112yes—
129C2×Dic71—44no—
130C2×Dic71—44no—
131C2×Dic71—44no—
132C71⋊C8—82no—
133C22×C71—21no—
134C22×C71—21no—
135C22×C71—21no—
136C22×C71—21no—
137C22×C71—21no—
138C22×C71—21no—
139C22×C71—21no—
140C22×C71—44no—
141C22×C71#38611yescharacteristic
142C22×C71C2×#9912yes—
143C22×C71#45612yes—
144C4×C71—22no—
145C4×C71—24no—
146C4×C71—22no—
147C4×C71—44no—
148C4×C71—42no—
149C4×C71—81no—
150C4×C71—81no—
151Dic71—84no—
152#9462—711nomaximal, Sylow
153C2×C71—22no—
154C2×C71—22no—
155C2×C71—41no—
156C2×C71#188611yescharacteristic
157C2×C71#189811yescharacteristic
158C2×C71#191211yescharacteristic
159#2434D7111yescharacteristic
160#2310—711no—
161#1784—711no—
162#1511—712no—
163#1284—712no—
164C71#946211yescharacteristic, complemented, Sylow
165C23×Q8C2×D7111yescharacteristic
166C2×C4⋊Q8Dic7112yescomplemented
167C2×C22⋊Q8C2×D7111yescharacteristic
168C2×#117—1421no—
169C2×#101—1422no—
170#407—712no—
171#395—22no—
172#391—1424no—
173#391Dic7112yescomplemented
174#385—21no—
175#385—21no—
176#381—712no—
177#381—1421no—
178C4×C22⋊C4C2×D7111yescharacteristic
179#328—1424no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—44no—
183C22×Q8C2×Dic7111yescharacteristic
184C22×Q8C2×Dic7112yes—
185C22×Q8C22⋊D7112yes—
186C23×C4C22⋊D7111yescharacteristic
187C23×C4C22⋊D7111yescharacteristic
188C23×C4C22×D7111yescharacteristic
189C2×M16—1422no—
190C4⋊Q8—44no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193#117—2844no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—1422no—
198C2×C4⋊C4C2×Dic7111yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—1422no—
202C2×C42—1422no—
203C2×C42C2×Dic7112yes—
204#101—2844no—
205C22⋊C8—1422no—
206C22⋊C8—1422no—
207#95—22no—
208#95—22no—
209#95—21no—
210#95—714no—
211#95—1424no—
212#95—1422no—
213C24C2×C22⋊D7111yescharacteristic
214C2×Q8—24no—
215C2×Q8—24no—
216C2×Q8—24no—
217C2×Q8—24no—
218C2×Q8—22no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—44no—
222C2×Q8—44no—
223C2×Q8—48no—
224C2×Q8—42no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—1424no—
236C22×C4C22⋊Dic7112yes—
237C22×C4C22⋊Dic7112yes—
238C22×C4C22×Dic7111yescharacteristic
239C22×C4D4×D7112yes—
240C22×C4C2×C22⋊D7111yescharacteristic
241C22×C4#1176586412yes—
242M16—2844no—
243C2×C8—2842no—
244C4⋊C4—22no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—42no—
248C4⋊C4—2844no—
249C22⋊C4—24no—
250C22⋊C4—41no—
251C22⋊C4—41no—
252C22⋊C4—2844no—
253C42—42no—
254C42—2844no—
255C23—22no—
256C23—22no—
257C23—41no—
258C23C2×C22⋊Dic7111yescharacteristic
259C23#1455609611yescharacteristic
260C23#1455610911yescharacteristic
261Q8—48no—
262Q8—48no—
263Q8—82no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—2844no—
286C2×C4—2844no—
287C2×C4—2844no—
288C2×C4#1455604314yes—
289C2×C4D4×Dic7112yes—
290C2×C4#1455610012yes—
291C8—5682no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—44no—
300C22#1798587911yescharacteristic
301C22C2×#1455604312yes—
302C22#1798618112yes—
303C4—22no—
304C4—24no—
305C4—22no—
306C4—44no—
307C4—42no—
308C4—81no—
309C4—81no—
310C4—5684no—
311C2—22no—
312C2—22no—
313C2—41no—
314C2#2258873411yescharacteristic
315C2#2258879411yescharacteristic
316C2#2258887811yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2434C2
socleC22×C71#386
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenter#2434D71
hypocenterC71#9462
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2434

Complements

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

NH ≅ G/Ntype
14. C2×C4⋊Q8×C71C4semidirect
17. #391×C71C4semidirect
164. C71#9462semidirect
166. C2×C4⋊Q8Dic71semidirect
173. #391Dic71semidirect

Maximal subgroups

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

subgroupindex
#225909262
#225882642
#225868732
#225861932
C71×#24342
#946271