Finite Group Databasebeta

← #31890918

Subgroups — #31890918

Order 18176.

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

normal subgroups only

This group has 418 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#22590943C211yesmaximal, characteristic
2C2×#17985725C211yesmaximal, characteristic
3#22586187C212yesmaximal
4#22586186C212yesmaximal
5C2×#324×C71C211yesmaximal, characteristic
6C2×C22⋊(C71⋊C8)C2212yes—
7#17985885—21no—
8#17985885—21no—
9#17985885—21no—
10#17985885—21no—
11#17985862C411yescharacteristic
12#17985862C411yescharacteristic
13#17985859C411yescharacteristic
14#17985859C411yescharacteristic
15C4×C22⋊Dic71C2211yescharacteristic
16C4×C22⋊Dic71C2211yescharacteristic
17#17985725—21no—
18#17985725—21no—
19C22×C4⋊C4×C71C2211yescharacteristic
20C2×C22⋊C8×C71C2212yes—
21#324×C71—21no—
22#324×C71—21no—
23C22×C71⋊C8—22no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71—21no—
26C2×C22⋊Dic71C2×C411yescharacteristic
27C2×C22⋊Dic71C2×C411yescharacteristic
28C2×C22⋊Dic71C2×C411yescharacteristic
29C2×C22⋊Dic71C2×C411yescharacteristic
30C2×C4×Dic71—21no—
31C2×C4×Dic71—21no—
32C22⋊(C71⋊C8)—22no—
33C22⋊(C71⋊C8)—22no—
34C71⋊#95—21no—
35C71⋊#95—21no—
36C71⋊#95—21no—
37C71⋊#95—21no—
38C71⋊#95—21no—
39C71⋊#95—21no—
40C71⋊#95—21no—
41C71⋊#95—21no—
42C23×C4×C71C2×C411yescharacteristic
43C23×C4×C71C2×C411yescharacteristic
44C23×C4×C71C2311yescharacteristic
45C22×C8×C71—22no—
46C2×C4⋊C4×C71—22no—
47C2×C4⋊C4×C71—22no—
48C2×C4⋊C4×C71D411yescharacteristic
49C2×C4⋊C4×C71D411yescharacteristic
50C2×C4⋊C4×C71D411yescharacteristic
51C2×C4⋊C4×C71D411yescharacteristic
52C22⋊C8×C71—22no—
53C22⋊C8×C71—22no—
54C22×Dic71—21no—
55C22×Dic71—21no—
56C22×Dic71—21no—
57C22×Dic71—21no—
58C2×C71⋊C8—42no—
59C2×C71⋊C8—82no—
60C4⋊Dic71—81no—
61C4⋊Dic71—81no—
62C22⋊Dic71—22no—
63C22⋊Dic71—22no—
64C22⋊Dic71—22no—
65C22⋊Dic71—22no—
66C4×Dic71—81no—
67C4×Dic71—81no—
68C24×C71C22×C411yescharacteristic
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—22no—
72C22×C4×C71—22no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—81no—
76C22×C4×C71—81no—
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22⋊C411yescharacteristic
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C2×D411yescharacteristic
82C22×C4×C71C2×D411yescharacteristic
83C22×C4×C71#4112yes—
84C22×C4×C71#4112yes—
85C2×C8×C71—42no—
86C2×C8×C71—82no—
87C4⋊C4×C71—22no—
88C4⋊C4×C71—22no—
89C4⋊C4×C71—22no—
90C4⋊C4×C71—22no—
91C23×C71—41no—
92C23×C71—41no—
93C23×C71C2×C22⋊C411yescharacteristic
94C23×C71#11711yescharacteristic
95C23×C71#11711yescharacteristic
96C23×C71#13112yes—
97C23×C71#13112yes—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—21no—
101C2×C4×C71—21no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—21no—
106C2×C4×C71—42no—
107C2×C4×C71—41no—
108C2×C4×C71—41no—
109C2×C4×C71—42no—
110C2×C4×C71—42no—
111C2×C4×C71—42no—
112C2×C4×C71—41no—
113C2×C4×C71—41no—
114C2×C4×C71—81no—
115C2×C4×C71—81no—
116C2×C4×C71#12312yes—
117C2×C4×C71#12312yes—
118C2×Dic71—41no—
119C2×Dic71—41no—
120C2×Dic71—41no—
121C2×Dic71—41no—
122C2×Dic71—41no—
123C2×Dic71—41no—
124C2×Dic71—41no—
125C2×Dic71—41no—
126C2×Dic71—41no—
127C2×Dic71—41no—
128C2×Dic71—41no—
129C2×Dic71—41no—
130C8×C71—82no—
131C71⋊C8—82no—
132C22×C71—21no—
133C22×C71—21no—
134C22×C71—21no—
135C22×C71—21no—
136C22×C71—21no—
137C22×C71—21no—
138C22×C71—42no—
139C22×C71—42no—
140C22×C71#38111yescharacteristic
141C22×C71#40811yescharacteristic
142C22×C71#40811yescharacteristic
143C22×C71#43312yes—
144C22×C71#43312yes—
145C4×C71—22no—
146C4×C71—22no—
147C4×C71—41no—
148C4×C71—41no—
149C4×C71—41no—
150C4×C71—41no—
151Dic71—81no—
152Dic71—81no—
153Dic71—81no—
154Dic71—81no—
155#8737—711nomaximal, Sylow
156C2×C71—21no—
157C2×C71—21no—
158C2×C71—22no—
159C2×C71—22no—
160C2×C71#181811yescharacteristic
161C2×C71#182811yescharacteristic
162C2×C71#183011yescharacteristic
163#2313—711no—
164C2×#324—711no—
165C2×#324D7111yescharacteristic
166#1282—712no—
167#1282—712no—
168C71#873711yescharacteristic, complemented, Sylow
169C22×C4⋊C4C2×D7111yescharacteristic
170C2×C22⋊C8—712no—
171C2×C22⋊C8C2×D7112yes—
172#387—1421no—
173#387—1421no—
174#387—1421no—
175#387—1421no—
176#381—711no—
177#381—711no—
178#380—711no—
179#380—711no—
180C4×C22⋊C4—711no—
181C4×C22⋊C4—711no—
182#324—21no—
183#324—21no—
184#324—1421no—
185#324—1421no—
186C23×C4C4×D7111yescharacteristic
187C23×C4C4×D7111yescharacteristic
188C23×C4C22×D7111yescharacteristic
189C22×C8—22no—
190C22×C8—1422no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—1421no—
194C2×C4⋊C4—1421no—
195C2×C4⋊C4D28411yescharacteristic
196C2×C4⋊C4D28411yescharacteristic
197C2×C4⋊C4C22⋊D7111yescharacteristic
198C2×C4⋊C4C22⋊D7111yescharacteristic
199C2×C22⋊C4—711no—
200C2×C22⋊C4—711no—
201C2×C22⋊C4—711no—
202C2×C22⋊C4—711no—
203C2×C42—1421no—
204C2×C42—1421no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—1422no—
208C22⋊C8—1422no—
209#95—1421no—
210#95—1421no—
211#95—1421no—
212#95—1421no—
213#95—1421no—
214#95—1421no—
215#95—1421no—
216#95—1421no—
217C24C2×C4×D7111yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—1421no—
227C22×C4—1421no—
228C22×C4—1421no—
229C22×C4—1421no—
230C22×C4(C2×D71)⋊C411yescharacteristic
231C22×C4(C2×D71)⋊C411yescharacteristic
232C22×C4(C2×D71)⋊C411yescharacteristic
233C22×C4(C2×D71)⋊C411yescharacteristic
234C22×C4C2×C22⋊D7111yescharacteristic
235C22×C4C2×D28411yescharacteristic
236C22×C4#1176586412yes—
237C22×C4#1176586412yes—
238C2×C8—42no—
239C2×C8—82no—
240C2×C8—2842no—
241C2×C8—5682no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—5681no—
247C4⋊C4—5681no—
248C22⋊C4—1422no—
249C22⋊C4—1422no—
250C22⋊C4—1422no—
251C22⋊C4—1422no—
252C42—5681no—
253C42—5681no—
254C23—41no—
255C23—41no—
256C23C2×(C2×D71)⋊C411yescharacteristic
257C23#1455608411yescharacteristic
258C23#1455608411yescharacteristic
259C23#1455613412yes—
260C23#1455613612yes—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—2841no—
280C2×C4—2841no—
281C2×C4—2841no—
282C2×C4—2841no—
283C2×C4—2841no—
284C2×C4—2841no—
285C2×C4—2841no—
286C2×C4—2841no—
287C2×C4—2841no—
288C2×C4—2841no—
289C2×C4—2841no—
290C2×C4—2841no—
291C2×C4#1455610912yes—
292C2×C4#1455611212yes—
293C8—82no—
294C8—5682no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—42no—
302C22—42no—
303C22#1798586011yescharacteristic
304C22#1798596511yescharacteristic
305C22#1798596711yescharacteristic
306C22#1798605712yes—
307C22#1798605712yes—
308C4—22no—
309C4—22no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—5681no—
315C4—5681no—
316C4—5681no—
317C4—5681no—
318C2—21no—
319C2—21no—
320C2—22no—
321C2—22no—
322C2#2258840911yescharacteristic
323C2#2258844611yescharacteristic
324C2#2258845211yescharacteristic
subgroupG/H
centerC22#17985860
commutator, prime radicalC24×C71C22×C4
FrattiniC23×C4C22×D71
FittingC2×#324×C71C2
socleC22×C71#381
cosocleC23×C4×C71C23
hypercenterC2×#324D71
hypocenterC71#8737
LCSG ▷ C24×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
168. C71#8737semidirect

Maximal subgroups

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

subgroupindex
#225909432
C2×#179857252
#225861872
#225861862
C2×#324×C712
#873771