Finite Group Databasebeta

← #31891074

Subgroups — #31891074

Order 18176.

384 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 95 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#22591305C211yesmaximal, characteristic
2#22586684C211yesmaximal, characteristic
3#22586234C212yesmaximal
4#22586205C212yesmaximal
5C4×#101×C71C211yesmaximal, characteristic
6#17986056C2212yes—
7#17985906—21no—
8#17985885—21no—
9#17985885—21no—
10#17985867C411yescharacteristic
11#17985867C2211yescharacteristic
12#17985865C411yescharacteristic
13C4⋊C4×Dic71—21no—
14C4×C4⋊Dic71C411yescharacteristic
15#17985845C411yescharacteristic
16#17985845C2211yescharacteristic
17#17985725—21no—
18#17985725—21no—
19C2×C4×Q8×C71C2211yescharacteristic
20C2×#101×C71—21no—
21C4×M16×C71C2212yes—
22#343×C71—21no—
23C2×C4⋊Dic71—21no—
24C2×C4⋊Dic71—21no—
25C2×C4×Dic71—21no—
26C2×C4×Dic71—21no—
27C2×C4×Dic71—21no—
28C4⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)—22no—
30#14556039—22no—
31C71⋊#95—21no—
32C71⋊#95—21no—
33C71⋊#95—21no—
34C71⋊#95—21no—
35C71⋊#95—21no—
36C71⋊#95—21no—
37C71⋊#95—21no—
38C71⋊#95C2×C411yescharacteristic
39C71⋊#95C2×C411yescharacteristic
40C71⋊#95C2×C411yescharacteristic
41C71⋊#95C2×C411yescharacteristic
42C22×Q8×C71D411yescharacteristic
43C2×M16×C71—22no—
44C4×Q8×C71—22no—
45C4×Q8×C71—22no—
46C2×C4⋊C4×C71D411yescharacteristic
47C2×C4⋊C4×C71D411yescharacteristic
48C2×C4⋊C4×C71D411yescharacteristic
49C2×C42×C71C2×C411yescharacteristic
50C2×C42×C71C2×C411yescharacteristic
51C2×C42×C71C2311yescharacteristic
52#101×C71—24no—
53#97×C71—22no—
54C4×C8×C71—22no—
55C22×Dic71—21no—
56C22×Dic71—21no—
57C22×Dic71—21no—
58C22×Dic71—21no—
59C2×C71⋊C8—42no—
60C4⋊Dic71—42no—
61C4⋊Dic71—42no—
62C4×Dic71—42no—
63C4×Dic71—42no—
64C4×Dic71—42no—
65C2×Q8×C71—22no—
66C2×Q8×C71—22no—
67C2×Q8×C71—41no—
68C22×C4×C71C22⋊C411yescharacteristic
69C22×C4×C71C22⋊C411yescharacteristic
70C22×C4×C71C22⋊C411yescharacteristic
71C22×C4×C71C22⋊C411yescharacteristic
72C22×C4×C71C22×C411yescharacteristic
73C22×C4×C71C2×D411yescharacteristic
74C22×C4×C71C2×D411yescharacteristic
75M16×C71—24no—
76M16×C71—42no—
77C2×C8×C71—42no—
78C4⋊C4×C71—22no—
79C4⋊C4×C71—22no—
80C4⋊C4×C71—22no—
81C42×C71—21no—
82C42×C71—21no—
83C42×C71—21no—
84C42×C71—21no—
85C42×C71#4112yes—
86C42×C71#4112yes—
87C23×C71C2×C22⋊C411yescharacteristic
88Q8×C71—44no—
89C2×C4×C71—21no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—21no—
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—21no—
98C2×C4×C71—41no—
99C2×C4×C71—41no—
100C2×C4×C71—41no—
101C2×C4×C71—41no—
102C2×C4×C71#10412yes—
103C2×C4×C71#11711yescharacteristic
104C2×C4×C71#11711yescharacteristic
105C2×C4×C71#12312yes—
106C2×C4×C71#12312yes—
107C2×Dic71—22no—
108C2×Dic71—41no—
109C2×Dic71—41no—
110C2×Dic71—41no—
111C2×Dic71—41no—
112C2×Dic71—41no—
113C2×Dic71—41no—
114C2×Dic71—41no—
115C2×Dic71—41no—
116C2×Dic71—41no—
117C2×Dic71—41no—
118C2×Dic71—41no—
119C8×C71—44no—
120C71⋊C8—82no—
121C22×C71—21no—
122C22×C71—21no—
123C22×C71#38111yescharacteristic
124C22×C71C2×#10411yescharacteristic
125C22×C71#42111yescharacteristic
126C4×C71—22no—
127C4×C71—22no—
128C4×C71—22no—
129C4×C71—21no—
130C4×C71—21no—
131C4×C71—41no—
132C4×C71—41no—
133C4×C71—41no—
134Dic71—42no—
135Dic71—81no—
136Dic71—81no—
137Dic71—81no—
138#8775—711nomaximal, Sylow
139C2×C71—21no—
140C2×C71—21no—
141C2×C71#182611yescharacteristic
142C2×C71#182711yescharacteristic
143C2×C71#183011yescharacteristic
144#2356—711no—
145C4×#101D7111yescharacteristic
146#1467—711no—
147#1300—712no—
148#1288—712no—
149C71#877511yescharacteristic, complemented, Sylow
150C2×C4×Q8C2×D7111yescharacteristic
151#432—712no—
152C2×#101—21no—
153C4×M16C2×D7112yes—
154#391—1421no—
155#387—1421no—
156#387—1421no—
157#383—711no—
158#383—711no—
159#382—711no—
160C4×C4⋊C4—711no—
161C4×C4⋊C4—1421no—
162#376—711no—
163#376—711no—
164#343—21no—
165#324—1421no—
166#324—1421no—
167C22×Q8C22⋊D7111yescharacteristic
168C2×M16—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C2×C4⋊C4—1421no—
172C2×C4⋊C4—1421no—
173C2×C4⋊C4D28411yescharacteristic
174C2×C4⋊C4D28411yescharacteristic
175C2×C4⋊C4C22⋊D7111yescharacteristic
176C2×C42—1421no—
177C2×C42—1421no—
178C2×C42—1421no—
179C2×C42C4×D7111yescharacteristic
180C2×C42C4×D7111yescharacteristic
181C2×C42C22×D7111yescharacteristic
182C4⋊C8—1422no—
183#101—24no—
184C22⋊C8—1422no—
185#97—22no—
186#97—1422no—
187C4×C8—22no—
188#95—711no—
189#95—711no—
190#95—711no—
191#95—711no—
192#95—1421no—
193#95—1421no—
194#95—1421no—
195#95—1421no—
196#95—1421no—
197#95—1421no—
198#95—1421no—
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—41no—
202C22×C4—1421no—
203C22×C4—1421no—
204C22×C4—1421no—
205C22×C4—1421no—
206C22×C4(C2×D71)⋊C411yescharacteristic
207C22×C4(C2×D71)⋊C411yescharacteristic
208C22×C4(C2×D71)⋊C411yescharacteristic
209C22×C4(C2×D71)⋊C411yescharacteristic
210C22×C4C2×C4×D7111yescharacteristic
211C22×C4C2×C22⋊D7111yescharacteristic
212C22×C4C2×D28411yescharacteristic
213M16—24no—
214M16—42no—
215C2×C8—42no—
216C2×C8—2842no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—2842no—
221C4⋊C4—2842no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—2842no—
227C42—2842no—
228C42—2842no—
229C42#1176586412yes—
230C42#1176586412yes—
231C23C2×(C2×D71)⋊C411yescharacteristic
232Q8—44no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—2841no—
247C2×C4—2841no—
248C2×C4—2841no—
249C2×C4—2841no—
250C2×C4—2841no—
251C2×C4—2841no—
252C2×C4—2841no—
253C2×C4—2841no—
254C2×C4—2841no—
255C2×C4—2841no—
256C2×C4—2841no—
257C2×C4—1422no—
258C2×C4#1455605612yes—
259C2×C4#1455608411yescharacteristic
260C2×C4#1455608411yescharacteristic
261C2×C4#1455610912yes—
262C2×C4#1455611212yes—
263C8—44no—
264C8—5682no—
265C22—21no—
266C22—21no—
267C22#1798586011yescharacteristic
268C22C2×#1455605611yescharacteristic
269C22#1798601711yescharacteristic
270C4—22no—
271C4—22no—
272C4—22no—
273C4—21no—
274C4—21no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—2842no—
279C4—5681no—
280C4—5681no—
281C4—5681no—
282C2—21no—
283C2—21no—
284C2#2258844011yescharacteristic
285C2#2258844511yescharacteristic
286C2#2258845211yescharacteristic
subgroupG/H
centerC22#17985860
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC4×#101×C71C2
socleC22×C71#381
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenterC4×#101D71
hypocenterC71#8775
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
149. C71#8775semidirect

Maximal subgroups

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

subgroupindex
#225913052
#225866842
#225862342
#225862052
C4×#101×C712
#877571