Finite Group Databasebeta

← #31895267

Subgroups — #31895267

Order 18176.

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

normal subgroups only

This group has 610 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#22590697C211yesmaximal, characteristic
2#22588439C212yesmaximal
3#22586670C211yesmaximal, characteristic
4#22586192C212yesmaximal
5C71×#2484C211yesmaximal, characteristic
6C2×#14556087—21no—
7#17985963C2212yes—
8C4⋊C4×Dic71—24no—
9C4×C22⋊Dic71—21no—
10C4×C22⋊Dic71C2212yes—
11#17985758—24no—
12#17985725—22no—
13C2×#117×C71C2211yescharacteristic
14C22×C4⋊C4×C71C2211yescharacteristic
15#400×C71C412yescomplemented
16#398×C71C412yescomplemented
17#387×C71—22no—
18#385×C71—21no—
19#381×C71—21no—
20#381×C71C2211yescharacteristic
21C2×#11765849—22no—
22#14556087—28no—
23C2×C4⋊Dic71—22no—
24C2×C22⋊Dic71—22no—
25C2×C4×Dic71—22no—
26C2×C4×Dic71—22no—
27C2×C4×Dic71D414yes—
28C22⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)—22no—
30C71⋊#95—22no—
31C71⋊#95—22no—
32C23×C4×C71D411yescharacteristic
33C23×C4×C71D411yescharacteristic
34C23×C4×C71C2311yescharacteristic
35#117×C71—22no—
36#117×C71—22no—
37C2×C4⋊C4×C71—22no—
38C2×C4⋊C4×C71—22no—
39C2×C4⋊C4×C71—22no—
40C2×C4⋊C4×C71C2×C411yescharacteristic
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C4⋊C4×C71C2×C412yes—
43C2×C4⋊C4×C71D412yes—
44C2×C22⋊C4×C71—21no—
45C2×C22⋊C4×C71—21no—
46C2×C42×C71—21no—
47#95×C71—22no—
48#95×C71—22no—
49#95×C71—22no—
50#95×C71—21no—
51#95×C71C2×C412yes—
52C22×Dic71—24no—
53#11765849—44no—
54C2×C71⋊C8—42no—
55C4⋊Dic71—28no—
56C22⋊Dic71—28no—
57C4×Dic71—28no—
58C4×Dic71—44no—
59C4×Dic71—82no—
60C24×C71C2×D411yescharacteristic
61C22×C4×C71—21no—
62C22×C4×C71—21no—
63C22×C4×C71—24no—
64C22×C4×C71—21no—
65C22×C4×C71—21no—
66C22×C4×C71—22no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—42no—
70C22×C4×C71—42no—
71C22×C4×C71C22⋊C412yes—
72C22×C4×C71C22⋊C412yes—
73C22×C4×C71C22×C411yescharacteristic
74C22×C4×C71C2×D412yes—
75C22×C4×C71C2×D411yescharacteristic
76C22×C4×C71#4112yes—
77C4⋊C4×C71—24no—
78C4⋊C4×C71—24no—
79C4⋊C4×C71—22no—
80C4⋊C4×C71—22no—
81C4⋊C4×C71—44no—
82C22⋊C4×C71—42no—
83C22⋊C4×C71—41no—
84C22⋊C4×C71—41no—
85C42×C71—42no—
86C23×C71—22no—
87C23×C71—22no—
88C23×C71—41no—
89C23×C71C2×C22⋊C411yescharacteristic
90C23×C71#12011yescharacteristic
91C23×C71#12311yescharacteristic
92C2×C4×C71—24no—
93C2×C4×C71—22no—
94C2×C4×C71—22no—
95C2×C4×C71—22no—
96C2×C4×C71—22no—
97C2×C4×C71—24no—
98C2×C4×C71—22no—
99C2×C4×C71—41no—
100C2×C4×C71—41no—
101C2×C4×C71—42no—
102C2×C4×C71—42no—
103C2×C4×C71—42no—
104C2×C4×C71—42no—
105C2×C4×C71—42no—
106C2×C4×C71—42no—
107C2×C4×C71—41no—
108C2×C4×C71—41no—
109C2×C4×C71—42no—
110C2×C4×C71—42no—
111C2×C4×C71—81no—
112C2×C4×C71#10414yes—
113C2×C4×C71C4×D412yes—
114C2×C4×C71#12112yes—
115C2×Dic71—28no—
116C2×Dic71—44no—
117C2×Dic71—44no—
118C71⋊C8—82no—
119C22×C71—21no—
120C22×C71—21no—
121C22×C71—21no—
122C22×C71—21no—
123C22×C71—21no—
124C22×C71—21no—
125C22×C71—21no—
126C22×C71—44no—
127C22×C71#38611yescharacteristic
128C22×C71C2×#10412yes—
129C22×C71#45812yes—
130C4×C71—22no—
131C4×C71—22no—
132C4×C71—24no—
133C4×C71—42no—
134C4×C71—82no—
135C4×C71—81no—
136C4×C71—81no—
137Dic71—48no—
138#9478—711nomaximal, Sylow
139C2×C71—22no—
140C2×C71—22no—
141C2×C71—41no—
142C2×C71#189011yescharacteristic
143C2×C71#189611yescharacteristic
144C2×C71#191211yescharacteristic
145#2484D7111yescharacteristic
146#2274—711no—
147#1826—712no—
148#1463—711no—
149#1284—712no—
150C71#947811yescharacteristic, complemented, Sylow
151C2×#117—1421no—
152C2×#117C2×D7111yescharacteristic
153C22×C4⋊C4C2×D7111yescharacteristic
154#407—712no—
155#400Dic7112yescomplemented
156#398Dic7112yescomplemented
157#387—22no—
158#385—21no—
159#381—21no—
160#381C2×D7111yescharacteristic
161C4×C4⋊C4—1424no—
162C4×C22⋊C4—712no—
163C4×C22⋊C4—1421no—
164#339—1424no—
165#324—1422no—
166C23×C4C22⋊D7111yescharacteristic
167C23×C4C22⋊D7111yescharacteristic
168C23×C4C22×D7111yescharacteristic
169C2×M16—1422no—
170#117—22no—
171#117—22no—
172#117—1428no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—1422no—
177C2×C4⋊C4C2×Dic7111yescharacteristic
178C2×C4⋊C4C2×Dic7111yescharacteristic
179C2×C4⋊C4C2×Dic7112yes—
180C2×C4⋊C4C22⋊D7112yes—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—1422no—
184C2×C42—21no—
185C2×C42—714no—
186C2×C42—1422no—
187C2×C42—1422no—
188C22⋊C8—1422no—
189C22⋊C8—1422no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—21no—
194#95—1422no—
195#95—1422no—
196#95C2×Dic7112yes—
197C24C2×C22⋊D7111yescharacteristic
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—24no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—1424no—
209C22×C4C22⋊Dic7112yes—
210C22×C4C22⋊Dic7112yes—
211C22×C4C22×Dic7111yescharacteristic
212C22×C4D4×D7112yes—
213C22×C4C2×C22⋊D7111yescharacteristic
214C22×C4#1176586412yes—
215M16—2844no—
216C2×C8—2842no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—44no—
222C4⋊C4—1428no—
223C22⋊C4—42no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—1428no—
227C42—42no—
228C42—2844no—
229C42—5682no—
230C42—1428no—
231C23—22no—
232C23—22no—
233C23—41no—
234C23C2×C22⋊Dic7111yescharacteristic
235C23#1455609611yescharacteristic
236C23#1455610911yescharacteristic
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—81no—
257C2×C4—2844no—
258C2×C4—2844no—
259C2×C4—1428no—
260C2×C4#1455605514yes—
261C2×C4D4×Dic7112yes—
262C2×C4#1455610012yes—
263C8—5682no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22#1798587911yescharacteristic
273C22C2×#1455605512yes—
274C22#1798618812yes—
275C4—22no—
276C4—22no—
277C4—24no—
278C4—42no—
279C4—82no—
280C4—81no—
281C4—81no—
282C4—2848no—
283C2—22no—
284C2—22no—
285C2—41no—
286C2#2258875411yescharacteristic
287C2#2258878411yescharacteristic
288C2#2258887811yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2484C2
socleC22×C71#386
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenter#2484D71
hypocenterC71#9478
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2484

Complements

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

NH ≅ G/Ntype
15. #400×C71C4semidirect
16. #398×C71C4semidirect
150. C71#9478semidirect
155. #400Dic71semidirect
156. #398Dic71semidirect

Maximal subgroups

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

subgroupindex
#225906972
#225884392
#225866702
#225861922
C71×#24842
#947871