Finite Group Databasebeta

← #31889698

Subgroups — #31889698

Order 18176.

682 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 139 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#22589330C212yesmaximal, complemented
2#22587078C211yesmaximal, characteristic, complemented
3#22586935C211yesmaximal, characteristic, complemented
4#22586779C212yesmaximal
5C71×#3509C211yesmaximal, characteristic
6#17986023C2212yes—
7C4×#11765849C2212yes—
8#17985768—24no—
9#17985745—22no—
10#17985741—22no—
11Q8⋊(C71⋊C8)—22no—
12D4⋊(C71⋊C8)—22no—
13#557×C71—22no—
14#556×C71—22no—
15#552×C71—24no—
16#541×C71—22no—
17#536×C71C412yes—
18#533×C71—22no—
19#533×C71C2211yescharacteristic
20C2×C4⋊Q8×C71C2211yescharacteristic
21#519×C71C412yes—
22C4×#41×C71C2211yescharacteristic
23C2×#11765849—22no—
24C2×#11765849—22no—
25C4⋊(C71⋊C8)—22no—
26C4⋊(C71⋊C8)—22no—
27#14556039—22no—
28C4×C71⋊C8—22no—
29C2×#41×C71C2×C411yescharacteristic
30C22×Q8×C71C2×C412yes—
31C4⋊Q8×C71—22no—
32C4⋊Q8×C71—21no—
33C4⋊Q8×C71—21no—
34C4⋊Q8×C71—22no—
35C4⋊Q8×C71D412yes—
36#126×C71—28no—
37#125×C71—24no—
38#125×C71—28no—
39#125×C71D412yes—
40#124×C71—22no—
41#123×C71—24no—
42C22⋊Q8×C71—24no—
43C22⋊Q8×C71—22no—
44C22⋊Q8×C71—24no—
45C4×Q8×C71—24no—
46C4×Q8×C71—24no—
47C4×Q8×C71—24no—
48C4×Q8×C71—22no—
49C4×Q8×C71D412yes—
50C4×D4×C71—22no—
51C4×D4×C71D412yes—
52#117×C71—24no—
53#117×C71—24no—
54#117×C71—21no—
55#117×C71C2×C411yescharacteristic
56#117×C71C2×C412yes—
57C2×C4⋊C4×C71D414yes—
58C2×C42×C71—21no—
59C2×C42×C71C2311yescharacteristic
60#11765849—24no—
61#11765849—42no—
62#11765849—44no—
63C2×C71⋊C8—42no—
64C2×C71⋊C8—42no—
65#41×C71—42no—
66C2×Q8×C71—22no—
67C2×Q8×C71—22no—
68C2×Q8×C71—22no—
69C2×Q8×C71—44no—
70C2×Q8×C71C22⋊C411yescharacteristic
71C2×D4×C71—21no—
72C2×D4×C71C22⋊C411yescharacteristic
73C22×C4×C71—21no—
74C22×C4×C71—22no—
75C22×C4×C71C22⋊C414yes—
76C22×C4×C71C22×C411yescharacteristic
77C22×C4×C71C2×D412yes—
78C4⋊C4×C71—28no—
79C4⋊C4×C71—28no—
80C4⋊C4×C71—24no—
81C4⋊C4×C71—24no—
82C4⋊C4×C71—22no—
83C4⋊C4×C71—22no—
84C4⋊C4×C71—24no—
85C4⋊C4×C71C22⋊C412yes—
86C4⋊C4×C71C22⋊C414yes—
87C22⋊C4×C71—28no—
88C22⋊C4×C71—21no—
89C22⋊C4×C71—22no—
90C22⋊C4×C71—22no—
91C42×C71—28no—
92C42×C71—21no—
93C42×C71—22no—
94C42×C71—22no—
95C42×C71C2×D411yescharacteristic
96C42×C71C2×D411yescharacteristic
97C42×C71C2×D412yes—
98C23×C71—21no—
99C23×C71C2×C22⋊C411yescharacteristic
100Q8×C71—22no—
101Q8×C71—44no—
102Q8×C71—44no—
103Q8×C71—42no—
104Q8×C71—42no—
105D4×C71—22no—
106D4×C71—42no—
107C2×C4×C71—22no—
108C2×C4×C71—22no—
109C2×C4×C71—22no—
110C2×C4×C71—22no—
111C2×C4×C71—24no—
112C2×C4×C71—24no—
113C2×C4×C71—21no—
114C2×C4×C71—42no—
115C2×C4×C71—42no—
116C2×C4×C71—42no—
117C2×C4×C71—44no—
118C2×C4×C71C2×C22⋊C411yescharacteristic
119C2×C4×C71C2×C22⋊C411yescharacteristic
120C2×C4×C71#12012yes—
121C2×C4×C71#12012yes—
122C71⋊C8—44no—
123C71⋊C8—82no—
124C22×C71—22no—
125C22×C71—21no—
126C22×C71—42no—
127C22×C71—41no—
128C22×C71#37911yescharacteristic
129C4×C71—22no—
130C4×C71—21no—
131C4×C71—42no—
132C4×C71—44no—
133C4×C71—41no—
134C4×C71—44no—
135C4×C71#41312yes—
136C4×C71#45412yes—
137C4×C71#45612yes—
138#8474—711nomaximal, Sylow
139C2×C71—41no—
140C2×C71—81no—
141C2×C71#178511yescharacteristic
142C2×C71#179312yes—
143#3509D7111yescharacteristic
144#1997—712no—
145#1561—711no—
146#1526—711no—
147#1489—712no—
148C71#847411yescharacteristic, complemented, Sylow
149#557—22no—
150#556—22no—
151#552—24no—
152#541—22no—
153#536Dic7112yes—
154#533—22no—
155#533C2×D7111yescharacteristic
156C2×C4⋊Q8C2×D7111yescharacteristic
157#519Dic7112yes—
158C4×#41C2×D7111yescharacteristic
159#423—712no—
160C4×M16—712no—
161#343—1424no—
162#332—1422no—
163#330—1422no—
164Q8⋊C8—1422no—
165D4⋊C8—1422no—
166C2×#41C2×Dic7111yescharacteristic
167C22×Q8C2×Dic7112yes—
168C2×M16—1422no—
169C2×M16—1422no—
170C4⋊Q8—22no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—22no—
174C4⋊Q8C22⋊D7112yes—
175#126—28no—
176#125—24no—
177#125—28no—
178#125C22⋊D7112yes—
179#124—22no—
180#123—24no—
181C22⋊Q8—24no—
182C22⋊Q8—22no—
183C22⋊Q8—24no—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—22no—
188C4×Q8C22⋊D7112yes—
189C4×D4—22no—
190C4×D4C22⋊D7112yes—
191#117—24no—
192#117—24no—
193#117—21no—
194#117C2×Dic7111yescharacteristic
195#117C2×Dic7112yes—
196C2×C4⋊C4C22⋊D7114yes—
197C2×C42—21no—
198C2×C42C22×D7111yescharacteristic
199C4⋊C8—1422no—
200C4⋊C8—1422no—
201#97—1422no—
202C4×C8—1422no—
203#41—42no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—44no—
208C2×Q8C22⋊Dic7111yescharacteristic
209C2×D4—21no—
210C2×D4C22⋊Dic7111yescharacteristic
211C22×C4—21no—
212C22×C4—22no—
213C22×C4C22⋊Dic7114yes—
214C22×C4C22×Dic7111yescharacteristic
215C22×C4C2×C22⋊D7112yes—
216M16—2842no—
217M16—2844no—
218M16—1424no—
219C2×C8—2842no—
220C2×C8—2842no—
221C4⋊C4—28no—
222C4⋊C4—28no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—24no—
228C4⋊C4C22⋊Dic7112yes—
229C4⋊C4C22⋊Dic7114yes—
230C22⋊C4—28no—
231C22⋊C4—21no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C42—28no—
235C42—21no—
236C42—22no—
237C42—22no—
238C42C2×C22⋊D7112yes—
239C42C2×C22⋊D7111yescharacteristic
240C42C2×C22⋊D7111yescharacteristic
241C23—21no—
242C23C2×C22⋊Dic7111yescharacteristic
243Q8—22no—
244Q8—44no—
245Q8—42no—
246Q8—42no—
247Q8—44no—
248D4—22no—
249D4—42no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4C2×C22⋊Dic7111yescharacteristic
262C2×C4C2×C22⋊Dic7111yescharacteristic
263C2×C4#1455609712yes—
264C2×C4#1455609712yes—
265C8—2844no—
266C8—5682no—
267C22—21no—
268C22—22no—
269C22—41no—
270C22—42no—
271C22#1798585511yescharacteristic
272C4—21no—
273C4—22no—
274C4—41no—
275C4—44no—
276C4—42no—
277C4—44no—
278C4#1798598312yes—
279C4#1798617212yes—
280C4#1798618212yes—
281C2—41no—
282C2—81no—
283C2#2258826811yescharacteristic
284C2#2258829912yes—
subgroupG/H
centerC22#17985855
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#3509C2
socleC22×C71#379
prime radicalC2×#41×C71C2×C4
cosocleC2×C42×C71C23
hypercenter#3509D71
hypocenterC71#8474
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3509

Complements

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

NH ≅ G/Ntype
1. #22589330C2semidirect
2. #22587078C2semidirect
3. #22586935C2semidirect
148. C71#8474semidirect

Maximal subgroups

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

subgroupindex
#225893302
#225870782
#225869352
#225867792
C71×#35092
#847471