Finite Group Databasebeta

← #31893461

Subgroups — #31893461

Order 18176.

458 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 119 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#22596978C211yesmaximal, characteristic
2#22587021C212yesmaximal, complemented
3#22586654C211yesmaximal, characteristic
4#22586198C212yesmaximal, complemented
5C71×#2628C211yesmaximal, characteristic
6#17986060—24no—
7#17986052C2212yes—
8#17986023C2212yes—
9#17985963—21no—
10C4×#11765849—21no—
11#17985748—24no—
12#17985725—21no—
13#17985723—21no—
14C2×C4⋊Q8×C71C2211yescharacteristic
15C2×#117×C71C2211yescharacteristic
16#400×C71—21no—
17#399×C71—21no—
18#395×C71C412yes—
19#393×C71C412yes—
20#389×C71—21no—
21#382×C71C2211yescharacteristic
22#380×C71—21no—
23C2×#11765849—22no—
24C4⋊(C71⋊C8)—22no—
25C4⋊(C71⋊C8)—22no—
26C4⋊(C71⋊C8)D414yes—
27C22⋊(C71⋊C8)—24no—
28C22⋊(C71⋊C8)—22no—
29#14556039—22no—
30C4×C71⋊C8—22no—
31C4×C71⋊C8—22no—
32C22×Q8×C71C2×C412yes—
33C23×C4×C71D411yescharacteristic
34C4⋊Q8×C71—22no—
35C4⋊Q8×C71—22no—
36#117×C71—22no—
37#117×C71—22no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71—22no—
42C2×C4⋊C4×C71C2×C411yescharacteristic
43C2×C4⋊C4×C71D411yescharacteristic
44C2×C22⋊C4×C71—22no—
45C2×C22⋊C4×C71C2×C411yescharacteristic
46C2×C22⋊C4×C71D411yescharacteristic
47C2×C42×C71D411yescharacteristic
48C2×C42×C71C2311yescharacteristic
49#95×C71—22no—
50#95×C71—22no—
51#95×C71—21no—
52#95×C71C2×C412yes—
53#11765849—42no—
54#11765849—42no—
55C2×C71⋊C8—24no—
56C2×C71⋊C8—42no—
57C24×C71C22⋊C411yescharacteristic
58C2×Q8×C71—24no—
59C2×Q8×C71—82no—
60C22×C4×C71—21no—
61C22×C4×C71—21no—
62C22×C4×C71—22no—
63C22×C4×C71—22no—
64C22×C4×C71—22no—
65C22×C4×C71—22no—
66C22×C4×C71C22⋊C411yescharacteristic
67C22×C4×C71C22⋊C411yescharacteristic
68C22×C4×C71C22⋊C411yescharacteristic
69C22×C4×C71C22×C411yescharacteristic
70C22×C4×C71C2×D411yescharacteristic
71C22×C4×C71C2×D411yescharacteristic
72C4⋊C4×C71—22no—
73C4⋊C4×C71—22no—
74C4⋊C4×C71—41no—
75C4⋊C4×C71—42no—
76C4⋊C4×C71—41no—
77C4⋊C4×C71—42no—
78C4⋊C4×C71—82no—
79C22⋊C4×C71—22no—
80C22⋊C4×C71—22no—
81C22⋊C4×C71—82no—
82C42×C71—21no—
83C42×C71—21no—
84C42×C71C2×D412yes—
85C42×C71#4112yes—
86C23×C71—21no—
87C23×C71—21no—
88C23×C71—21no—
89C23×C71—41no—
90C23×C71—41no—
91C23×C71C2×C22⋊C411yescharacteristic
92Q8×C71—84no—
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—42no—
101C2×C4×C71—42no—
102C2×C4×C71—41no—
103C2×C4×C71—41no—
104C2×C4×C71—42no—
105C2×C4×C71—42no—
106C2×C4×C71—42no—
107C2×C4×C71—42no—
108C2×C4×C71—42no—
109C2×C4×C71—42no—
110C2×C4×C71C4×D412yes—
111C2×C4×C71#12011yescharacteristic
112C2×C4×C71#12112yes—
113C2×C4×C71#12311yescharacteristic
114C2×C4×C71#13512yes—
115C2×C4×C71#13712yes—
116C71⋊C8—44no—
117C71⋊C8—82no—
118C22×C71—21no—
119C22×C71—21no—
120C22×C71—42no—
121C22×C71—41no—
122C22×C71—41no—
123C22×C71—41no—
124C22×C71—81no—
125C22×C71#38611yescharacteristic
126C22×C71#41611yescharacteristic
127C22×C71#41911yescharacteristic
128C4×C71—22no—
129C4×C71—22no—
130C4×C71—41no—
131C4×C71—41no—
132C4×C71—41no—
133C4×C71—82no—
134C4×C71—82no—
135C4×C71#45212yes—
136C4×C71#48512yes—
137#9192—711nomaximal, Sylow
138C2×C71—21no—
139C2×C71—21no—
140C2×C71—81no—
141C2×C71#187311yescharacteristic
142C2×C71#190911yescharacteristic
143C2×C71#191411yescharacteristic
144#2921—711no—
145#2628D7111yescharacteristic
146#1548—712no—
147#1458—711no—
148#1286—712no—
149C71#919211yescharacteristic, complemented, Sylow
150C2×C4⋊Q8C2×D7111yescharacteristic
151C2×#117C2×D7111yescharacteristic
152#433—1424no—
153#431—712no—
154#423—712no—
155#407—1421no—
156C4×M16—1421no—
157#400—21no—
158#399—21no—
159#395Dic7112yes—
160#393Dic7112yes—
161#389—21no—
162#382C2×D7111yescharacteristic
163#380—21no—
164#333—1424no—
165#324—1421no—
166#323—1421no—
167C22×Q8C2×Dic7112yes—
168C23×C4C22⋊D7111yescharacteristic
169C2×M16—1422no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172#117—22no—
173#117—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4C2×Dic7111yescharacteristic
179C2×C4⋊C4C22⋊D7111yescharacteristic
180C2×C22⋊C4—22no—
181C2×C22⋊C4C2×Dic7111yescharacteristic
182C2×C22⋊C4C22⋊D7111yescharacteristic
183C2×C42C22⋊D7111yescharacteristic
184C2×C42C22×D7111yescharacteristic
185C4⋊C8—714no—
186C4⋊C8—1422no—
187C4⋊C8—1422no—
188C22⋊C8—1422no—
189C22⋊C8—1424no—
190#97—1422no—
191C4×C8—1422no—
192C4×C8—1422no—
193#95—22no—
194#95—22no—
195#95—21no—
196#95C2×Dic7112yes—
197C24C22⋊Dic7111yescharacteristic
198C2×Q8—24no—
199C2×Q8—82no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4C22⋊Dic7111yescharacteristic
207C22×C4C22⋊Dic7111yescharacteristic
208C22×C4C22⋊Dic7111yescharacteristic
209C22×C4C22×Dic7111yescharacteristic
210C22×C4C2×C22⋊D7111yescharacteristic
211C22×C4C2×C22⋊D7111yescharacteristic
212M16—2842no—
213M16—2842no—
214C2×C8—2842no—
215C2×C8—1424no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—82no—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C22⋊C4—82no—
226C42—21no—
227C42—21no—
228C42D4×D7112yes—
229C42#1176586412yes—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—41no—
234C23—41no—
235C23C2×C22⋊Dic7111yescharacteristic
236Q8—84no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4D4×Dic7112yes—
255C2×C4#1455609611yescharacteristic
256C2×C4#1455610012yes—
257C2×C4#1455610911yescharacteristic
258C2×C4#1455614812yes—
259C2×C4#1455615912yes—
260C8—2844no—
261C8—5682no—
262C22—21no—
263C22—21no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22#1798587911yescharacteristic
270C22#1798599511yescharacteristic
271C22#1798600811yescharacteristic
272C4—22no—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—82no—
278C4—82no—
279C4#1798615912yes—
280C4#1798633712yes—
281C2—21no—
282C2—21no—
283C2—81no—
284C2#2258865311yescharacteristic
285C2#2258886311yescharacteristic
286C2#2258888811yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2628C2
socleC22×C71#386
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2628D71
hypocenterC71#9192
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2628

Complements

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

NH ≅ G/Ntype
2. #22587021C2semidirect
4. #22586198C2semidirect
149. C71#9192semidirect

Maximal subgroups

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

subgroupindex
#225969782
#225870212
#225866542
#225861982
C71×#26282
#919271