Finite Group Databasebeta

← #31901386

Subgroups — #31901386

Order 18176.

372 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 85 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#22589185C211yesmaximal, characteristic
2#22587128C211yesmaximal, characteristic
3#22587078C211yesmaximal, characteristic
4#22587021C211yesmaximal, characteristic
5#22586761C211yesmaximal, characteristic
6#22586231C211yesmaximal, characteristic
7C71×#3209C211yesmaximal, characteristic
8#17986052C2211yescharacteristic
9#17986023C2211yescharacteristic
10#17986023C2211yescharacteristic
11C2×#14556047—22no—
12C4×#11765849C2211yescharacteristic
13#17985751—21no—
14#17985749—21no—
15#17985748—22no—
16#17985745—21no—
17#17985745—21no—
18Q8⋊(C71⋊C8)—22no—
19#533×C71C2211yescharacteristic
20C2×C4⋊Q8×C71C2211yescharacteristic
21#494×C71—21no—
22#487×C71—21no—
23#462×C71—21no—
24#462×C71—21no—
25#451×C71—22no—
26#431×C71C2211yescharacteristic
27C2×#11765849—21no—
28C2×#11765849—21no—
29C2×#11765849—21no—
30C4⋊(C71⋊C8)—21no—
31C4⋊(C71⋊C8)—21no—
32C4⋊(C71⋊C8)—21no—
33C4⋊(C71⋊C8)—21no—
34C4⋊(C71⋊C8)D412yes—
35#14556047—44no—
36C22⋊(C71⋊C8)—21no—
37#14556039—21no—
38C4×C71⋊C8—21no—
39C4×C71⋊C8—21no—
40C22×Q8×C71D412yes—
41C2×Q16×C71—22no—
42C2×Q16×C71—22no—
43C4⋊Q8×C71—21no—
44C4⋊Q8×C71—21no—
45C4⋊Q8×C71—21no—
46C4⋊Q8×C71—21no—
47C4⋊Q8×C71D411yescharacteristic
48C4⋊Q8×C71D411yescharacteristic
49#125×C71—21no—
50C22⋊Q8×C71—21no—
51C22⋊Q8×C71—21no—
52C4×Q8×C71—21no—
53C4×Q8×C71—21no—
54#117×C71—21no—
55C2×C4⋊C4×C71—21no—
56C2×C4⋊C4×C71—21no—
57C2×C42×C71C2311yescharacteristic
58C4⋊C8×C71D411yescharacteristic
59C4⋊C8×C71D411yescharacteristic
60#103×C71—22no—
61#103×C71—22no—
62C22⋊C8×C71—21no—
63C4×C8×C71—21no—
64#11765849—22no—
65#11765849—41no—
66#11765849—42no—
67#11765849—81no—
68C2×C71⋊C8—22no—
69C2×C71⋊C8—41no—
70C2×C71⋊C8—41no—
71C2×C71⋊C8—41no—
72C2×Q8×C71—22no—
73C2×Q8×C71—21no—
74C2×Q8×C71—21no—
75C2×Q8×C71—22no—
76C2×Q8×C71—42no—
77C2×Q8×C71—42no—
78C22×C4×C71—22no—
79C22×C4×C71C2×D411yescharacteristic
80C22×C4×C71#4111yescharacteristic
81C22×C4×C71#4111yescharacteristic
82Q16×C71—82no—
83Q16×C71—82no—
84C2×C8×C71—21no—
85C2×C8×C71—21no—
86C4⋊C4×C71—21no—
87C4⋊C4×C71—21no—
88C4⋊C4×C71—21no—
89C4⋊C4×C71—21no—
90C4⋊C4×C71—41no—
91C4⋊C4×C71—41no—
92C4⋊C4×C71—41no—
93C4⋊C4×C71—41no—
94C4⋊C4×C71—41no—
95C4⋊C4×C71—41no—
96C22⋊C4×C71—41no—
97C42×C71—41no—
98C42×C71C2×D411yescharacteristic
99C42×C71C2×D411yescharacteristic
100C42×C71C2×D411yescharacteristic
101C42×C71#4111yescharacteristic
102C23×C71#12311yescharacteristic
103Q8×C71—41no—
104Q8×C71—42no—
105Q8×C71—41no—
106Q8×C71—42no—
107Q8×C71—41no—
108Q8×C71—41no—
109Q8×C71—82no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—22no—
114C2×C4×C71—21no—
115C2×C4×C71—44no—
116C2×C4×C71—41no—
117C2×C4×C71—41no—
118C2×C4×C71—42no—
119C2×C4×C71#12011yescharacteristic
120C2×C4×C71#12111yescharacteristic
121C2×C4×C71#12111yescharacteristic
122C2×C4×C71#12111yescharacteristic
123C2×C4×C71#12311yescharacteristic
124C2×C4×C71#12411yescharacteristic
125C8×C71—41no—
126C8×C71—41no—
127C71⋊C8—42no—
128C71⋊C8—42no—
129C71⋊C8—81no—
130C71⋊C8—81no—
131C22×C71—21no—
132C22×C71—21no—
133C22×C71—21no—
134C22×C71#39411yescharacteristic
135C4×C71—21no—
136C4×C71—21no—
137C4×C71—21no—
138C4×C71—81no—
139C4×C71—82no—
140C4×C71—81no—
141C4×C71#45311yescharacteristic
142C4×C71#45511yescharacteristic
143C4×C71#47111yescharacteristic
144C4×C71#47312yes—
145C4×C71#47311yescharacteristic
146#10493—711nomaximal, Sylow
147C2×C71—41no—
148C2×C71#203711yescharacteristic
149C2×C71#204711yescharacteristic
150C2×C71#205211yescharacteristic
151#3209D7111yescharacteristic
152#1966—711no—
153#1574—711no—
154#1561—711no—
155#1548—711no—
156#1485—711no—
157#1299—711no—
158C71#1049311yescharacteristic, complemented, Sylow
159#533C2×D7111yescharacteristic
160C2×C4⋊Q8C2×D7111yescharacteristic
161#494—21no—
162#487—21no—
163#462—21no—
164#462—21no—
165#451—22no—
166#431—711no—
167#431C2×D7111yescharacteristic
168#423—711no—
169#423—711no—
170C2×#101—1422no—
171C4×M16—711no—
172#335—1421no—
173#334—1421no—
174#333—1422no—
175#332—1421no—
176#332—1421no—
177Q8⋊C8—1422no—
178C22×Q8C22⋊D7112yes—
179C2×Q16—22no—
180C2×Q16—22no—
181C2×M16—1421no—
182C2×M16—1421no—
183C2×M16—1421no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8C22⋊D7111yescharacteristic
189C4⋊Q8C22⋊D7111yescharacteristic
190#125—21no—
191C22⋊Q8—21no—
192C22⋊Q8—21no—
193C4×Q8—21no—
194C4×Q8—21no—
195#117—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C42C22×D7111yescharacteristic
199C4⋊C8—712no—
200C4⋊C8—1421no—
201C4⋊C8—1421no—
202C4⋊C8—1421no—
203C4⋊C8—1421no—
204C4⋊C8C22⋊D7111yescharacteristic
205C4⋊C8C22⋊D7111yescharacteristic
206#103—22no—
207#103—22no—
208#101—2844no—
209C22⋊C8—21no—
210C22⋊C8—1421no—
211#97—1421no—
212C4×C8—21no—
213C4×C8—1421no—
214C4×C8—1421no—
215C2×Q8—22no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—22no—
219C2×Q8—42no—
220C2×Q8—42no—
221C22×C4—22no—
222C22×C4C2×C22⋊D7111yescharacteristic
223C22×C4#1176586411yescharacteristic
224C22×C4#1176586411yescharacteristic
225Q16—82no—
226Q16—82no—
227M16—2841no—
228M16—2842no—
229M16—5681no—
230M16—1422no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—2841no—
234C2×C8—2841no—
235C2×C8—2841no—
236C2×C8—1422no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C22⋊C4—41no—
248C42—41no—
249C42D4×D7111yescharacteristic
250C42C2×C22⋊D7111yescharacteristic
251C42C2×C22⋊D7111yescharacteristic
252C42#1176586411yescharacteristic
253C23#1455610911yescharacteristic
254Q8—41no—
255Q8—42no—
256Q8—41no—
257Q8—42no—
258Q8—41no—
259Q8—41no—
260Q8—82no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—44no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4#1455609711yescharacteristic
271C2×C4#1455609911yescharacteristic
272C2×C4#1455610011yescharacteristic
273C2×C4#1455610011yescharacteristic
274C2×C4#1455610911yescharacteristic
275C2×C4#1455611511yescharacteristic
276C8—41no—
277C8—41no—
278C8—2842no—
279C8—2842no—
280C8—5681no—
281C8—5681no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22#1798591311yescharacteristic
286C4—21no—
287C4—21no—
288C4—21no—
289C4—82no—
290C4—81no—
291C4—81no—
292C4#1798616711yescharacteristic
293C4#1798617711yescharacteristic
294C4#1798626411yescharacteristic
295C4#1798627112yes—
296C4#1798627211yescharacteristic
297C2—41no—
298C2#2258954711yescharacteristic
299C2#2258960511yescharacteristic
300C2#2258963611yescharacteristic
subgroupG/H
centerC22#17985913
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#3209C2
socleC22×C71#394
prime radicalC23×C71#123
hypercenter#3209D71
hypocenterC71#10493
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
158. C71#10493semidirect

Maximal subgroups

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

subgroupindex
#225891852
#225871282
#225870782
#225870212
#225867612
#225862312
C71×#32092
#1049371