Finite Group Databasebeta

← #31901393

Subgroups — #31901393

Order 18176.

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

normal subgroups only

This group has 344 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#22589330C211yesmaximal, characteristic
2#22587128C211yesmaximal, characteristic
3#22587078C211yesmaximal, characteristic
4#22587025C211yesmaximal, characteristic
5#22586761C211yesmaximal, characteristic
6#22586231C211yesmaximal, characteristic
7C71×#3229C211yesmaximal, characteristic
8#17986052C2211yescharacteristic
9#17986023C2211yescharacteristic
10#17986023C2211yescharacteristic
11C4×#11765849C2211yescharacteristic
12#17985768—22no—
13#17985751—21no—
14#17985749—21no—
15#17985745—22no—
16#17985745—21no—
17#17985745—21no—
18Q8⋊(C71⋊C8)—22no—
19#533×C71C2211yescharacteristic
20C2×C4⋊Q8×C71C2211yescharacteristic
21#498×C71—21no—
22#487×C71—21no—
23#483×C71—22no—
24#475×C71—21no—
25#475×C71—21no—
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—
35C22⋊(C71⋊C8)—21no—
36#14556039—21no—
37C4×C71⋊C8—21no—
38C4×C71⋊C8—21no—
39C22×Q8×C71—21no—
40C4⋊Q8×C71—21no—
41C4⋊Q8×C71—21no—
42C4⋊Q8×C71—21no—
43C4⋊Q8×C71—21no—
44C4⋊Q8×C71D411yescharacteristic
45C4⋊Q8×C71D411yescharacteristic
46#125×C71—21no—
47C22⋊Q8×C71—21no—
48C22⋊Q8×C71—21no—
49C4×Q8×C71—21no—
50C4×Q8×C71—21no—
51#117×C71—21no—
52C2×C4⋊C4×C71—21no—
53C2×C4⋊C4×C71D412yes—
54C2×C42×C71C2311yescharacteristic
55#106×C71—22no—
56#106×C71—22no—
57C4⋊C8×C71D411yescharacteristic
58C4⋊C8×C71D411yescharacteristic
59#103×C71—22no—
60#103×C71—22no—
61C22⋊C8×C71—21no—
62C4×C8×C71—21no—
63#11765849—22no—
64#11765849—41no—
65#11765849—42no—
66#11765849—81no—
67C2×C71⋊C8—22no—
68C2×C71⋊C8—41no—
69C2×C71⋊C8—41no—
70C2×C71⋊C8—41no—
71C2×Q8×C71—21no—
72C2×Q8×C71—21no—
73C2×Q8×C71—41no—
74C2×Q8×C71—41no—
75C2×Q8×C71—82no—
76C22×C4×C71—22no—
77C22×C4×C71C2×D411yescharacteristic
78C22×C4×C71#4111yescharacteristic
79C22×C4×C71#4111yescharacteristic
80C2×C8×C71—21no—
81C2×C8×C71—21no—
82C4⋊C4×C71—22no—
83C4⋊C4×C71—21no—
84C4⋊C4×C71—22no—
85C4⋊C4×C71—21no—
86C4⋊C4×C71—21no—
87C4⋊C4×C71—21no—
88C4⋊C4×C71—41no—
89C4⋊C4×C71—41no—
90C4⋊C4×C71—41no—
91C4⋊C4×C71—41no—
92C22⋊C4×C71—41no—
93C42×C71—41no—
94C42×C71C2×D411yescharacteristic
95C42×C71C2×D411yescharacteristic
96C42×C71C2×D411yescharacteristic
97C42×C71#4111yescharacteristic
98C23×C71#12311yescharacteristic
99Q8×C71—41no—
100Q8×C71—41no—
101Q8×C71—41no—
102Q8×C71—41no—
103Q8×C71—82no—
104Q8×C71—82no—
105C2×C4×C71—21no—
106C2×C4×C71—21no—
107C2×C4×C71—22no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—41no—
111C2×C4×C71—41no—
112C2×C4×C71—42no—
113C2×C4×C71—82no—
114C2×C4×C71#12011yescharacteristic
115C2×C4×C71#12111yescharacteristic
116C2×C4×C71#12111yescharacteristic
117C2×C4×C71#12111yescharacteristic
118C2×C4×C71#12311yescharacteristic
119C2×C4×C71#12411yescharacteristic
120C8×C71—41no—
121C8×C71—41no—
122C71⋊C8—42no—
123C71⋊C8—42no—
124C71⋊C8—81no—
125C71⋊C8—81no—
126C22×C71—21no—
127C22×C71—21no—
128C22×C71—21no—
129C22×C71#39411yescharacteristic
130C4×C71—21no—
131C4×C71—21no—
132C4×C71—21no—
133C4×C71—81no—
134C4×C71—81no—
135C4×C71—82no—
136C4×C71#45411yescharacteristic
137C4×C71#45611yescharacteristic
138C4×C71#47112yes—
139C4×C71#47111yescharacteristic
140C4×C71#47311yescharacteristic
141#10494—711nomaximal, Sylow
142C2×C71—41no—
143C2×C71#203811yescharacteristic
144C2×C71#204611yescharacteristic
145C2×C71#205211yescharacteristic
146#3229D7111yescharacteristic
147#1997—711no—
148#1574—711no—
149#1561—711no—
150#1549—711no—
151#1485—711no—
152#1299—711no—
153C71#1049411yescharacteristic, complemented, Sylow
154#533C2×D7111yescharacteristic
155C2×C4⋊Q8C2×D7111yescharacteristic
156#498—21no—
157#487—21no—
158#483—22no—
159#475—21no—
160#475—21no—
161#431—711no—
162#431C2×D7111yescharacteristic
163#423—711no—
164#423—711no—
165C4×M16—711no—
166#343—1422no—
167#335—1421no—
168#334—1421no—
169#332—1422no—
170#332—1421no—
171#332—1421no—
172Q8⋊C8—1422no—
173C22×Q8—21no—
174C2×M16—1421no—
175C2×M16—1421no—
176C2×M16—1421no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8C22⋊D7111yescharacteristic
182C4⋊Q8C22⋊D7111yescharacteristic
183#125—21no—
184C22⋊Q8—21no—
185C22⋊Q8—21no—
186C4×Q8—21no—
187C4×Q8—21no—
188#117—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4C22⋊D7112yes—
191C2×C42C22×D7111yescharacteristic
192#106—22no—
193#106—22no—
194C4⋊C8—712no—
195C4⋊C8—1421no—
196C4⋊C8—1421no—
197C4⋊C8—1421no—
198C4⋊C8—1421no—
199C4⋊C8C22⋊D7111yescharacteristic
200C4⋊C8C22⋊D7111yescharacteristic
201#103—22no—
202#103—22no—
203C22⋊C8—21no—
204C22⋊C8—1421no—
205#97—1421no—
206C4×C8—21no—
207C4×C8—1421no—
208C4×C8—1421no—
209C2×Q8—21no—
210C2×Q8—21no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—82no—
214C22×C4—22no—
215C22×C4C2×C22⋊D7111yescharacteristic
216C22×C4#1176586411yescharacteristic
217C22×C4#1176586411yescharacteristic
218M16—2841no—
219M16—2842no—
220M16—5681no—
221M16—1422no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—2841no—
225C2×C8—2841no—
226C2×C8—2841no—
227C2×C8—1422no—
228C4⋊C4—22no—
229C4⋊C4—21no—
230C4⋊C4—22no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C22⋊C4—41no—
239C42—41no—
240C42D4×D7111yescharacteristic
241C42C2×C22⋊D7111yescharacteristic
242C42C2×C22⋊D7111yescharacteristic
243C42#1176586411yescharacteristic
244C23#1455610911yescharacteristic
245Q8—41no—
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8—82no—
250Q8—82no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—82no—
260C2×C4#1455609711yescharacteristic
261C2×C4#1455609911yescharacteristic
262C2×C4#1455610011yescharacteristic
263C2×C4#1455610011yescharacteristic
264C2×C4#1455610911yescharacteristic
265C2×C4#1455611511yescharacteristic
266C8—41no—
267C8—41no—
268C8—2842no—
269C8—2842no—
270C8—5681no—
271C8—5681no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22#1798591311yescharacteristic
276C4—21no—
277C4—21no—
278C4—21no—
279C4—81no—
280C4—82no—
281C4—81no—
282C4#1798617211yescharacteristic
283C4#1798618211yescharacteristic
284C4#1798625912yes—
285C4#1798626411yescharacteristic
286C4#1798627211yescharacteristic
287C2—41no—
288C2#2258955411yescharacteristic
289C2#2258959811yescharacteristic
290C2#2258963611yescharacteristic
subgroupG/H
centerC22#17985913
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#3229C2
socleC22×C71#394
prime radicalC23×C71#123
hypercenter#3229D71
hypocenterC71#10494
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

Complements

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

NH ≅ G/Ntype
153. C71#10494semidirect

Maximal subgroups

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

subgroupindex
#225893302
#225871282
#225870782
#225870252
#225867612
#225862312
C71×#32292
#1049471