Finite Group Databasebeta

← #31900476

Subgroups — #31900476

Order 18176.

500 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 125 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#22589267C212yesmaximal
2#22589145C211yesmaximal, characteristic
3#22586223C211yesmaximal, characteristic, complemented
4#22586181C212yesmaximal, complemented
5C71×#2571C211yesmaximal, characteristic, complemented
6C2×#14556125C2211yescharacteristic
7C2×C4⋊(C71⋊C8)C2212yes—
8C2×#14556048—22no—
9C2×#14556048—22no—
10C2×#14556048—22no—
11C2×C4×C71⋊C8C2211yescharacteristic
12#400×C71—22no—
13#389×C71—22no—
14#384×C71C2211yescharacteristic
15#382×C71—22no—
16C4×C4⋊C4×C71C2212yes—
17C22×D284—21no—
18C22×D284—22no—
19C22×C71⋊C8—21no—
20C22×C71⋊C8—22no—
21#14556125—22no—
22#14556125—21no—
23#14556125Q812yes—
24C4⋊(C71⋊C8)—22no—
25C4⋊(C71⋊C8)—22no—
26#14556048—44no—
27#14556048—44no—
28#14556048—44no—
29C4×C71⋊C8—22no—
30C2×C4⋊C4×C71—22no—
31C2×C4⋊C4×C71—22no—
32C2×C4⋊C4×C71D412yes—
33C2×C4⋊C4×C71D412yes—
34C2×C4⋊C4×C71D412yes—
35C2×C42×C71—22no—
36C2×C42×C71C2311yescharacteristic
37#95×C71—22no—
38#95×C71—24no—
39#95×C71—22no—
40C23×D71—41no—
41C2×D284—22no—
42C2×D284—24no—
43C2×D284—42no—
44C2×D284—42no—
45C2×D284—42no—
46C2×D284—42no—
47C2×D284—42no—
48C2×D284—42no—
49C2×D284—41no—
50C2×D284—42no—
51C2×C71⋊C8—22no—
52C2×C71⋊C8—41no—
53C2×C71⋊C8—41no—
54C2×C71⋊C8—42no—
55C2×C71⋊C8—42no—
56C2×C71⋊C8—42no—
57C22×C4×C71—22no—
58C22×C4×C71—22no—
59C22×C4×C71—22no—
60C22×C4×C71C2×D411yescharacteristic
61C22×C4×C71C2×D412yes—
62C4⋊C4×C71—22no—
63C4⋊C4×C71—22no—
64C4⋊C4×C71—22no—
65C4⋊C4×C71—22no—
66C4⋊C4×C71—42no—
67C4⋊C4×C71—42no—
68C4⋊C4×C71—42no—
69C4⋊C4×C71—82no—
70C42×C71—42no—
71C42×C71—42no—
72C42×C71C2×Q811yescharacteristic
73C42×C71#4112yes—
74C42×C71#4111yescharacteristic
75C23×C71#12011yescharacteristic
76C22×D71—42no—
77C22×D71—82no—
78C22×D71—82no—
79C22×D71—81no—
80C22×D71—81no—
81C2×C4×C71—21no—
82C2×C4×C71—21no—
83C2×C4×C71—42no—
84C2×C4×C71—42no—
85C2×C4×C71—42no—
86C2×C4×C71—42no—
87C2×C4×C71—44no—
88C2×C4×C71—42no—
89C2×C4×C71—42no—
90C2×C4×C71—42no—
91C2×C4×C71C22⋊Q811yescharacteristic
92C2×C4×C71C22⋊Q812yes—
93C2×C4×C71#12411yescharacteristic
94C2×C4×C71#12412yes—
95C2×C4×C71#13512yes—
96C2×C4×C71#13512yes—
97C2×C4×C71AGL(1,ℤ/8)12yes—
98C2×C4×C71AGL(1,ℤ/8)12yes—
99D284—44no—
100D284—44no—
101D284—44no—
102D284—82no—
103D284—82no—
104D284—82no—
105C71⋊C8—42no—
106C71⋊C8—82no—
107C22×C71#39311yescharacteristic
108C22×C71#44912yes—
109C22×C71#44912yes—
110C22×C71#45311yescharacteristic
111C22×C71#45411yescharacteristic
112C2×D71—84no—
113C2×D71—82no—
114C2×D71—161no—
115C2×D71—161no—
116C2×D71—162no—
117C4×C71—21no—
118C4×C71—21no—
119C4×C71—82no—
120C4×C71—82no—
121C4×C71—82no—
122C4×C71#47812yes—
123C4×C71#47812yes—
124C4×C71#48812yes—
125C4×C71#49012yes—
126#10351—711nomaximal, Sylow
127C2×C71#162011yescharacteristic
128C2×C71#162411yescharacteristic
129C2×C71#162812yes—
130C2×C71#202512yes—
131C2×C71#202711yescharacteristic
132D71—162no—
133#2571D7111yescharacteristic, complemented
134#1982—712no—
135#1957—711no—
136#1296—711no—
137#1279—712no—
138C71#1035111yescharacteristic, complemented, Sylow
139C2×#127—711no—
140C2×C4⋊C8—712no—
141C2×#102—1422no—
142C2×#102—1422no—
143C2×#102—1422no—
144C2×C4×C8—711no—
145#400—22no—
146#389—22no—
147#384C2×D7111yescharacteristic
148#382—22no—
149C4×C4⋊C4C2×D7112yes—
150C22×D4—1421no—
151C22×D4—1422no—
152C22×C8—1422no—
153C22×C8—1421no—
154#127—712no—
155#127—1422no—
156#127—1421no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4C22⋊D7112yes—
160C2×C4⋊C4C22⋊D7112yes—
161C2×C4⋊C4C22⋊D7112yes—
162C2×C42—22no—
163C2×C42C22×D7111yescharacteristic
164C4⋊C8—1422no—
165C4⋊C8—1422no—
166#102—2844no—
167#102—2844no—
168#102—2844no—
169C4×C8—1422no—
170#95—22no—
171#95—24no—
172#95—22no—
173C24—2841no—
174C2×D4—2842no—
175C2×D4—2842no—
176C2×D4—2841no—
177C2×D4—2842no—
178C2×D4—2842no—
179C2×D4—2842no—
180C2×D4—2842no—
181C2×D4—2842no—
182C2×D4—1422no—
183C2×D4—1424no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4C2×C22⋊D7111yescharacteristic
188C22×C4C2×C22⋊D7112yes—
189C2×C8—2841no—
190C2×C8—2841no—
191C2×C8—2842no—
192C2×C8—2842no—
193C2×C8—2842no—
194C2×C8—1422no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—82no—
203C42—42no—
204C42—42no—
205C42Q8×D7111yescharacteristic
206C42#1176586612yes—
207C42#1176586611yescharacteristic
208C23—2842no—
209C23—5682no—
210C23—5682no—
211C23—5681no—
212C23—5681no—
213C23#1455609711yescharacteristic
214D4—2844no—
215D4—2844no—
216D4—2844no—
217D4—5682no—
218D4—5682no—
219D4—5682no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4#1455610411yescharacteristic
231C2×C4#1455610412yes—
232C2×C4#1455611711yescharacteristic
233C2×C4#1455611712yes—
234C2×C4#1455614812yes—
235C2×C4#1455614812yes—
236C2×C4#1455615312yes—
237C2×C4#1455615312yes—
238C8—2842no—
239C8—5682no—
240C22—5684no—
241C22—5682no—
242C22—11361no—
243C22—11361no—
244C22—11362no—
245C22#1798591011yescharacteristic
246C22#1798614412yes—
247C22#1798614412yes—
248C22#1798616711yescharacteristic
249C22#1798617211yescharacteristic
250C4—21no—
251C4—21no—
252C4—82no—
253C4—82no—
254C4—82no—
255C4#1798629612yes—
256C4#1798629612yes—
257C4#1798635612yes—
258C4#1798636512yes—
259C2—11362no—
260C2#2258739911yescharacteristic
261C2#2258742311yescharacteristic
262C2#2258745012yes—
263C2#2258948312yes—
264C2#2258949311yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2571C2
socleC23×C71#120
prime radicalC2×#14556125C22
hypercenter#2571D71
hypocenterC71#10351
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2571

Complements

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

NH ≅ G/Ntype
3. #22586223C2semidirect
4. #22586181C2semidirect
5. C71×#2571C2semidirect
133. #2571D71semidirect
138. C71#10351semidirect

Maximal subgroups

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

subgroupindex
#225892672
#225891452
#225862232
#225861812
C71×#25712
#1035171