Finite Group Databasebeta

← #31900525

Subgroups — #31900525

Order 18176.

400 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 91 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#22596612C211yesmaximal, characteristic
2#22589314C212yesmaximal
3#22586222C212yesmaximal
4#22586221C211yesmaximal, characteristic
5C71×#1960C211yesmaximal, characteristic
6C2×#14556039C2212yes—
7#17985941—22no—
8#17985938—21no—
9#17985920—22no—
10#17985919—21no—
11#17985865C2212yes—
12#17985761—22no—
13#17985761—22no—
14C2×C4⋊Q8×C71C2211yescharacteristic
15C2×#103×C71—21no—
16C2×#103×C71—21no—
17C2×#97×C71C2211yescharacteristic
18#382×C71C2211yescharacteristic
19C22×Dic142—21no—
20C22×C71⋊C8—22no—
21C2×C4⋊Dic71—21no—
22C2×C4⋊Dic71D412yes—
23C2×C4⋊Dic71D412yes—
24C2×Dic71⋊C4—22no—
25#14556039—22no—
26#14556039—22no—
27C71⋊#95—21no—
28C71⋊#95—22no—
29C71⋊#95—21no—
30C71⋊#95—22no—
31C71⋊#95—22no—
32C22×Q8×C71—21no—
33C22×C8×C71—21no—
34C4⋊Q8×C71—22no—
35C4⋊Q8×C71—21no—
36C4⋊Q8×C71Q812yescomplemented
37C2×C4⋊C4×C71—22no—
38C2×C4⋊C4×C71D411yescharacteristic
39C2×C4⋊C4×C71D411yescharacteristic
40C2×C42×C71C2311yescharacteristic
41#103×C71—42no—
42#103×C71—42no—
43#97×C71—22no—
44#95×C71—22no—
45C2×Dic142—81no—
46C2×Dic142—82no—
47C22×Dic71—22no—
48C22×Dic71—22no—
49C2×C71⋊C8—24no—
50C2×C71⋊C8—42no—
51C2×C71⋊C8—42no—
52C4⋊Dic71—42no—
53C4⋊Dic71—42no—
54C4⋊Dic71—81no—
55Dic71⋊C4—82no—
56C2×Q8×C71—22no—
57C2×Q8×C71—41no—
58C2×Q8×C71—41no—
59C2×Q8×C71—41no—
60C2×Q8×C71—41no—
61C2×Q8×C71—41no—
62C22×C4×C71—21no—
63C22×C4×C71—21no—
64C22×C4×C71—41no—
65C22×C4×C71C2×D412yes—
66C22×C4×C71C2×D411yescharacteristic
67C2×C8×C71—22no—
68C2×C8×C71—41no—
69C2×C8×C71—41no—
70C4⋊C4×C71—24no—
71C4⋊C4×C71—41no—
72C4⋊C4×C71—41no—
73C4⋊C4×C71—42no—
74C42×C71C2×Q811yescharacteristic
75C42×C71#4112yes—
76C42×C71#4111yescharacteristic
77C23×C71#12011yescharacteristic
78Q8×C71—42no—
79Q8×C71—42no—
80Q8×C71—81no—
81Q8×C71—81no—
82C2×C4×C71—21no—
83C2×C4×C71—21no—
84C2×C4×C71—22no—
85C2×C4×C71—22no—
86C2×C4×C71—42no—
87C2×C4×C71—41no—
88C2×C4×C71—41no—
89C2×C4×C71—42no—
90C2×C4×C71—42no—
91C2×C4×C71—81no—
92C2×C4×C71—81no—
93C2×C4×C71C22⋊Q812yes—
94C2×C4×C71C22⋊Q811yescharacteristic
95C2×C4×C71#12412yes—
96C2×C4×C71#12411yescharacteristic
97C2×Dic71—42no—
98C2×Dic71—42no—
99C2×Dic71—42no—
100C2×Dic71—42no—
101C2×Dic71—42no—
102C2×Dic71—42no—
103Dic142—84no—
104C8×C71—42no—
105C71⋊C8—44no—
106C22×C71#39311yescharacteristic
107C22×C71#45311yescharacteristic
108C22×C71#45411yescharacteristic
109C22×C71#45512yes—
110C22×C71#45612yes—
111C4×C71—21no—
112C4×C71—21no—
113C4×C71—22no—
114C4×C71—22no—
115C4×C71—81no—
116C4×C71—81no—
117C4×C71—82no—
118Dic71—82no—
119Dic71—82no—
120#10359—711nomaximal, Sylow
121C2×C71#165911yescharacteristic
122C2×C71#166011yescharacteristic
123C2×C71#166412yes—
124C2×C71#202711yescharacteristic
125C2×C71#202812yes—
126#2844—711no—
127#1993—712no—
128#1960D7111yescharacteristic
129#1295—712no—
130#1295—711no—
131C71#1035911yescharacteristic, complemented, Sylow
132C2×C4⋊Q8C2×D7111yescharacteristic
133C2×#103—21no—
134C2×#103—21no—
135C2×#97—712no—
136C2×#97C2×D7111yescharacteristic
137#400—1422no—
138#400—1421no—
139#395—1422no—
140#395—1421no—
141#382—712no—
142#382C2×D7111yescharacteristic
143#340—1422no—
144#340—1422no—
145C22×Q8—21no—
146C22×Q8—1421no—
147C22×C8—21no—
148C22×C8—1422no—
149C4⋊Q8—22no—
150C4⋊Q8—21no—
151C4⋊Q8Dic14212yescomplemented
152C2×C4⋊C4—22no—
153C2×C4⋊C4—712no—
154C2×C4⋊C4—712no—
155C2×C4⋊C4—1422no—
156C2×C4⋊C4—1421no—
157C2×C4⋊C4C22⋊D7111yescharacteristic
158C2×C4⋊C4C22⋊D7111yescharacteristic
159C2×C42C22×D7111yescharacteristic
160#103—42no—
161#103—42no—
162#97—22no—
163#97—1422no—
164#97—1422no—
165#95—22no—
166#95—1422no—
167#95—1421no—
168#95—1421no—
169#95—1422no—
170#95—1422no—
171C2×Q8—22no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—5681no—
178C2×Q8—5682no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—1422no—
183C22×C4—1422no—
184C22×C4D4×D7112yes—
185C22×C4C2×C22⋊D7111yescharacteristic
186C2×C8—22no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—2842no—
190C2×C8—2842no—
191C2×C8—1424no—
192C4⋊C4—24no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C4⋊C4—2842no—
197C4⋊C4—2842no—
198C4⋊C4—5682no—
199C4⋊C4—5681no—
200C42C2×Dic14211yescharacteristic
201C42#1176586412yes—
202C42#1176586511yescharacteristic
203C23#1455609611yescharacteristic
204Q8—42no—
205Q8—42no—
206Q8—81no—
207Q8—81no—
208Q8—5684no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—2842no—
221C2×C4—2842no—
222C2×C4—2842no—
223C2×C4—2842no—
224C2×C4—2842no—
225C2×C4—2842no—
226C2×C4#1455610512yes—
227C2×C4#1455610711yescharacteristic
228C2×C4#1455611412yes—
229C2×C4#1455611511yescharacteristic
230C8—42no—
231C8—2844no—
232C22#1798591111yescharacteristic
233C22#1798616611yescharacteristic
234C22#1798617111yescharacteristic
235C22#1798617512yes—
236C22#1798618012yes—
237C4—21no—
238C4—21no—
239C4—22no—
240C4—22no—
241C4—81no—
242C4—82no—
243C4—81no—
244C4—5682no—
245C4—5682no—
246C2#2258764411yescharacteristic
247C2#2258764911yescharacteristic
248C2#2258767112yes—
249C2#2258949611yescharacteristic
250C2#2258949912yes—
subgroupG/H
centerC23#14556096
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1960C2
socle, prime radicalC23×C71#120
hypercenter#1960D71
hypocenterC71#10359
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
36. C4⋊Q8×C71Q8semidirect
131. C71#10359semidirect
151. C4⋊Q8Dic142semidirect

Maximal subgroups

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

subgroupindex
#225966122
#225893142
#225862222
#225862212
C71×#19602
#1035971