Finite Group Databasebeta

← #31900481

Subgroups — #31900481

Order 18176.

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

normal subgroups only

This group has 416 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#22589288C212yesmaximal
2#22589141C211yesmaximal, characteristic
3#22586223C211yesmaximal, characteristic
4#22586181C212yesmaximal
5C71×#2571C211yesmaximal, characteristic
6C2×#14556128C2211yescharacteristic
7C2×C4⋊(C71⋊C8)C2212yes—
8C2×#14556051—22no—
9C2×C4×C71⋊C8C2211yescharacteristic
10#17985761—22no—
11#17985761—22no—
12#400×C71—22no—
13#389×C71—22no—
14#384×C71C2211yescharacteristic
15#382×C71—22no—
16C4×C4⋊C4×C71C2212yes—
17C22×Dic142—21no—
18C22×C71⋊C8—21no—
19C22×C71⋊C8—22no—
20#14556128—22no—
21#14556128—21no—
22#14556128Q812yes—
23C2×C4⋊Dic71—22no—
24C4⋊(C71⋊C8)—22no—
25C4⋊(C71⋊C8)—22no—
26#14556051—44no—
27C4×C71⋊C8—22no—
28C2×C4⋊C4×C71—22no—
29C2×C4⋊C4×C71—22no—
30C2×C4⋊C4×C71D412yes—
31C2×C4⋊C4×C71D412yes—
32C2×C4⋊C4×C71D412yes—
33C2×C42×C71—22no—
34C2×C42×C71C2311yescharacteristic
35#95×C71—22no—
36#95×C71—24no—
37#95×C71—22no—
38C2×Dic142—22no—
39C2×Dic142—42no—
40C2×Dic142—42no—
41C2×Dic142—41no—
42C22×Dic71—41no—
43C2×C71⋊C8—22no—
44C2×C71⋊C8—41no—
45C2×C71⋊C8—41no—
46C2×C71⋊C8—42no—
47C2×C71⋊C8—42no—
48C2×C71⋊C8—42no—
49C4⋊Dic71—24no—
50C4⋊Dic71—42no—
51C22×C4×C71—22no—
52C22×C4×C71—22no—
53C22×C4×C71—22no—
54C22×C4×C71C2×D411yescharacteristic
55C22×C4×C71C2×D412yes—
56C4⋊C4×C71—22no—
57C4⋊C4×C71—22no—
58C4⋊C4×C71—22no—
59C4⋊C4×C71—22no—
60C4⋊C4×C71—42no—
61C4⋊C4×C71—42no—
62C4⋊C4×C71—42no—
63C4⋊C4×C71—82no—
64C42×C71—42no—
65C42×C71—42no—
66C42×C71C2×Q811yescharacteristic
67C42×C71#4112yes—
68C42×C71#4111yescharacteristic
69C23×C71#12011yescharacteristic
70C2×C4×C71—21no—
71C2×C4×C71—21no—
72C2×C4×C71—42no—
73C2×C4×C71—42no—
74C2×C4×C71—42no—
75C2×C4×C71—42no—
76C2×C4×C71—44no—
77C2×C4×C71—42no—
78C2×C4×C71—42no—
79C2×C4×C71—42no—
80C2×C4×C71C22⋊Q811yescharacteristic
81C2×C4×C71C22⋊Q812yes—
82C2×C4×C71#12411yescharacteristic
83C2×C4×C71#12412yes—
84C2×C4×C71#13512yes—
85C2×C4×C71#13512yes—
86C2×C4×C71#13712yes—
87C2×C4×C71#13712yes—
88C2×Dic71—42no—
89C2×Dic71—81no—
90C2×Dic71—81no—
91Dic142—44no—
92Dic142—82no—
93C71⋊C8—42no—
94C71⋊C8—82no—
95C22×C71#39311yescharacteristic
96C22×C71#45212yes—
97C22×C71#45212yes—
98C22×C71#45311yescharacteristic
99C22×C71#45411yescharacteristic
100C4×C71—21no—
101C4×C71—21no—
102C4×C71—82no—
103C4×C71—82no—
104C4×C71—82no—
105C4×C71#47912yes—
106C4×C71#47912yes—
107C4×C71#48812yes—
108C4×C71#49112yes—
109Dic71—82no—
110#10352—711nomaximal, Sylow
111C2×C71#163011yescharacteristic
112C2×C71#163411yescharacteristic
113C2×C71#163812yes—
114C2×C71#202612yes—
115C2×C71#202711yescharacteristic
116#2571D7111yescharacteristic
117#1987—712no—
118#1956—711no—
119#1296—711no—
120#1279—712no—
121C71#1035211yescharacteristic, complemented, Sylow
122C2×C4⋊Q8—711no—
123C2×C4⋊C8—712no—
124C2×#103—1422no—
125C2×C4×C8—711no—
126#400—22no—
127#389—22no—
128#384C2×D7111yescharacteristic
129#382—22no—
130C4×C4⋊C4C2×D7112yes—
131#340—1422no—
132#340—1422no—
133C22×Q8—1421no—
134C22×C8—1422no—
135C22×C8—1421no—
136C4⋊Q8—712no—
137C4⋊Q8—1421no—
138C4⋊Q8—1422no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—1422no—
142C2×C4⋊C4C22⋊D7112yes—
143C2×C4⋊C4C22⋊D7112yes—
144C2×C4⋊C4C22⋊D7112yes—
145C2×C42—22no—
146C2×C42C22×D7111yescharacteristic
147C4⋊C8—1422no—
148C4⋊C8—1422no—
149#103—2844no—
150C4×C8—1422no—
151#95—22no—
152#95—24no—
153#95—22no—
154C2×Q8—2842no—
155C2×Q8—2842no—
156C2×Q8—2841no—
157C2×Q8—1422no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—2841no—
162C22×C4C2×C22⋊D7111yescharacteristic
163C22×C4C2×C22⋊D7112yes—
164C2×C8—2841no—
165C2×C8—2841no—
166C2×C8—2842no—
167C2×C8—2842no—
168C2×C8—2842no—
169C2×C8—1422no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—82no—
178C4⋊C4—2842no—
179C4⋊C4—1424no—
180C42—42no—
181C42—42no—
182C42Q8×D7111yescharacteristic
183C42#1176586612yes—
184C42#1176586611yescharacteristic
185C23#1455609711yescharacteristic
186Q8—2844no—
187Q8—5682no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—44no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—2842no—
199C2×C4—5681no—
200C2×C4—5681no—
201C2×C4#1455610411yescharacteristic
202C2×C4#1455610412yes—
203C2×C4#1455611711yescharacteristic
204C2×C4#1455611712yes—
205C2×C4#1455614812yes—
206C2×C4#1455614812yes—
207C2×C4#1455615812yes—
208C2×C4#1455615812yes—
209C8—2842no—
210C8—5682no—
211C22#1798591011yescharacteristic
212C22#1798616112yes—
213C22#1798616112yes—
214C22#1798616711yescharacteristic
215C22#1798617211yescharacteristic
216C4—21no—
217C4—21no—
218C4—82no—
219C4—82no—
220C4—82no—
221C4—5682no—
222C4#1798630312yes—
223C4#1798630312yes—
224C4#1798635612yes—
225C4#1798636912yes—
226C2#2258746111yescharacteristic
227C2#2258748511yescharacteristic
228C2#2258751212yes—
229C2#2258948812yes—
230C2#2258949311yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2571C2
socle, prime radicalC23×C71#120
hypercenter#2571D71
hypocenterC71#10352
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
121. C71#10352semidirect

Maximal subgroups

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

subgroupindex
#225892882
#225891412
#225862232
#225861812
C71×#25712
#1035271