Finite Group Databasebeta

← #31885753

Subgroups — #31885753

Order 18176.

490 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 123 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#22588408C212yesmaximal
2#22586163C214yesmaximal, complemented
3C71×#2296C211yesmaximal, characteristic
4C2×C22⋊(C71⋊C8)—22no—
5#17985754—22no—
6#17985754C2214yes—
7C2×#117×C71C411yescharacteristic
8C2×#117×C71C411yescharacteristic
9C22×C42×C71C2211yescharacteristic
10#389×C71C412yes—
11#389×C71C2212yes—
12#385×C71—22no—
13C4×C4⋊C4×C71—22no—
14C22×C71⋊C8—24no—
15C22⋊(C71⋊C8)—44no—
16C23×C4×C71C2×C411yescharacteristic
17C23×C4×C71D412yes—
18#117×C71—24no—
19#117×C71—42no—
20C2×C4⋊C4×C71—22no—
21C2×C4⋊C4×C71—22no—
22C2×C4⋊C4×C71C2×C412yes—
23C2×C4⋊C4×C71C2×C412yes—
24C2×C22⋊C4×C71—21no—
25C2×C22⋊C4×C71—21no—
26C2×C42×C71—21no—
27C2×C42×C71—21no—
28C2×C42×C71—42no—
29C2×C42×C71C2×C411yescharacteristic
30C2×C42×C71D412yes—
31C2×C42×C71C2311yescharacteristic
32#95×C71—22no—
33#95×C71—22no—
34C2×C71⋊C8—44no—
35C2×C71⋊C8—44no—
36C2×C71⋊C8—44no—
37C24×C71C22⋊C411yescharacteristic
38C22×C4×C71—22no—
39C22×C4×C71—22no—
40C22×C4×C71—22no—
41C22×C4×C71—24no—
42C22×C4×C71—22no—
43C22×C4×C71—24no—
44C22×C4×C71—22no—
45C22×C4×C71—42no—
46C22×C4×C71C22⋊C412yes—
47C22×C4×C71C22⋊C411yescharacteristic
48C22×C4×C71C22×C411yescharacteristic
49C22×C4×C71C2×D412yes—
50C4⋊C4×C71—22no—
51C4⋊C4×C71—42no—
52C4⋊C4×C71—44no—
53C4⋊C4×C71—44no—
54C4⋊C4×C71M1614yes—
55C22⋊C4×C71—24no—
56C22⋊C4×C71—42no—
57C42×C71—24no—
58C42×C71—41no—
59C42×C71—42no—
60C42×C71—41no—
61C42×C71—42no—
62C23×C71—21no—
63C23×C71—21no—
64C23×C71—22no—
65C23×C71—22no—
66C23×C71#9912yes—
67C23×C71C2×C22⋊C411yescharacteristic
68C2×C4×C71—28no—
69C2×C4×C71—21no—
70C2×C4×C71—21no—
71C2×C4×C71—21no—
72C2×C4×C71—22no—
73C2×C4×C71—21no—
74C2×C4×C71—22no—
75C2×C4×C71—24no—
76C2×C4×C71—22no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71—48no—
80C2×C4×C71—42no—
81C2×C4×C71—42no—
82C2×C4×C71—42no—
83C2×C4×C71—42no—
84C2×C4×C71—42no—
85C2×C4×C71#10112yes—
86C2×C4×C71C2×M1612yes—
87C2×C4×C71#13514yes—
88C71⋊C8—84no—
89C22×C71—22no—
90C22×C71—24no—
91C22×C71—41no—
92C22×C71—41no—
93C22×C71—42no—
94C22×C71#40711yescharacteristic
95C22×C71C2×#9911yescharacteristic
96C22×C71C2×#10111yescharacteristic
97C22×C71#41612yes—
98C22×C71#42112yes—
99C4×C71—22no—
100C4×C71—22no—
101C4×C71—24no—
102C4×C71—44no—
103C4×C71—42no—
104C4×C71—82no—
105#7646—711nomaximal, Sylow
106C2×C71—42no—
107C2×C71#146311yescharacteristic
108C2×C71#148812yes—
109C2×C71#151612yes—
110C2×C71#153711yescharacteristic
111C2×C71#154411yescharacteristic
112#2296D7111yescharacteristic
113#1817—712no—
114#1271—714no—
115C71#764611yescharacteristic, complemented, Sylow
116C2×#117Dic7111yescharacteristic
117C2×#117Dic7111yescharacteristic
118C22×C42C2×D7111yescharacteristic
119C2×C22⋊C8—1422no—
120#389Dic7112yes—
121#389C2×D7112yes—
122#385—22no—
123C4×C4⋊C4—22no—
124#336—714no—
125#336—1422no—
126C23×C4C2×Dic7111yescharacteristic
127C23×C4C22⋊D7112yes—
128C22×C8—1424no—
129#117—24no—
130#117—42no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4C2×Dic7112yes—
134C2×C4⋊C4C2×Dic7112yes—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—42no—
140C2×C42C2×Dic7111yescharacteristic
141C2×C42C22⋊D7112yes—
142C2×C42C22×D7111yescharacteristic
143C22⋊C8—2844no—
144#95—22no—
145#95—22no—
146C24C22⋊Dic7111yescharacteristic
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—24no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—22no—
154C22×C4—42no—
155C22×C4C22⋊Dic7112yes—
156C22×C4C22⋊Dic7111yescharacteristic
157C22×C4C22×Dic7111yescharacteristic
158C22×C4C2×C22⋊D7112yes—
159C2×C8—2844no—
160C2×C8—2844no—
161C2×C8—2844no—
162C4⋊C4—22no—
163C4⋊C4—42no—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C4⋊C4#1176584914yes—
167C22⋊C4—24no—
168C22⋊C4—42no—
169C42—24no—
170C42—41no—
171C42—42no—
172C42—41no—
173C42—42no—
174C23—21no—
175C23—21no—
176C23—22no—
177C23—22no—
178C23#1455604312yes—
179C23C2×C22⋊Dic7111yescharacteristic
180C2×C4—28no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—48no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4#1455604712yes—
198C2×C4C2×#1176584912yes—
199C2×C4#1455614814yes—
200C8—5684no—
201C22—24no—
202C22—22no—
203C22—41no—
204C22—41no—
205C22—42no—
206C22#1798596311yescharacteristic
207C22C2×#1455604311yescharacteristic
208C22C2×#1455604711yescharacteristic
209C22#1798599512yes—
210C22#1798601612yes—
211C4—24no—
212C4—22no—
213C4—22no—
214C4—42no—
215C4—44no—
216C4—82no—
217C2—42no—
218C2#2258667011yescharacteristic
219C2#2258677412yes—
220C2#2258689612yes—
221C2#2258697811yescharacteristic
222C2#2258700511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2296C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2296D71
hypocenterC71#7646
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2296

Complements

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

NH ≅ G/Ntype
2. #22586163C2semidirect
115. C71#7646semidirect

Maximal subgroups

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

subgroupindex
#225884082
#225861632
C71×#22962
#764671