Finite Group Databasebeta

← #31890851

Subgroups — #31890851

Order 18176.

554 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 139 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#22588102C212yesmaximal
2#22586221C214yesmaximal, complemented
3C2×#382×C71C211yesmaximal, characteristic
4C2×C22⋊(C71⋊C8)—22no—
5C2×#14556039C2214yes—
6#17985754—22no—
7C22×C4⋊C4×C71C411yescharacteristic
8C22×C4⋊C4×C71C411yescharacteristic
9C22×C42×C71C2211yescharacteristic
10#382×C71—22no—
11#382×C71C412yes—
12#382×C71C2212yes—
13C2×#95×C71—22no—
14C22×C71⋊C8—24no—
15C22⋊(C71⋊C8)—28no—
16#14556039—28no—
17C23×C4×C71—21no—
18C23×C4×C71—21no—
19C23×C4×C71C2×C411yescharacteristic
20C23×C4×C71D412yes—
21C2×C4⋊C4×C71—21no—
22C2×C4⋊C4×C71—21no—
23C2×C4⋊C4×C71—42no—
24C2×C4⋊C4×C71—42no—
25C2×C4⋊C4×C71C2×C412yes—
26C2×C4⋊C4×C71C2×C412yes—
27C2×C42×C71—24no—
28C2×C42×C71C2×C411yescharacteristic
29C2×C42×C71D412yes—
30C2×C42×C71C2311yescharacteristic
31#95×C71—24no—
32#95×C71—24no—
33C2×C71⋊C8—28no—
34C2×C71⋊C8—44no—
35C2×C71⋊C8—44no—
36C24×C71C22⋊C411yescharacteristic
37C22×C4×C71—22no—
38C22×C4×C71—22no—
39C22×C4×C71—22no—
40C22×C4×C71—24no—
41C22×C4×C71—24no—
42C22×C4×C71—22no—
43C22×C4×C71—22no—
44C22×C4×C71—42no—
45C22×C4×C71—44no—
46C22×C4×C71—42no—
47C22×C4×C71—42no—
48C22×C4×C71—44no—
49C22×C4×C71C22⋊C412yes—
50C22×C4×C71C22⋊C411yescharacteristic
51C22×C4×C71C22×C411yescharacteristic
52C22×C4×C71C2×D412yes—
53C4⋊C4×C71—42no—
54C4⋊C4×C71—42no—
55C4⋊C4×C71—42no—
56C4⋊C4×C71—42no—
57C42×C71—22no—
58C42×C71—24no—
59C42×C71#4114yes—
60C23×C71—21no—
61C23×C71—21no—
62C23×C71—22no—
63C23×C71#10012yes—
64C23×C71#10414yes—
65C23×C71C2×C22⋊C411yescharacteristic
66C2×C4×C71—28no—
67C2×C4×C71—21no—
68C2×C4×C71—21no—
69C2×C4×C71—21no—
70C2×C4×C71—21no—
71C2×C4×C71—22no—
72C2×C4×C71—22no—
73C2×C4×C71—22no—
74C2×C4×C71—22no—
75C2×C4×C71—22no—
76C2×C4×C71—48no—
77C2×C4×C71—42no—
78C2×C4×C71—42no—
79C2×C4×C71—44no—
80C2×C4×C71—44no—
81C2×C4×C71—84no—
82C2×C4×C71—84no—
83C2×C4×C71#10112yes—
84C2×C4×C71#11712yes—
85C2×C4×C71#12314yes—
86C71⋊C8—48no—
87C22×C71—22no—
88C22×C71—24no—
89C22×C71—24no—
90C22×C71—22no—
91C22×C71#32914yes—
92C22×C71#38111yescharacteristic
93C22×C71C2×#10011yescharacteristic
94C22×C71C2×#10111yescharacteristic
95C22×C71C2×#10412yes—
96C22×C71#42112yes—
97C4×C71—22no—
98C4×C71—22no—
99C4×C71—24no—
100C4×C71—42no—
101C4×C71—82no—
102C4×C71—82no—
103#8718—711nomaximal, Sylow
104C2×C71—24no—
105C2×C71C2×#32911yescharacteristic
106C2×C71#152812yes—
107C2×C71#153311yescharacteristic
108C2×C71#182111yescharacteristic
109C2×C71#182612yes—
110C2×#382D7111yescharacteristic
111#1749—712no—
112#1295—714no—
113C71#871811yescharacteristic, complemented, Sylow
114C22×C4⋊C4Dic7111yescharacteristic
115C22×C4⋊C4Dic7111yescharacteristic
116C22×C42C2×D7111yescharacteristic
117C2×C22⋊C8—1422no—
118C2×#97—714no—
119#382—22no—
120#382Dic7112yes—
121#382C2×D7112yes—
122C2×#95—22no—
123#336—1422no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4C2×Dic7111yescharacteristic
127C23×C4C22⋊D7112yes—
128C22×C8—1424no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4C2×Dic7112yes—
134C2×C4⋊C4C2×Dic7112yes—
135C2×C42—24no—
136C2×C42C2×Dic7111yescharacteristic
137C2×C42C22⋊D7112yes—
138C2×C42C22×D7111yescharacteristic
139C22⋊C8—1428no—
140#97—1428no—
141#95—24no—
142#95—24no—
143C24C22⋊Dic7111yescharacteristic
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—42no—
152C22×C4—44no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—44no—
156C22×C4C22⋊Dic7112yes—
157C22×C4C22⋊Dic7111yescharacteristic
158C22×C4C22×Dic7111yescharacteristic
159C22×C4C2×C22⋊D7112yes—
160C2×C8—2844no—
161C2×C8—2844no—
162C2×C8—1428no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C42—22no—
168C42—24no—
169C42#1176586514yes—
170C23—21no—
171C23—21no—
172C23—22no—
173C23#1455604512yes—
174C23#1455605514yes—
175C23C2×C22⋊Dic7111yescharacteristic
176C2×C4—28no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—48no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—44no—
190C2×C4—44no—
191C2×C4—84no—
192C2×C4—84no—
193C2×C4#1455604712yes—
194C2×C4#1455608712yes—
195C2×C4#1455611114yes—
196C8—2848no—
197C22—22no—
198C22—24no—
199C22—24no—
200C22—22no—
201C22#1798573914yes—
202C22#1798586211yescharacteristic
203C22C2×#1455604511yescharacteristic
204C22C2×#1455604711yescharacteristic
205C22C2×#1455605512yes—
206C22#1798601612yes—
207C4—22no—
208C4—22no—
209C4—24no—
210C4—42no—
211C4—82no—
212C4—82no—
213C2—24no—
214C2C2×#1798573911yescharacteristic
215C2#2258694212yes—
216C2#2258696411yescharacteristic
217C2#2258842011yescharacteristic
218C2#2258843912yes—
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC2×#382×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenterC2×#382D71
hypocenterC71#8718
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
2. #22586221C2semidirect
113. C71#8718semidirect

Maximal subgroups

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

subgroupindex
#225881022
#225862212
C2×#382×C712
#871871