Finite Group Databasebeta

← #31913724

Subgroups — #31913724

Order 18176.

566 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 107 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#22588264C212yesmaximal
2#22586467C214yesmaximal, complemented
3C71×#2310C211yesmaximal, characteristic
4C2×#14556047—22no—
5C2×#14556047C2214yes—
6#17985963—22no—
7C23×Q8×C71C2211yescharacteristic
8C2×#117×C71C411yescharacteristic
9C2×#117×C71C411yescharacteristic
10#391×C71—22no—
11#391×C71C412yes—
12#391×C71C2212yes—
13#381×C71—22no—
14C2×#11765849—24no—
15#14556047—28no—
16#14556047—44no—
17C22⋊(C71⋊C8)—44no—
18C22×Q8×C71—24no—
19C22×Q8×C71—42no—
20C22×Q8×C71—42no—
21C22×Q8×C71C2×C411yescharacteristic
22C22×Q8×C71D412yes—
23C22×Q8×C71C2311yescharacteristic
24C23×C4×C71C2×C411yescharacteristic
25C23×C4×C71D412yes—
26#117×C71—42no—
27#117×C71—42no—
28C2×C4⋊C4×C71—21no—
29C2×C4⋊C4×C71C2×C412yes—
30C2×C22⋊C4×C71—21no—
31C2×C22⋊C4×C71—21no—
32C2×C42×C71—21no—
33C2×C42×C71C2×C412yes—
34#95×C71—24no—
35#95×C71—24no—
36#11765849—44no—
37#11765849—44no—
38C2×C71⋊C8—44no—
39C24×C71C22⋊C411yescharacteristic
40C2×Q8×C71—24no—
41C2×Q8×C71—22no—
42C2×Q8×C71—24no—
43C2×Q8×C71—41no—
44C2×Q8×C71—41no—
45C2×Q8×C71—42no—
46C2×Q8×C71—41no—
47C2×Q8×C71—41no—
48C2×Q8×C71—416no—
49C2×Q8×C71#4114yes—
50C22×C4×C71—22no—
51C22×C4×C71—22no—
52C22×C4×C71—22no—
53C22×C4×C71—24no—
54C22×C4×C71—24no—
55C22×C4×C71—22no—
56C22×C4×C71—22no—
57C22×C4×C71—42no—
58C22×C4×C71C22⋊C412yes—
59C22×C4×C71C22⋊C411yescharacteristic
60C22×C4×C71C22×C411yescharacteristic
61C22×C4×C71C2×D412yes—
62C4⋊C4×C71—42no—
63C4⋊C4×C71—42no—
64C22⋊C4×C71—42no—
65C22⋊C4×C71—42no—
66C42×C71—42no—
67C42×C71—42no—
68C23×C71—21no—
69C23×C71—21no—
70C23×C71—42no—
71C23×C71#9912yes—
72C23×C71C2×C22⋊C411yescharacteristic
73Q8×C71—44no—
74Q8×C71—48no—
75Q8×C71—44no—
76C2×C4×C71—28no—
77C2×C4×C71—21no—
78C2×C4×C71—21no—
79C2×C4×C71—21no—
80C2×C4×C71—21no—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71—22no—
84C2×C4×C71—22no—
85C2×C4×C71—22no—
86C2×C4×C71—48no—
87C2×C4×C71—44no—
88C2×C4×C71—44no—
89C2×C4×C71—42no—
90C2×C4×C71—42no—
91C2×C4×C71#9912yes—
92C2×C4×C71#11712yes—
93C2×C4×C71#12314yes—
94C71⋊C8—84no—
95C22×C71—22no—
96C22×C71—22no—
97C22×C71—24no—
98C22×C71—42no—
99C22×C71—42no—
100C22×C71#38111yescharacteristic
101C22×C71C2×#9911yescharacteristic
102C22×C71C2×#9911yescharacteristic
103C4×C71—24no—
104C4×C71—22no—
105C4×C71—22no—
106C4×C71—42no—
107C4×C71—82no—
108C4×C71—82no—
109#12862—711nomaximal, Sylow
110C2×C71—21no—
111C2×C71—21no—
112C2×C71—42no—
113C2×C71#181911yescharacteristic
114C2×C71#213312yes—
115#2310D7111yescharacteristic
116#1784—712no—
117#1394—714no—
118C71#1286211yescharacteristic, complemented, Sylow
119C23×Q8C2×D7111yescharacteristic
120C2×#117Dic7111yescharacteristic
121C2×#117Dic7111yescharacteristic
122C2×#101—714no—
123C2×#101—1422no—
124#407—1422no—
125#391—22no—
126#391Dic7112yes—
127#391C2×D7112yes—
128#381—22no—
129C22×Q8—24no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8C2×Dic7111yescharacteristic
133C22×Q8C22⋊D7112yes—
134C22×Q8C22×D7111yescharacteristic
135C23×C4C2×Dic7111yescharacteristic
136C23×C4C22⋊D7112yes—
137C2×M16—1424no—
138#117—42no—
139#117—42no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4C2×Dic7112yes—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C42—21no—
145C2×C42C2×Dic7112yes—
146#101—2844no—
147#101—1428no—
148C22⋊C8—2844no—
149#95—24no—
150#95—24no—
151C24C22⋊Dic7111yescharacteristic
152C2×Q8—24no—
153C2×Q8—22no—
154C2×Q8—24no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—42no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—416no—
161C2×Q8#1176586514yes—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—42no—
170C22×C4C22⋊Dic7112yes—
171C22×C4C22⋊Dic7111yescharacteristic
172C22×C4C22×Dic7111yescharacteristic
173C22×C4C2×C22⋊D7112yes—
174M16—2844no—
175M16—2844no—
176C2×C8—2844no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C22⋊C4—42no—
180C22⋊C4—42no—
181C42—42no—
182C42—42no—
183C23—21no—
184C23—21no—
185C23—42no—
186C23#1455604312yes—
187C23C2×C22⋊Dic7111yescharacteristic
188Q8—44no—
189Q8—48no—
190Q8—44no—
191C2×C4—28no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—48no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4#1455604312yes—
207C2×C4#1455608712yes—
208C2×C4#1455611114yes—
209C8—5684no—
210C22—22no—
211C22—24no—
212C22—22no—
213C22—42no—
214C22—42no—
215C22#1798586211yescharacteristic
216C22C2×#1455604311yescharacteristic
217C22C2×#1455604311yescharacteristic
218C4—24no—
219C4—22no—
220C4—22no—
221C4—42no—
222C4—82no—
223C4—82no—
224C2—21no—
225C2—21no—
226C2—42no—
227C2#2258841411yescharacteristic
228C2#2259002912yes—
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC22×Q8C22×D71
FittingC71×#2310C2
socleC22×C71C2×#99
prime radicalC24×C71C22⋊C4
cosocleC22×Q8×C71C23
hypercenter#2310D71
hypocenterC71#12862
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ #2310

Complements

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

NH ≅ G/Ntype
2. #22586467C2semidirect
118. C71#12862semidirect

Maximal subgroups

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

subgroupindex
#225882642
#225864672
C71×#23102
#1286271