Finite Group Databasebeta

← #31906614

Subgroups — #31906614

Order 18176.

454 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#17985725C212yesmaximal
2#22586286C212yesmaximal, complemented
3#22586284C212yesmaximal, complemented
4C71×#2317C211yesmaximal, characteristic
5C2×C22⋊(C71⋊C8)—22no—
6#17985725—22no—
7#17985725C2214yes—
8C2×#117×C71C411yescharacteristic
9C2×#117×C71C411yescharacteristic
10C22×C4⋊C4×C71C2211yescharacteristic
11#389×C71—22no—
12#385×C71—22no—
13#384×C71C411yescharacteristic
14#384×C71C2211yescharacteristic
15#382×C71C411yescharacteristic
16#382×C71C2211yescharacteristic
17C22×C71⋊C8—42no—
18C22⋊(C71⋊C8)—24no—
19C22⋊(C71⋊C8)—42no—
20C23×C4×C71C2×C411yescharacteristic
21C23×C4×C71D412yes—
22#117×C71—22no—
23#117×C71—42no—
24#117×C71C814yescomplemented
25C2×C4⋊C4×C71—21no—
26C2×C4⋊C4×C71—22no—
27C2×C4⋊C4×C71—22no—
28C2×C4⋊C4×C71—42no—
29C2×C4⋊C4×C71C2×C411yescharacteristic
30C2×C4⋊C4×C71C2×C411yescharacteristic
31C2×C4⋊C4×C71C2×C411yescharacteristic
32C2×C4⋊C4×C71D412yes—
33C2×C4⋊C4×C71C2311yescharacteristic
34C2×C22⋊C4×C71—21no—
35C2×C22⋊C4×C71—21no—
36C2×C42×C71—21no—
37C2×C42×C71C2×C411yescharacteristic
38C2×C42×C71C2×C411yescharacteristic
39#95×C71—22no—
40#95×C71—22no—
41C2×C71⋊C8—44no—
42C2×C71⋊C8—84no—
43C24×C71C22⋊C411yescharacteristic
44C22×C4×C71—22no—
45C22×C4×C71—22no—
46C22×C4×C71—22no—
47C22×C4×C71—21no—
48C22×C4×C71—21no—
49C22×C4×C71—22no—
50C22×C4×C71—21no—
51C22×C4×C71—21no—
52C22×C4×C71—82no—
53C22×C4×C71C22⋊C411yescharacteristic
54C22×C4×C71C22⋊C412yes—
55C22×C4×C71C2×C812yes—
56C22×C4×C71M1612yes—
57C22×C4×C71C22×C411yescharacteristic
58C22×C4×C71C2×D412yes—
59C4⋊C4×C71—24no—
60C4⋊C4×C71—41no—
61C4⋊C4×C71—42no—
62C4⋊C4×C71—41no—
63C4⋊C4×C71—41no—
64C4⋊C4×C71—41no—
65C4⋊C4×C71—82no—
66C4⋊C4×C71—82no—
67C22⋊C4×C71—24no—
68C22⋊C4×C71—42no—
69C42×C71—21no—
70C42×C71—21no—
71C42×C71—42no—
72C42×C71C2×C812yescomplemented
73C42×C71C2×C812yescomplemented
74C23×C71—41no—
75C23×C71—41no—
76C23×C71C22⋊C812yes—
77C23×C71#9912yes—
78C23×C71#10112yes—
79C23×C71C2×C22⋊C411yescharacteristic
80C2×C4×C71—24no—
81C2×C4×C71—24no—
82C2×C4×C71—21no—
83C2×C4×C71—21no—
84C2×C4×C71—21no—
85C2×C4×C71—21no—
86C2×C4×C71—22no—
87C2×C4×C71—22no—
88C2×C4×C71—22no—
89C2×C4×C71—22no—
90C2×C4×C71—41no—
91C2×C4×C71—41no—
92C2×C4×C71—41no—
93C2×C4×C71—41no—
94C2×C4×C71—44no—
95C2×C4×C71—42no—
96C2×C4×C71—42no—
97C2×C4×C71—42no—
98C2×C4×C71—42no—
99C2×C4×C71—41no—
100C2×C4×C71—41no—
101C2×C4×C71—82no—
102C2×C4×C71C22⋊C812yes—
103C2×C4×C71C22×C811yescharacteristic
104C2×C4×C71C2×M1611yescharacteristic
105C71⋊C8—84no—
106C22×C71—21no—
107C22×C71—22no—
108C22×C71—21no—
109C22×C71—22no—
110C22×C71—42no—
111C22×C71—42no—
112C22×C71#32414yes—
113C22×C71C2×C22⋊C811yescharacteristic
114C22×C71C2×#9911yescharacteristic
115C22×C71C2×#10111yescharacteristic
116C4×C71—22no—
117C4×C71—21no—
118C4×C71—21no—
119C4×C71—42no—
120C4×C71—42no—
121C4×C71—42no—
122C4×C71—42no—
123C4×C71—81no—
124C4×C71—81no—
125#11448—711nomaximal, Sylow
126C2×C71—21no—
127C2×C71—24no—
128C2×C71—21no—
129C2×C71C2×#32411yescharacteristic
130C2×C71#212711yescharacteristic
131C2×C71#213311yescharacteristic
132#2317D7111yescharacteristic
133C2×#324—712no—
134#1325—712no—
135#1324—712no—
136C71#1144811yescharacteristic, complemented, Sylow
137C2×#117Dic7111yescharacteristic
138C2×#117Dic7111yescharacteristic
139C22×C4⋊C4C2×D7111yescharacteristic
140C2×C22⋊C8—1422no—
141#389—22no—
142#385—22no—
143#384Dic7111yescharacteristic
144#384C2×D7111yescharacteristic
145#382Dic7111yescharacteristic
146#382C2×D7111yescharacteristic
147#324—714no—
148#324—1422no—
149C23×C4C2×Dic7111yescharacteristic
150C23×C4C22⋊D7112yes—
151C22×C8—2842no—
152#117—22no—
153#117—42no—
154#117C71⋊C814yescomplemented
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4C2×Dic7111yescharacteristic
160C2×C4⋊C4C2×Dic7111yescharacteristic
161C2×C4⋊C4C2×Dic7111yescharacteristic
162C2×C4⋊C4C22⋊D7112yes—
163C2×C4⋊C4C22×D7111yescharacteristic
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C42—21no—
167C2×C42C2×Dic7111yescharacteristic
168C2×C42C2×Dic7111yescharacteristic
169C22⋊C8—2842no—
170C22⋊C8—1424no—
171#95—22no—
172#95—22no—
173C24C22⋊Dic7111yescharacteristic
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—82no—
183C22×C4C22⋊Dic7111yescharacteristic
184C22×C4C22⋊Dic7112yes—
185C22×C4C2×C71⋊C812yes—
186C22×C4#1176584912yes—
187C22×C4C22×Dic7111yescharacteristic
188C22×C4C2×C22⋊D7112yes—
189C2×C8—2844no—
190C2×C8—5684no—
191C4⋊C4—24no—
192C4⋊C4—41no—
193C4⋊C4—42no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—82no—
198C4⋊C4—82no—
199C22⋊C4—24no—
200C22⋊C4—42no—
201C42—21no—
202C42—21no—
203C42—42no—
204C42C2×C71⋊C812yescomplemented
205C42C2×C71⋊C812yescomplemented
206C23—41no—
207C23—41no—
208C23C22⋊(C71⋊C8)12yes—
209C23#1455604312yes—
210C23#1455604712yes—
211C23C2×C22⋊Dic7111yescharacteristic
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—44no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—82no—
234C2×C4C22⋊(C71⋊C8)12yes—
235C2×C4C22×C71⋊C811yescharacteristic
236C2×C4C2×#1176584911yescharacteristic
237C8—5684no—
238C22—21no—
239C22—22no—
240C22—21no—
241C22—22no—
242C22—42no—
243C22—42no—
244C22#1798572514yes—
245C22C2×C22⋊(C71⋊C8)11yescharacteristic
246C22C2×#1455604311yescharacteristic
247C22C2×#1455604711yescharacteristic
248C4—22no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—42no—
254C4—42no—
255C4—81no—
256C4—81no—
257C2—21no—
258C2—21no—
259C2—24no—
260C2C2×#1798572511yescharacteristic
261C2#2259000911yescharacteristic
262C2#2259002911yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC2×C4×C71C22×C8
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2317C2
socleC22×C71C2×#99
prime radicalC24×C71C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2317D71
hypocenterC71#11448
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. #22586286C2semidirect
3. #22586284C2semidirect
24. #117×C71C8semidirect
72. C42×C71C2×C8semidirect
73. C42×C71C2×C8semidirect
136. C71#11448semidirect
154. #117C71⋊C8semidirect
204. C42C2×C71⋊C8semidirect
205. C42C2×C71⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×#179857252
#225862862
#225862842
C71×#23172
#1144871