Finite Group Databasebeta

← #34009863

Subgroups — #34009863

Order 18688.

454 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 104 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×#17989432C212yesmaximal
2#22624874C212yesmaximal, complemented
3#22624872C212yesmaximal, complemented
4#22612551C211yesmaximal, characteristic
5C2×C22⋊#10596833—22no—
6#17989432—22no—
7#17989432C2214yes—
8C2×#14556550C411yescharacteristic
9C22×C4⋊Dic73C2211yescharacteristic
10#17988485—22no—
11#17988459—22no—
12#17988456C411yescharacteristic
13#17988456C2211yescharacteristic
14#17988452C411yescharacteristic
15#17988452C2211yescharacteristic
16C2×#117×C73C411yescharacteristic
17C22×#10596833—42no—
18C22⋊#10596833—24no—
19C22⋊#10596833—42no—
20C23×Dic73D412yes—
21#14556550—22no—
22#14556550C814yescomplemented
23C2×C4⋊Dic73—21no—
24C2×C4⋊Dic73—42no—
25C2×C4⋊Dic73C2×C411yescharacteristic
26C2×C4⋊Dic73D412yes—
27C2×C4⋊Dic73C2311yescharacteristic
28C2×#11924007—22no—
29C2×#11924007—22no—
30C2×C22⋊Dic73—21no—
31C2×C4×Dic73C2×C411yescharacteristic
32C2×C4×Dic73C2×C411yescharacteristic
33#14556498—22no—
34#14556498—22no—
35C23×C4×C73C2×C411yescharacteristic
36#117×C73—42no—
37C2×C4⋊C4×C73C2×C411yescharacteristic
38C2×C4⋊C4×C73C2×C411yescharacteristic
39C2×C22⋊C4×C73—21no—
40C2×C42×C73—21no—
41C2×#10596833—44no—
42C2×#10596833—84no—
43C22×Dic73—22no—
44C22×Dic73—21no—
45C22×Dic73—21no—
46C22×Dic73—22no—
47C22×Dic73—82no—
48C22×Dic73C22⋊C412yes—
49C22×Dic73C2×D412yes—
50C4⋊Dic73—24no—
51C4⋊Dic73—41no—
52C4⋊Dic73—42no—
53C4⋊Dic73—41no—
54#11924007—82no—
55#11924007—82no—
56C22⋊Dic73—24no—
57C4×Dic73—21no—
58C4×Dic73—21no—
59C4×Dic73C2×C812yescomplemented
60C4×Dic73C2×C812yescomplemented
61C24×C73C22⋊C411yescharacteristic
62C22×C4×C73—22no—
63C22×C4×C73—22no—
64C22×C4×C73—21no—
65C22×C4×C73—21no—
66C22×C4×C73C22⋊C411yescharacteristic
67C22×C4×C73C2×C812yes—
68C22×C4×C73M1612yes—
69C22×C4×C73C22×C411yescharacteristic
70C4⋊C4×C73—41no—
71C4⋊C4×C73—41no—
72C22⋊C4×C73—42no—
73C42×C73—42no—
74C23×C73—41no—
75C23×C73—41no—
76C23×C73C22⋊C812yes—
77C23×C73#9912yes—
78C23×C73#10112yes—
79C23×C73C2×C22⋊C411yescharacteristic
80C2×C4×C73—24no—
81C2×C4×C73—24no—
82C2×C4×C73—21no—
83C2×C4×C73—21no—
84C2×C4×C73—21no—
85C2×C4×C73—21no—
86C2×C4×C73—42no—
87C2×C4×C73—42no—
88C2×C4×C73—41no—
89C2×C4×C73—41no—
90C2×C4×C73C22⋊C812yes—
91C2×C4×C73C22×C811yescharacteristic
92C2×C4×C73C2×M1611yescharacteristic
93C2×Dic73—22no—
94C2×Dic73—22no—
95C2×Dic73—22no—
96C2×Dic73—22no—
97C2×Dic73—41no—
98C2×Dic73—41no—
99C2×Dic73—41no—
100C2×Dic73—41no—
101C2×Dic73—44no—
102C2×Dic73—42no—
103C2×Dic73—42no—
104C2×Dic73—82no—
105#10596833—84no—
106C22×C73—21no—
107C22×C73—22no—
108C22×C73—21no—
109C22×C73—22no—
110C22×C73—42no—
111C22×C73—42no—
112C22×C73#32414yes—
113C22×C73C2×C22⋊C811yescharacteristic
114C22×C73C2×#9911yescharacteristic
115C22×C73C2×#10111yescharacteristic
116C4×C73—22no—
117C4×C73—21no—
118C4×C73—21no—
119C4×C73—81no—
120C4×C73—81no—
121Dic73—42no—
122Dic73—42no—
123Dic73—42no—
124Dic73—42no—
125#11448—731nomaximal, Sylow
126C2×C73—21no—
127C2×C73—24no—
128C2×C73—21no—
129C2×C73C2×#32411yescharacteristic
130C2×C73#212711yescharacteristic
131C2×C73#213311yescharacteristic
132#2317—731no—
133C2×#324—732no—
134#1325—732no—
135#1324—732no—
136C73#1144811yescharacteristic, complemented, Sylow
137C2×#117—731no—
138C2×#117#6440811yescharacteristic
139C22×C4⋊C4—731no—
140C2×C22⋊C8—1462no—
141#389—1462no—
142#385—1462no—
143#384—731no—
144#384—731no—
145#382—731no—
146#382—731no—
147#324—734no—
148#324—1462no—
149C23×C4—732no—
150C23×C4C2×#6440811yescharacteristic
151C22×C8—2922no—
152#117—42no—
153#117—734no—
154#117—1462no—
155C2×C4⋊C4—2922no—
156C2×C4⋊C4—732no—
157C2×C4⋊C4—731no—
158C2×C4⋊C4—731no—
159C2×C4⋊C4—1462no—
160C2×C4⋊C4—1462no—
161C2×C4⋊C4—1461no—
162C2×C4⋊C4C2×#6440811yescharacteristic
163C2×C4⋊C4C2×#6440811yescharacteristic
164C2×C22⋊C4—21no—
165C2×C22⋊C4—1461no—
166C2×C42—21no—
167C2×C42—731no—
168C2×C42—731no—
169C22⋊C8—2922no—
170C22⋊C8—1464no—
171#95—1462no—
172#95—1462no—
173C24C22⋊#6440811yescharacteristic
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—5842no—
179C22×C4—732no—
180C22×C4—732no—
181C22×C4—1462no—
182C22×C4—1461no—
183C22×C4—1461no—
184C22×C4—1462no—
185C22×C4C22⋊#6440811yescharacteristic
186C22×C4#1192403612yes—
187C22×C4#1192403812yes—
188C22×C4C22×#6440811yescharacteristic
189C2×C8—2924no—
190C2×C8—5844no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—2922no—
194C4⋊C4—2921no—
195C4⋊C4—2921no—
196C4⋊C4—5842no—
197C4⋊C4—5842no—
198C4⋊C4—1464no—
199C22⋊C4—42no—
200C22⋊C4—1464no—
201C42—42no—
202C42—732no—
203C42—732no—
204C42—1461no—
205C42—1461no—
206C23—41no—
207C23—41no—
208C23#1455664812yes—
209C23#1455664912yes—
210C23#1455665212yes—
211C23C2×C22⋊#6440811yescharacteristic
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—2921no—
223C2×C4—2921no—
224C2×C4—2924no—
225C2×C4—2921no—
226C2×C4—2921no—
227C2×C4—2922no—
228C2×C4—2922no—
229C2×C4—5842no—
230C2×C4—1462no—
231C2×C4—1462no—
232C2×C4—1462no—
233C2×C4—1462no—
234C2×C4#1455664812yes—
235C2×C4C2×#1192403611yescharacteristic
236C2×C4C2×#1192403811yescharacteristic
237C8—5844no—
238C22—21no—
239C22—22no—
240C22—21no—
241C22—22no—
242C22—42no—
243C22—42no—
244C22#1798943314yes—
245C22C2×#1455664811yescharacteristic
246C22C2×#1455664911yescharacteristic
247C22C2×#1455665211yescharacteristic
248C4—21no—
249C4—21no—
250C4—22no—
251C4—81no—
252C4—81no—
253C4—2922no—
254C4—2922no—
255C4—2922no—
256C4—2922no—
257C2—21no—
258C2—24no—
259C2—21no—
260C2C2×#1798943311yescharacteristic
261C2#2262620611yescharacteristic
262C2#2262622611yescharacteristic
subgroupG/H
centerC22C2×#14556649
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC2×#117×C73C4
socleC22×C73C2×#99
prime radicalC24×C73C22⋊C4
cosocleC2×C4⋊Dic73C23
hypercenterC2×#117#64408
hypocenterC73#11448
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #22624874C2semidirect
3. #22624872C2semidirect
22. #14556550C8semidirect
59. C4×Dic73C2×C8semidirect
60. C4×Dic73C2×C8semidirect
136. C73#11448semidirect

Maximal subgroups

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

subgroupindex
C2×#179894322
#226248742
#226248722
#226125512
#1144873