Finite Group Databasebeta

← #34005267

Subgroups — #34005267

Order 18688.

522 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 120 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#22625162C212yesmaximal
2#22624759C212yesmaximal, complemented
3#22624755C212yesmaximal, complemented
4#117×Dic73C211yesmaximal, characteristic
5#17989603C2214yes—
6#17989432—22no—
7#17989430—22no—
8C2×#14556550C2211yescharacteristic
9C22×C4×Dic73C411yescharacteristic
10C4⋊C4×Dic73—22no—
11C22⋊C4×Dic73—22no—
12#17988433C411yescharacteristic
13#17988433C2211yescharacteristic
14C42×Dic73C411yescharacteristic
15C42×Dic73C2211yescharacteristic
16C2×#117×C73C411yescharacteristic
17C4⋊#10596833—24no—
18C22⋊#10596833—24no—
19C4×#10596833—24no—
20C23×Dic73—21no—
21#14556550—22no—
22#14556550—22no—
23C2×C4⋊Dic73D412yes—
24C2×C22⋊Dic73D412yes—
25C2×C4×Dic73—22no—
26C2×C4×Dic73—22no—
27C2×C4×Dic73—21no—
28C2×C4×Dic73—22no—
29C2×C4×Dic73C814yescomplemented
30C2×C4×Dic73C2×C411yescharacteristic
31C2×C4×Dic73C2×C411yescharacteristic
32C2×C4×Dic73C2×C411yescharacteristic
33C2×C4×Dic73C2311yescharacteristic
34#14556498—22no—
35#14556498—22no—
36C23×C4×C73C2×C411yescharacteristic
37#117×C73—22no—
38#117×C73—22no—
39C2×C4⋊C4×C73—21no—
40C2×C22⋊C4×C73—21no—
41C2×C42×C73C2×C411yescharacteristic
42C2×C42×C73C2×C411yescharacteristic
43C2×#10596833—44no—
44C22×Dic73—21no—
45C22×Dic73—24no—
46C22×Dic73—21no—
47C22×Dic73—44no—
48C22×Dic73C22⋊C412yes—
49C22×Dic73C2×D412yes—
50C4⋊Dic73—24no—
51C22⋊Dic73—24no—
52C4×Dic73—24no—
53C4×Dic73—21no—
54C4×Dic73—21no—
55C4×Dic73—21no—
56C4×Dic73—21no—
57C4×Dic73—44no—
58C4×Dic73—44no—
59C4×Dic73C2×C812yes—
60C4×Dic73C2×C812yes—
61C4×Dic73#4112yes—
62C4×Dic73#4112yes—
63C24×C73C22⋊C411yescharacteristic
64C22×C4×C73—21no—
65C22×C4×C73—21no—
66C22×C4×C73—21no—
67C22×C4×C73—21no—
68C22×C4×C73—21no—
69C22×C4×C73—21no—
70C22×C4×C73C22⋊C411yescharacteristic
71C22×C4×C73C2×C812yes—
72C22×C4×C73M1612yes—
73C22×C4×C73C22×C411yescharacteristic
74C4⋊C4×C73—24no—
75C22⋊C4×C73—24no—
76C42×C73—21no—
77C42×C73—21no—
78C42×C73M1612yes—
79C42×C73M1612yes—
80C23×C73—21no—
81C23×C73—21no—
82C23×C73—42no—
83C23×C73C22⋊C812yes—
84C23×C73C2×C22⋊C411yescharacteristic
85C2×C4×C73—21no—
86C2×C4×C73—21no—
87C2×C4×C73—22no—
88C2×C4×C73—22no—
89C2×C4×C73—22no—
90C2×C4×C73—22no—
91C2×C4×C73—22no—
92C2×C4×C73—22no—
93C2×C4×C73—42no—
94C2×C4×C73—42no—
95C2×C4×C73C22⋊C812yes—
96C2×C4×C73#11711yescharacteristic
97C2×C4×C73#11711yescharacteristic
98C2×C4×C73C22×C811yescharacteristic
99C2×C4×C73C2×M1611yescharacteristic
100C2×C4×C73C2×M1611yescharacteristic
101C2×C4×C73C2×M1611yescharacteristic
102C2×Dic73—24no—
103C2×Dic73—22no—
104C2×Dic73—22no—
105C2×Dic73—22no—
106C2×Dic73—42no—
107C2×Dic73—48no—
108C2×Dic73—42no—
109C2×Dic73—42no—
110C2×Dic73#12314yes—
111#10596833—84no—
112C22×C73—22no—
113C22×C73—22no—
114C22×C73—42no—
115C22×C73—42no—
116C22×C73—42no—
117C22×C73#38111yescharacteristic
118C22×C73C2×C22⋊C811yescharacteristic
119C22×C73#40711yescharacteristic
120C4×C73—22no—
121C4×C73—42no—
122C4×C73—42no—
123C4×C73#43111yescharacteristic
124C4×C73#43111yescharacteristic
125C4×C73#43211yescharacteristic
126C4×C73#43211yescharacteristic
127Dic73—24no—
128Dic73—24no—
129Dic73—42no—
130Dic73—42no—
131#8730—731nomaximal, Sylow
132C2×C73—21no—
133C2×C73—21no—
134C2×C73—42no—
135C2×C73#181711yescharacteristic
136C2×C73#182712yes—
137C4×#117—731no—
138#1458—732no—
139#1276—732no—
140#1274—732no—
141C73#873011yescharacteristic, complemented, Sylow
142C2×#117—731no—
143C2×#117#6440811yescharacteristic
144C22×C42—731no—
145#431—734no—
146C4×C4⋊C4—1462no—
147C4×C22⋊C4—1462no—
148#376—731no—
149#376—731no—
150C43—731no—
151C43—731no—
152#324—1462no—
153#323—1462no—
154C23×C4—1461no—
155C23×C4C2×#6440811yescharacteristic
156#117—22no—
157#117—22no—
158#117—1462no—
159#117—1462no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—732no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—732no—
164C2×C42—731no—
165C2×C42—734no—
166C2×C42—731no—
167C2×C42—731no—
168C2×C42—731no—
169C2×C42—1462no—
170C2×C42—1462no—
171C2×C42—1461no—
172C2×C42—1462no—
173C2×C42C2×#6440811yescharacteristic
174C2×C42C2×#6440811yescharacteristic
175C4⋊C8—1464no—
176C22⋊C8—1464no—
177C4×C8—1464no—
178#95—1462no—
179#95—1462no—
180C24C22⋊#6440811yescharacteristic
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—2924no—
188C22×C4—732no—
189C22×C4—732no—
190C22×C4—1464no—
191C22×C4—1461no—
192C22×C4—1461no—
193C22×C4C22⋊#6440811yescharacteristic
194C22×C4#1192403612yes—
195C22×C4#1192403812yes—
196C22×C4C22×#6440811yescharacteristic
197C2×C8—2924no—
198C4⋊C4—24no—
199C4⋊C4—1464no—
200C22⋊C4—24no—
201C22⋊C4—1464no—
202C42—21no—
203C42—21no—
204C42—2924no—
205C42—2924no—
206C42—732no—
207C42—732no—
208C42—732no—
209C42—732no—
210C42—1461no—
211C42—1461no—
212C42—1464no—
213C42—1461no—
214C42—1461no—
215C42#1192403712yes—
216C42#1192403712yes—
217C23—21no—
218C23—21no—
219C23—42no—
220C23#1455664812yes—
221C23C2×C22⋊#6440811yescharacteristic
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—2922no—
233C2×C4—2928no—
234C2×C4—2922no—
235C2×C4—2922no—
236C2×C4—734no—
237C2×C4—1462no—
238C2×C4—1462no—
239C2×C4—1464no—
240C2×C4—1462no—
241C2×C4#1455664812yes—
242C2×C4#1455667011yescharacteristic
243C2×C4#1455667011yescharacteristic
244C2×C4C2×#1192403611yescharacteristic
245C2×C4C2×#1192403711yescharacteristic
246C2×C4C2×#1192403711yescharacteristic
247C2×C4C2×#1192403811yescharacteristic
248C8—5844no—
249C22—22no—
250C22—22no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22#1798953011yescharacteristic
255C22C2×#1455664811yescharacteristic
256C22#1798955711yescharacteristic
257C4—22no—
258C4—42no—
259C4—42no—
260C4—2922no—
261C4—2922no—
262C4—1464no—
263C4—1464no—
264C4#1798960411yescharacteristic
265C4#1798960411yescharacteristic
266C4#1798960811yescharacteristic
267C4#1798960811yescharacteristic
268C2—21no—
269C2—21no—
270C2—42no—
271C2#2262582211yescharacteristic
272C2#2262584112yes—
subgroupG/H
centerC2×C4#14556670
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC2×#117×C73C4
socleC22×C73#381
prime radicalC24×C73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenterC2×#117#64408
hypocenterC73#8730
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #22624759C2semidirect
3. #22624755C2semidirect
29. C2×C4×Dic73C8semidirect
141. C73#8730semidirect

Maximal subgroups

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

subgroupindex
#226251622
#226247592
#226247552
#117×Dic732
#873073