Finite Group Databasebeta

← #34001728

Subgroups — #34001728

Order 18688.

432 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 114 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#22625717C211yesmaximal, characteristic
2#22625646C211yesmaximal, characteristic
3#22624749C214yesmaximal
4#22618292C211yesmaximal, characteristic
5C2×C4⋊#10596833—21no—
6C2×#14556646—21no—
7C2×C4×#10596833—21no—
8#17989453—21no—
9#17989453C2212yes—
10#17989453C2212yes—
11C2×#14556582—21no—
12#17988486—21no—
13#17988486C2212yes—
14#17988483—21no—
15#17988483C412yes—
16#17988457C411yescharacteristic
17#17988452—21no—
18C42×Dic73C2211yescharacteristic
19#384×C73C411yescharacteristic
20C22×#10596833—22no—
21C22×#10596833—22no—
22C4⋊#10596833—24no—
23#14556646—42no—
24C4×#10596833—24no—
25#14556582—42no—
26#14556582—42no—
27C2×C4⋊Dic73—22no—
28C2×C4⋊Dic73—21no—
29C2×C4⋊Dic73—21no—
30C2×C4⋊Dic73C2×C412yes—
31C2×#11924007—22no—
32C2×C4×Dic73C2×C411yescharacteristic
33C2×C4×Dic73D411yescharacteristic
34C2×C4×Dic73D411yescharacteristic
35C2×C4×Dic73D411yescharacteristic
36C2×C4×Dic73D411yescharacteristic
37C2×C4×Dic73C2311yescharacteristic
38#14556498—22no—
39#14556498—22no—
40C2×C4⋊C4×C73—21no—
41C2×C4⋊C4×C73—21no—
42C2×C4⋊C4×C73C2×C412yes—
43C2×C42×C73C2×C411yescharacteristic
44C2×#10596833—24no—
45C2×#10596833—24no—
46C2×#10596833—42no—
47C2×#10596833—42no—
48C2×#10596833—42no—
49C2×#10596833—42no—
50C22×Dic73—22no—
51C22×Dic73C22⋊C411yescharacteristic
52C22×Dic73C22⋊C411yescharacteristic
53C22×Dic73C2×D411yescharacteristic
54C22×Dic73C2×D411yescharacteristic
55C4⋊Dic73—42no—
56C4⋊Dic73—42no—
57C4⋊Dic73—42no—
58C4⋊Dic73—42no—
59C4⋊Dic73—42no—
60#11924007—44no—
61C4×Dic73—22no—
62C4×Dic73—22no—
63C4×Dic73—22no—
64C4×Dic73—22no—
65C4×Dic73—41no—
66C4×Dic73D812yes—
67C4×Dic73SD1612yes—
68C22×C4×C73—22no—
69C22×C4×C73C22⋊C411yescharacteristic
70C22×C4×C73C22⋊C411yescharacteristic
71C22×C4×C73C22×C411yescharacteristic
72C4⋊C4×C73—24no—
73C4⋊C4×C73—42no—
74C4⋊C4×C73—42no—
75C42×C73—22no—
76C23×C73C2×C22⋊C411yescharacteristic
77C2×C4×C73—21no—
78C2×C4×C73—21no—
79C2×C4×C73—21no—
80C2×C4×C73—21no—
81C2×C4×C73—42no—
82C2×C4×C73—42no—
83C2×C4×C73—42no—
84C2×C4×C73#10112yes—
85C2×C4×C73#10212yes—
86C2×C4×C73#10412yes—
87C2×C4×C73#10412yes—
88C2×C4×C73#13112yes—
89C2×Dic73—21no—
90C2×Dic73—21no—
91C2×Dic73—21no—
92C2×Dic73—21no—
93C2×Dic73—21no—
94C2×Dic73—21no—
95C2×Dic73—21no—
96C2×Dic73—21no—
97C2×Dic73—21no—
98C2×Dic73—42no—
99C2×Dic73—42no—
100C2×Dic73—42no—
101C2×Dic73#10212yes—
102C2×Dic73C2×D811yescharacteristic
103C2×Dic73C2×SD1611yescharacteristic
104C2×Dic73#13712yes—
105#10596833—44no—
106#10596833—44no—
107C22×C73#40811yescharacteristic
108C22×C73#41011yescharacteristic
109C22×C73C2×#10111yescharacteristic
110C22×C73C2×#10211yescharacteristic
111C22×C73#41911yescharacteristic
112C22×C73C2×#10411yescharacteristic
113C22×C73C2×#10411yescharacteristic
114C4×C73—22no—
115C4×C73—22no—
116C4×C73—82no—
117C4×C73#33014yes—
118Dic73—22no—
119Dic73—21no—
120Dic73—21no—
121Dic73—22no—
122Dic73—41no—
123Dic73—82no—
124#7680—731nomaximal, Sylow
125C2×C73#147011yescharacteristic
126C2×C73#148511yescharacteristic
127C2×C73#149211yescharacteristic
128C2×C73#150211yescharacteristic
129C2×C73#150611yescharacteristic
130C2×C73C2×#33011yescharacteristic
131C2×C73#154611yescharacteristic
132#2859—731no—
133#1767—731no—
134#1744—731no—
135#1271—734no—
136C73#768011yescharacteristic, complemented, Sylow
137C2×#125—1461no—
138C2×C4⋊C8—1461no—
139C2×#97—1461no—
140C2×C4×C8—1461no—
141#389—732no—
142#389—732no—
143#389—1461no—
144#389—1461no—
145#384—731no—
146#384#6440811yescharacteristic
147#382—1461no—
148C43—731no—
149#336—732no—
150#336—732no—
151#336—1461no—
152C22×C8—1462no—
153C22×C8—1462no—
154#125—2922no—
155#125—2922no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—732no—
159C2×C4⋊C4—1462no—
160C2×C4⋊C4—1462no—
161C2×C4⋊C4—1461no—
162C2×C4⋊C4—1461no—
163C2×C4⋊C4C2×#6440812yes—
164C2×C42—731no—
165C2×C42—731no—
166C2×C42—731no—
167C2×C42—731no—
168C2×C42—731no—
169C2×C42—731no—
170C2×C42C2×#6440811yescharacteristic
171C4⋊C8—1464no—
172#97—2922no—
173C4×C8—1464no—
174#95—1462no—
175#95—1462no—
176C22×C4—22no—
177C22×C4—731no—
178C22×C4—731no—
179C22×C4—731no—
180C22×C4—731no—
181C22×C4—1462no—
182C22×C4C22⋊#6440811yescharacteristic
183C22×C4C22⋊#6440811yescharacteristic
184C22×C4C22×#6440811yescharacteristic
185C2×C8—2922no—
186C2×C8—2922no—
187C2×C8—2922no—
188C2×C8—2922no—
189C2×C8—1464no—
190C2×C8—1464no—
191C4⋊C4—24no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—2922no—
195C4⋊C4—2924no—
196C4⋊C4—2922no—
197C4⋊C4—2922no—
198C4⋊C4—2922no—
199C4⋊C4—2922no—
200C42—22no—
201C42—2921no—
202C42—732no—
203C42—732no—
204C42—1462no—
205C42—1462no—
206C42—1462no—
207C42—1462no—
208C23C2×C22⋊#6440811yescharacteristic
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—2922no—
217C2×C4—2922no—
218C2×C4—2922no—
219C2×C4—731no—
220C2×C4—732no—
221C2×C4—731no—
222C2×C4—732no—
223C2×C4—1461no—
224C2×C4—1461no—
225C2×C4—1461no—
226C2×C4—1461no—
227C2×C4—1461no—
228C2×C4—1461no—
229C2×C4—1461no—
230C2×C4—1461no—
231C2×C4—1461no—
232C2×C4#1455665212yes—
233C2×C4#1455665312yes—
234C2×C4#1455665512yes—
235C2×C4#1455665612yes—
236C2×C4#1455667812yes—
237C8—2924no—
238C8—2924no—
239C22#1798956111yescharacteristic
240C22#1798956811yescharacteristic
241C22C2×#1455665211yescharacteristic
242C22C2×#1455665311yescharacteristic
243C22#1798958011yescharacteristic
244C22C2×#1455665511yescharacteristic
245C22C2×#1455665611yescharacteristic
246C4—22no—
247C4—22no—
248C4—82no—
249C4—2921no—
250C4—5842no—
251C4—1461no—
252C4—1461no—
253C4—1462no—
254C4—1462no—
255C4#1798944414yes—
256C2#2262520411yescharacteristic
257C2#2262525411yescharacteristic
258C2#2262527711yescharacteristic
259C2#2262531211yescharacteristic
260C2#2262532311yescharacteristic
261C2C2×#1798944411yescharacteristic
262C2#2262544011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
Fitting#384×C73C4
socle, prime radicalC23×C73C2×C22⋊C4
cosocleC2×C4×Dic73C23
hypercenter#384#64408
hypocenterC73#7680
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
136. C73#7680semidirect

Maximal subgroups

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

subgroupindex
#226257172
#226256462
#226247492
#226182922
#768073