Finite Group Databasebeta

← #34009882

Subgroups — #34009882

Order 18688.

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

normal subgroups only

This group has 434 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
1#22625162C211yesmaximal, characteristic
2#22625162C211yesmaximal, characteristic
3#22624876C214yesmaximal
4#22612555C211yesmaximal, characteristic
5#17989603—21no—
6#17989603—21no—
7#17989432—21no—
8#17989432—21no—
9#17989430C2212yes—
10#17989430C2212yes—
11C2×#14556550C411yescharacteristic
12C2×#14556550C2211yescharacteristic
13#17988483—21no—
14#17988483—21no—
15#17988460C412yes—
16#17988460C2212yes—
17#17988456—21no—
18#17988452—21no—
19C22×C4⋊C4×C73C411yescharacteristic
20C4⋊#10596833—41no—
21C4⋊#10596833—41no—
22C22⋊#10596833—22no—
23C22⋊#10596833—22no—
24C4×#10596833—41no—
25C4×#10596833—41no—
26#14556550—22no—
27#14556550—41no—
28#14556550—41no—
29#14556550C814yescomplemented
30C2×C4⋊Dic73—21no—
31C2×C4⋊Dic73D411yescharacteristic
32C2×C4⋊Dic73D411yescharacteristic
33C2×#11924007—22no—
34C2×#11924007—22no—
35C2×C22⋊Dic73C2×C412yes—
36C2×C22⋊Dic73C2×C411yescharacteristic
37C2×C22⋊Dic73C2311yescharacteristic
38C2×C4×Dic73—21no—
39C2×C4×Dic73D411yescharacteristic
40C2×C4×Dic73D411yescharacteristic
41#14556498—22no—
42#14556498—22no—
43C23×C4×C73C2×C411yescharacteristic
44C23×C4×C73C2×C412yes—
45C2×C4⋊C4×C73—21no—
46C2×C4⋊C4×C73—21no—
47C2×C4⋊C4×C73—41no—
48C2×C4⋊C4×C73—41no—
49C2×#10596833—42no—
50C2×#10596833—42no—
51C22×Dic73—22no—
52C22×Dic73C22⋊C411yescharacteristic
53C22×Dic73C22⋊C411yescharacteristic
54C22×Dic73C2×D411yescharacteristic
55C22×Dic73C2×D411yescharacteristic
56C4⋊Dic73—24no—
57C4⋊Dic73—41no—
58C4⋊Dic73—41no—
59#11924007—82no—
60#11924007—82no—
61C22⋊Dic73—22no—
62C22⋊Dic73—41no—
63C22⋊Dic73—41no—
64C22⋊Dic73C2×C814yes—
65C4×Dic73—24no—
66C4×Dic73—21no—
67C4×Dic73—21no—
68C4×Dic73—21no—
69C4×Dic73—21no—
70C24×C73C22×C411yescharacteristic
71C22×C4×C73—21no—
72C22×C4×C73—21no—
73C22×C4×C73—22no—
74C22×C4×C73—22no—
75C22×C4×C73—42no—
76C22×C4×C73—82no—
77C22×C4×C73C22⋊C411yescharacteristic
78C22×C4×C73C22⋊C411yescharacteristic
79C22×C4×C73C2×C812yes—
80C22×C4×C73M1612yes—
81C4⋊C4×C73—42no—
82C4⋊C4×C73—41no—
83C4⋊C4×C73—41no—
84C23×C73—21no—
85C23×C73—21no—
86C23×C73—41no—
87C23×C73—41no—
88C23×C73C2×C22⋊C411yescharacteristic
89C23×C73C22×C811yescharacteristic
90C23×C73C2×M1611yescharacteristic
91C2×C4×C73—21no—
92C2×C4×C73—21no—
93C2×C4×C73—22no—
94C2×C4×C73—22no—
95C2×C4×C73—22no—
96C2×C4×C73—22no—
97C2×C4×C73—42no—
98C2×C4×C73—42no—
99C2×C4×C73—42no—
100C2×C4×C73—84no—
101C2×C4×C73C22⋊C812yes—
102C2×C4×C73C22⋊C812yes—
103C2×C4×C73#9911yescharacteristic
104C2×C4×C73#9911yescharacteristic
105C2×C4×C73#10112yes—
106C2×Dic73—24no—
107C2×Dic73—21no—
108C2×Dic73—21no—
109C2×Dic73—21no—
110C2×Dic73—21no—
111C2×Dic73—42no—
112C2×Dic73—42no—
113C2×Dic73—41no—
114C2×Dic73—41no—
115C2×Dic73—41no—
116C2×Dic73—41no—
117#10596833—82no—
118#10596833—82no—
119C22×C73—21no—
120C22×C73—21no—
121C22×C73—21no—
122C22×C73—21no—
123C22×C73—41no—
124C22×C73—41no—
125C22×C73—42no—
126C22×C73—41no—
127C22×C73—41no—
128C22×C73C2×C22⋊C811yescharacteristic
129C22×C73C2×#9911yescharacteristic
130C22×C73C2×#10111yescharacteristic
131C4×C73—22no—
132C4×C73—82no—
133C4×C73#32412yes—
134C4×C73#32412yes—
135Dic73—44no—
136Dic73—41no—
137Dic73—41no—
138Dic73—41no—
139Dic73—41no—
140#11454—731nomaximal, Sylow
141C2×C73—21no—
142C2×C73—21no—
143C2×C73—41no—
144C2×C73—41no—
145C2×C73C2×#32411yescharacteristic
146C2×C73#212211yescharacteristic
147C2×C73#212311yescharacteristic
148#2317—731no—
149#1458—731no—
150#1458—731no—
151#1326—734no—
152C73#1145411yescharacteristic, complemented, Sylow
153C2×#117—731no—
154C2×#117—731no—
155C22×C4⋊C4#6440811yescharacteristic
156#431—1461no—
157#431—1461no—
158#389—1461no—
159#389—1461no—
160#385—732no—
161#385—732no—
162#384—1461no—
163#382—1461no—
164#324—1461no—
165#324—1461no—
166#323—732no—
167#323—732no—
168C23×C4C2×#6440812yes—
169C23×C4C2×#6440811yescharacteristic
170#117—2921no—
171#117—2921no—
172#117—734no—
173#117—1462no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—731no—
179C2×C4⋊C4—731no—
180C2×C4⋊C4—1462no—
181C2×C4⋊C4—1462no—
182C2×C4⋊C4—1461no—
183C2×C22⋊C4—731no—
184C2×C22⋊C4—731no—
185C2×C22⋊C4—732no—
186C2×C42—731no—
187C2×C42—731no—
188C2×C42—1461no—
189C4⋊C8—2921no—
190C4⋊C8—2921no—
191C22⋊C8—1462no—
192C22⋊C8—1462no—
193C4×C8—2921no—
194C4×C8—2921no—
195#95—1462no—
196#95—1462no—
197C24C22×#6440811yescharacteristic
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—42no—
203C22×C4—82no—
204C22×C4—731no—
205C22×C4—731no—
206C22×C4—731no—
207C22×C4—731no—
208C22×C4—1462no—
209C22×C4C22⋊#6440811yescharacteristic
210C22×C4C22⋊#6440811yescharacteristic
211C22×C4#1192403612yes—
212C22×C4#1192403812yes—
213C2×C8—2922no—
214C2×C8—2922no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—2921no—
219C4⋊C4—2921no—
220C4⋊C4—5842no—
221C4⋊C4—5842no—
222C4⋊C4—1464no—
223C22⋊C4—2921no—
224C22⋊C4—2921no—
225C22⋊C4—734no—
226C22⋊C4—1462no—
227C42—1464no—
228C42—1461no—
229C42—1461no—
230C42—1461no—
231C42—1461no—
232C23—21no—
233C23—21no—
234C23—41no—
235C23—41no—
236C23C2×C22⋊#6440811yescharacteristic
237C23C2×#1192403611yescharacteristic
238C23C2×#1192403811yescharacteristic
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—84no—
249C2×C4—2922no—
250C2×C4—2922no—
251C2×C4—2921no—
252C2×C4—2921no—
253C2×C4—2921no—
254C2×C4—2921no—
255C2×C4—1464no—
256C2×C4—1461no—
257C2×C4—1461no—
258C2×C4—1461no—
259C2×C4—1461no—
260C2×C4#1455664812yes—
261C2×C4#1455664812yes—
262C2×C4#1455664911yescharacteristic
263C2×C4#1455664911yescharacteristic
264C2×C4#1455665212yes—
265C8—5842no—
266C8—5842no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—42no—
274C22—41no—
275C22—41no—
276C22C2×#1455664811yescharacteristic
277C22C2×#1455664911yescharacteristic
278C22C2×#1455665211yescharacteristic
279C4—22no—
280C4—82no—
281C4—2924no—
282C4—2921no—
283C4—2921no—
284C4—2921no—
285C4—2921no—
286C4#1798943312yes—
287C4#1798943312yes—
288C2—21no—
289C2—21no—
290C2—41no—
291C2—41no—
292C2C2×#1798943311yescharacteristic
293C2#2262619011yescharacteristic
294C2#2262619311yescharacteristic
subgroupG/H
centerC22C2×#14556649
commutatorC23×C73C22×C8
FrattiniC24C22×#64408
FittingC22×C4⋊C4×C73C4
socleC22×C73C2×#99
prime radicalC24×C73C22×C4
cosocleC2×C22⋊Dic73C23
hypercenterC22×C4⋊C4#64408
hypocenterC73#11454
LCSG ▷ C23×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
29. #14556550C8semidirect
152. C73#11454semidirect

Maximal subgroups

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

subgroupindex
#226251622
#226248762
#226125552
#1145473