Finite Group Databasebeta

← #34005209

Subgroups — #34005209

Order 18688.

540 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 126 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#22625786C211yesmaximal, characteristic
2C4×#14556648C211yesmaximal, characteristic
3#22624767C214yesmaximal, complemented
4#22612336C211yesmaximal, characteristic
5C2×C4⋊#10596833C2212yes—
6C2×#14556648—21no—
7C2×#14556648—21no—
8C2×C4×#10596833C2212yes—
9#17989453—21no—
10#17989453—21no—
11C2×#14556549—21no—
12C2×C42×D73C2211yescharacteristic
13#17988457C411yescharacteristic
14#17988452—21no—
15#17988444—21no—
16#17988444C412yes—
17C4⋊C4×Dic73—21no—
18C4⋊C4×Dic73C2212yes—
19#384×C73C411yescharacteristic
20C22×#10596833—22no—
21C22×#10596833—22no—
22C4⋊#10596833—24no—
23#14556648—24no—
24#14556648—42no—
25C4×#10596833—24no—
26C22×C4×D73C2×C411yescharacteristic
27C22×C4×D73D411yescharacteristic
28C22×C4×D73D411yescharacteristic
29#14556549—42no—
30#14556549—42no—
31C2×C4⋊Dic73—21no—
32C2×C4⋊Dic73—21no—
33C2×C4⋊Dic73C2×C412yes—
34C2×#11924005—22no—
35C2×C4×Dic73—22no—
36C2×C4×Dic73D411yescharacteristic
37C2×C4×Dic73D411yescharacteristic
38C2×C4×Dic73C2311yescharacteristic
39C42×D73—24no—
40#14556498—22no—
41#14556498—22no—
42C2×C4⋊C4×C73—21no—
43C2×C4⋊C4×C73—21no—
44C2×C4⋊C4×C73C2×C412yes—
45C2×C42×C73C2×C411yescharacteristic
46C2×#10596833—24no—
47C2×#10596833—42no—
48C2×#10596833—42no—
49C2×#10596833—42no—
50C2×#10596833—42no—
51C2×#10596833—42no—
52C23×D73C22⋊C411yescharacteristic
53C22×Dic73—22no—
54C22×Dic73C22⋊C411yescharacteristic
55C22×Dic73C2×D411yescharacteristic
56C22×Dic73C2×D411yescharacteristic
57C2×C4×D73—22no—
58C2×C4×D73—22no—
59C2×C4×D73—22no—
60C2×C4×D73—22no—
61C2×C4×D73—22no—
62C2×C4×D73—22no—
63C2×C4×D73—22no—
64C2×C4×D73—41no—
65C2×C4×D73D812yes—
66C2×C4×D73SD1612yes—
67C4⋊Dic73—24no—
68C4⋊Dic73—42no—
69C4⋊Dic73—42no—
70#11924005—44no—
71C4×Dic73—22no—
72C4×Dic73—22no—
73C4×Dic73—42no—
74C4×Dic73—42no—
75C4×Dic73#4114yes—
76C22×C4×C73—22no—
77C22×C4×C73C22⋊C411yescharacteristic
78C22×C4×C73C22⋊C411yescharacteristic
79C22×C4×C73C22×C411yescharacteristic
80C4⋊C4×C73—24no—
81C4⋊C4×C73—42no—
82C4⋊C4×C73—42no—
83C42×C73—22no—
84C23×C73C2×C22⋊C411yescharacteristic
85C22×D73—21no—
86C22×D73—21no—
87C22×D73—21no—
88C22×D73—21no—
89C22×D73—21no—
90C22×D73—21no—
91C22×D73#10212yes—
92C2×C4×C73—21no—
93C2×C4×C73—21no—
94C2×C4×C73—21no—
95C2×C4×C73—21no—
96C2×C4×C73—42no—
97C2×C4×C73—42no—
98C2×C4×C73—42no—
99C2×C4×C73#10212yes—
100C2×C4×C73#10412yes—
101C2×C4×C73#11712yes—
102C2×C4×C73#13112yes—
103C2×C4×C73#13112yes—
104C2×Dic73—21no—
105C2×Dic73—21no—
106C2×Dic73—21no—
107C2×Dic73—21no—
108C2×Dic73—21no—
109C2×Dic73—42no—
110C2×Dic73—42no—
111C2×Dic73—42no—
112C2×Dic73#12312yes—
113C2×Dic73#12312yes—
114C2×Dic73C2×D811yescharacteristic
115C2×Dic73C2×SD1611yescharacteristic
116C2×Dic73#13712yes—
117C4×D73—24no—
118C4×D73—24no—
119C4×D73—44no—
120C4×D73—44no—
121#10596833—44no—
122#10596833—82no—
123C22×C73#38111yescharacteristic
124C22×C73#40811yescharacteristic
125C22×C73#40811yescharacteristic
126C22×C73C2×#10211yescharacteristic
127C22×C73#41911yescharacteristic
128C22×C73C2×#10411yescharacteristic
129C22×C73#42111yescharacteristic
130C2×D73—22no—
131C2×D73—22no—
132C2×D73—22no—
133C2×D73—41no—
134C2×D73—41no—
135C2×D73—41no—
136C2×D73—41no—
137C4×C73—22no—
138C4×C73—22no—
139C4×C73—82no—
140C4×C73#43314yes—
141Dic73—22no—
142Dic73—41no—
143Dic73—82no—
144Dic73#48012yes—
145Dic73#48312yes—
146#8709—731nomaximal, Sylow
147C2×C73#148011yescharacteristic
148C2×C73#148511yescharacteristic
149C2×C73#149211yescharacteristic
150C2×C73#149711yescharacteristic
151C2×C73#181811yescharacteristic
152C2×C73#182311yescharacteristic
153C2×C73#182611yescharacteristic
154D73—42no—
155#2285—731no—
156#1797—731no—
157C4×C22⋊C8—731no—
158#1279—734no—
159C73#870911yescharacteristic, complemented, Sylow
160C2×#117—1461no—
161C22×C42—731no—
162C2×C4⋊C8—732no—
163C2×C22⋊C8—1461no—
164C2×C22⋊C8—1461no—
165C2×C4×C8—732no—
166#384—731no—
167#384#6440811yescharacteristic
168#382—1461no—
169#380—732no—
170#380—1461no—
171C4×C4⋊C4—732no—
172C4×C4⋊C4—1461no—
173#336—1461no—
174#336—1461no—
175C23×C4—731no—
176C23×C4—731no—
177C23×C4—731no—
178C22×C8—1462no—
179C22×C8—1462no—
180#117—2922no—
181#117—2922no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—732no—
185C2×C4⋊C4—1461no—
186C2×C4⋊C4—1461no—
187C2×C4⋊C4C2×#6440812yes—
188C2×C22⋊C4—1462no—
189C2×C42—731no—
190C2×C42—731no—
191C2×C42—731no—
192C2×C42—1464no—
193C2×C42—1462no—
194C2×C42C2×#6440811yescharacteristic
195C4⋊C8—1464no—
196C22⋊C8—2922no—
197C22⋊C8—1464no—
198C4×C8—1464no—
199#95—1462no—
200#95—1462no—
201C24—731no—
202C22×C4—22no—
203C22×C4—2921no—
204C22×C4—731no—
205C22×C4—731no—
206C22×C4—732no—
207C22×C4—732no—
208C22×C4—731no—
209C22×C4—1462no—
210C22×C4—1462no—
211C22×C4—1462no—
212C22×C4—1462no—
213C22×C4—1462no—
214C22×C4—1462no—
215C22×C4—1462no—
216C22×C4—1462no—
217C22×C4C22⋊#6440811yescharacteristic
218C22×C4C22⋊#6440811yescharacteristic
219C22×C4C22×#6440811yescharacteristic
220C2×C8—2922no—
221C2×C8—2922no—
222C2×C8—2922no—
223C2×C8—2922no—
224C2×C8—2922no—
225C2×C8—1464no—
226C4⋊C4—24no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—2922no—
230C4⋊C4—2922no—
231C4⋊C4—1464no—
232C22⋊C4—2924no—
233C42—22no—
234C42—2922no—
235C42—2922no—
236C42—734no—
237C42—1462no—
238C42—1462no—
239C23—732no—
240C23—1461no—
241C23—1461no—
242C23—1461no—
243C23—1461no—
244C23—1461no—
245C23—1461no—
246C23C2×C22⋊#6440811yescharacteristic
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—2924no—
255C2×C4—2924no—
256C2×C4—2922no—
257C2×C4—2922no—
258C2×C4—2922no—
259C2×C4—732no—
260C2×C4—731no—
261C2×C4—731no—
262C2×C4—732no—
263C2×C4—732no—
264C2×C4—1464no—
265C2×C4—1464no—
266C2×C4—1461no—
267C2×C4—1461no—
268C2×C4—1461no—
269C2×C4—1461no—
270C2×C4—1461no—
271C2×C4#1455665312yes—
272C2×C4#1455665612yes—
273C2×C4#1455667012yes—
274C2×C4#1455667812yes—
275C2×C4#1455667912yes—
276C8—2924no—
277C8—5842no—
278C22—2921no—
279C22—2921no—
280C22—2921no—
281C22—2921no—
282C22—1462no—
283C22—1462no—
284C22—1462no—
285C22#1798953011yescharacteristic
286C22#1798956111yescharacteristic
287C22#1798956211yescharacteristic
288C22C2×#1455665311yescharacteristic
289C22#1798958011yescharacteristic
290C22C2×#1455665611yescharacteristic
291C22#1798958511yescharacteristic
292C4—22no—
293C4—22no—
294C4—82no—
295C4—2921no—
296C4—5842no—
297C4—732no—
298C4—732no—
299C4—1462no—
300C4#1798961214yes—
301C2—2922no—
302C2#2262523711yescharacteristic
303C2#2262525411yescharacteristic
304C2#2262527811yescharacteristic
305C2#2262529511yescharacteristic
306C2#2262582511yescharacteristic
307C2#2262583411yescharacteristic
308C2#2262583811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
Fitting#384×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC23×D73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenter#384#64408
hypocenterC73#8709
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
3. #22624767C2semidirect
159. C73#8709semidirect

Maximal subgroups

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

subgroupindex
#226257862
C4×#145566482
#226247672
#226123362
#870973