Finite Group Databasebeta

← #34005213

Subgroups — #34005213

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—44no—
30C2×C4⋊Dic73—21no—
31C2×C4⋊Dic73—21no—
32C2×C4⋊Dic73C2×C412yes—
33C2×#11924005—22no—
34C2×C4×Dic73—22no—
35C2×C4×Dic73D411yescharacteristic
36C2×C4×Dic73D411yescharacteristic
37C2×C4×Dic73C2311yescharacteristic
38C42×D73—24no—
39#14556498—22no—
40#14556498—22no—
41C2×C4⋊C4×C73—21no—
42C2×C4⋊C4×C73—21no—
43C2×C4⋊C4×C73C2×C412yes—
44C2×C42×C73C2×C411yescharacteristic
45C2×#10596833—24no—
46C2×#10596833—42no—
47C2×#10596833—42no—
48C2×#10596833—42no—
49C2×#10596833—42no—
50C2×#10596833—42no—
51C23×D73C22⋊C411yescharacteristic
52C22×Dic73—22no—
53C22×Dic73C22⋊C411yescharacteristic
54C22×Dic73C2×D411yescharacteristic
55C22×Dic73C2×D411yescharacteristic
56C2×C4×D73—22no—
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—41no—
64C2×C4×D73SD1612yes—
65C2×C4×D73Q1612yes—
66C4⋊Dic73—24no—
67C4⋊Dic73—42no—
68C4⋊Dic73—42no—
69#11924005—44no—
70C4×Dic73—22no—
71C4×Dic73—22no—
72C4×Dic73—44no—
73C4×Dic73#4114yes—
74C22×C4×C73—22no—
75C22×C4×C73C22⋊C411yescharacteristic
76C22×C4×C73C22⋊C411yescharacteristic
77C22×C4×C73C22×C411yescharacteristic
78C4⋊C4×C73—24no—
79C4⋊C4×C73—42no—
80C4⋊C4×C73—42no—
81C42×C73—22no—
82C23×C73C2×C22⋊C411yescharacteristic
83C22×D73—21no—
84C22×D73—21no—
85C22×D73—21no—
86C22×D73—21no—
87C22×D73—21no—
88C22×D73—21no—
89C22×D73#10312yes—
90C2×C4×C73—21no—
91C2×C4×C73—21no—
92C2×C4×C73—21no—
93C2×C4×C73—21no—
94C2×C4×C73—42no—
95C2×C4×C73—42no—
96C2×C4×C73—42no—
97C2×C4×C73#10312yes—
98C2×C4×C73#10412yes—
99C2×C4×C73#11712yes—
100C2×C4×C73#13112yes—
101C2×C4×C73#13112yes—
102C2×Dic73—21no—
103C2×Dic73—21no—
104C2×Dic73—21no—
105C2×Dic73—21no—
106C2×Dic73—21no—
107C2×Dic73—42no—
108C2×Dic73—42no—
109C2×Dic73—42no—
110C2×Dic73#12312yes—
111C2×Dic73#12312yes—
112C2×Dic73C2×SD1611yescharacteristic
113C2×Dic73C2×Q1611yescharacteristic
114C2×Dic73AGL(1,ℤ/8)12yes—
115C4×D73—24no—
116C4×D73—24no—
117C4×D73—44no—
118C4×D73—44no—
119#10596833—44no—
120#10596833—82no—
121C22×C73#38111yescharacteristic
122C22×C73#40811yescharacteristic
123C22×C73#40811yescharacteristic
124C22×C73C2×#10311yescharacteristic
125C22×C73#41811yescharacteristic
126C22×C73C2×#10411yescharacteristic
127C22×C73#42111yescharacteristic
128C2×D73—22no—
129C2×D73—22no—
130C2×D73—22no—
131C2×D73—41no—
132C2×D73—41no—
133C2×D73—41no—
134C2×D73—41no—
135C4×C73—22no—
136C4×C73—22no—
137C4×C73—82no—
138C4×C73#43314yes—
139Dic73—22no—
140Dic73—41no—
141Dic73—82no—
142Dic73#48112yes—
143Dic73#48412yes—
144#8710—731nomaximal, Sylow
145C2×C73#148111yescharacteristic
146C2×C73#148411yescharacteristic
147C2×C73#149311yescharacteristic
148C2×C73#149611yescharacteristic
149C2×C73#181811yescharacteristic
150C2×C73#182411yescharacteristic
151C2×C73#182611yescharacteristic
152D73—42no—
153#2285—731no—
154#1797—731no—
155C4×C22⋊C8—731no—
156#1279—734no—
157C73#871011yescharacteristic, complemented, Sylow
158C2×#117—1461no—
159C22×C42—731no—
160C2×C4⋊C8—732no—
161C2×C22⋊C8—1461no—
162C2×C22⋊C8—1461no—
163C2×C4×C8—732no—
164#384—731no—
165#384#6440811yescharacteristic
166#382—1461no—
167#380—732no—
168#380—1461no—
169C4×C4⋊C4—732no—
170C4×C4⋊C4—1461no—
171#336—1461no—
172#336—1461no—
173C23×C4—731no—
174C23×C4—731no—
175C23×C4—731no—
176C22×C8—1462no—
177C22×C8—1462no—
178#117—2924no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—732no—
182C2×C4⋊C4—1461no—
183C2×C4⋊C4—1461no—
184C2×C4⋊C4C2×#6440812yes—
185C2×C22⋊C4—1462no—
186C2×C42—731no—
187C2×C42—731no—
188C2×C42—731no—
189C2×C42—1464no—
190C2×C42—1462no—
191C2×C42C2×#6440811yescharacteristic
192C4⋊C8—1464no—
193C22⋊C8—2922no—
194C22⋊C8—1464no—
195C4×C8—1464no—
196#95—1462no—
197#95—1462no—
198C24—731no—
199C22×C4—22no—
200C22×C4—2921no—
201C22×C4—731no—
202C22×C4—731no—
203C22×C4—732no—
204C22×C4—732no—
205C22×C4—731no—
206C22×C4—1462no—
207C22×C4—1462no—
208C22×C4—1462no—
209C22×C4—1462no—
210C22×C4—1462no—
211C22×C4—1462no—
212C22×C4—1462no—
213C22×C4—1462no—
214C22×C4C22⋊#6440811yescharacteristic
215C22×C4C22⋊#6440811yescharacteristic
216C22×C4C22×#6440811yescharacteristic
217C2×C8—2922no—
218C2×C8—2922no—
219C2×C8—2922no—
220C2×C8—2922no—
221C2×C8—2922no—
222C2×C8—1464no—
223C4⋊C4—24no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—2922no—
227C4⋊C4—2922no—
228C4⋊C4—1464no—
229C22⋊C4—2924no—
230C42—22no—
231C42—2924no—
232C42—734no—
233C42—1462no—
234C42—1462no—
235C23—732no—
236C23—1461no—
237C23—1461no—
238C23—1461no—
239C23—1461no—
240C23—1461no—
241C23—1461no—
242C23C2×C22⋊#6440811yescharacteristic
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—2924no—
251C2×C4—2924no—
252C2×C4—2922no—
253C2×C4—2922no—
254C2×C4—2922no—
255C2×C4—732no—
256C2×C4—731no—
257C2×C4—731no—
258C2×C4—732no—
259C2×C4—732no—
260C2×C4—1464no—
261C2×C4—1464no—
262C2×C4—1461no—
263C2×C4—1461no—
264C2×C4—1461no—
265C2×C4—1461no—
266C2×C4—1461no—
267C2×C4#1455665412yes—
268C2×C4#1455665612yes—
269C2×C4#1455667012yes—
270C2×C4#1455667812yes—
271C2×C4#1455667912yes—
272C8—2924no—
273C8—5842no—
274C22—2921no—
275C22—2921no—
276C22—2921no—
277C22—2921no—
278C22—1462no—
279C22—1462no—
280C22—1462no—
281C22#1798953011yescharacteristic
282C22#1798956111yescharacteristic
283C22#1798956211yescharacteristic
284C22C2×#1455665411yescharacteristic
285C22#1798957911yescharacteristic
286C22C2×#1455665611yescharacteristic
287C22#1798958511yescharacteristic
288C4—22no—
289C4—22no—
290C4—82no—
291C4—2921no—
292C4—5842no—
293C4—732no—
294C4—732no—
295C4—1462no—
296C4#1798961214yes—
297C2—2922no—
298C2#2262524111yescharacteristic
299C2#2262525211yescharacteristic
300C2#2262528011yescharacteristic
301C2#2262529111yescharacteristic
302C2#2262582511yescharacteristic
303C2#2262583511yescharacteristic
304C2#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#8710
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
157. C73#8710semidirect

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
#871073