Finite Group Databasebeta

← #34001680

Subgroups — #34001680

Order 18688.

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

normal subgroups only

This group has 568 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
2#22625592C211yesmaximal, characteristic
3#22624752C214yesmaximal, complemented
4#22618284C211yesmaximal, characteristic
5C22×#11924036—21no—
6C2×C4⋊#10596833—21no—
7C2×#14556648—21no—
8#17989453—21no—
9#17989453C2212yes—
10#17989453C2212yes—
11C2×#14556591C411yescharacteristic
12C2×#14556571C412yes—
13C2×C42×D73C2211yescharacteristic
14#17988492C2212yes—
15#17988486—21no—
16#17988486—21no—
17#17988444—21no—
18#17988444—21no—
19#384×C73C411yescharacteristic
20C2×#11924036—24no—
21C2×#11924036—42no—
22C2×#11924036—42no—
23C22×#10596833—22no—
24C22×#10596833—22no—
25C4⋊#10596833—42no—
26#14556648—42no—
27C22×Dic146C2×C412yes—
28C22×C4×D73C2×C411yescharacteristic
29C22×C4×D73D411yescharacteristic
30C22×C4×D73D411yescharacteristic
31#14556591—42no—
32#14556571—44no—
33C2×C4⋊Dic73—22no—
34C2×C4⋊Dic73—21no—
35C2×C4⋊Dic73—21no—
36C2×#11924005—22no—
37C2×C4×Dic73D411yescharacteristic
38C2×C4×Dic73D411yescharacteristic
39C2×C4×Dic73C2311yescharacteristic
40C42×D73—42no—
41#14556498—22no—
42#14556498—22no—
43C2×C4⋊C4×C73—21no—
44C2×C4⋊C4×C73—21no—
45C2×C4⋊C4×C73C2×C412yes—
46C2×C42×C73C2×C411yescharacteristic
47#11924036—48no—
48C2×#10596833—24no—
49C2×#10596833—42no—
50C2×#10596833—42no—
51C2×#10596833—42no—
52C2×#10596833—42no—
53C2×#10596833—42no—
54C23×D73C22⋊C411yescharacteristic
55C2×Dic146—24no—
56C2×Dic146—42no—
57C2×Dic146—42no—
58C22×Dic73—22no—
59C22×Dic73C22⋊C411yescharacteristic
60C22×Dic73C2×D411yescharacteristic
61C22×Dic73C2×D411yescharacteristic
62C2×C4×D73—22no—
63C2×C4×D73—22no—
64C2×C4×D73—22no—
65C2×C4×D73—22no—
66C2×C4×D73—41no—
67C2×C4×D73—41no—
68C2×C4×D73—41no—
69C2×C4×D73SD1612yes—
70C2×C4×D73SD1612yes—
71C2×C4×D73Q1612yes—
72C2×C4×D73Q1612yes—
73C4⋊Dic73—44no—
74C4⋊Dic73—42no—
75C4⋊Dic73—42no—
76#11924005—44no—
77C4×Dic73—22no—
78C4×Dic73—41no—
79C4×Dic73—41no—
80C22×C4×C73—22no—
81C22×C4×C73C22⋊C411yescharacteristic
82C22×C4×C73C22⋊C411yescharacteristic
83C22×C4×C73C22×C411yescharacteristic
84C4⋊C4×C73—42no—
85C4⋊C4×C73—42no—
86C4⋊C4×C73—42no—
87C42×C73—22no—
88C23×C73C2×C22⋊C411yescharacteristic
89C22×D73—21no—
90C22×D73—21no—
91C22×D73—21no—
92C22×D73—21no—
93C22×D73#10012yes—
94C22×D73#10312yes—
95C22×D73#10312yes—
96C2×C4×C73—21no—
97C2×C4×C73—21no—
98C2×C4×C73—21no—
99C2×C4×C73—21no—
100C2×C4×C73—42no—
101C2×C4×C73—42no—
102C2×C4×C73—42no—
103C2×C4×C73#9912yes—
104C2×C4×C73#10312yes—
105C2×C4×C73#10312yes—
106C2×C4×C73#10412yes—
107C2×C4×C73#13112yes—
108C2×Dic73—21no—
109C2×Dic73—21no—
110C2×Dic73—21no—
111C2×Dic73—21no—
112C2×Dic73—21no—
113C2×Dic73—21no—
114C2×Dic73—21no—
115C2×Dic73—42no—
116C2×Dic73—42no—
117C2×Dic73—42no—
118C2×Dic73C2×SD1611yescharacteristic
119C2×Dic73C2×SD1611yescharacteristic
120C2×Dic73C2×Q1611yescharacteristic
121C2×Dic73C2×Q1611yescharacteristic
122C4×D73—24no—
123C4×D73—24no—
124C4×D73—42no—
125C4×D73—42no—
126C4×D73—44no—
127Dic146—84no—
128#10596833—44no—
129#10596833—82no—
130C22×C73#40811yescharacteristic
131C22×C73C2×#9911yescharacteristic
132C22×C73C2×#10011yescharacteristic
133C22×C73C2×#10311yescharacteristic
134C22×C73C2×#10311yescharacteristic
135C22×C73C2×#10411yescharacteristic
136C22×C73#42111yescharacteristic
137C2×D73—22no—
138C2×D73—22no—
139C2×D73—22no—
140C2×D73—22no—
141C2×D73—22no—
142C2×D73—22no—
143C2×D73#33314yes—
144C4×C73—22no—
145C4×C73—22no—
146C4×C73—82no—
147C4×C73#32814yes—
148Dic73—21no—
149Dic73—21no—
150Dic73—21no—
151Dic73—21no—
152Dic73—41no—
153Dic73—82no—
154#7667—731nomaximal, Sylow
155C2×C73#147111yescharacteristic
156C2×C73#148111yescharacteristic
157C2×C73#149311yescharacteristic
158C2×C73C2×#32811yescharacteristic
159C2×C73#152011yescharacteristic
160C2×C73#153011yescharacteristic
161C2×C73C2×#33311yescharacteristic
162D73—24no—
163#2857—731no—
164#1797—731no—
165C2×#336—731no—
166#1272—734no—
167C73#766711yescharacteristic, complemented, Sylow
168C23×C8—1461no—
169C2×C4⋊Q8—731no—
170C2×C22⋊Q8—732no—
171C22×C42—731no—
172C2×C4⋊C8—1461no—
173C2×C22⋊C8—1461no—
174#391—732no—
175#389—1461no—
176#389—1461no—
177#384#6440811yescharacteristic
178#380—1461no—
179#380—1461no—
180#336—732no—
181#336—732no—
182#336—1461no—
183C22×Q8—732no—
184C23×C4—731no—
185C23×C4—731no—
186C23×C4—731no—
187C22×C8—2922no—
188C22×C8—2922no—
189C22×C8—1462no—
190C22×C8—1462no—
191C22×C8—1464no—
192C4⋊Q8—2922no—
193C22⋊Q8—2924no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—1462no—
197C2×C4⋊C4—1461no—
198C2×C4⋊C4—1461no—
199C2×C4⋊C4C2×#6440812yes—
200C2×C22⋊C4—1462no—
201C2×C42—2922no—
202C2×C42—731no—
203C2×C42—731no—
204C2×C42—731no—
205C2×C42C2×#6440811yescharacteristic
206C4⋊C8—2922no—
207C22⋊C8—2922no—
208#95—1462no—
209#95—1462no—
210C24—731no—
211C2×Q8—2922no—
212C2×Q8—2922no—
213C2×Q8—1464no—
214C22×C4—22no—
215C22×C4—2921no—
216C22×C4—2921no—
217C22×C4—2921no—
218C22×C4—732no—
219C22×C4—732no—
220C22×C4—731no—
221C22×C4—731no—
222C22×C4—732no—
223C22×C4—732no—
224C22×C4—731no—
225C22×C4—1462no—
226C22×C4—1462no—
227C22×C4—1462no—
228C22×C4—1462no—
229C22×C4—1462no—
230C22×C4C22⋊#6440811yescharacteristic
231C22×C4C22⋊#6440811yescharacteristic
232C22×C4C22×#6440811yescharacteristic
233C2×C8—2922no—
234C2×C8—2922no—
235C2×C8—2928no—
236C2×C8—2922no—
237C2×C8—2922no—
238C2×C8—2922no—
239C2×C8—1464no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—2922no—
244C4⋊C4—2924no—
245C4⋊C4—2922no—
246C22⋊C4—2924no—
247C42—22no—
248C42—2921no—
249C42—2921no—
250C42—1462no—
251C23—732no—
252C23—732no—
253C23—732no—
254C23—1461no—
255C23—1461no—
256C23—1461no—
257C23—1461no—
258C23C2×C22⋊#6440811yescharacteristic
259Q8—5844no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—2924no—
268C2×C4—2922no—
269C2×C4—2922no—
270C2×C4—2922no—
271C2×C4—2922no—
272C2×C4—2922no—
273C2×C4—731no—
274C2×C4—731no—
275C2×C4—731no—
276C2×C4—731no—
277C2×C4—1461no—
278C2×C4—1461no—
279C2×C4—1461no—
280C2×C4—1461no—
281C2×C4—1461no—
282C2×C4—1461no—
283C2×C4—1461no—
284C2×C4—1464no—
285C2×C4—1464no—
286C2×C4#1455665012yes—
287C2×C4#1455665412yes—
288C2×C4#1455665412yes—
289C2×C4#1455665512yes—
290C2×C4#1455667912yes—
291C8—2924no—
292C8—5842no—
293C22—734no—
294C22—1462no—
295C22—1462no—
296C22—1462no—
297C22—1462no—
298C22—1462no—
299C22—1462no—
300C22#1798956211yescharacteristic
301C22C2×#1455665011yescharacteristic
302C22D73⋊#10011yescharacteristic
303C22C2×#1455665411yescharacteristic
304C22C2×#1455665411yescharacteristic
305C22C2×#1455665511yescharacteristic
306C22#1798958511yescharacteristic
307C4—22no—
308C4—22no—
309C4—82no—
310C4—2921no—
311C4—5842no—
312C4—1461no—
313C4—1461no—
314C4—1461no—
315C4—1461no—
316C4#1798944014yes—
317C2—1464no—
318C2#2262520811yescharacteristic
319C2#2262524111yescharacteristic
320C2#2262528111yescharacteristic
321C2C2×#1798944011yescharacteristic
322C2#2262537011yescharacteristic
323C2#2262540111yescharacteristic
324C2D73⋊#33311yescharacteristic
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#7667
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
3. #22624752C2semidirect
167. C73#7667semidirect

Maximal subgroups

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

subgroupindex
#226257862
#226255922
#226247522
#226182842
#766773