Finite Group Databasebeta

← #34001671

Subgroups — #34001671

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#22624749C214yesmaximal, complemented
4#22618284C211yesmaximal, characteristic
5C22×#11924036—21no—
6C2×C4⋊#10596833—21no—
7C2×#14556648—21no—
8#17989453—21no—
9#17989453C2212yes—
10#17989453C2212yes—
11C2×#14556591C411yescharacteristic
12C2×#14556571—21no—
13C2×C42×D73C2211yescharacteristic
14#17988492—21no—
15#17988486—21no—
16#17988486C2212yes—
17#17988444—21no—
18#17988444C412yes—
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×Dic146—21no—
28C22×C4×D73C2×C411yescharacteristic
29C22×C4×D73D411yescharacteristic
30C22×C4×D73D411yescharacteristic
31#14556591—42no—
32#14556571—42no—
33#14556571—42no—
34C2×C4⋊Dic73—22no—
35C2×C4⋊Dic73—21no—
36C2×C4⋊Dic73C2×C412yes—
37C2×#11924005—22no—
38C2×C4×Dic73D411yescharacteristic
39C2×C4×Dic73D411yescharacteristic
40C2×C4×Dic73C2311yescharacteristic
41C42×D73—42no—
42#14556498—22no—
43#14556498—22no—
44C2×C4⋊C4×C73—21no—
45C2×C4⋊C4×C73—21no—
46C2×C4⋊C4×C73C2×C412yes—
47C2×C42×C73C2×C411yescharacteristic
48#11924036—48no—
49C2×#10596833—24no—
50C2×#10596833—42no—
51C2×#10596833—42no—
52C2×#10596833—42no—
53C2×#10596833—42no—
54C2×#10596833—42no—
55C23×D73C22⋊C411yescharacteristic
56C2×Dic146—42no—
57C2×Dic146—42no—
58C2×Dic146—42no—
59C22×Dic73—22no—
60C22×Dic73C22⋊C411yescharacteristic
61C22×Dic73C2×D411yescharacteristic
62C22×Dic73C2×D411yescharacteristic
63C2×C4×D73—22no—
64C2×C4×D73—22no—
65C2×C4×D73—22no—
66C2×C4×D73—22no—
67C2×C4×D73—41no—
68C2×C4×D73—41no—
69C2×C4×D73—41no—
70C2×C4×D73D812yes—
71C2×C4×D73SD1612yes—
72C2×C4×D73SD1612yes—
73C2×C4×D73Q1612yes—
74C4⋊Dic73—42no—
75C4⋊Dic73—42no—
76C4⋊Dic73—42no—
77C4⋊Dic73—42no—
78#11924005—44no—
79C4×Dic73—22no—
80C4×Dic73—41no—
81C4×Dic73—41no—
82C22×C4×C73—22no—
83C22×C4×C73C22⋊C411yescharacteristic
84C22×C4×C73C22⋊C411yescharacteristic
85C22×C4×C73C22×C411yescharacteristic
86C4⋊C4×C73—24no—
87C4⋊C4×C73—42no—
88C4⋊C4×C73—42no—
89C42×C73—22no—
90C23×C73C2×C22⋊C411yescharacteristic
91C22×D73—21no—
92C22×D73—21no—
93C22×D73—21no—
94C22×D73—21no—
95C22×D73#10112yes—
96C22×D73#10212yes—
97C22×D73#10312yes—
98C2×C4×C73—21no—
99C2×C4×C73—21no—
100C2×C4×C73—21no—
101C2×C4×C73—21no—
102C2×C4×C73—42no—
103C2×C4×C73—42no—
104C2×C4×C73—42no—
105C2×C4×C73#9912yes—
106C2×C4×C73#10212yes—
107C2×C4×C73#10312yes—
108C2×C4×C73#10412yes—
109C2×C4×C73#13112yes—
110C2×Dic73—21no—
111C2×Dic73—21no—
112C2×Dic73—21no—
113C2×Dic73—21no—
114C2×Dic73—21no—
115C2×Dic73—21no—
116C2×Dic73—21no—
117C2×Dic73—42no—
118C2×Dic73—42no—
119C2×Dic73—42no—
120C2×Dic73C2×D811yescharacteristic
121C2×Dic73C2×SD1611yescharacteristic
122C2×Dic73C2×SD1611yescharacteristic
123C2×Dic73C2×Q1611yescharacteristic
124C4×D73—24no—
125C4×D73—24no—
126C4×D73—42no—
127C4×D73—42no—
128C4×D73—44no—
129Dic146—84no—
130#10596833—44no—
131#10596833—82no—
132C22×C73#40811yescharacteristic
133C22×C73C2×#9911yescharacteristic
134C22×C73C2×#10111yescharacteristic
135C22×C73C2×#10211yescharacteristic
136C22×C73C2×#10311yescharacteristic
137C22×C73C2×#10411yescharacteristic
138C22×C73#42111yescharacteristic
139C2×D73—22no—
140C2×D73—22no—
141C2×D73—22no—
142C2×D73—22no—
143C2×D73—22no—
144C2×D73—22no—
145C2×D73#33214yes—
146C4×C73—22no—
147C4×C73—22no—
148C4×C73—82no—
149C4×C73#32814yes—
150Dic73—21no—
151Dic73—21no—
152Dic73—21no—
153Dic73—21no—
154Dic73—41no—
155Dic73—82no—
156#7665—731nomaximal, Sylow
157C2×C73#147211yescharacteristic
158C2×C73#148111yescharacteristic
159C2×C73#149211yescharacteristic
160C2×C73C2×#32811yescharacteristic
161C2×C73#151911yescharacteristic
162C2×C73#153111yescharacteristic
163C2×C73C2×#33211yescharacteristic
164D73—24no—
165#2857—731no—
166#1797—731no—
167C2×#336—731no—
168#1271—734no—
169C73#766511yescharacteristic, complemented, Sylow
170C23×C8—1461no—
171C2×C4⋊Q8—731no—
172C2×C22⋊Q8—1461no—
173C22×C42—731no—
174C2×C4⋊C8—1461no—
175C2×C22⋊C8—1461no—
176#391—1461no—
177#389—732no—
178#389—1461no—
179#384#6440811yescharacteristic
180#380—732no—
181#380—1461no—
182#336—732no—
183#336—732no—
184#336—1461no—
185C22×Q8—1461no—
186C23×C4—731no—
187C23×C4—731no—
188C23×C4—731no—
189C22×C8—2922no—
190C22×C8—2922no—
191C22×C8—1462no—
192C22×C8—1462no—
193C22×C8—1464no—
194C4⋊Q8—2922no—
195C22⋊Q8—2922no—
196C22⋊Q8—2922no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—732no—
200C2×C4⋊C4—1462no—
201C2×C4⋊C4—1461no—
202C2×C4⋊C4C2×#6440812yes—
203C2×C22⋊C4—1462no—
204C2×C42—2922no—
205C2×C42—731no—
206C2×C42—731no—
207C2×C42—731no—
208C2×C42C2×#6440811yescharacteristic
209C4⋊C8—2922no—
210C22⋊C8—2922no—
211#95—1462no—
212#95—1462no—
213C24—731no—
214C2×Q8—2922no—
215C2×Q8—2922no—
216C2×Q8—2922no—
217C22×C4—22no—
218C22×C4—2921no—
219C22×C4—2921no—
220C22×C4—2921no—
221C22×C4—732no—
222C22×C4—732no—
223C22×C4—731no—
224C22×C4—731no—
225C22×C4—732no—
226C22×C4—732no—
227C22×C4—731no—
228C22×C4—1462no—
229C22×C4—1462no—
230C22×C4—1462no—
231C22×C4—1462no—
232C22×C4—1462no—
233C22×C4C22⋊#6440811yescharacteristic
234C22×C4C22⋊#6440811yescharacteristic
235C22×C4C22×#6440811yescharacteristic
236C2×C8—2922no—
237C2×C8—2922no—
238C2×C8—2928no—
239C2×C8—2922no—
240C2×C8—2922no—
241C2×C8—2922no—
242C2×C8—1464no—
243C4⋊C4—24no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—2922no—
247C4⋊C4—2922no—
248C4⋊C4—2922no—
249C4⋊C4—2922no—
250C22⋊C4—2924no—
251C42—22no—
252C42—2921no—
253C42—2921no—
254C42—1462no—
255C23—732no—
256C23—732no—
257C23—732no—
258C23—1461no—
259C23—1461no—
260C23—1461no—
261C23—1461no—
262C23C2×C22⋊#6440811yescharacteristic
263Q8—5844no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—2924no—
272C2×C4—2922no—
273C2×C4—2922no—
274C2×C4—2922no—
275C2×C4—2922no—
276C2×C4—2922no—
277C2×C4—731no—
278C2×C4—731no—
279C2×C4—731no—
280C2×C4—731no—
281C2×C4—1461no—
282C2×C4—1461no—
283C2×C4—1461no—
284C2×C4—1461no—
285C2×C4—1461no—
286C2×C4—1461no—
287C2×C4—1461no—
288C2×C4—1464no—
289C2×C4—1464no—
290C2×C4#1455665012yes—
291C2×C4#1455665312yes—
292C2×C4#1455665412yes—
293C2×C4#1455665512yes—
294C2×C4#1455667812yes—
295C8—2924no—
296C8—5842no—
297C22—734no—
298C22—1462no—
299C22—1462no—
300C22—1462no—
301C22—1462no—
302C22—1462no—
303C22—1462no—
304C22#1798956111yescharacteristic
305C22C2×#1455665011yescharacteristic
306C22D73⋊#10111yescharacteristic
307C22C2×#1455665311yescharacteristic
308C22C2×#1455665411yescharacteristic
309C22C2×#1455665511yescharacteristic
310C22#1798958611yescharacteristic
311C4—22no—
312C4—22no—
313C4—82no—
314C4—2921no—
315C4—5842no—
316C4—1461no—
317C4—1461no—
318C4—1461no—
319C4—1461no—
320C4#1798944014yes—
321C2—1464no—
322C2#2262520911yescharacteristic
323C2#2262524011yescharacteristic
324C2#2262527711yescharacteristic
325C2C2×#1798944011yescharacteristic
326C2#2262536811yescharacteristic
327C2#2262540411yescharacteristic
328C2D73⋊#33211yescharacteristic
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#7665
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. #22624749C2semidirect
169. C73#7665semidirect

Maximal subgroups

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

subgroupindex
#226257862
#226255922
#226247492
#226182842
#766573