Finite Group Databasebeta

← #34023854

Subgroups — #34023854

Order 18688.

1050 conjugacy classes of subgroups; only the 220 normal subgroups are shown.

normal subgroups only

This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 220 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#22626378C212yesmaximal
2#22626122C218yesmaximal
3#22625829C214yesmaximal
4C2×#17989143C211yesmaximal, characteristic
5#17989532—22no—
6#17989531—22no—
7#17989530C2214yes—
8#17989529C2214yes—
9C4×C4⋊#64408—22no—
10C4×C22⋊#64408C2214yes—
11#17989523—22no—
12#17989465C22116yes—
13#17989143—22no—
14#17989143—22no—
15#17989143—22no—
16#17989143—22no—
17C2×#14556590C412yescomplemented
18C2×#14556583C411yescharacteristic, complemented
19C2×#14556569C2214yes—
20C2×#14556556C414yescomplemented
21C2×#14556549C2212yes—
22C2×C42×D73C2211yescharacteristic
23C2×#125×C73C411yescharacteristic, complemented
24C2×C4⋊#64408—24no—
25C2×C22⋊#64408C2318yes—
26C2×C4×#64408—24no—
27(C2×#64408)⋊C4—24no—
28(C2×#64408)⋊C4—24no—
29(C2×#64408)⋊C4—24no—
30(C2×#64408)⋊C4—24no—
31C22×Dic146C2×C414yes—
32C22×C4×D73C2311yescharacteristic
33C22×C4×D73C2312yes—
34#14556590—22no—
35#14556590—22no—
36#14556590—22no—
37#14556590—22no—
38#14556583—21no—
39#14556583—22no—
40#14556583—21no—
41#14556569—24no—
42#14556569—24no—
43#14556569—24no—
44#14556569—24no—
45#14556556—24no—
46#14556556—24no—
47#14556556—24no—
48#14556556—24no—
49#14556549—24no—
50#14556549—24no—
51C2×C4⋊Dic73C2×C412yes—
52C2×#11924007C2×C414yes—
53C2×#11924007C2×C414yes—
54C2×#11924005C2314yes—
55C2×C4×Dic73C2×C414yes—
56C2×C4×Dic73C2×C411yescharacteristic
57C2×C4×Dic73C2×C412yes—
58C42×D73D412yes—
59C42×D73D412yes—
60C42×D73D412yes—
61C42×D73D412yes—
62#125×C73—21no—
63#125×C73—22no—
64#125×C73—21no—
65C2×C4⋊C4×C73C2×C412yes—
66C2×C4⋊C4×C73C2×C414yes—
67C2×C42×C73C2×C411yescharacteristic
68C22×#64408—28no—
69C4⋊#64408—48no—
70C22⋊#64408—216no—
71C4×#64408—48no—
72C23×D73C2411yescharacteristic
73C2×Dic146—24no—
74C2×Dic146—24no—
75C2×Dic146—48no—
76C22×Dic73C22×C411yescharacteristic
77C22×Dic73C22×C412yes—
78C22×Dic73C22×C414yes—
79C2×C4×D73—24no—
80C2×C4×D73—44no—
81C2×C4×D73C2×D412yes—
82C2×C4×D73C2×D412yes—
83C2×C4×D73C2×D412yes—
84C2×C4×D73C2×D412yes—
85C2×C4×D73C2×D412yes—
86C2×C4×D73C2×D412yes—
87C4⋊Dic73—22no—
88C4⋊Dic73—22no—
89#11924007—24no—
90#11924007—24no—
91#11924007—24no—
92#11924007—24no—
93#11924005—28no—
94C4×Dic73—28no—
95C4×Dic73C22⋊C412yes—
96C4×Dic73C22⋊C412yes—
97C4×Dic73C22⋊C412yes—
98C4×Dic73C22⋊C412yes—
99C4×Dic73C22⋊C411yescharacteristic
100C4×Dic73C22⋊C412yes—
101C4×Dic73C22⋊C411yescharacteristic
102C22×C4×C73C22×C411yescharacteristic
103C22×C4×C73C22×C414yes—
104C22×C4×C73C22×C412yes—
105C4⋊C4×C73—22no—
106C4⋊C4×C73—22no—
107C4⋊C4×C73—24no—
108C4⋊C4×C73—24no—
109C42×C73C22⋊C411yescharacteristic
110C42×C73C22⋊C412yes—
111C42×C73C22⋊C411yescharacteristic
112C23×C73C23×C411yescharacteristic
113C22×D73—42no—
114C22×D73C22×D412yes—
115C22×D732−1+414yes—
116C2×#64408—48no—
117C2×#64408—48no—
118C2×#64408—48no—
119C2×C4×C73—22no—
120C2×C4×C73—24no—
121C2×C4×C73—42no—
122C2×C4×C73—44no—
123C2×C4×C73C2×C22⋊C412yes—
124C2×C4×C73C2×C22⋊C412yes—
125C2×C4×C73C2×C22⋊C411yescharacteristic
126C2×C4×C73C2×C22⋊C411yescharacteristic
127C2×Dic73—22no—
128C2×Dic73—24no—
129C2×Dic73—24no—
130C2×Dic73—44no—
131C2×Dic73C2×C22⋊C412yes—
132C2×Dic73C2×C22⋊C412yes—
133C2×Dic73C2×C22⋊C412yes—
134C4×D73—24no—
135C4×D73—24no—
136C4×D73—24no—
137C4×D73—24no—
138C4×D73—42no—
139C4×D73—42no—
140Dic146—48no—
141Dic146—48no—
142C22×C73—22no—
143C22×C73C22×C22⋊C411yescharacteristic
144C22×C73#51912yes—
145C2×D73—22no—
146C2×D73—22no—
147C2×D73—44no—
148C2×D73#53614yes—
149#64408—88no—
150C4×C73—21no—
151C4×C73—21no—
152C4×C73—22no—
153C4×C73—22no—
154C4×C73—44no—
155Dic73—22no—
156Dic73—22no—
157Dic73—44no—
158Dic73#41014yes—
159#21826—731nomaximal, Sylow
160C2×C73—22no—
161C2×C73#231011yescharacteristic
162C2×C73C2×#41012yes—
163D73—24no—
164C2×#533—731no—
165#2283—732no—
166#1996—738no—
167#1820—734no—
168C73#2182611yescharacteristic, complemented, Sylow
169#533—1462no—
170#533—1462no—
171#533—1462no—
172#533—1462no—
173C2×C4⋊Q8—732no—
174C2×#125—731no—
175C2×#125#6440811yescharacteristic, complemented
176C2×C22⋊Q8—734no—
177C2×C4×Q8—734no—
178C2×#117—732no—
179C22×C42—731no—
180#383—1462no—
181#382—1462no—
182#381—734no—
183#380—734no—
184C4×C4⋊C4—1462no—
185C4×C22⋊C4—734no—
186#376—1462no—
187#342—7316no—
188C22×Q8—734no—
189C23×C4—732no—
190C23×C4—731no—
191C4⋊Q8—1462no—
192C4⋊Q8—1462no—
193C4⋊Q8—1462no—
194C4⋊Q8—1462no—
195#125—21no—
196#125—22no—
197#125—21no—
198#125—1461no—
199#125—1462no—
200#125—1461no—
201C22⋊Q8—1464no—
202C22⋊Q8—1464no—
203C22⋊Q8—1464no—
204C22⋊Q8—1464no—
205C4×Q8—1464no—
206C4×Q8—1464no—
207C4×Q8—1464no—
208C4×Q8—1464no—
209#117—1464no—
210#117—1464no—
211C2×C4⋊C4—732no—
212C2×C4⋊C4—734no—
213C2×C4⋊C4—734no—
214C2×C4⋊C4—1464no—
215C2×C4⋊C4C2×#6440812yes—
216C2×C4⋊C4C2×#6440814yes—
217C2×C22⋊C4—734no—
218C2×C22⋊C4—738no—
219C2×C42—732no—
220C2×C42—731no—
221C2×C42—732no—
222C2×C42—732no—
223C2×C42—732no—
224C2×C42—732no—
225C2×C42—734no—
226C2×C42—1464no—
227C2×C42C2×#6440811yescharacteristic
228#95—1464no—
229#95—1464no—
230#95—1464no—
231#95—1464no—
232C24—731no—
233C2×Q8—2928no—
234C2×Q8—1464no—
235C2×Q8—1464no—
236C22×C4—2924no—
237C22×C4—732no—
238C22×C4—732no—
239C22×C4—734no—
240C22×C4—731no—
241C22×C4—732no—
242C22×C4—732no—
243C22×C4—732no—
244C22×C4—732no—
245C22×C4—732no—
246C22×C4—1464no—
247C22×C4—1468no—
248C22×C4C22×#6440811yescharacteristic
249C22×C4C22×#6440814yes—
250C22×C4C22×#6440812yes—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—2928no—
256C4⋊C4—1464no—
257C4⋊C4—1464no—
258C4⋊C4—1464no—
259C4⋊C4—1464no—
260C4⋊C4—1462no—
261C4⋊C4—1462no—
262C22⋊C4—1468no—
263C22⋊C4—14616no—
264C42—2928no—
265C42—732no—
266C42—732no—
267C42—732no—
268C42—732no—
269C42—731no—
270C42—732no—
271C42—731no—
272C42—1468no—
273C42C22⋊#6440811yescharacteristic
274C42C22⋊#6440812yes—
275C42C22⋊#6440811yescharacteristic
276C23—2922no—
277C23—734no—
278C23—732no—
279C23C23×#6440811yescharacteristic
280Q8—2928no—
281Q8—2928no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—2922no—
287C2×C4—2922no—
288C2×C4—2924no—
289C2×C4—2928no—
290C2×C4—2928no—
291C2×C4—2928no—
292C2×C4—732no—
293C2×C4—732no—
294C2×C4—732no—
295C2×C4—1464no—
296C2×C4—1464no—
297C2×C4—1464no—
298C2×C4—1464no—
299C2×C4—1464no—
300C2×C4—1464no—
301C2×C4—1462no—
302C2×C4C2×C22⋊#6440811yescharacteristic
303C2×C4C2×C22⋊#6440811yescharacteristic
304C2×C4C2×C22⋊#6440812yes—
305C2×C4C2×C22⋊#6440812yes—
306C22—22no—
307C22—2924no—
308C22—734no—
309C22—1462no—
310C22—1462no—
311C22C22×C22⋊#6440811yescharacteristic
312C22#1798965512yes—
313C4—21no—
314C4—21no—
315C4—22no—
316C4—22no—
317C4—44no—
318C4—2924no—
319C4—5848no—
320C4—734no—
321C4—1462no—
322C4—1462no—
323C2—22no—
324C2—1464no—
325C2#2262640511yescharacteristic
326C2D73⋊#41012yes—
subgroupG/H
centerC22C22×C22⋊#64408
commutatorC23×C73C23×C4
FrattiniC23C23×#64408
FittingC2×#125×C73C4
socleC22×C73C22×C22⋊C4
prime radical, cosocleC23×D73C24
hypercenterC2×#125#64408
hypocenterC73#21826
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
17. C2×#14556590C4semidirect
18. C2×#14556583C4semidirect
20. C2×#14556556C4semidirect
23. C2×#125×C73C4semidirect
168. C73#21826semidirect
175. C2×#125#64408semidirect

Maximal subgroups

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

subgroupindex
#226263782
#226261222
#226258292
C2×#179891432
#2182673