Finite Group Databasebeta

← #34011044

Subgroups — #34011044

Order 18688.

462 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C4×#14556649C211yesmaximal, characteristic
2#22625190C211yesmaximal, characteristic
3#22624928C214yesmaximal, complemented
4#22614242C211yesmaximal, characteristic
5#17989608—21no—
6C2×#14556649—21no—
7C4×C22⋊#64408—21no—
8#17989465C2212yes—
9#17989431C2212yes—
10#17989430—21no—
11C2×C4×C22⋊D73C2211yescharacteristic
12#17988525C412yescomplemented
13#17988522C2212yes—
14#17988483—21no—
15#17988466—21no—
16#17988460—21no—
17#17988447C411yescharacteristic, complemented
18#17988444—21no—
19C22×C4⋊C4×C73C411yescharacteristic, complemented
20C2×C22⋊#64408—22no—
21C2×C4×#64408—41no—
22C4⋊#10596833—41no—
23#14556649—24no—
24C22⋊#10596833—22no—
25#14556646—41no—
26(C2×#64408)⋊C4—41no—
27C22×C22⋊D73D411yescharacteristic
28C22×C4×D73D411yescharacteristic
29C4×C22⋊D73—41no—
30C4×C22⋊D73—41no—
31C2×C4⋊Dic73—21no—
32C2×#11924007—22no—
33C2×#11924007C2×C411yescharacteristic
34C2×C22⋊Dic73—21no—
35C2×C22⋊Dic73D411yescharacteristic
36C2×#11924005—22no—
37C2×#11924005C2311yescharacteristic
38C2×C4×Dic73D411yescharacteristic
39#14556498—22no—
40#14556498—22no—
41#14556498C2×C412yes—
42C23×C4×C73—21no—
43C23×C4×C73C2×C411yescharacteristic
44C2×C4⋊C4×C73—21no—
45C2×C4⋊C4×C73—41no—
46C2×C4⋊C4×C73—41no—
47C2×C4⋊C4×C73C2×C412yes—
48C22×#64408—42no—
49C2×#10596833—42no—
50C22⋊#64408—24no—
51C22⋊#64408—42no—
52C4×#64408—82no—
53C23×D73C2×D411yescharacteristic
54C2×C22⋊D73—22no—
55C2×C22⋊D73—22no—
56C2×C22⋊D73—41no—
57C22×Dic73—22no—
58C22×Dic73C22⋊C411yescharacteristic
59C22×Dic73C22⋊C411yescharacteristic
60C22×Dic73C2×D411yescharacteristic
61C2×C4×D73—21no—
62C2×C4×D73—21no—
63C2×C4×D73—42no—
64C4⋊Dic73—42no—
65#11924007—41no—
66#11924007—82no—
67C22⋊Dic73—41no—
68C22⋊Dic73—42no—
69#11924005—41no—
70#11924005—82no—
71C4×Dic73—21no—
72C4×Dic73—21no—
73C24×C73C22⋊C411yescharacteristic
74C22×C4×C73—21no—
75C22×C4×C73—21no—
76C22×C4×C73—22no—
77C22×C4×C73—42no—
78C22×C4×C73—42no—
79C22×C4×C73—82no—
80C22×C4×C73C22⋊C411yescharacteristic
81C22×C4×C73C22×C411yescharacteristic
82C4⋊C4×C73—42no—
83C4⋊C4×C73—41no—
84C4⋊C4×C73—41no—
85C23×C73—21no—
86C23×C73—21no—
87C23×C73—41no—
88C23×C73—41no—
89C23×C73#10412yes—
90C23×C73C2×C22⋊C411yescharacteristic
91C22×D73—21no—
92C22×D73—21no—
93C22×D73—41no—
94C22×D73—41no—
95C22×D73#13712yes—
96C2×#64408—44no—
97C2×#64408—82no—
98C2×#64408—82no—
99C2×C4×C73—22no—
100C2×C4×C73—21no—
101C2×C4×C73—42no—
102C2×C4×C73—42no—
103C2×C4×C73—41no—
104C2×C4×C73—42no—
105C2×C4×C73—42no—
106C2×C4×C73—42no—
107C2×C4×C73—84no—
108C2×C4×C73#9911yescharacteristic
109C2×C4×C73#9911yescharacteristic
110C22⋊D73—44no—
111C2×Dic73—21no—
112C2×Dic73—21no—
113C2×Dic73—21no—
114C2×Dic73—21no—
115C2×Dic73—21no—
116C2×Dic73—41no—
117C2×Dic73—41no—
118C2×Dic73—42no—
119C2×Dic73—42no—
120C2×Dic73—42no—
121C4×D73—42no—
122C4×D73—81no—
123#10596833—82no—
124C22×C73—21no—
125C22×C73—21no—
126C22×C73—22no—
127C22×C73—42no—
128C22×C73—42no—
129C22×C73—41no—
130C22×C73—41no—
131C22×C73C2×#9911yescharacteristic
132C22×C73#41911yescharacteristic
133C22×C73C2×#10411yescharacteristic
134C2×D73—22no—
135C2×D73—41no—
136C2×D73—41no—
137C2×D73—42no—
138C2×D73—81no—
139#64408—84no—
140C4×C73—21no—
141C4×C73—41no—
142C4×C73—82no—
143C4×C73#35212yes—
144Dic73—41no—
145Dic73—41no—
146Dic73—41no—
147Dic73—82no—
148#11786—731nomaximal, Sylow
149C2×C73—21no—
150C2×C73—21no—
151C2×C73—42no—
152C2×C73#150811yescharacteristic
153C2×C73C2×#35211yescharacteristic
154C2×C73#212811yescharacteristic
155D73—42no—
156#2487—731no—
157C4×#99—731no—
158#1466—731no—
159#1352—734no—
160C73#1178611yescharacteristic, complemented, Sylow
161C2×C4×D4—731no—
162C22×C4⋊C4#6440811yescharacteristic, complemented
163#432—1461no—
164C2×#99—1461no—
165#400—732no—
166#399—732no—
167#389—1461no—
168#386—1461no—
169#385—1461no—
170#380—731no—
171#380—1461no—
172C4×C22⋊C4—1461no—
173#342—732no—
174#323—732no—
175#323—1461no—
176C22×D4—731no—
177C23×C4—21no—
178C23×C4—731no—
179C23×C4C2×#6440811yescharacteristic
180C4×D4—2921no—
181C4×D4—2921no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—731no—
186C2×C4⋊C4—1461no—
187C2×C4⋊C4—1462no—
188C2×C4⋊C4C2×#6440812yes—
189C2×C22⋊C4—731no—
190C2×C22⋊C4—731no—
191C2×C22⋊C4—1461no—
192C2×C22⋊C4—1462no—
193C2×C22⋊C4—1462no—
194C2×C42—2921no—
195C2×C42—731no—
196C4⋊C8—2921no—
197#99—1464no—
198C22⋊C8—1462no—
199#97—2921no—
200#95—2921no—
201#95—732no—
202#95—1462no—
203#95—1462no—
204C24—731no—
205C24C22⋊#6440811yescharacteristic
206C2×D4—2921no—
207C2×D4—1462no—
208C2×D4—1462no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—82no—
215C22×C4—2922no—
216C22×C4—2922no—
217C22×C4—731no—
218C22×C4—731no—
219C22×C4—731no—
220C22×C4—1462no—
221C22×C4—1461no—
222C22×C4—1461no—
223C22×C4C22⋊#6440811yescharacteristic
224C22×C4C22×#6440811yescharacteristic
225C2×C8—2922no—
226C4⋊C4—42no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—2922no—
230C4⋊C4—2921no—
231C4⋊C4—5842no—
232C22⋊C4—2921no—
233C22⋊C4—2922no—
234C22⋊C4—2922no—
235C22⋊C4—2921no—
236C22⋊C4—5842no—
237C22⋊C4—1464no—
238C42—5842no—
239C42—1461no—
240C42—1461no—
241C23—21no—
242C23—21no—
243C23—41no—
244C23—41no—
245C23—2921no—
246C23—2921no—
247C23—732no—
248C23—1461no—
249C23—1461no—
250C23#1455665612yes—
251C23C2×C22⋊#6440811yescharacteristic
252D4—2924no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—84no—
262C2×C4—2922no—
263C2×C4—2921no—
264C2×C4—2921no—
265C2×C4—2922no—
266C2×C4—2922no—
267C2×C4—2924no—
268C2×C4—2922no—
269C2×C4—5841no—
270C2×C4—5842no—
271C2×C4—5842no—
272C2×C4—1461no—
273C2×C4—1461no—
274C2×C4—1461no—
275C2×C4—1461no—
276C2×C4—1461no—
277C2×C4#1455664911yescharacteristic
278C2×C4#1455664911yescharacteristic
279C8—5842no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—42no—
284C22—42no—
285C22—41no—
286C22—41no—
287C22—2922no—
288C22—2921no—
289C22—2921no—
290C22—5841no—
291C22—1462no—
292C22C2×#1455664911yescharacteristic
293C22#1798958011yescharacteristic
294C22C2×#1455665611yescharacteristic
295C4—21no—
296C4—41no—
297C4—82no—
298C4—2921no—
299C4—2921no—
300C4—2921no—
301C4—5842no—
302C4—5844no—
303C4#1798948912yes—
304C2—21no—
305C2—21no—
306C2—42no—
307C2—2922no—
308C2#2262533111yescharacteristic
309C2C2×#1798948911yescharacteristic
310C2#2262621011yescharacteristic
subgroupG/H
centerC22C2×#14556649
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC22×C4⋊C4×C73C4
socleC22×C73C2×#99
prime radicalC22×C22⋊D73D4
cosocleC2×#11924005C23
hypercenterC22×C4⋊C4#64408
hypocenterC73#11786
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #22624928C2semidirect
12. #17988525C4semidirect
17. #17988447C4semidirect
19. C22×C4⋊C4×C73C4semidirect
160. C73#11786semidirect
162. C22×C4⋊C4#64408semidirect

Maximal subgroups

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

subgroupindex
C4×#145566492
#226251902
#226249282
#226142422
#1178673