Finite Group Databasebeta

← #34012279

Subgroups — #34012279

Order 18688.

826 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 152 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
1C2×#17989566C212yesmaximal
2#22625026C214yesmaximal, complemented
3#22612524C211yesmaximal, characteristic
4C2×#14556670—22no—
5#17989566—24no—
6#17989566C418yescomplemented
7C2×#14556650C2214yes—
8C2×#14556649—22no—
9C22×#11924028C2211yescharacteristic
10C2×#14556550C411yescharacteristic, complemented
11#17988492C412yescomplemented
12#17988488C2212yes—
13#17988466—22no—
14#17988444—22no—
15C22×C4⋊C4×C73C411yescharacteristic, complemented
16#14556670—28no—
17#14556670—44no—
18C2×C4⋊#64408—42no—
19C2×C22⋊#64408—24no—
20C2×C4×#64408—42no—
21#14556650—24no—
22#14556650C2×C418yescomplemented
23#14556649—28no—
24C2×#11924028—42no—
25C2×#11924028—42no—
26C2×#11924028C2×C414yes—
27C2×#11924028D412yes—
28C2×#11924028Q812yes—
29C22×Dic146C2×C411yescharacteristic
30C22×D292C2311yescharacteristic
31C22×C22⋊D73D412yes—
32C22×C4×D73D412yes—
33#14556550—42no—
34C2×C4⋊Dic73—21no—
35C2×C22⋊Dic73—21no—
36C2×#11924005—24no—
37C2×C4×Dic73C2×C412yes—
38#14556498—24no—
39C23×C4×C73—21no—
40C23×C4×C73C2×C411yescharacteristic
41C2×C4⋊C4×C73—21no—
42C2×C4⋊C4×C73—42no—
43C2×C4⋊C4×C73C2×C412yes—
44C22×#64408—44no—
45C4⋊#64408—48no—
46C22⋊#64408—28no—
47C22⋊#64408—44no—
48C4×#64408—48no—
49C23×D73C2×D412yes—
50#11924028—44no—
51#11924028—44no—
52#11924028—44no—
53#11924028—44no—
54C2×Dic146—41no—
55C2×Dic146—41no—
56C2×Dic146C4212yes—
57C2×Dic146C4⋊C412yes—
58C2×D292—41no—
59C2×D292—41no—
60C2×D292C22×C412yes—
61C2×D292C2×D411yescharacteristic
62C2×D292C2×Q811yescharacteristic
63C2×C22⋊D73—42no—
64C2×C22⋊D73—42no—
65C2×C22⋊D73C22⋊C414yes—
66C2×C22⋊D73C4⋊C414yes—
67C22×Dic73—22no—
68C22×Dic73C22⋊C412yes—
69C2×C4×D73—22no—
70C2×C4×D73—22no—
71C2×C4×D73—24no—
72C2×C4×D73C22⋊C414yes—
73C2×C4×D73C4⋊C414yes—
74C4⋊Dic73—42no—
75C22⋊Dic73—42no—
76#11924005—84no—
77C4×Dic73—42no—
78C24×C73C22⋊C411yescharacteristic
79C22×C4×C73—22no—
80C22×C4×C73—22no—
81C22×C4×C73—22no—
82C22×C4×C73—42no—
83C22×C4×C73—82no—
84C22×C4×C73C4212yes—
85C22×C4×C73C22⋊C411yescharacteristic
86C22×C4×C73C4⋊C412yes—
87C22×C4×C73C22×C411yescharacteristic
88C4⋊C4×C73—42no—
89C4⋊C4×C73—42no—
90C23×C73—21no—
91C23×C73—21no—
92C23×C73—41no—
93C23×C73—41no—
94C23×C73#9512yes—
95C23×C73C2×C22⋊C411yescharacteristic
96C22×D73—22no—
97C22×D73—22no—
98C22×D73—42no—
99C22×D73—42no—
100C22×D73C2×C22⋊C412yes—
101C22×D73C2×C4⋊C412yes—
102C2×#64408—48no—
103C2×#64408—84no—
104C2×#64408—84no—
105C2×C4×C73—21no—
106C2×C4×C73—21no—
107C2×C4×C73—24no—
108C2×C4×C73—24no—
109C2×C4×C73—44no—
110C2×C4×C73—42no—
111C2×C4×C73—84no—
112C2×C4×C73#9512yes—
113C2×C4×C73#9912yes—
114C2×C4×C73#9912yes—
115C2×C4×C73C2×C4211yescharacteristic
116C2×C4×C73C2×C4⋊C411yescharacteristic
117C22⋊D73—44no—
118C22⋊D73—44no—
119C2×Dic73—22no—
120C2×Dic73—22no—
121C2×Dic73—42no—
122C2×Dic73—42no—
123C2×Dic73—42no—
124C2×Dic73#9514yes—
125D292—42no—
126D292—41no—
127D292—41no—
128C4×D73—24no—
129C4×D73—24no—
130C4×D73—24no—
131C4×D73—24no—
132Dic146—42no—
133Dic146—42no—
134C22×C73—21no—
135C22×C73—21no—
136C22×C73—22no—
137C22×C73—42no—
138C22×C73—42no—
139C22×C73—41no—
140C22×C73—41no—
141C22×C73C2×#9511yescharacteristic
142C22×C73C2×#9911yescharacteristic
143C22×C73C2×#9911yescharacteristic
144C2×D73—22no—
145C2×D73—22no—
146C2×D73—42no—
147C2×D73—42no—
148C2×D73—42no—
149C2×D73—42no—
150C2×D73—82no—
151#64408—88no—
152C4×C73—21no—
153C4×C73—21no—
154C4×C73—82no—
155C4×C73#34214yes—
156Dic73—24no—
157Dic73—82no—
158#12583—731nomaximal, Sylow
159C2×C73—21no—
160C2×C73—21no—
161C2×C73—42no—
162C2×C73C2×#34211yescharacteristic
163C2×C73#212811yescharacteristic
164C2×C73#212911yescharacteristic
165D73—42no—
166D73—42no—
167C2×#410—732no—
168#2311—731no—
169#1393—734no—
170C73#1258311yescharacteristic, complemented, Sylow
171C22×#41—731no—
172C2×#117—731no—
173C2×#117—1462no—
174C22×C4⋊C4#6440811yescharacteristic, complemented
175#410—738no—
176#410—1464no—
177C2×#99—734no—
178C2×#99—1462no—
179#391—732no—
180#390—732no—
181#386—1462no—
182#380—1462no—
183C2×#41—2922no—
184C2×#41—2922no—
185C2×#41—734no—
186C2×#41—732no—
187C2×#41—732no—
188C22×Q8—731no—
189C22×D4—732no—
190C22×D4—731no—
191C23×C4—21no—
192C23×C4—732no—
193C23×C4C2×#6440811yescharacteristic
194#117—2924no—
195#117—2922no—
196#117—1468no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—2922no—
200C2×C4⋊C4—1461no—
201C2×C4⋊C4C2×#6440812yes—
202C2×C22⋊C4—1461no—
203C2×C22⋊C4—1464no—
204C2×C22⋊C4—1464no—
205C2×C42—2922no—
206C2×C42—732no—
207#99—738no—
208#99—1464no—
209#99—1468no—
210#95—1464no—
211C24—732no—
212C24C22⋊#6440811yescharacteristic
213#41—2924no—
214#41—2924no—
215#41—2924no—
216#41—2924no—
217C2×Q8—2921no—
218C2×Q8—2921no—
219C2×Q8—732no—
220C2×Q8—732no—
221C2×D4—2921no—
222C2×D4—2921no—
223C2×D4—2922no—
224C2×D4—2922no—
225C2×D4—734no—
226C2×D4—734no—
227C2×D4—732no—
228C2×D4—731no—
229C2×D4—731no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—42no—
234C22×C4—82no—
235C22×C4—2924no—
236C22×C4—732no—
237C22×C4—734no—
238C22×C4—734no—
239C22×C4—1462no—
240C22×C4—1462no—
241C22×C4—1462no—
242C22×C4—1464no—
243C22×C4C4×#6440812yes—
244C22×C4C22⋊#6440811yescharacteristic
245C22×C4C4⋊#6440812yes—
246C22×C4C22×#6440811yescharacteristic
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—2928no—
250C4⋊C4—2922no—
251C22⋊C4—2922no—
252C22⋊C4—2924no—
253C22⋊C4—5844no—
254C22⋊C4—1468no—
255C42—2922no—
256C42—2928no—
257C23—21no—
258C23—21no—
259C23—41no—
260C23—41no—
261C23—2922no—
262C23—2922no—
263C23—732no—
264C23—732no—
265C23—1462no—
266C23—1462no—
267C23(C2×#64408)⋊C412yes—
268C23C2×C22⋊#6440811yescharacteristic
269Q8—2922no—
270Q8—2922no—
271D4—2922no—
272D4—2921no—
273D4—2921no—
274D4—2924no—
275D4—2924no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—84no—
283C2×C4—2922no—
284C2×C4—2922no—
285C2×C4—2928no—
286C2×C4—2922no—
287C2×C4—5844no—
288C2×C4—5844no—
289C2×C4—734no—
290C2×C4—1462no—
291C2×C4—1462no—
292C2×C4—1464no—
293C2×C4—1464no—
294C2×C4—1464no—
295C2×C4—1464no—
296C2×C4(C2×#64408)⋊C412yes—
297C2×C4#1455664912yes—
298C2×C4#1455665012yes—
299C2×C4C2×C4×#6440811yescharacteristic
300C2×C4C2×C4⋊#6440811yescharacteristic
301C22—21no—
302C22—21no—
303C22—22no—
304C22—42no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—2922no—
309C22—2922no—
310C22—2922no—
311C22—2922no—
312C22—5842no—
313C22—1462no—
314C22—1462no—
315C22C2×(C2×#64408)⋊C411yescharacteristic
316C22C2×#1455664911yescharacteristic
317C22C2×#1455665011yescharacteristic
318C4—21no—
319C4—21no—
320C4—82no—
321C4—5842no—
322C4—5848no—
323C4—1464no—
324C4#1798946514yes—
325C2—21no—
326C2—21no—
327C2—42no—
328C2—2922no—
329C2—2922no—
330C2C2×#1798946511yescharacteristic
331C2#2262621011yescharacteristic
332C2#2262621311yescharacteristic
subgroupG/H
centerC22C2×#14556649
commutatorC2×C4×C73C2×C42
FrattiniC22×C4C22×#64408
FittingC22×C4⋊C4×C73C4
socleC22×C73C2×#99
prime radicalC22×#11924028C22
cosocleC22×D292C23
hypercenterC22×C4⋊C4#64408
hypocenterC73#12583
LCSG ▷ C2×C4×C73 ▷ C22×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
2. #22625026C2semidirect
6. #17989566C4semidirect
10. C2×#14556550C4semidirect
11. #17988492C4semidirect
15. C22×C4⋊C4×C73C4semidirect
22. #14556650C2×C4semidirect
170. C73#12583semidirect
174. C22×C4⋊C4#64408semidirect

Maximal subgroups

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

subgroupindex
C2×#179895662
#226250262
#226125242
#1258373