Finite Group Databasebeta

← #33062066

Subgroups — #33062066

Order 18688.

1068 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 179 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#22620381C211yesmaximal, characteristic
2#22620053C212yesmaximal, complemented
3#22609052C214yesmaximal, complemented
4#22608990C214yesmaximal, complemented
5#22608645C211yesmaximal, characteristic, complemented
6#22608378C212yesmaximal, complemented
7C2×#559×C73C211yesmaximal, characteristic, complemented
8C2×#14556617—22no—
9#17989137C2211yescharacteristic
10#17988974—22no—
11#17988963C2212yes—
12#17988959C2212yes—
13#17988900—22no—
14#17988882C2214yescomplemented
15#17988787C2214yescomplemented
16#17988726—22no—
17#17988685—24no—
18#17988644C2212yes—
19C4×#11924012C2211yescharacteristic
20#17988331C2214yes—
21Q8⋊#10596831C2218yescomplemented
22#559×C73—22no—
23#559×C73—24no—
24#559×C73C2214yes—
25C2×#127×C73C2211yescharacteristic
26C2×#124×C73—22no—
27C2×#123×C73—22no—
28C2×C4×Q8×C73C2212yes—
29C2×C4×D4×C73—22no—
30C2×#11924028—21no—
31#14556617—48no—
32C2×#11924016—24no—
33C2×#11924015—24no—
34C2×#11924013—24no—
35C2×#11924012—21no—
36#14556591C2311yescharacteristic
37#14556588C2311yescharacteristic
38#14556579—21no—
39#14556565—22no—
40C4×D292—22no—
41#14556524—24no—
42C4⋊#10596831C2314yes—
43#14556515—24no—
44#14556511—24no—
45C22⋊#10596831—22no—
46#14556502—22no—
47#14556502—21no—
48C4×#10596831C2312yes—
49C22×Q8×C73D412yes—
50C22×D4×C73—22no—
51C22×D4×C73D412yes—
52C23×C4×C73—22no—
53#127×C73—22no—
54#127×C73D412yes—
55#127×C73C2312yes—
56#124×C73—28no—
57#124×C73—28no—
58#123×C73—28no—
59#123×C73—28no—
60C4×Q8×C73—22no—
61C4×Q8×C73—22no—
62C4×Q8×C73D414yes—
63C4×Q8×C73C2314yes—
64C4×D4×C73—28no—
65C4×D4×C73—28no—
66C2×C4⋊C4×C73—22no—
67C2×C4⋊C4×C73D412yes—
68C2×C22⋊C4×C73—24no—
69C2×C22⋊C4×C73—22no—
70C2×C42×C73—22no—
71C2×C42×C73C2311yescharacteristic
72#11924028—44no—
73C2×Dic146—21no—
74C2×D292—22no—
75C2×D292—21no—
76C2×C22⋊D73—41no—
77C2×C4×D73—41no—
78#11924016—48no—
79#11924015—48no—
80#11924013—48no—
81#11924012—24no—
82C2×#10596831—22no—
83C2×#10596831—24no—
84C4⋊Dic73—22no—
85#11924005—42no—
86C24×C73—22no—
87C2×Q8×C73—22no—
88C2×Q8×C73—44no—
89C2×Q8×C73C2×D414yes—
90C2×D4×C73—24no—
91C2×D4×C73—22no—
92C2×D4×C73—24no—
93C2×D4×C73—48no—
94C2×D4×C73—44no—
95C2×D4×C73C2×D414yes—
96C22×C4×C73—22no—
97C22×C4×C73—22no—
98C22×C4×C73—28no—
99C22×C4×C73—84no—
100C22×C4×C73C2×D412yes—
101C22×C4×C73C2×D411yescharacteristic
102C4⋊C4×C73—24no—
103C4⋊C4×C73—22no—
104C4⋊C4×C73—24no—
105C4⋊C4×C73C2×D414yes—
106C22⋊C4×C73—216no—
107C22⋊C4×C73—28no—
108C42×C73—24no—
109C42×C73—24no—
110C42×C73C2×D411yescharacteristic
111C42×C73C2×D412yes—
112C42×C73C2411yescharacteristic
113C23×C73—24no—
114C23×C73—48no—
115C23×C73—44no—
116C23×C73#12011yescharacteristic
117C22×D73—41no—
118Q8×C73—24no—
119Q8×C73—24no—
120Q8×C73—44no—
121D4×C73—44no—
122D4×C73—48no—
123D4×C73—48no—
124D4×C73—82no—
125C2×C4×C73—22no—
126C2×C4×C73—24no—
127C2×C4×C73—24no—
128C2×C4×C73—24no—
129C2×C4×C73—44no—
130C2×C4×C73—44no—
131C2×C4×C73—88no—
132C2×C4×C73#12012yes—
133C2×C4×C73#12011yescharacteristic
134C2×C4×C73C22×D411yescharacteristic
135C2×C4×C73C22×D412yes—
136C22⋊D73—82no—
137C2×Dic73—41no—
138D292—42no—
139D292—44no—
140C4×D73—82no—
141Dic146—42no—
142#10596831—24no—
143#10596831—44no—
144C22×C73—22no—
145C22×C73—48no—
146C22×C73—44no—
147C22×C73—88no—
148C22×C73#45312yes—
149C22×C73C2×#12011yescharacteristic
150C2×D73—82no—
151C2×D73—81no—
152C4×C73—22no—
153C4×C73—21no—
154C4×C73—21no—
155C4×C73—44no—
156C4×C73—44no—
157C4×C73#57514yes—
158Dic73—81no—
159#18526—731nomaximal, Sylow
160C2×C73—22no—
161C2×C73—84no—
162C2×C73#300512yes—
163C2×C73C2×#45311yescharacteristic
164D73—161no—
165C2×#559D7311yescharacteristic, complemented
166#3151—731no—
167#3112—732no—
168#1630—734no—
169#1620—734no—
170#1555—731no—
171#1491—732no—
172C73#1852611yescharacteristic, complemented, Sylow
173C2×AGL(1,ℤ/8)—1462no—
174#559—22no—
175#559—24no—
176#559C2×D7314yes—
177#532—731no—
178C2×#127C2×D7311yescharacteristic
179C2×#124—22no—
180C2×#123—22no—
181C2×C4×Q8C2×D7312yes—
182C2×C4×D4—22no—
183#496—1462no—
184#493—732no—
185#492—732no—
186#480—1462no—
187#477—734no—
188#460—734no—
189#448—1462no—
190#440—1464no—
191#432—732no—
192C4×M16—731no—
193#331—734no—
194Q8⋊C8—738no—
195C2×#41—1461no—
196C22×Q8C22⋊D7312yes—
197C22×D4—22no—
198C22×D4C22⋊D7312yes—
199C23×C4—22no—
200AGL(1,ℤ/8)—2928no—
201C2×SD16—1464no—
202C2×SD16—1464no—
203C2×D8—1464no—
204C2×M16—1461no—
205C4⋊Q8—731no—
206#127—22no—
207#127—731no—
208#127C22⋊D7312yes—
209#127C22×D7312yes—
210#124—28no—
211#124—28no—
212#124—1461no—
213#123—28no—
214#123—28no—
215#121—1462no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8C22⋊D7314yes—
219C4×Q8C22×D7314yes—
220C4×D4—28no—
221C4×D4—28no—
222C4×D4—1462no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4C22⋊D7312yes—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—22no—
227C2×C42—22no—
228C2×C42C22×D7311yescharacteristic
229#107—1464no—
230C4⋊C8—734no—
231#103—1464no—
232#102—1464no—
233C22⋊C8—1462no—
234#97—1461no—
235#97—1462no—
236C4×C8—732no—
237C24—22no—
238#41—2924no—
239C2×Q8—22no—
240C2×Q8—44no—
241C2×Q8—1461no—
242C2×Q8C2×C22⋊D7314yes—
243C2×D4—24no—
244C2×D4—22no—
245C2×D4—24no—
246C2×D4—48no—
247C2×D4—44no—
248C2×D4—2921no—
249C2×D4—1462no—
250C2×D4—1461no—
251C2×D4C2×C22⋊D7314yes—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—28no—
255C22×C4—84no—
256C22×C4—2921no—
257C22×C4C2×C22⋊D7312yes—
258C22×C4C2×C22⋊D7311yescharacteristic
259SD16—2928no—
260SD16—2928no—
261D8—2928no—
262M16—1464no—
263C2×C8—1464no—
264C2×C8—1462no—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—1462no—
269C4⋊C4C2×C22⋊D7314yes—
270C22⋊C4—216no—
271C22⋊C4—28no—
272C22⋊C4—2922no—
273C42—24no—
274C42—24no—
275C42C2×C22⋊D7312yes—
276C42C2×C22⋊D7311yescharacteristic
277C42C23×D7311yescharacteristic
278C23—24no—
279C23—48no—
280C23—44no—
281C23—2921no—
282C23#1455656011yescharacteristic
283Q8—24no—
284Q8—24no—
285Q8—44no—
286Q8—2922no—
287D4—44no—
288D4—48no—
289D4—48no—
290D4—82no—
291D4—2922no—
292D4—2924no—
293D4—5842no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—88no—
301C2×C4—2921no—
302C2×C4—5842no—
303C2×C4#1455656012yes—
304C2×C4#1455656011yescharacteristic
305C2×C4C22×C22⋊D7311yescharacteristic
306C2×C4C22×C22⋊D7312yes—
307C8—2924no—
308C8—1464no—
309C22—22no—
310C22—48no—
311C22—44no—
312C22—88no—
313C22—5842no—
314C22—5841no—
315C22#1798875512yes—
316C22C2×#1455656011yescharacteristic
317C4—22no—
318C4—21no—
319C4—21no—
320C4—44no—
321C4—44no—
322C4—5841no—
323C4#1798938514yes—
324C2—22no—
325C2—84no—
326C2—11681no—
327C2#2261917312yes—
328C2C2×#1798875511yescharacteristic
subgroupG/H
centerC22C2×#14556560
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC2×#559×C73C2
socleC22×C73C2×#120
prime radical#22620381C2
hypercenterC2×#559D73
hypocenterC73#18526
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
2. #22620053C2semidirect
3. #22609052C2semidirect
4. #22608990C2semidirect
5. #22608645C2semidirect
6. #22608378C2semidirect
7. C2×#559×C73C2semidirect
14. #17988882C22semidirect
15. #17988787C22semidirect
21. Q8⋊#10596831C22semidirect
165. C2×#559D73semidirect
172. C73#18526semidirect

Maximal subgroups

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

subgroupindex
#226203812
#226200532
#226090522
#226089902
#226086452
#226083782
C2×#559×C732
#1852673