Finite Group Databasebeta

← #32988730

Subgroups — #32988730

Order 18688.

576 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 109 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#22621072C211yesmaximal, characteristic
2#22618327C211yesmaximal, characteristic
3#22610909C211yesmaximal, characteristic
4#22608603C211yesmaximal, characteristic
5#22608602C211yesmaximal, characteristic
6#22607763C211yesmaximal, characteristic
7C73×#3229C211yesmaximal, characteristic
8C2×#14556591C2211yescharacteristic
9C2×#14556589—22no—
10#17988983—21no—
11#17988937—21no—
12#17988915—22no—
13#17988872—22no—
14#17988640C2211yescharacteristic
15#17988611C2211yescharacteristic
16#17988456—22no—
17#17988456—22no—
18C42×Dic73C2211yescharacteristic
19#17988346—22no—
20#17988333—22no—
21#17988332—22no—
22#533×C73C2211yescharacteristic
23C2×C4⋊Q8×C73C2211yescharacteristic
24#498×C73—21no—
25#487×C73—21no—
26#483×C73—22no—
27#475×C73—22no—
28#431×C73C2211yescharacteristic
29C22×Dic146—21no—
30C2×#11924012—21no—
31#14556591—21no—
32#14556591—21no—
33#14556591—21no—
34#14556591—21no—
35#14556589—44no—
36#14556589—44no—
37C2×C4⋊Dic73—21no—
38C2×C4⋊Dic73D412yes—
39C2×#11924007—28no—
40C2×C4×Dic73—21no—
41C2×C4×Dic73—21no—
42C2×C4×Dic73D412yes—
43#14556522—24no—
44C4⋊#10596831—21no—
45C4⋊#10596831—21no—
46C4⋊#10596831D412yes—
47#14556515—24no—
48C22⋊#10596831—21no—
49C4×#10596831—21no—
50C22×Q8×C73—21no—
51C4⋊Q8×C73—21no—
52C4⋊Q8×C73—21no—
53C4⋊Q8×C73—21no—
54C4⋊Q8×C73—21no—
55C4⋊Q8×C73D411yescharacteristic
56C4⋊Q8×C73D411yescharacteristic
57#125×C73—21no—
58C22⋊Q8×C73—21no—
59C22⋊Q8×C73—21no—
60C4×Q8×C73—21no—
61C4×Q8×C73—21no—
62#117×C73—21no—
63C2×C4⋊C4×C73—21no—
64C2×C4⋊C4×C73D412yes—
65C2×C42×C73C2311yescharacteristic
66#106×C73—24no—
67C4⋊C8×C73D412yes—
68#103×C73—24no—
69C22⋊C8×C73—21no—
70C4×C8×C73—21no—
71C2×Dic146—41no—
72C2×Dic146—41no—
73C2×Dic146—42no—
74C2×Dic146—42no—
75C22×Dic73—22no—
76C22×Dic73—22no—
77#11924012—42no—
78C2×#10596831—22no—
79C2×#10596831—41no—
80C4⋊Dic73—22no—
81C4⋊Dic73—22no—
82C4⋊Dic73—41no—
83C4⋊Dic73—41no—
84#11924007—416no—
85C4×Dic73—24no—
86C4×Dic73—24no—
87C4×Dic73—22no—
88C4×Dic73SD1614yes—
89C2×Q8×C73—21no—
90C2×Q8×C73—21no—
91C2×Q8×C73—41no—
92C2×Q8×C73—41no—
93C2×Q8×C73—42no—
94C2×Q8×C73—42no—
95C22×C4×C73—22no—
96C22×C4×C73C2×D411yescharacteristic
97C22×C4×C73C2×D411yescharacteristic
98C22×C4×C73C2×D411yescharacteristic
99C2×C8×C73—22no—
100C4⋊C4×C73—22no—
101C4⋊C4×C73—21no—
102C4⋊C4×C73—21no—
103C4⋊C4×C73—22no—
104C4⋊C4×C73—21no—
105C4⋊C4×C73—21no—
106C4⋊C4×C73—41no—
107C4⋊C4×C73—41no—
108C4⋊C4×C73—41no—
109C4⋊C4×C73—41no—
110C22⋊C4×C73—41no—
111C42×C73—41no—
112C42×C73C2×D411yescharacteristic
113C42×C73C2×D411yescharacteristic
114C42×C73C2×D411yescharacteristic
115C42×C73#4111yescharacteristic
116C23×C73#12011yescharacteristic
117Q8×C73—42no—
118Q8×C73—42no—
119Q8×C73—82no—
120Q8×C73—81no—
121Q8×C73—81no—
122C2×C4×C73—22no—
123C2×C4×C73—22no—
124C2×C4×C73—41no—
125C2×C4×C73—41no—
126C2×C4×C73—42no—
127C2×C4×C73—44no—
128C2×C4×C73#12011yescharacteristic
129C2×C4×C73#12011yescharacteristic
130C2×C4×C73#12111yescharacteristic
131C2×C4×C73#12111yescharacteristic
132C2×C4×C73#12111yescharacteristic
133C2×C4×C73#12711yescharacteristic
134C2×C4×C73C2×SD1612yes—
135C2×C4×C73AGL(1,ℤ/8)12yes—
136C2×Dic73—24no—
137C2×Dic73—24no—
138C2×Dic73—24no—
139C2×Dic73—42no—
140C2×Dic73—44no—
141Dic146—82no—
142Dic146—81no—
143Dic146—81no—
144C8×C73—42no—
145#10596831—42no—
146#10596831—81no—
147C22×C73—21no—
148C22×C73—21no—
149C22×C73#39211yescharacteristic
150C22×C73#45012yes—
151C4×C73—21no—
152C4×C73—21no—
153C4×C73—81no—
154C4×C73—81no—
155C4×C73—82no—
156C4×C73#45612yes—
157C4×C73#45612yes—
158C4×C73#46012yes—
159C4×C73#49211yescharacteristic
160C4×C73#49611yescharacteristic
161Dic73—28no—
162Dic73—82no—
163#10299—731nomaximal, Sylow
164C2×C73—22no—
165C2×C73#200511yescharacteristic
166C2×C73#200511yescharacteristic
167C2×C73#201411yescharacteristic
168#3229—731no—
169#3229D7311yescharacteristic
170#2868—731no—
171#1994—731no—
172#1546—731no—
173#1546—731no—
174#1276—731no—
175C73#1029911yescharacteristic, complemented, Sylow
176#533C2×D7311yescharacteristic
177C2×C4⋊Q8—731no—
178C2×C4⋊Q8—1462no—
179C2×C4⋊Q8C2×D7311yescharacteristic
180#498—21no—
181#498—1461no—
182#487—21no—
183#487—1461no—
184#483—22no—
185#483—1462no—
186#475—22no—
187#475—1462no—
188#431—731no—
189#431C2×D7311yescharacteristic
190#423—731no—
191#384—1462no—
192#384—1462no—
193C43—731no—
194#339—1462no—
195#332—1462no—
196#332—1462no—
197C22×Q8—21no—
198C22×Q8—1461no—
199C2×M16—1461no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—2924no—
205C4⋊Q8—2924no—
206C4⋊Q8—1461no—
207C4⋊Q8—1461no—
208C4⋊Q8—1461no—
209C4⋊Q8—1461no—
210C4⋊Q8C22⋊D7311yescharacteristic
211C4⋊Q8C22⋊D7311yescharacteristic
212#125—21no—
213C22⋊Q8—21no—
214C22⋊Q8—21no—
215C4×Q8—21no—
216C4×Q8—21no—
217#117—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—732no—
220C2×C4⋊C4—1468no—
221C2×C4⋊C4—1461no—
222C2×C4⋊C4C22⋊D7312yes—
223C2×C42—732no—
224C2×C42—1461no—
225C2×C42—1461no—
226C2×C42C22×D7311yescharacteristic
227#106—24no—
228#106—1464no—
229C4⋊C8—732no—
230C4⋊C8—1461no—
231C4⋊C8—1461no—
232C4⋊C8C22⋊D7312yes—
233#103—24no—
234#103—1464no—
235C22⋊C8—21no—
236C22⋊C8—1461no—
237C4×C8—21no—
238C4×C8—1461no—
239C2×Q8—21no—
240C2×Q8—21no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—2921no—
246C2×Q8—2921no—
247C2×Q8—2922no—
248C2×Q8—2922no—
249C22×C4—22no—
250C22×C4—1462no—
251C22×C4—1462no—
252C22×C4D4×D7311yescharacteristic
253C22×C4D4×D7311yescharacteristic
254C22×C4C2×C22⋊D7311yescharacteristic
255M16—2922no—
256C2×C8—22no—
257C2×C8—2921no—
258C2×C8—1462no—
259C4⋊C4—21no—
260C4⋊C4—22no—
261C4⋊C4—21no—
262C4⋊C4—22no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—29216no—
270C4⋊C4—2921no—
271C4⋊C4—2921no—
272C4⋊C4—1462no—
273C4⋊C4—1462no—
274C22⋊C4—41no—
275C42—41no—
276C42—734no—
277C42—1464no—
278C42—1464no—
279C42—1462no—
280C42D4×D7311yescharacteristic
281C42C2×C22⋊D7311yescharacteristic
282C42C2×C22⋊D7311yescharacteristic
283C42#1192402711yescharacteristic
284C23#1455655911yescharacteristic
285Q8—42no—
286Q8—42no—
287Q8—81no—
288Q8—81no—
289Q8—82no—
290Q8—5841no—
291Q8—5841no—
292Q8—5842no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—44no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—2924no—
300C2×C4—2922no—
301C2×C4—1464no—
302C2×C4—1464no—
303C2×C4—1464no—
304C2×C4#1455655911yescharacteristic
305C2×C4#1455656011yescharacteristic
306C2×C4#1455656211yescharacteristic
307C2×C4#1455656311yescharacteristic
308C2×C4#1455656311yescharacteristic
309C2×C4#1455658711yescharacteristic
310C2×C4SD16×D7312yes—
311C2×C4#1455661812yes—
312C8—42no—
313C8—2922no—
314C8—5841no—
315C22—21no—
316C22—21no—
317C22#1798849511yescharacteristic
318C22#1798873612yes—
319C4—21no—
320C4—21no—
321C4—81no—
322C4—81no—
323C4—82no—
324C4—5842no—
325C4—1468no—
326C4#1798876812yes—
327C4#1798877012yes—
328C4#1798878512yes—
329C4#1798895811yescharacteristic
330C4#1798897311yescharacteristic
331C2—22no—
332C2#2261096411yescharacteristic
333C2#2261096511yescharacteristic
334C2#2261101711yescharacteristic
subgroupG/H
centerC22#17988495
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#3229C2
socleC22×C73#392
prime radicalC23×C73#120
hypercenter#3229D73
hypocenterC73#10299
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

Complements

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

NH ≅ G/Ntype
175. C73#10299semidirect

Maximal subgroups

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

subgroupindex
#226210722
#226183272
#226109092
#226086032
#226086022
#226077632
C73×#32292
#1029973