Finite Group Databasebeta

← #32982032

Subgroups — #32982032

Order 18688.

560 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 131 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#22615744C211yesmaximal, characteristic
2#22608502C212yesmaximal
3C2×#17988313C211yesmaximal, characteristic
4#22607798C212yesmaximal
5C73×#2316C211yesmaximal, characteristic
6C2×#14556570C2211yescharacteristic
7C2×C22⋊#10596831C2212yes—
8#17988526C412yescomplemented
9#17988520—22no—
10#17988507C412yescomplemented
11#17988460—21no—
12#17988450—21no—
13#17988447C2211yescharacteristic
14#17988323—24no—
15#17988313—22no—
16C2×#117×C73—21no—
17C22×C4⋊C4×C73C2211yescharacteristic
18#389×C73—24no—
19#380×C73—21no—
20#380×C73C2212yes—
21C22×Dic146C2×C411yescharacteristic
22C22×#10596831—22no—
23#14556570—22no—
24#14556570—22no—
25C2×C4⋊Dic73—21no—
26C2×C4⋊Dic73C2×C411yescharacteristic
27C2×#11924007—22no—
28C2×#11924007—21no—
29C2×C22⋊Dic73—21no—
30C2×C22⋊Dic73—21no—
31C22⋊#10596831—22no—
32C22⋊#10596831—22no—
33#14556498—22no—
34#14556498—22no—
35#14556498—22no—
36#14556498C2×C412yes—
37C23×C4×C73D411yescharacteristic
38C23×C4×C73D411yescharacteristic
39C23×C4×C73C2311yescharacteristic
40#117×C73—42no—
41#117×C73—42no—
42C2×C4⋊C4×C73—22no—
43C2×C4⋊C4×C73—22no—
44C2×C4⋊C4×C73—22no—
45C2×C4⋊C4×C73—22no—
46C2×C4⋊C4×C73—22no—
47C2×C4⋊C4×C73C2×C412yes—
48C2×C4⋊C4×C73D412yes—
49C2×C22⋊C4×C73—22no—
50C2×C42×C73—22no—
51#95×C73—22no—
52#95×C73D414yes—
53C2×Dic146—22no—
54C2×Dic146—81no—
55C22×Dic73—21no—
56C22×Dic73—22no—
57C22×Dic73—21no—
58C2×#10596831—42no—
59C2×#10596831—82no—
60C4⋊Dic73—22no—
61C4⋊Dic73—81no—
62#11924007—42no—
63#11924007—82no—
64C22⋊Dic73—41no—
65C22⋊Dic73—41no—
66C22⋊Dic73—41no—
67C22⋊Dic73—41no—
68C24×C73C2×D411yescharacteristic
69C22×C4×C73—21no—
70C22×C4×C73—21no—
71C22×C4×C73—21no—
72C22×C4×C73—21no—
73C22×C4×C73—21no—
74C22×C4×C73—21no—
75C22×C4×C73—24no—
76C22×C4×C73—81no—
77C22×C4×C73—81no—
78C22×C4×C73C22⋊C412yes—
79C22×C4×C73C22⋊C412yes—
80C22×C4×C73SD1612yes—
81C22×C4×C73Q1612yes—
82C22×C4×C73C22×C411yescharacteristic
83C22×C4×C73C2×D412yes—
84C22×C4×C73C2×D411yescharacteristic
85C22×C4×C73#4112yes—
86C4⋊C4×C73—24no—
87C4⋊C4×C73—24no—
88C4⋊C4×C73—44no—
89C4⋊C4×C73—82no—
90C4⋊C4×C73—82no—
91C22⋊C4×C73—42no—
92C22⋊C4×C73—42no—
93C42×C73—44no—
94C23×C73—41no—
95C23×C73—41no—
96C23×C73C2×C22⋊C411yescharacteristic
97C23×C73#12011yescharacteristic
98C23×C73#12311yescharacteristic
99C23×C73C2×SD1611yescharacteristic
100C23×C73C2×Q1611yescharacteristic
101C23×C73AGL(1,ℤ/8)11yescharacteristic
102C23×C73#13711yescharacteristic
103C2×C4×C73—22no—
104C2×C4×C73—22no—
105C2×C4×C73—22no—
106C2×C4×C73—22no—
107C2×C4×C73—22no—
108C2×C4×C73—22no—
109C2×C4×C73—22no—
110C2×C4×C73—42no—
111C2×C4×C73—41no—
112C2×C4×C73—41no—
113C2×C4×C73—42no—
114C2×C4×C73—42no—
115C2×C4×C73—42no—
116C2×C4×C73—44no—
117C2×C4×C73—44no—
118C2×C4×C73—44no—
119C2×C4×C73—81no—
120C2×C4×C73#10314yes—
121C2×C4×C73C4×D412yes—
122C2×C4×C73#12112yes—
123C2×Dic73—42no—
124C2×Dic73—42no—
125C2×Dic73—42no—
126C2×Dic73—42no—
127C2×Dic73—41no—
128C2×Dic73—41no—
129C2×Dic73—42no—
130Dic146—82no—
131#10596831—82no—
132C22×C73—21no—
133C22×C73—21no—
134C22×C73—21no—
135C22×C73—21no—
136C22×C73—21no—
137C22×C73—21no—
138C22×C73—41no—
139C22×C73—41no—
140C22×C73—41no—
141C22×C73—41no—
142C22×C73#38611yescharacteristic
143C22×C73C2×#10311yescharacteristic
144C22×C73#41711yescharacteristic
145C22×C73#45011yescharacteristic
146C22×C73#45111yescharacteristic
147C22×C73#48311yescharacteristic
148C22×C73#48411yescharacteristic
149C4×C73—24no—
150C4×C73—22no—
151C4×C73—22no—
152C4×C73—42no—
153C4×C73—84no—
154Dic73—82no—
155Dic73—81no—
156Dic73—81no—
157#9200—731nomaximal, Sylow
158C2×C73—21no—
159C2×C73—21no—
160C2×C73—21no—
161C2×C73—21no—
162C2×C73—21no—
163C2×C73—21no—
164C2×C73#187211yescharacteristic
165C2×C73#191211yescharacteristic
166C2×C73#191311yescharacteristic
167#2627—731no—
168#2316D7311yescharacteristic
169#1520—732no—
170C2×#324—731no—
171#1289—732no—
172C73#920011yescharacteristic, complemented, Sylow
173C2×C22⋊Q8—731no—
174C2×#117—21no—
175C22×C4⋊C4C2×D7311yescharacteristic
176C2×C22⋊C8—732no—
177#400—732no—
178#398—1462no—
179#395—732no—
180#389—24no—
181#385—1461no—
182#381—1461no—
183#380—21no—
184#380—731no—
185#380C2×D7312yes—
186#328—1464no—
187#324—1462no—
188C22×Q8—731no—
189C23×C4C22⋊D7311yescharacteristic
190C23×C4C22⋊D7311yescharacteristic
191C23×C4C22×D7311yescharacteristic
192C22×C8—1462no—
193C22⋊Q8—1462no—
194C22⋊Q8—1462no—
195#117—42no—
196#117—42no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—731no—
203C2×C4⋊C4—1461no—
204C2×C4⋊C4—1462no—
205C2×C4⋊C4—1461no—
206C2×C4⋊C4C4×D7312yes—
207C2×C4⋊C4D29212yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—1461no—
210C2×C22⋊C4—1461no—
211C2×C42—22no—
212C22⋊C8—1462no—
213C22⋊C8—1462no—
214#95—22no—
215#95—732no—
216#95—1462no—
217#95—1462no—
218#95—1462no—
219#95C22⋊D7314yes—
220C24C2×C22⋊D7311yescharacteristic
221C2×Q8—5841no—
222C2×Q8—1462no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—24no—
230C22×C4—81no—
231C22×C4—81no—
232C22×C4—1462no—
233C22×C4—1461no—
234C22×C4—1461no—
235C22×C4#1192400512yes—
236C22×C4#1192400512yes—
237C22×C4#1192401612yes—
238C22×C4#1192401812yes—
239C22×C4C2×C4×D7311yescharacteristic
240C22×C4C2×C22⋊D7312yes—
241C22×C4C2×D29211yescharacteristic
242C22×C4#1192402812yes—
243C2×C8—2922no—
244C2×C8—5842no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—44no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—2922no—
251C4⋊C4—5842no—
252C4⋊C4—5841no—
253C4⋊C4—1462no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—2921no—
257C22⋊C4—2921no—
258C22⋊C4—2921no—
259C22⋊C4—2921no—
260C42—44no—
261C23—41no—
262C23—41no—
263C23C2×#1192400511yescharacteristic
264C23#1455656011yescharacteristic
265C23#1455657411yescharacteristic
266C23C2×#1192401611yescharacteristic
267C23C2×#1192401811yescharacteristic
268C23#1455661711yescharacteristic
269C23#1455662211yescharacteristic
270Q8—5842no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—81no—
288C2×C4—2922no—
289C2×C4—2922no—
290C2×C4—2922no—
291C2×C4—2922no—
292C2×C4—2921no—
293C2×C4—2921no—
294C2×C4—2922no—
295C2×C4#1455651414yes—
296C2×C4C4×C22⋊D7312yes—
297C2×C4#1455656512yes—
298C8—5842no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22#1798846611yescharacteristic
310C22C2×#1455651411yescharacteristic
311C22#1798858611yescharacteristic
312C22#1798873811yescharacteristic
313C22#1798874311yescharacteristic
314C22#1798891711yescharacteristic
315C22#1798892211yescharacteristic
316C4—22no—
317C4—22no—
318C4—24no—
319C4—42no—
320C4—84no—
321C4—5842no—
322C4—5841no—
323C4—5841no—
324C2—21no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2—21no—
329C2—21no—
330C2#2261023811yescharacteristic
331C2#2261046811yescharacteristic
332C2#2261047311yescharacteristic
subgroupG/H
centerC22#17988466
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#2316C2
socleC22×C73#386
prime radicalC24×C73C2×D4
cosocleC23×C4×C73C23
hypercenter#2316D73
hypocenterC73#9200
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2316

Complements

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

NH ≅ G/Ntype
8. #17988526C4semidirect
10. #17988507C4semidirect
172. C73#9200semidirect

Maximal subgroups

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

subgroupindex
#226157442
#226085022
C2×#179883132
#226077982
C73×#23162
#920073