Finite Group Databasebeta

← #32983958

Subgroups — #32983958

Order 18688.

522 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 107 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#22614248C211yesmaximal, characteristic
2#22610009C212yesmaximal, complemented
3#22608269C211yesmaximal, characteristic
4#22607836C212yesmaximal, complemented
5C73×#2975C211yesmaximal, characteristic
6C22×C4⋊Dic73C2211yescharacteristic
7#17988551C2212yes—
8#17988526—21no—
9#17988523—21no—
10#17988485—21no—
11#17988467C412yes—
12#17988459C412yes—
13#17988447—21no—
14#17988447C2211yescharacteristic
15#17988355—24no—
16#17988313—21no—
17#17988311—21no—
18C2×C4×D4×C73C2211yescharacteristic
19#436×C73—24no—
20#431×C73—21no—
21#407×C73—21no—
22#407×C73C2212yes—
23C23×Dic73C2×C412yes—
24C2×#11924012—22no—
25C2×C4⋊Dic73—22no—
26C2×C4⋊Dic73—21no—
27C2×C4⋊Dic73—21no—
28C2×C4⋊Dic73—21no—
29C2×C4⋊Dic73—22no—
30C2×#11924007—22no—
31C2×C22⋊Dic73—22no—
32C2×C22⋊Dic73C2×C412yes—
33C22⋊#10596831—22no—
34C22⋊#10596831—22no—
35#14556498—22no—
36#14556498—22no—
37#14556498—21no—
38C22×D4×C73C2×C411yescharacteristic
39C23×C4×C73D411yescharacteristic
40C23×C4×C73C2311yescharacteristic
41C2×M16×C73—22no—
42C2×M16×C73D414yes—
43C4×D4×C73—22no—
44C4×D4×C73—22no—
45C2×C4⋊C4×C73D411yescharacteristic
46C2×C22⋊C4×C73C2×C411yescharacteristic
47C2×C22⋊C4×C73D411yescharacteristic
48C2×C42×C73D411yescharacteristic
49C4⋊C8×C73—22no—
50C22⋊C8×C73—22no—
51C22⋊C8×C73—22no—
52C22⋊C8×C73—22no—
53C4×C8×C73—22no—
54C22×Dic73—22no—
55C22×Dic73—22no—
56C22×Dic73—24no—
57C22×Dic73—82no—
58#11924012—44no—
59C2×#10596831—42no—
60C4⋊Dic73—42no—
61C4⋊Dic73—41no—
62C4⋊Dic73—42no—
63C4⋊Dic73—41no—
64#11924007—82no—
65C22⋊Dic73—42no—
66C22⋊Dic73—82no—
67C24×C73C22⋊C411yescharacteristic
68C24×C73C22×C411yescharacteristic
69C2×D4×C73—22no—
70C2×D4×C73—22no—
71C2×D4×C73—41no—
72C22×C4×C73—21no—
73C22×C4×C73—21no—
74C22×C4×C73—42no—
75C22×C4×C73—42no—
76C22×C4×C73C22⋊C411yescharacteristic
77C22×C4×C73C22⋊C411yescharacteristic
78C22×C4×C73C22⋊C411yescharacteristic
79C22×C4×C73C2×D412yes—
80C22×C4×C73C2×D411yescharacteristic
81C22×C4×C73C2×D411yescharacteristic
82C22×C4×C73#4112yes—
83M16×C73—44no—
84M16×C73—82no—
85C2×C8×C73—24no—
86C4⋊C4×C73—22no—
87C22⋊C4×C73—22no—
88C22⋊C4×C73—22no—
89C42×C73—21no—
90C42×C73—21no—
91C23×C73—21no—
92C23×C73—21no—
93C23×C73—21no—
94C23×C73—21no—
95C23×C73—21no—
96C23×C73—41no—
97C23×C73—41no—
98C23×C73—41no—
99C23×C73—41no—
100C23×C73C2×C22⋊C411yescharacteristic
101C23×C73C4×D412yes—
102D4×C73—44no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—24no—
106C2×C4×C73—22no—
107C2×C4×C73—21no—
108C2×C4×C73—21no—
109C2×C4×C73—41no—
110C2×C4×C73—42no—
111C2×C4×C73—42no—
112C2×C4×C73—41no—
113C2×C4×C73—41no—
114C2×C4×C73—81no—
115C2×C4×C73#10112yes—
116C2×C4×C73#12011yescharacteristic
117C2×C4×C73#12112yes—
118C2×C4×C73#12311yescharacteristic
119C2×Dic73—42no—
120C2×Dic73—42no—
121C2×Dic73—42no—
122C2×Dic73—42no—
123C2×Dic73—42no—
124C2×Dic73—42no—
125C2×Dic73—84no—
126C8×C73—44no—
127#10596831—82no—
128C22×C73—21no—
129C22×C73—21no—
130C22×C73—22no—
131C22×C73—22no—
132C22×C73—22no—
133C22×C73—42no—
134C22×C73—42no—
135C22×C73—42no—
136C22×C73—41no—
137C22×C73—41no—
138C22×C73—41no—
139C22×C73—81no—
140C22×C73#38611yescharacteristic
141C22×C73C2×#10111yescharacteristic
142C22×C73#41311yescharacteristic
143C4×C73—22no—
144C4×C73—21no—
145C4×C73—22no—
146C4×C73—41no—
147C4×C73—41no—
148C4×C73#45812yes—
149Dic73—82no—
150Dic73—82no—
151#9510—731nomaximal, Sylow
152C2×C73—21no—
153C2×C73—21no—
154C2×C73—42no—
155C2×C73—81no—
156C2×C73#189011yescharacteristic
157C2×C73#190111yescharacteristic
158C2×C73#191411yescharacteristic
159#2975D7311yescharacteristic
160#2487—731no—
161#1821—732no—
162#1465—731no—
163#1305—732no—
164C73#951011yescharacteristic, complemented, Sylow
165C2×C4×D4C2×D7311yescharacteristic
166C22×C4⋊C4—731no—
167#436—24no—
168#431—21no—
169#407—21no—
170#407—732no—
171#407C2×D7312yes—
172#400—1461no—
173#399—1461no—
174#389—1461no—
175#386—732no—
176#385—732no—
177#380—731no—
178#380—1461no—
179#343—1464no—
180#324—1461no—
181#323—1461no—
182C22×D4C4×D7311yescharacteristic
183C23×C4—732no—
184C23×C4C22⋊D7311yescharacteristic
185C23×C4C22×D7311yescharacteristic
186C2×M16—22no—
187C2×M16—1462no—
188C2×M16C22⋊D7314yes—
189C4×D4—22no—
190C4×D4—22no—
191C2×C4⋊C4—1462no—
192C2×C4⋊C4—1461no—
193C2×C4⋊C4—1461no—
194C2×C4⋊C4—1461no—
195C2×C4⋊C4—1462no—
196C2×C4⋊C4—1462no—
197C2×C4⋊C4D29211yescharacteristic
198C2×C22⋊C4—732no—
199C2×C22⋊C4—1462no—
200C2×C22⋊C4C4×D7311yescharacteristic
201C2×C22⋊C4D29211yescharacteristic
202C2×C42C22⋊D7311yescharacteristic
203C4⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—1462no—
208C22⋊C8—1462no—
209C4×C8—22no—
210#95—1462no—
211#95—1462no—
212#95—1461no—
213C24#1192400511yescharacteristic
214C24C2×C4×D7311yescharacteristic
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—41no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—5842no—
223C22×C4—1462no—
224C22×C4—1462no—
225C22×C4—1464no—
226C22×C4#1192400511yescharacteristic
227C22×C4#1192400511yescharacteristic
228C22×C4#1192400511yescharacteristic
229C22×C4C2×C22⋊D7312yes—
230C22×C4C2×C22⋊D7311yescharacteristic
231C22×C4C2×D29211yescharacteristic
232C22×C4#1192402812yes—
233M16—44no—
234M16—82no—
235M16—2924no—
236C2×C8—24no—
237C2×C8—2922no—
238C4⋊C4—22no—
239C4⋊C4—2922no—
240C4⋊C4—2921no—
241C4⋊C4—2922no—
242C4⋊C4—2921no—
243C4⋊C4—5842no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—2922no—
247C22⋊C4—5842no—
248C42—21no—
249C42—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23C2×#1192400511yescharacteristic
260C23C4×C22⋊D7312yes—
261D4—44no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—81no—
274C2×C4—2922no—
275C2×C4—2922no—
276C2×C4—2922no—
277C2×C4—2922no—
278C2×C4—2922no—
279C2×C4—2922no—
280C2×C4—5844no—
281C2×C4#1455650912yes—
282C2×C4#1455656011yescharacteristic
283C2×C4#1455656512yes—
284C2×C4#1455657411yescharacteristic
285C8—44no—
286C8—5842no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—81no—
299C22#1798846611yescharacteristic
300C22C2×#1455650911yescharacteristic
301C22#1798856911yescharacteristic
302C4—22no—
303C4—21no—
304C4—22no—
305C4—41no—
306C4—41no—
307C4—5842no—
308C4—5842no—
309C4#1798877712yes—
310C2—21no—
311C2—21no—
312C2—42no—
313C2—81no—
314C2#2261034411yescharacteristic
315C2#2261039911yescharacteristic
316C2#2261047811yescharacteristic
subgroupG/H
centerC22#17988466
commutatorC24×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#2975C2
socleC22×C73#386
prime radicalC22×D4×C73C2×C4
cosocleC23×C4×C73C23
hypercenter#2975D73
hypocenterC73#9510
LCSG ▷ C24×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2975

Complements

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

NH ≅ G/Ntype
2. #22610009C2semidirect
4. #22607836C2semidirect
164. C73#9510semidirect

Maximal subgroups

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

subgroupindex
#226142482
#226100092
#226082692
#226078362
C73×#29752
#951073