Finite Group Databasebeta

← #33001990

Subgroups — #33001990

Order 18688.

480 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 93 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#22612645C211yesmaximal, characteristic
2C2×#17988383C212yesmaximal
3#22609925C212yesmaximal, complemented
4#22608058C211yesmaximal, characteristic, complemented
5C73×#1394C211yesmaximal, characteristic, complemented
6C2×#14556580C2211yescharacteristic
7C2×Q8×Dic73C2211yescharacteristic
8C2×#14556553C411yescharacteristic
9#17988610—22no—
10C2×#14556510C2212yes—
11#17988489C411yescharacteristic
12#17988476—22no—
13#17988461—22no—
14#17988456C411yescharacteristic
15#17988433C411yescharacteristic
16#17988383—24no—
17#17988355—22no—
18C2×#101×C73C2212yes—
19#391×C73C2211yescharacteristic
20C22×D292C2×C411yescharacteristic
21C22×C4×D73—21no—
22C2×#11924012—22no—
23#14556580—22no—
24#14556580D412yes—
25#14556580Q812yes—
26Q8×Dic73—42no—
27#14556553—42no—
28C2×C4⋊Dic73C2×C411yescharacteristic
29C2×C4⋊Dic73D412yes—
30C2×#11924007—22no—
31C2×#11924005—21no—
32C2×#11924005—22no—
33C2×#11924005—22no—
34C2×C4×Dic73C2×C411yescharacteristic
35C2×C4×Dic73C2×C411yescharacteristic
36C2×C4×Dic73D412yes—
37#14556520—44no—
38#14556510—22no—
39#14556510—22no—
40#14556498—21no—
41#14556498—21no—
42C22×Q8×C73C2311yescharacteristic
43C2×M16×C73—22no—
44C2×C4⋊C4×C73C2×C411yescharacteristic
45C2×C42×C73C2×C411yescharacteristic
46#101×C73—22no—
47#101×C73—22no—
48#95×C73—22no—
49C23×D73—21no—
50C2×D292—21no—
51C2×D292—41no—
52C2×D292—41no—
53C2×D292C4⋊C412yes—
54C22×Dic73—21no—
55C22×Dic73C22⋊C411yescharacteristic
56C22×Dic73C22⋊C412yes—
57C22×Dic73C22⋊C411yescharacteristic
58C2×C4×D73—42no—
59C2×C4×D73—82no—
60#11924012—42no—
61#11924012—42no—
62C2×#10596831—42no—
63C4⋊Dic73—41no—
64C4⋊Dic73—42no—
65#11924007—82no—
66#11924005—24no—
67#11924005—24no—
68#11924005—42no—
69#11924005—82no—
70C4×Dic73—21no—
71C4×Dic73—24no—
72C4×Dic73—22no—
73C4×Dic73C4⋊C412yes—
74C2×Q8×C73—41no—
75C2×Q8×C73—41no—
76C2×Q8×C73C2×D411yescharacteristic
77C2×Q8×C73C2×Q811yescharacteristic
78C2×Q8×C73#4112yes—
79C22×C4×C73—21no—
80C22×C4×C73—21no—
81C22×C4×C73C22×C411yescharacteristic
82C22×C4×C73C2×D412yes—
83M16×C73—42no—
84M16×C73—42no—
85C2×C8×C73—42no—
86C4⋊C4×C73—41no—
87C42×C73—41no—
88C23×C73C2×C22⋊C411yescharacteristic
89C22×D73—22no—
90C22×D73—42no—
91C22×D73—42no—
92C22×D73—41no—
93C22×D73—41no—
94Q8×C73—41no—
95Q8×C73—41no—
96Q8×C73—42no—
97C2×C4×C73—21no—
98C2×C4×C73—22no—
99C2×C4×C73—21no—
100C2×C4×C73—22no—
101C2×C4×C73—41no—
102C2×C4×C73—41no—
103C2×C4×C73—42no—
104C2×C4×C73—41no—
105C2×C4×C73—41no—
106C2×C4×C73C2×C4⋊C411yescharacteristic
107C2×C4×C73#11711yescharacteristic
108C2×C4×C73#12112yes—
109C2×C4×C73C22⋊Q812yes—
110C2×Dic73—22no—
111C2×Dic73—21no—
112C2×Dic73—22no—
113C2×Dic73—41no—
114C2×Dic73—41no—
115C2×Dic73—41no—
116C2×Dic73—42no—
117C2×Dic73—41no—
118C2×Dic73#9912yes—
119D292—42no—
120D292—42no—
121C4×D73—84no—
122C8×C73—82no—
123#10596831—82no—
124C22×C73—21no—
125C22×C73—21no—
126C22×C73#38011yescharacteristic
127C22×C73C2×#9911yescharacteristic
128C22×C73#41011yescharacteristic
129C2×D73—44no—
130C2×D73—42no—
131C2×D73—82no—
132C2×D73—81no—
133C2×D73—81no—
134C4×C73—22no—
135C4×C73—21no—
136C4×C73—21no—
137C4×C73—22no—
138C4×C73—81no—
139C4×C73—81no—
140Dic73—22no—
141Dic73—42no—
142Dic73—41no—
143Dic73—42no—
144#12803—731nomaximal, Sylow
145C2×C73—21no—
146C2×C73—21no—
147C2×C73#179911yescharacteristic
148C2×C73#213311yescharacteristic
149C2×C73#213411yescharacteristic
150D73—82no—
151#2326—731no—
152C2×#355—732no—
153#1801—732no—
154#1394—731no—
155#1394D7311yescharacteristic, complemented
156C73#1280311yescharacteristic, complemented, Sylow
157C2×#124—731no—
158C2×C4×Q8—731no—
159C2×C4×D4—731no—
160#423—1462no—
161C2×#101—732no—
162C2×#101C2×D7312yes—
163#391C2×D7311yescharacteristic
164#390—731no—
165#388—1462no—
166#385—1462no—
167#384—731no—
168#376—731no—
169#355—1464no—
170#343—1462no—
171C22×Q8C22×D7311yescharacteristic
172C22×D4—731no—
173C23×C4—1461no—
174C2×M16—22no—
175C2×M16—1462no—
176#124—732no—
177#124—732no—
178#124—1462no—
179C4×Q8—2922no—
180C4×D4—2922no—
181C2×C4⋊C4—731no—
182C2×C4⋊C4—732no—
183C2×C4⋊C4—1462no—
184C2×C4⋊C4C4×D7311yescharacteristic
185C2×C22⋊C4—1461no—
186C2×C22⋊C4—1462no—
187C2×C22⋊C4—1462no—
188C2×C42—731no—
189C2×C42—731no—
190C2×C42—732no—
191C2×C42C4×D7311yescharacteristic
192C4⋊C8—2924no—
193#101—22no—
194#101—22no—
195#101—1462no—
196#101—1462no—
197#95—22no—
198#95—1461no—
199#95—1461no—
200C24—1461no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8D4×D7311yescharacteristic
204C2×Q8Q8×D7311yescharacteristic
205C2×Q8#1192402812yes—
206C2×D4—2921no—
207C2×D4—2921no—
208C2×D4—732no—
209C2×D4—1461no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—2922no—
213C22×C4—5842no—
214C22×C4—731no—
215C22×C4—731no—
216C22×C4—732no—
217C22×C4—1461no—
218C22×C4C2×C4×D7311yescharacteristic
219C22×C4D4×D7312yes—
220M16—42no—
221M16—42no—
222M16—2922no—
223M16—2922no—
224C2×C8—42no—
225C2×C8—2922no—
226C4⋊C4—41no—
227C4⋊C4—2921no—
228C4⋊C4—2922no—
229C4⋊C4—5842no—
230C22⋊C4—2922no—
231C22⋊C4—5842no—
232C22⋊C4—1464no—
233C22⋊C4—1464no—
234C42—41no—
235C42—732no—
236C42—1464no—
237C42—1462no—
238C42—1461no—
239C23—2922no—
240C23—2922no—
241C23—2921no—
242C23—2921no—
243C23—1462no—
244C23C22⋊C4×D7311yescharacteristic
245Q8—41no—
246Q8—41no—
247Q8—42no—
248D4—2922no—
249D4—2922no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—2921no—
260C2×C4—2921no—
261C2×C4—2922no—
262C2×C4—2921no—
263C2×C4—2921no—
264C2×C4—5844no—
265C2×C4—732no—
266C2×C4—1462no—
267C2×C4—1461no—
268C2×C4—1462no—
269C2×C4C4⋊C4×D7311yescharacteristic
270C2×C4#1455654811yescharacteristic
271C2×C4#1455656412yes—
272C2×C4#1455656912yes—
273C8—82no—
274C8—5842no—
275C22—21no—
276C22—21no—
277C22—2924no—
278C22—2922no—
279C22—5842no—
280C22—5841no—
281C22—5841no—
282C22#1798844611yescharacteristic
283C22#99×D7311yescharacteristic
284C22#1798855911yescharacteristic
285C4—22no—
286C4—21no—
287C4—21no—
288C4—22no—
289C4—81no—
290C4—81no—
291C4—2922no—
292C4—2921no—
293C4—2922no—
294C4—1462no—
295C2—21no—
296C2—21no—
297C2—5842no—
298C2#2260991811yescharacteristic
299C2#2261161911yescharacteristic
300C2#2261162111yescharacteristic
subgroupG/H
centerC22#99×D73
commutatorC22×C4×C73C22×C4
FrattiniC22×Q8C22×D73
FittingC73×#1394C2
socleC22×C73C2×#99
prime radicalC22×D292C2×C4
cosocleC22×Q8×C73C23
hypercenter#1394D73
hypocenterC73#12803
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
3. #22609925C2semidirect
4. #22608058C2semidirect
5. C73×#1394C2semidirect
155. #1394D73semidirect
156. C73#12803semidirect

Maximal subgroups

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

subgroupindex
#226126452
C2×#179883832
#226099252
#226080582
C73×#13942
#1280373