Finite Group Databasebeta

← #32978746

Subgroups — #32978746

Order 18688.

540 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 117 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#22612470C211yesmaximal, characteristic
2#22609785C212yesmaximal, complemented
3C2×#17988326C212yesmaximal
4#22607812C211yesmaximal, characteristic, complemented
5C73×#1295C211yesmaximal, characteristic, complemented
6C2×#14556579C2211yescharacteristic
7C2×#14556556C411yescharacteristic
8C2×#14556553C411yescharacteristic
9C2×#14556520—22no—
10C2×#14556502C2212yes—
11#17988492C411yescharacteristic
12#17988488C411yescharacteristic
13#17988476—22no—
14#17988434—22no—
15C42×Dic73C2211yescharacteristic
16#17988341—22no—
17#17988326—24no—
18C2×#97×C73C2212yes—
19#382×C73C2211yescharacteristic
20C22×Dic146C2×C411yescharacteristic
21C22×D292C2×C411yescharacteristic
22C22×C4×D73—21no—
23C22×#10596831—22no—
24#14556579—22no—
25#14556579D412yes—
26#14556579Q812yes—
27#14556556—42no—
28#14556553—42no—
29C2×#11924007—21no—
30C2×#11924005—22no—
31C2×#11924005—21no—
32C2×#11924005—22no—
33C2×C4×Dic73—21no—
34C2×C4×Dic73C2×C411yescharacteristic
35C2×C4×Dic73C2×C411yescharacteristic
36C2×C4×Dic73D412yes—
37C2×C4×Dic73D412yes—
38#14556520—28no—
39#14556502—22no—
40#14556502—22no—
41#14556498—22no—
42C22×C8×C73—22no—
43C2×C4⋊C4×C73C2×C411yescharacteristic
44C2×C4⋊C4×C73C2×C411yescharacteristic
45C2×C42×C73C2311yescharacteristic
46#97×C73—22no—
47#97×C73—22no—
48#95×C73—22no—
49C23×D73—21no—
50C2×Dic146—21no—
51C2×Dic146—41no—
52C2×Dic146—41no—
53C2×Dic146C4⋊C412yes—
54C2×D292—21no—
55C2×D292—41no—
56C2×D292—41no—
57C2×D292C4⋊C412yes—
58C22×Dic73—21no—
59C22×Dic73C22⋊C412yes—
60C22×Dic73C22⋊C411yescharacteristic
61C22×Dic73C22⋊C411yescharacteristic
62C2×C4×D73—42no—
63C2×C4×D73—44no—
64C2×#10596831—24no—
65C2×#10596831—42no—
66C2×#10596831—42no—
67#11924007—42no—
68#11924005—24no—
69#11924005—24no—
70#11924005—44no—
71#11924005—42no—
72C4×Dic73—22no—
73C4×Dic73—24no—
74C4×Dic73—41no—
75C4×Dic73—42no—
76C4×Dic73—42no—
77C22×C4×C73—21no—
78C22×C4×C73—21no—
79C22×C4×C73C22×C411yescharacteristic
80C22×C4×C73C2×D412yes—
81C2×C8×C73—24no—
82C2×C8×C73—42no—
83C2×C8×C73—42no—
84C4⋊C4×C73—41no—
85C4⋊C4×C73—41no—
86C42×C73C2×D411yescharacteristic
87C42×C73C2×Q811yescharacteristic
88C42×C73#4112yes—
89C23×C73C2×C22⋊C411yescharacteristic
90C22×D73—22no—
91C22×D73—41no—
92C22×D73—41no—
93C22×D73—42no—
94C22×D73—42no—
95C2×C4×C73—21no—
96C2×C4×C73—21no—
97C2×C4×C73—22no—
98C2×C4×C73—22no—
99C2×C4×C73—41no—
100C2×C4×C73—41no—
101C2×C4×C73—42no—
102C2×C4×C73—42no—
103C2×C4×C73C2×C4⋊C411yescharacteristic
104C2×C4×C73#11711yescharacteristic
105C2×C4×C73#12112yes—
106C2×C4×C73C22⋊Q812yes—
107C2×Dic73—22no—
108C2×Dic73—21no—
109C2×Dic73—22no—
110C2×Dic73—22no—
111C2×Dic73—22no—
112C2×Dic73—22no—
113C2×Dic73—41no—
114C2×Dic73—41no—
115C2×Dic73#10012yes—
116C2×Dic73#10414yes—
117D292—42no—
118D292—42no—
119C4×D73—84no—
120Dic146—42no—
121Dic146—42no—
122C8×C73—44no—
123#10596831—44no—
124C22×C73#38011yescharacteristic
125C22×C73C2×#10011yescharacteristic
126C22×C73#41311yescharacteristic
127C22×C73C2×#10412yes—
128C22×C73#42112yes—
129C2×D73—42no—
130C2×D73—44no—
131C2×D73—81no—
132C2×D73—81no—
133C2×D73—82no—
134C4×C73—21no—
135C4×C73—21no—
136C4×C73—22no—
137C4×C73—22no—
138C4×C73—81no—
139C4×C73—81no—
140Dic73—41no—
141Dic73—42no—
142Dic73—42no—
143Dic73#32914yescomplemented
144#8579—731nomaximal, Sylow
145C2×C73C2×#32911yescharacteristic
146C2×C73#152912yes—
147C2×C73#153611yescharacteristic
148C2×C73#180011yescharacteristic
149C2×C73#180612yes—
150D73—82no—
151C4×#124—731no—
152#1767—732no—
153C2×#329—732no—
154#1295—731no—
155#1295D7311yescharacteristic, complemented
156C73#857911yescharacteristic, complemented, Sylow
157C2×#124—731no—
158C2×C4×Q8—731no—
159C2×C4×D4—731no—
160C2×C4⋊C8—1462no—
161C2×#97—732no—
162C2×#97C2×D7312yes—
163#391—731no—
164#390—731no—
165#388—1462no—
166#382C2×D7311yescharacteristic
167C4×C22⋊C4—1462no—
168C43—731no—
169#336—1462no—
170#329—1464no—
171C22×Q8—731no—
172C22×D4—731no—
173C23×C4—1461no—
174C22×C8—22no—
175C22×C8—1462no—
176#124—732no—
177#124—732no—
178#124—1462no—
179C4×Q8—2922no—
180C4×D4—2922no—
181C2×C4⋊C4—1461no—
182C2×C4⋊C4C4×D7311yescharacteristic
183C2×C4⋊C4C4×D7311yescharacteristic
184C2×C22⋊C4—1461no—
185C2×C22⋊C4—1462no—
186C2×C22⋊C4—1462no—
187C2×C42—731no—
188C2×C42—732no—
189C2×C42—731no—
190C2×C42—732no—
191C2×C42—1461no—
192C2×C42C22×D7311yescharacteristic
193C4⋊C8—1468no—
194#97—22no—
195#97—22no—
196#97—1462no—
197#97—1462no—
198#95—22no—
199#95—1462no—
200C24—1461no—
201C2×Q8—2921no—
202C2×Q8—2921no—
203C2×Q8—732no—
204C2×Q8—1461no—
205C2×D4—2921no—
206C2×D4—2921no—
207C2×D4—732no—
208C2×D4—1461no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—2922no—
212C22×C4—2924no—
213C22×C4—731no—
214C22×C4—732no—
215C22×C4—731no—
216C22×C4—1461no—
217C22×C4C2×C4×D7311yescharacteristic
218C22×C4D4×D7312yes—
219C2×C8—24no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—2922no—
223C2×C8—2922no—
224C2×C8—1464no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—2922no—
228C22⋊C4—2924no—
229C22⋊C4—2922no—
230C22⋊C4—1464no—
231C22⋊C4—1464no—
232C42—2921no—
233C42—2922no—
234C42—2922no—
235C42—1464no—
236C42—1462no—
237C42D4×D7311yescharacteristic
238C42Q8×D7311yescharacteristic
239C42#1192402812yes—
240C23—2921no—
241C23—2921no—
242C23—2922no—
243C23—2922no—
244C23—1462no—
245C23C22⋊C4×D7311yescharacteristic
246Q8—2922no—
247Q8—2922no—
248D4—2922no—
249D4—2922no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—2921no—
259C2×C4—2921no—
260C2×C4—5844no—
261C2×C4—732no—
262C2×C4—734no—
263C2×C4—1462no—
264C2×C4—1461no—
265C2×C4—1462no—
266C2×C4—1462no—
267C2×C4—1462no—
268C2×C4—1462no—
269C2×C4C4⋊C4×D7311yescharacteristic
270C2×C4#1455654811yescharacteristic
271C2×C4#1455656412yes—
272C2×C4#1455656912yes—
273C8—44no—
274C8—2924no—
275C22—2922no—
276C22—2924no—
277C22—5841no—
278C22—5842no—
279C22—5841no—
280C22#1798844611yescharacteristic
281C22#100×D7311yescharacteristic
282C22#1798857011yescharacteristic
283C22#104×D7312yes—
284C22#1798860212yes—
285C4—21no—
286C4—21no—
287C4—22no—
288C4—22no—
289C4—81no—
290C4—81no—
291C4—2922no—
292C4—2922no—
293C4—2921no—
294C4—734no—
295C2—5842no—
296C2#329×D7311yescharacteristic
297C2#2260853712yes—
298C2#2260856411yescharacteristic
299C2#2260992211yescharacteristic
300C2#2260995012yes—
subgroupG/H
centerC23C22⋊C4×D73
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1295C2
socleC23×C73C2×C22⋊C4
prime radicalC22×D292C2×C4
cosocleC2×C42×C73C23
hypercenter#1295D73
hypocenterC73#8579
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
2. #22609785C2semidirect
4. #22607812C2semidirect
5. C73×#1295C2semidirect
143. Dic73#329semidirect
155. #1295D73semidirect
156. C73#8579semidirect

Maximal subgroups

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

subgroupindex
#226124702
#226097852
C2×#179883262
#226078122
C73×#12952
#857973