Finite Group Databasebeta

← #34012518

Subgroups — #34012518

Order 18688.

628 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C2×#17989491C212yesmaximal
2#22625676C212yesmaximal, complemented
3#22625025C212yesmaximal, complemented
4#22612655C211yesmaximal, characteristic
5C2×#14556650C2214yes—
6C4×C22⋊#64408—22no—
7#17989491C418yescomplemented
8#17989465—22no—
9C2×#14556587C2211yescharacteristic
10C2×D4×Dic73C411yescharacteristic, complemented
11C2×#14556553C2211yescharacteristic
12#17988489C2211yescharacteristic
13#17988464—22no—
14#17988463—22no—
15#17988452C411yescharacteristic, complemented
16#17988433C411yescharacteristic, complemented
17#390×C73C411yescharacteristic, complemented
18C2×C22⋊#64408—24no—
19C2×C4×#64408—42no—
20#14556650—24no—
21#14556650C2×C418yescomplemented
22(C2×#64408)⋊C4—42no—
23C22×D292C2311yescharacteristic
24C22×C22⋊D73—22no—
25C23×Dic73—21no—
26C22×C4×D73D412yes—
27#14556587—22no—
28#14556587C2×C414yes—
29D4×Dic73—42no—
30#14556553—42no—
31C2×C4⋊Dic73C2×C411yescharacteristic
32C2×C22⋊Dic73—21no—
33C2×#11924005—22no—
34C2×#11924005D412yes—
35C2×C4×Dic73C2×C411yescharacteristic
36C2×C4×Dic73C2×C411yescharacteristic
37#14556498—21no—
38#14556498—21no—
39#14556498—22no—
40C22×D4×C73C2×C411yescharacteristic
41C2×C4⋊C4×C73C2×C411yescharacteristic
42C2×C22⋊C4×C73—22no—
43C2×C42×C73C2×C411yescharacteristic
44C22×#64408—44no—
45C22⋊#64408—28no—
46C22⋊#64408—44no—
47C4×#64408—84no—
48C23×D73C2×D412yes—
49C2×D292—41no—
50C2×D292—41no—
51C2×D292C22×C412yes—
52C2×D292#4112yes—
53C2×C22⋊D73—28no—
54C2×C22⋊D73—82no—
55C2×C22⋊D73—82no—
56C22×Dic73—21no—
57C22×Dic73—21no—
58C22×Dic73—42no—
59C22×Dic73—82no—
60C22×Dic73C22⋊C411yescharacteristic
61C22×Dic73C22⋊C411yescharacteristic
62C2×C4×D73—22no—
63C2×C4×D73—22no—
64C2×C4×D73—82no—
65C4⋊Dic73—41no—
66C22⋊Dic73—42no—
67#11924005—42no—
68#11924005—82no—
69C4×Dic73—21no—
70C4×Dic73—22no—
71C4×Dic73C4212yescomplemented
72C24×C73—21no—
73C2×D4×C73—21no—
74C2×D4×C73—41no—
75C2×D4×C73—41no—
76C2×D4×C73C4212yescomplemented
77C22×C4×C73—21no—
78C22×C4×C73C22⋊C412yes—
79C22×C4×C73C22×C411yescharacteristic
80C4⋊C4×C73—41no—
81C22⋊C4×C73—82no—
82C42×C73—41no—
83C23×C73—22no—
84C23×C73—42no—
85C23×C73—42no—
86C23×C73—41no—
87C23×C73—41no—
88C23×C73C2×C22⋊C411yescharacteristic
89C22×D73—22no—
90C22×D73—22no—
91C22×D73—42no—
92C22×D73—42no—
93C22×D73C4×D414yes—
94C2×#64408—48no—
95C2×#64408—84no—
96C2×#64408—84no—
97D4×C73—42no—
98D4×C73—42no—
99C2×C4×C73—21no—
100C2×C4×C73—21no—
101C2×C4×C73—22no—
102C2×C4×C73—42no—
103C2×C4×C73—41no—
104C2×C4×C73—41no—
105C2×C4×C73—41no—
106C2×C4×C73C2×C4211yescharacteristic
107C2×C4×C73#11711yescharacteristic
108C22⋊D73—88no—
109C2×Dic73—22no—
110C2×Dic73—21no—
111C2×Dic73—41no—
112C2×Dic73—41no—
113C2×Dic73—41no—
114C2×Dic73—41no—
115C2×Dic73—42no—
116C2×Dic73—41no—
117C2×Dic73—84no—
118C2×Dic73#9912yes—
119D292—42no—
120D292—42no—
121C4×D73—44no—
122C4×D73—82no—
123C22×C73—21no—
124C22×C73—21no—
125C22×C73—44no—
126C22×C73—42no—
127C22×C73—82no—
128C22×C73—81no—
129C22×C73—81no—
130C22×C73C4×C22⋊C411yescharacteristic
131C22×C73C2×#9911yescharacteristic
132C22×C73#41011yescharacteristic
133C2×D73—24no—
134C2×D73—42no—
135C2×D73—42no—
136C2×D73—44no—
137C2×D73—82no—
138#64408—88no—
139C4×C73—21no—
140C4×C73—21no—
141C4×C73—42no—
142C4×C73—81no—
143Dic73—22no—
144Dic73—42no—
145Dic73—41no—
146Dic73—81no—
147#12628—731nomaximal, Sylow
148C2×C73—21no—
149C2×C73—21no—
150C2×C73—82no—
151C2×C73C4×#9911yescharacteristic
152C2×C73#212711yescharacteristic
153C2×C73#212811yescharacteristic
154D73—44no—
155#2328—731no—
156C2×#353—732no—
157C4×#99—732no—
158#1393—732no—
159C73#1262811yescharacteristic, complemented, Sylow
160C2×#127—731no—
161C2×C4×D4—731no—
162C2×C4×D4—731no—
163C2×#99—734no—
164#390—731no—
165#390#6440811yescharacteristic, complemented
166#386—1462no—
167#386—1462no—
168#382—731no—
169C4×C22⋊C4—1462no—
170#376—731no—
171#353—738no—
172#342—1462no—
173C22×D4—731no—
174C22×D4—1462no—
175C22×D4C2×#6440811yescharacteristic
176C23×C4—732no—
177C23×C4—1461no—
178#127—734no—
179#127—1462no—
180C4×D4—2922no—
181C4×D4—2922no—
182C2×C4⋊C4—731no—
183C2×C4⋊C4C2×#6440811yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4—732no—
186C2×C22⋊C4—1461no—
187C2×C22⋊C4—1464no—
188C2×C22⋊C4—1462no—
189C2×C42—2922no—
190C2×C42—731no—
191C2×C42—731no—
192C2×C42C2×#6440811yescharacteristic
193#99—738no—
194#99—1464no—
195#95—2922no—
196#95—1461no—
197#95—1461no—
198#95—1462no—
199C24—21no—
200C24—732no—
201C2×D4—21no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—2921no—
205C2×D4—2921no—
206C2×D4—5842no—
207C2×D4—5842no—
208C2×D4—732no—
209C2×D4—732no—
210C2×D4—1468no—
211C2×D4C4×#6440812yescomplemented
212C22×C4—21no—
213C22×C4—2924no—
214C22×C4—2922no—
215C22×C4—5842no—
216C22×C4—5842no—
217C22×C4—731no—
218C22×C4—731no—
219C22×C4—1462no—
220C22×C4—1462no—
221C22×C4—1461no—
222C22×C4—1461no—
223C22×C4C22⋊#6440812yes—
224C22×C4C22×#6440811yescharacteristic
225C4⋊C4—41no—
226C4⋊C4—2921no—
227C22⋊C4—82no—
228C22⋊C4—2922no—
229C22⋊C4—2924no—
230C22⋊C4—2922no—
231C22⋊C4—5842no—
232C22⋊C4—1468no—
233C42—41no—
234C42—5844no—
235C42—732no—
236C42—1462no—
237C42—1461no—
238C23—22no—
239C23—42no—
240C23—42no—
241C23—41no—
242C23—41no—
243C23—2922no—
244C23—2922no—
245C23—734no—
246C23—1462no—
247C23—1462no—
248C23C2×C22⋊#6440811yescharacteristic
249D4—42no—
250D4—42no—
251D4—2922no—
252D4—2922no—
253D4—5848no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—2924no—
262C2×C4—2921no—
263C2×C4—2921no—
264C2×C4—2921no—
265C2×C4—2922no—
266C2×C4—2928no—
267C2×C4—2921no—
268C2×C4—2921no—
269C2×C4—5842no—
270C2×C4—5844no—
271C2×C4—5844no—
272C2×C4—5844no—
273C2×C4—732no—
274C2×C4—1462no—
275C2×C4—1461no—
276C2×C4C2×C4×#6440811yescharacteristic
277C2×C4#1455667011yescharacteristic
278C22—21no—
279C22—21no—
280C22—44no—
281C22—42no—
282C22—82no—
283C22—81no—
284C22—81no—
285C22—2924no—
286C22—2922no—
287C22—2922no—
288C22—5842no—
289C22—1464no—
290C22C4×C22⋊#6440811yescharacteristic
291C22D73⋊#9911yescharacteristic
292C22#1798956711yescharacteristic
293C4—21no—
294C4—21no—
295C4—42no—
296C4—81no—
297C4—2922no—
298C4—2921no—
299C4—5848no—
300C4—5841no—
301C4—1462no—
302C2—21no—
303C2—21no—
304C2—82no—
305C2—2924no—
306C2#2262567811yescharacteristic
307C2#2262620711yescharacteristic
308C2#2262621111yescharacteristic
subgroupG/H
centerC22D73⋊#99
commutatorC2×C4×C73C2×C42
FrattiniC22×C4C22×#64408
Fitting#390×C73C4
socleC22×C73C2×#99
prime radicalC2×#14556587C22
cosocleC22×D292C23
hypercenter#390#64408
hypocenterC73#12628
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #22625676C2semidirect
3. #22625025C2semidirect
7. #17989491C4semidirect
10. C2×D4×Dic73C4semidirect
15. #17988452C4semidirect
16. #17988433C4semidirect
17. #390×C73C4semidirect
21. #14556650C2×C4semidirect
71. C4×Dic73C42semidirect
76. C2×D4×C73C42semidirect
159. C73#12628semidirect
165. #390#64408semidirect
211. C2×D4C4×#64408semidirect

Maximal subgroups

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

subgroupindex
C2×#179894912
#226256762
#226250252
#226126552
#1262873