Finite Group Databasebeta

← #32996540

Subgroups — #32996540

Order 18688.

482 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×#14556506C211yesmaximal, characteristic
2#22608272C211yesmaximal, characteristic
3#22607931C214yesmaximal, complemented
4C73×#2486C211yesmaximal, characteristic
5#17988644—21no—
6C2×#14556506—21no—
7C4×C22⋊Dic73—21no—
8#17988354C2212yes—
9#17988311—21no—
10#17988311C2212yes—
11C2×#123×C73—21no—
12C2×C4×D4×C73C2211yescharacteristic
13C22×C4⋊C4×C73C411yescharacteristic, complemented
14#398×C73C412yescomplemented
15#397×C73C2212yes—
16#389×C73—21no—
17#381×C73—21no—
18#380×C73—21no—
19#380×C73C411yescharacteristic, complemented
20C2×C22⋊Dic73—22no—
21C2×C4×Dic73—41no—
22C4⋊#10596831—41no—
23#14556506—24no—
24C22⋊#10596831—22no—
25#14556502—41no—
26#14556498—41no—
27C22×D4×C73D411yescharacteristic
28C23×C4×C73—21no—
29C23×C4×C73C2×C411yescharacteristic
30C23×C4×C73D411yescharacteristic
31#123×C73—44no—
32C4×D4×C73—41no—
33C4×D4×C73—41no—
34C2×C4⋊C4×C73—21no—
35C2×C4⋊C4×C73—21no—
36C2×C4⋊C4×C73—22no—
37C2×C4⋊C4×C73—41no—
38C2×C4⋊C4×C73—41no—
39C2×C4⋊C4×C73C2×C412yes—
40C2×C4⋊C4×C73C2×C411yescharacteristic
41C2×C22⋊C4×C73—21no—
42C2×C22⋊C4×C73—22no—
43C2×C22⋊C4×C73D411yescharacteristic
44C2×C22⋊C4×C73C2311yescharacteristic
45C2×C42×C73D411yescharacteristic
46#95×C73—22no—
47#95×C73—22no—
48#95×C73C2×C412yes—
49C22×Dic73—42no—
50C2×#10596831—42no—
51C22⋊Dic73—24no—
52C22⋊Dic73—42no—
53C4×Dic73—82no—
54C24×C73C22⋊C411yescharacteristic
55C24×C73C2×D411yescharacteristic
56C2×D4×C73—22no—
57C2×D4×C73—22no—
58C2×D4×C73—41no—
59C22×C4×C73—21no—
60C22×C4×C73—21no—
61C22×C4×C73—21no—
62C22×C4×C73—21no—
63C22×C4×C73—22no—
64C22×C4×C73—22no—
65C22×C4×C73—42no—
66C22×C4×C73—42no—
67C22×C4×C73—42no—
68C22×C4×C73—42no—
69C22×C4×C73—42no—
70C22×C4×C73C22⋊C411yescharacteristic
71C22×C4×C73C22⋊C411yescharacteristic
72C22×C4×C73C22⋊C411yescharacteristic
73C22×C4×C73C22×C411yescharacteristic
74C22×C4×C73C2×D411yescharacteristic
75C4⋊C4×C73—41no—
76C4⋊C4×C73—42no—
77C4⋊C4×C73—42no—
78C4⋊C4×C73—44no—
79C4⋊C4×C73—41no—
80C4⋊C4×C73—41no—
81C22⋊C4×C73—41no—
82C22⋊C4×C73—41no—
83C22⋊C4×C73—42no—
84C22⋊C4×C73—44no—
85C42×C73—21no—
86C42×C73—21no—
87C23×C73—21no—
88C23×C73—21no—
89C23×C73—21no—
90C23×C73—21no—
91C23×C73—41no—
92C23×C73—41no—
93C23×C73—41no—
94C23×C73—41no—
95C23×C73#10412yes—
96C23×C73C2×C22⋊C411yescharacteristic
97C23×C73AGL(1,ℤ/8)12yes—
98D4×C73—44no—
99C2×C4×C73—22no—
100C2×C4×C73—21no—
101C2×C4×C73—21no—
102C2×C4×C73—21no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—21no—
106C2×C4×C73—42no—
107C2×C4×C73—42no—
108C2×C4×C73—42no—
109C2×C4×C73—41no—
110C2×C4×C73—41no—
111C2×C4×C73—41no—
112C2×C4×C73—42no—
113C2×C4×C73—42no—
114C2×C4×C73—42no—
115C2×C4×C73—42no—
116C2×C4×C73—42no—
117C2×C4×C73—42no—
118C2×C4×C73—81no—
119C2×C4×C73—84no—
120C2×C4×C73#9911yescharacteristic
121C2×C4×C73#9911yescharacteristic
122C2×Dic73—44no—
123C2×Dic73—82no—
124C2×Dic73—82no—
125#10596831—82no—
126C22×C73—21no—
127C22×C73—21no—
128C22×C73—22no—
129C22×C73—22no—
130C22×C73—42no—
131C22×C73—42no—
132C22×C73—42no—
133C22×C73—41no—
134C22×C73—41no—
135C22×C73—41no—
136C22×C73—41no—
137C22×C73—81no—
138C22×C73C2×#9911yescharacteristic
139C22×C73#41811yescharacteristic
140C22×C73C2×#10411yescharacteristic
141C4×C73—21no—
142C4×C73—41no—
143C4×C73—41no—
144C4×C73—41no—
145C4×C73—41no—
146C4×C73—82no—
147C4×C73—82no—
148C4×C73#35212yes—
149Dic73—84no—
150#11773—731nomaximal, Sylow
151C2×C73—21no—
152C2×C73—21no—
153C2×C73—42no—
154C2×C73—42no—
155C2×C73#150711yescharacteristic
156C2×C73C2×#35211yescharacteristic
157C2×C73#212911yescharacteristic
158#2486D7311yescharacteristic
159C4×#99—731no—
160#1466—731no—
161#1344—734no—
162C73#1177311yescharacteristic, complemented, Sylow
163C2×#123—21no—
164C2×C4×D4C2×D7311yescharacteristic
165C22×C4⋊C4Dic7311yescharacteristic, complemented
166#432—1461no—
167C2×#99—1461no—
168#398Dic7312yescomplemented
169#397C2×D7312yes—
170#389—21no—
171#381—21no—
172#380—21no—
173#380Dic7311yescharacteristic, complemented
174C4×C22⋊C4—1461no—
175#342—732no—
176#323—732no—
177#323—1461no—
178C22×D4C22⋊D7311yescharacteristic
179C23×C4—21no—
180C23×C4C2×Dic7311yescharacteristic
181C23×C4C22⋊D7311yescharacteristic
182#123—44no—
183C4×D4—41no—
184C4×D4—41no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4C2×Dic7312yes—
191C2×C4⋊C4C2×Dic7311yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—1462no—
195C2×C22⋊C4C22⋊D7311yescharacteristic
196C2×C22⋊C4C22×D7311yescharacteristic
197C2×C42—2921no—
198C2×C42C22⋊D7311yescharacteristic
199C4⋊C8—2921no—
200#99—1464no—
201C22⋊C8—1462no—
202#97—2921no—
203#95—22no—
204#95—22no—
205#95—2921no—
206#95C2×Dic7312yes—
207C24C22⋊Dic7311yescharacteristic
208C24C2×C22⋊D7311yescharacteristic
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—41no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—2922no—
224C22×C4C22⋊Dic7311yescharacteristic
225C22×C4C22⋊Dic7311yescharacteristic
226C22×C4C22⋊Dic7311yescharacteristic
227C22×C4C22×Dic7311yescharacteristic
228C22×C4C2×C22⋊D7311yescharacteristic
229C2×C8—2922no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—42no—
239C22⋊C4—44no—
240C22⋊C4—2922no—
241C22⋊C4—1464no—
242C42—21no—
243C42—21no—
244C42—5842no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23#1455651812yes—
254C23C2×C22⋊Dic7311yescharacteristic
255C23#1455661712yes—
256D4—44no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—81no—
277C2×C4—84no—
278C2×C4—2924no—
279C2×C4—5842no—
280C2×C4—5842no—
281C2×C4#1455650611yescharacteristic
282C2×C4#1455650611yescharacteristic
283C8—5842no—
284C22—21no—
285C22—21no—
286C22—22no—
287C22—22no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22C2×#1455650611yescharacteristic
297C22#1798859211yescharacteristic
298C22C2×#1455651811yescharacteristic
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—82no—
305C4—82no—
306C4—5844no—
307C4#1798837712yes—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2#2260844711yescharacteristic
313C2C2×#1798837711yescharacteristic
314C2#2261160611yescharacteristic
subgroupG/H
centerC22C2×#14556506
commutatorC22×C4×C73C22×C4
FrattiniC2×C22⋊C4C22×D73
FittingC73×#2486C2
socleC22×C73C2×#99
prime radicalC22×D4×C73D4
cosocleC2×C22⋊C4×C73C23
hypercenter#2486D73
hypocenterC73#11773
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
3. #22607931C2semidirect
13. C22×C4⋊C4×C73C4semidirect
14. #398×C73C4semidirect
19. #380×C73C4semidirect
162. C73#11773semidirect
165. C22×C4⋊C4Dic73semidirect
168. #398Dic73semidirect
173. #380Dic73semidirect

Maximal subgroups

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

subgroupindex
C4×#145565062
#226082722
#226079312
C73×#24862
#1177373