Finite Group Databasebeta

← #32995414

Subgroups — #32995414

Order 18688.

330 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 75 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#22618562C211yesmaximal, characteristic
2#22611743C211yesmaximal, characteristic, complemented
3#22607881C212yesmaximal, complemented
4#22607866C212yesmaximal
5C73×#2165C211yesmaximal, characteristic, complemented
6C2×C42×D73C411yescharacteristic
7#17988642—21no—
8#17988642—21no—
9#17988638—21no—
10#17988638—21no—
11#17988610C411yescharacteristic
12#17988610C2211yescharacteristic
13#17988550C411yescharacteristic
14#17988550C2211yescharacteristic
15C4×#11924012C411yescharacteristic
16#17988408—21no—
17#17988369C2212yes—
18#17988365—21no—
19C4×M16×C73C2211yescharacteristic
20#364×C73—21no—
21#349×C73C2212yes—
22#347×C73—21no—
23C22×C4×D73—21no—
24C22×C4×D73C2×C411yescharacteristic
25C2×#11924012C2×C411yescharacteristic
26C2×#11924012C2×C411yescharacteristic
27C2×C4×Dic73—21no—
28C2×C4×Dic73C2×C411yescharacteristic
29C42×D73—21no—
30C42×D73—21no—
31C42×D73—21no—
32C42×D73—21no—
33#14556531—22no—
34#14556520—21no—
35#14556520—21no—
36#14556520—21no—
37#14556520—21no—
38#14556503—21no—
39#14556503—21no—
40#14556503—21no—
41#14556503—21no—
42#14556502—21no—
43C4×#10596831—21no—
44C2×M16×C73C2×C411yescharacteristic
45C2×M16×C73C2311yescharacteristic
46C2×C42×C73C2×C411yescharacteristic
47M32×C73—22no—
48#97×C73D411yescharacteristic
49#97×C73D411yescharacteristic
50C4×C8×C73D411yescharacteristic
51C4×C8×C73D411yescharacteristic
52C23×D73M1611yescharacteristic
53C22×Dic73—21no—
54C22×Dic73M1611yescharacteristic
55C2×C4×D73—21no—
56C2×C4×D73—21no—
57C2×C4×D73—21no—
58C2×C4×D73—21no—
59C2×C4×D73—21no—
60C2×C4×D73—21no—
61C2×C4×D73—44no—
62C2×C4×D73—41no—
63C2×C4×D73—41no—
64#11924012—41no—
65#11924012—41no—
66C2×#10596831—21no—
67C2×#10596831—21no—
68C4×Dic73—21no—
69C4×Dic73—21no—
70C4×Dic73—21no—
71C4×Dic73—21no—
72C4×Dic73—21no—
73C4×Dic73M1611yescharacteristic
74C4×Dic73M1611yescharacteristic
75#11924003—42no—
76C22×C4×C73—21no—
77C22×C4×C73C22×C411yescharacteristic
78M16×C73—41no—
79M16×C73—41no—
80C2×C8×C73C22⋊C411yescharacteristic
81C2×C8×C73C22⋊C411yescharacteristic
82C2×C8×C73C2×D411yescharacteristic
83C2×C8×C73C2×D411yescharacteristic
84C42×C73—21no—
85C42×C73C22⋊C411yescharacteristic
86C42×C73C22⋊C411yescharacteristic
87C16×C73—42no—
88C23×C73C2×M1611yescharacteristic
89C22×D73—21no—
90C22×D73—42no—
91C22×D73—41no—
92C2×C4×C73—21no—
93C2×C4×C73—21no—
94C2×C4×C73—21no—
95C2×C4×C73—21no—
96C2×C4×C73—42no—
97C2×C4×C73C2×C22⋊C411yescharacteristic
98C2×C4×C73C2×M1611yescharacteristic
99C2×Dic73—21no—
100C2×Dic73—21no—
101C2×Dic73—21no—
102C2×Dic73—42no—
103C2×Dic73—41no—
104C4×D73—24no—
105C4×D73—42no—
106C4×D73—42no—
107C4×D73—42no—
108C4×D73—42no—
109C4×D73—41no—
110C4×D73—41no—
111C8×C73—21no—
112C8×C73—21no—
113C8×C73—21no—
114C8×C73—21no—
115#10596831—41no—
116#10596831—41no—
117C22×C73—21no—
118C22×C73—41no—
119C22×C73#40711yescharacteristic
120C2×D73—22no—
121C2×D73—41no—
122C2×D73—42no—
123C2×D73—41no—
124C2×D73—42no—
125C4×C73—22no—
126C4×C73—21no—
127C4×C73—41no—
128C4×C73#41011yescharacteristic
129C4×C73#41311yescharacteristic
130Dic73—22no—
131Dic73—41no—
132Dic73—41no—
133#11521—731nomaximal, Sylow
134C2×C73—21no—
135C2×C73—41no—
136C2×C73#146411yescharacteristic
137D73—42no—
138#2920—731no—
139#2165—731no—
140#2165D7311yescharacteristic, complemented
141#1327—732no—
142#1320—732no—
143C73#1152111yescharacteristic, complemented, Sylow
144C22×C42—731no—
145#432—1461no—
146#432—1461no—
147#431—1461no—
148#431—1461no—
149#423—731no—
150#423—731no—
151#407—731no—
152#407—731no—
153C4×M16—731no—
154C4×M16C2×D7311yescharacteristic
155#364—21no—
156#364—1461no—
157#349—732no—
158#349C2×D7312yes—
159#347—21no—
160#347—1461no—
161C23×C4—731no—
162C23×C4—1461no—
163C2×M16—731no—
164C2×M16—731no—
165C2×M16C4×D7311yescharacteristic
166C2×M16C22×D7311yescharacteristic
167C2×C42—731no—
168C2×C42—1461no—
169C2×C42—1461no—
170C2×C42—1461no—
171C2×C42—1461no—
172C2×C42—1461no—
173C2×C42C4×D7311yescharacteristic
174M32—22no—
175M32—1462no—
176C4⋊C8—1461no—
177C4⋊C8—1461no—
178C4⋊C8—1461no—
179C4⋊C8—1461no—
180C22⋊C8—1461no—
181C22⋊C8—1461no—
182C22⋊C8—1461no—
183C22⋊C8—1461no—
184#97—1461no—
185#97D29211yescharacteristic
186#97C22⋊D7311yescharacteristic
187C4×C8—1461no—
188C4×C8D29211yescharacteristic
189C4×C8C22⋊D7311yescharacteristic
190C24—731no—
191C22×C4—21no—
192C22×C4—2921no—
193C22×C4—2921no—
194C22×C4—2924no—
195C22×C4—731no—
196C22×C4—1461no—
197C22×C4—1461no—
198C22×C4—1461no—
199C22×C4—1461no—
200C22×C4—1461no—
201C22×C4—1461no—
202C22×C4—1461no—
203C22×C4C2×C4×D7311yescharacteristic
204M16—41no—
205M16—41no—
206M16—2921no—
207M16—2921no—
208C2×C8—1461no—
209C2×C8—1461no—
210C2×C8#1192400511yescharacteristic
211C2×C8#1192400511yescharacteristic
212C2×C8C2×C22⋊D7311yescharacteristic
213C2×C8C2×D29211yescharacteristic
214C42—21no—
215C42—731no—
216C42—731no—
217C42—1461no—
218C42—1461no—
219C42—1461no—
220C42—1461no—
221C42—1461no—
222C42#1192400511yescharacteristic
223C42#1192400511yescharacteristic
224C16—42no—
225C16—2922no—
226C23—2921no—
227C23—2922no—
228C23—1461no—
229C23M16×D7311yescharacteristic
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—2922no—
236C2×C4—2922no—
237C2×C4—2922no—
238C2×C4—2922no—
239C2×C4—2922no—
240C2×C4—2921no—
241C2×C4—2921no—
242C2×C4—2921no—
243C2×C4—1464no—
244C2×C4—1461no—
245C2×C4—1461no—
246C2×C4—1461no—
247C2×C4C2×#1192400511yescharacteristic
248C2×C4M16×D7311yescharacteristic
249C8—21no—
250C8—21no—
251C8—21no—
252C8—21no—
253C8—2921no—
254C8—2921no—
255C22—21no—
256C22—41no—
257C22—2921no—
258C22—2922no—
259C22—2922no—
260C22—2921no—
261C22—1462no—
262C22#1798855011yescharacteristic
263C4—22no—
264C4—21no—
265C4—41no—
266C4—2921no—
267C4—2921no—
268C4—1462no—
269C4#1798856011yescharacteristic
270C4#1798856911yescharacteristic
271C2—21no—
272C2—41no—
273C2—2922no—
274C2#2260826311yescharacteristic
subgroupG/H
centerC4#17988560
commutatorC22×C4×C73C22×C4
FrattiniC2×M16C22×D73
FittingC73×#2165C2
socleC2×C73#1464
prime radicalC23×D73M16
cosocleC2×M16×C73C23
hypercenter#2165D73
hypocenterC73#11521
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C42 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
2. #22611743C2semidirect
3. #22607881C2semidirect
5. C73×#2165C2semidirect
140. #2165D73semidirect
143. C73#11521semidirect

Maximal subgroups

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

subgroupindex
#226185622
#226117432
#226078812
#226078662
C73×#21652
#1152173