Finite Group Databasebeta

← #32995299

Subgroups — #32995299

Order 18688.

430 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 109 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×#17988650C211yesmaximal, characteristic
2#22611727C212yesmaximal, complemented
3#22607855C211yesmaximal, characteristic, complemented
4#22607854C212yesmaximal, complemented
5C73×#1314C211yesmaximal, characteristic, complemented
6C23×#10596831C411yescharacteristic
7C22×C8×D73C411yescharacteristic
8C2×C4×C22⋊D73C411yescharacteristic
9#17988650—22no—
10#17988650—22no—
11C2×#14556520C411yescharacteristic
12C2×#14556503C2211yescharacteristic
13C2×C8×Dic73C2211yescharacteristic
14#17988405—22no—
15C22⋊#11924003C2212yes—
16C16×Dic73—22no—
17C2×C22⋊C8×C73C2211yescharacteristic
18C22⋊C16×C73C2212yes—
19C22×C22⋊D73C811yescharacteristic
20C22×C4×D73C2×C411yescharacteristic
21C22×#10596831—21no—
22C22×#10596831—21no—
23C22×#10596831—42no—
24C22×#10596831C2×C411yescharacteristic
25C22×#10596831C2×C411yescharacteristic
26C2×C8×D73—21no—
27C2×C8×D73—21no—
28C2×C8×D73—42no—
29C4×C22⋊D73—22no—
30C4×C22⋊D73—22no—
31C2×#11924007C811yescharacteristic
32C2×C22⋊Dic73C811yescharacteristic
33C2×#11924005C811yescharacteristic
34C2×C4×Dic73C2×C411yescharacteristic
35C2×#11924003—22no—
36#14556520—22no—
37#14556503—22no—
38C8×Dic73D412yes—
39C8×Dic73D412yes—
40C23×C4×C73C2×C411yescharacteristic
41C22×C8×C73C2×C411yescharacteristic
42C22×C8×C73C2311yescharacteristic
43C2×C16×C73—22no—
44C22⋊C8×C73—22no—
45C23×D73C2×C811yescharacteristic
46C2×C22⋊D73—22no—
47C2×C22⋊D73—22no—
48C2×C22⋊D73C1614yescomplemented
49C22×Dic73C2×C811yescharacteristic
50C22×Dic73C2×C811yescharacteristic
51C22×Dic73C2×C811yescharacteristic
52C2×C4×D73—21no—
53C2×C4×D73—21no—
54C2×C4×D73—41no—
55C2×C4×D73—41no—
56C2×#10596831—22no—
57C2×#10596831—21no—
58C2×#10596831—42no—
59C2×#10596831—42no—
60C2×#10596831—41no—
61C2×#10596831C22⋊C411yescharacteristic
62C2×#10596831C22⋊C411yescharacteristic
63C8×D73—42no—
64C8×D73—42no—
65#11924007—22no—
66C22⋊Dic73—22no—
67#11924005—22no—
68C4×Dic73C22⋊C412yes—
69C4×Dic73M1612yes—
70#11924003—42no—
71C24×C73C2×C811yescharacteristic
72C22×C4×C73—21no—
73C22×C4×C73—21no—
74C22×C4×C73—41no—
75C22×C4×C73—41no—
76C22×C4×C73C2×C811yescharacteristic
77C22×C4×C73C22×C411yescharacteristic
78C2×C8×C73—21no—
79C2×C8×C73—41no—
80C2×C8×C73—41no—
81C2×C8×C73C2×D412yes—
82C16×C73—42no—
83C23×C73—21no—
84C23×C73—21no—
85C23×C73—41no—
86C23×C73—41no—
87C23×C73C2×C1612yes—
88C23×C73C22×C811yescharacteristic
89C22×D73—21no—
90C22×D73—21no—
91C22×D73—41no—
92C22×D73—41no—
93C22×D73C2×C1612yes—
94C2×C4×C73—21no—
95C2×C4×C73—21no—
96C2×C4×C73—21no—
97C2×C4×C73—42no—
98C2×C4×C73—41no—
99C2×C4×C73—41no—
100C2×C4×C73—41no—
101C2×C4×C73C2×C22⋊C411yescharacteristic
102C2×C4×C73C2×M1611yescharacteristic
103C22⋊D73—28no—
104C2×Dic73—21no—
105C2×Dic73—41no—
106C2×Dic73—41no—
107C2×Dic73C22⋊C812yes—
108C2×Dic73C22⋊C812yes—
109C2×Dic73C2×C1612yes—
110C2×Dic73M3212yes—
111C4×D73—42no—
112C4×D73—41no—
113C4×D73—41no—
114C8×C73—22no—
115C8×C73—41no—
116#10596831—21no—
117#10596831—41no—
118#10596831#10012yes—
119C22×C73—21no—
120C22×C73—21no—
121C22×C73—22no—
122C22×C73—42no—
123C22×C73—42no—
124C22×C73—41no—
125C22×C73—41no—
126C22×C73C2×C22⋊C811yescharacteristic
127C22×C73C22×C1611yescharacteristic
128C22×C73C2×M3211yescharacteristic
129C2×D73—22no—
130C2×D73—42no—
131C2×D73—41no—
132C2×D73—41no—
133C2×D73—42no—
134C4×C73—21no—
135C4×C73—41no—
136C4×C73#41011yescharacteristic
137C4×C73C2×#10011yescharacteristic
138Dic73—22no—
139Dic73—41no—
140Dic73C22⋊C1614yes—
141#11487—731nomaximal, Sylow
142C2×C73—21no—
143C2×C73—21no—
144C2×C73—42no—
145C2×C73#146011yescharacteristic
146C2×C73C2×C22⋊C1611yescharacteristic
147C2×C73#211511yescharacteristic
148D73—42no—
149C2×C8×D4—731no—
150#2162—732no—
151#1314—731no—
152#1314—732no—
153#1314D7311yescharacteristic, complemented
154C73#1148711yescharacteristic, complemented, Sylow
155C23×C8—731no—
156C23×C8—731no—
157C2×C4×D4—731no—
158C8×D4—1462no—
159C8×D4—1462no—
160C2×C4⋊C8—731no—
161C2×C22⋊C8—731no—
162C2×C22⋊C8C2×D7311yescharacteristic
163C2×C4×C8—731no—
164C4⋊C16—1462no—
165C22⋊C16—732no—
166C22⋊C16C2×D7312yes—
167C4×C16—1462no—
168C22×D4—731no—
169C23×C4—731no—
170C23×C4C4×D7311yescharacteristic
171C22×C8—2922no—
172C22×C8—2922no—
173C22×C8—731no—
174C22×C8—731no—
175C22×C8—1461no—
176C22×C8—1461no—
177C22×C8—1461no—
178C22×C8—1461no—
179C22×C8C4×D7311yescharacteristic
180C22×C8C22×D7311yescharacteristic
181C4×D4—1462no—
182C4×D4—1462no—
183C2×C4⋊C4—731no—
184C2×C22⋊C4—731no—
185C2×C22⋊C4—731no—
186C2×C42—731no—
187C2×C16—22no—
188C2×C16—1462no—
189C4⋊C8—1462no—
190C22⋊C8—22no—
191C22⋊C8—1462no—
192C4×C8—732no—
193C4×C8—732no—
194C24—731no—
195C24C8×D7311yescharacteristic
196C2×D4—734no—
197C2×D4—1462no—
198C2×D4—1462no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—2921no—
204C22×C4—2921no—
205C22×C4—731no—
206C22×C4—731no—
207C22×C4—731no—
208C22×C4—1461no—
209C22×C4—1461no—
210C22×C4C8×D7311yescharacteristic
211C22×C4C2×C4×D7311yescharacteristic
212C2×C8—21no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—2922no—
216C2×C8—2922no—
217C2×C8—2922no—
218C2×C8—2922no—
219C2×C8—2921no—
220C2×C8—731no—
221C2×C8—731no—
222C2×C8—1461no—
223C2×C8—1462no—
224C2×C8D4×D7312yes—
225C4⋊C4—1462no—
226C22⋊C4—1462no—
227C22⋊C4—1462no—
228C42—732no—
229C42—732no—
230C16—42no—
231C16—2922no—
232C23—21no—
233C23—21no—
234C23—41no—
235C23—41no—
236C23—2921no—
237C23—2921no—
238C23—732no—
239C23—1461no—
240C23—1461no—
241C23C16×D7312yes—
242C23C2×C8×D7311yescharacteristic
243D4—1468no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—2922no—
252C2×C4—2921no—
253C2×C4—2921no—
254C2×C4—2921no—
255C2×C4—2921no—
256C2×C4—732no—
257C2×C4—732no—
258C2×C4—732no—
259C2×C4—732no—
260C2×C4—1461no—
261C2×C4C22⋊C4×D7311yescharacteristic
262C2×C4M16×D7311yescharacteristic
263C8—22no—
264C8—41no—
265C8—2921no—
266C8—732no—
267C8—1461no—
268C22—21no—
269C22—21no—
270C22—22no—
271C22—42no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—2922no—
276C22—2922no—
277C22—2921no—
278C22—2921no—
279C22—1462no—
280C22C22⋊C8×D7311yescharacteristic
281C22C2×C16×D7311yescharacteristic
282C22M32×D7311yescharacteristic
283C4—21no—
284C4—41no—
285C4—2921no—
286C4—734no—
287C4—1462no—
288C4#1798855911yescharacteristic
289C4#100×D7311yescharacteristic
290C2—21no—
291C2—21no—
292C2—42no—
293C2—2922no—
294C2#2260825011yescharacteristic
295C2C22⋊C16×D7311yescharacteristic
296C2#2261155011yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D73
commutatorC22×C73C22×C16
FrattiniC22×C8C22×D73
FittingC73×#1314C2
socleC22×C73C2×C22⋊C8
prime radicalC22×C22⋊D73C8
cosocleC22×C8×C73C23
hypercenter#1314D73
hypocenterC73#11487
LCSG ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1314

Complements

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

NH ≅ G/Ntype
2. #22611727C2semidirect
3. #22607855C2semidirect
4. #22607854C2semidirect
5. C73×#1314C2semidirect
48. C2×C22⋊D73C16semidirect
153. #1314D73semidirect
154. C73#11487semidirect

Maximal subgroups

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

subgroupindex
C2×#179886502
#226117272
#226078552
#226078542
C73×#13142
#1148773