Finite Group Databasebeta

← #32995209

Subgroups — #32995209

Order 18688.

438 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 97 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#22611540C212yesmaximal, complemented
3#22607870C212yesmaximal, complemented
4#22607855C211yesmaximal, characteristic, complemented
5C73×#1314C211yesmaximal, characteristic, complemented
6C23×#10596831C411yescharacteristic
7C22×C8×D73C2211yescharacteristic
8C2×C4×C22⋊D73C411yescharacteristic
9C2×#14556530—22no—
10#17988650—22no—
11#17988650—22no—
12C2×#14556520C2211yescharacteristic
13C2×#14556503C411yescharacteristic
14C2×C8×Dic73C411yescharacteristic
15C22⋊#11924003C2212yes—
16#17988366—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—42no—
27C2×C8×D73D412yes—
28C2×C8×D73D412yes—
29C4×C22⋊D73—22no—
30C4×C22⋊D73—22no—
31C2×#11924007C811yescharacteristic
32C2×C22⋊Dic73C811yescharacteristic
33C2×#11924005C811yescharacteristic
34C2×C4×Dic73C2×C411yescharacteristic
35#14556530—44no—
36C2×#11924003—22no—
37#14556520—22no—
38#14556503—22no—
39C8×Dic73—21no—
40C8×Dic73—21no—
41C23×C4×C73C2×C411yescharacteristic
42C22×C8×C73C2×C411yescharacteristic
43C22×C8×C73C2311yescharacteristic
44C2×C16×C73—22no—
45C22⋊C8×C73—22no—
46C23×D73C2×C811yescharacteristic
47C2×C22⋊D73—22no—
48C2×C22⋊D73—22no—
49C2×C22⋊D73—22no—
50C22×Dic73C2×C811yescharacteristic
51C22×Dic73C2×C811yescharacteristic
52C22×Dic73C2×C811yescharacteristic
53C2×C4×D73—41no—
54C2×C4×D73—41no—
55C2×C4×D73C22⋊C412yes—
56C2×C4×D73M1612yes—
57C2×#10596831—22no—
58C2×#10596831—21no—
59C2×#10596831—42no—
60C2×#10596831—42no—
61C2×#10596831—41no—
62C2×#10596831C22⋊C411yescharacteristic
63C2×#10596831C22⋊C411yescharacteristic
64C8×D73—24no—
65C8×D73—42no—
66#11924007—22no—
67C22⋊Dic73—22no—
68#11924005—22no—
69C4×Dic73—21no—
70C4×Dic73—21no—
71#11924003—42no—
72C24×C73C2×C811yescharacteristic
73C22×C4×C73—21no—
74C22×C4×C73—21no—
75C22×C4×C73—41no—
76C22×C4×C73—41no—
77C22×C4×C73C2×C811yescharacteristic
78C22×C4×C73C22×C411yescharacteristic
79C2×C8×C73—21no—
80C2×C8×C73—41no—
81C2×C8×C73—41no—
82C2×C8×C73C2×D412yes—
83C16×C73—42no—
84C23×C73—21no—
85C23×C73—21no—
86C23×C73—41no—
87C23×C73—41no—
88C23×C73M3212yescomplemented
89C23×C73C22×C811yescharacteristic
90C22×D73—21no—
91C22×D73—21no—
92C22×D73—41no—
93C22×D73—41no—
94C22×D73C22⋊C812yes—
95C2×C4×C73—21no—
96C2×C4×C73—21no—
97C2×C4×C73—21no—
98C2×C4×C73—42no—
99C2×C4×C73—41no—
100C2×C4×C73—41no—
101C2×C4×C73—41no—
102C2×C4×C73C2×C22⋊C411yescharacteristic
103C2×C4×C73C2×M1611yescharacteristic
104C22⋊D73—28no—
105C2×Dic73—21no—
106C2×Dic73—21no—
107C2×Dic73—21no—
108C2×Dic73—41no—
109C2×Dic73—41no—
110C2×Dic73C22⋊C812yes—
111C2×Dic73M3212yescomplemented
112C4×D73—22no—
113C4×D73—22no—
114C4×D73—42no—
115C8×C73—22no—
116C8×C73—41no—
117#10596831—21no—
118#10596831—41no—
119#10596831#9912yes—
120C22×C73—21no—
121C22×C73—21no—
122C22×C73—22no—
123C22×C73—42no—
124C22×C73—42no—
125C22×C73—41no—
126C22×C73—41no—
127C22×C73C2×C22⋊C811yescharacteristic
128C22×C73C2×M3211yescharacteristic
129C22×C73C2×M3211yescharacteristic
130C2×D73—22no—
131C2×D73—22no—
132C2×D73—22no—
133C2×D73—42no—
134C2×D73—42no—
135C4×C73—21no—
136C4×C73—41no—
137C4×C73C2×#9911yescharacteristic
138C4×C73#41311yescharacteristic
139Dic73—22no—
140Dic73—22no—
141Dic73—41no—
142#11466—731nomaximal, Sylow
143C2×C73—21no—
144C2×C73—21no—
145C2×C73—42no—
146C2×C73#145911yescharacteristic
147C2×C73#211211yescharacteristic
148C2×C73#211511yescharacteristic
149D73—42no—
150C2×C8×D4—731no—
151#2112—732no—
152#1322—732no—
153#1314—731no—
154#1314D7311yescharacteristic, complemented
155C73#1146611yescharacteristic, complemented, Sylow
156C23×C8—731no—
157C23×C8—731no—
158C2×C4×D4—731no—
159C2×M32—1462no—
160C8×D4—1462no—
161C8×D4—1462no—
162C2×C4⋊C8—731no—
163C2×C22⋊C8—731no—
164C2×C22⋊C8C2×D7311yescharacteristic
165C2×C4×C8—731no—
166C22⋊C16—732no—
167C22⋊C16—1462no—
168C22⋊C16C2×D7312yes—
169C22×D4—731no—
170C23×C4—731no—
171C23×C4C4×D7311yescharacteristic
172C22×C8—2922no—
173C22×C8—2922no—
174C22×C8—732no—
175C22×C8—731no—
176C22×C8—731no—
177C22×C8—732no—
178C22×C8—1461no—
179C22×C8—1461no—
180C22×C8C4×D7311yescharacteristic
181C22×C8C22×D7311yescharacteristic
182C4×D4—1462no—
183C4×D4—1462no—
184C2×C4⋊C4—731no—
185C2×C22⋊C4—731no—
186C2×C22⋊C4—731no—
187C2×C42—731no—
188M32—2924no—
189C2×C16—22no—
190C2×C16—1462no—
191C4⋊C8—1462no—
192C22⋊C8—22no—
193C22⋊C8—1462no—
194C4×C8—1461no—
195C4×C8—1461no—
196C24—731no—
197C24C8×D7311yescharacteristic
198C2×D4—1462no—
199C2×D4—1462no—
200C2×D4—1462no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—2921no—
206C22×C4—2921no—
207C22×C4—731no—
208C22×C4—732no—
209C22×C4—731no—
210C22×C4—731no—
211C22×C4—732no—
212C22×C4C8×D7311yescharacteristic
213C22×C4C2×C4×D7311yescharacteristic
214C2×C8—21no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—2922no—
218C2×C8—2922no—
219C2×C8—2922no—
220C2×C8—2921no—
221C2×C8—731no—
222C2×C8—731no—
223C2×C8—1464no—
224C2×C8—1461no—
225C2×C8—1462no—
226C2×C8D4×D7312yes—
227C4⋊C4—1462no—
228C22⋊C4—1462no—
229C22⋊C4—1462no—
230C42—1461no—
231C42—1461no—
232C16—42no—
233C16—2922no—
234C23—21no—
235C23—21no—
236C23—41no—
237C23—41no—
238C23—2921no—
239C23—2921no—
240C23—732no—
241C23—1461no—
242C23—1461no—
243C23#1455653012yescomplemented
244C23C2×C8×D7311yescharacteristic
245D4—1468no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—2922no—
254C2×C4—2921no—
255C2×C4—2921no—
256C2×C4—732no—
257C2×C4—732no—
258C2×C4—1461no—
259C2×C4—1461no—
260C2×C4—1461no—
261C2×C4—1462no—
262C2×C4—1462no—
263C2×C4C22⋊C4×D7311yescharacteristic
264C2×C4M16×D7311yescharacteristic
265C8—22no—
266C8—41no—
267C8—2921no—
268C8—732no—
269C8—1461no—
270C22—21no—
271C22—21no—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—2922no—
278C22—2922no—
279C22—1462no—
280C22—1462no—
281C22—1462no—
282C22C22⋊C8×D7311yescharacteristic
283C22M32×D7311yescharacteristic
284C22C2×#1455653011yescharacteristic
285C4—21no—
286C4—41no—
287C4—2921no—
288C4—1462no—
289C4—1462no—
290C4#99×D7311yescharacteristic
291C4#1798856811yescharacteristic
292C2—21no—
293C2—21no—
294C2—42no—
295C2—2922no—
296C2#2260824611yescharacteristic
297C2#2261154011yescharacteristic
298C2#2261155011yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D73
commutatorC23×C73C22×C8
FrattiniC22×C8C22×D73
FittingC73×#1314C2
socleC22×C73C2×C22⋊C8
prime radicalC22×C22⋊D73C8
cosocleC22×C8×C73C23
hypercenter#1314D73
hypocenterC73#11466
LCSG ▷ C23×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. #22611540C2semidirect
3. #22607870C2semidirect
4. #22607855C2semidirect
5. C73×#1314C2semidirect
88. C23×C73M32semidirect
111. C2×Dic73M32semidirect
154. #1314D73semidirect
155. C73#11466semidirect
243. C23#14556530semidirect

Maximal subgroups

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

subgroupindex
C2×#179886502
#226115402
#226078702
#226078552
C73×#13142
#1146673