Finite Group Databasebeta

← #32995181

Subgroups — #32995181

Order 18688.

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

normal subgroups only

This group has 386 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×#17988639C211yesmaximal, characteristic
2#22611727C212yesmaximal, complemented
3#22607871C211yesmaximal, characteristic, complemented
4#22607854C212yesmaximal
5C73×#1322C211yesmaximal, characteristic, complemented
6C2×C42×D73C411yescharacteristic
7#17988639—22no—
8#17988639—22no—
9C2×C4⋊#10596831C411yescharacteristic
10C2×#14556503C411yescharacteristic
11C2×#14556503C2211yescharacteristic
12C2×C8×Dic73C411yescharacteristic
13C2×C8×Dic73C2211yescharacteristic
14#17988405—22no—
15C22⋊#11924003C2212yes—
16C16×Dic73—22no—
17C2×C4⋊C8×C73C2211yescharacteristic
18C22⋊C16×C73C2212yes—
19C22×C4×D73C811yescharacteristic
20C22×C4×D73C811yescharacteristic
21C22×C4×D73C2×C411yescharacteristic
22C22×#10596831C2×C411yescharacteristic
23C22×#10596831C2×C411yescharacteristic
24C2×C4×Dic73C811yescharacteristic
25C2×C4×Dic73C811yescharacteristic
26C2×C4×Dic73C2×C411yescharacteristic
27C42×D73—21no—
28C42×D73—21no—
29C42×D73—21no—
30C42×D73—21no—
31C2×#11924003—22no—
32C4⋊#10596831—22no—
33#14556503—22no—
34#14556503—22no—
35C8×Dic73—21no—
36C8×Dic73—21no—
37C8×Dic73D412yes—
38C8×Dic73D412yes—
39C22×C8×C73C2×C411yescharacteristic
40C22×C8×C73C2311yescharacteristic
41C2×C42×C73C2×C411yescharacteristic
42C2×C16×C73—22no—
43C4⋊C8×C73—22no—
44C23×D73C2×C811yescharacteristic
45C22×Dic73C2×C811yescharacteristic
46C22×Dic73C2×C811yescharacteristic
47C22×Dic73C2×C811yescharacteristic
48C2×C4×D73—21no—
49C2×C4×D73—21no—
50C2×C4×D73—22no—
51C2×C4×D73—21no—
52C2×C4×D73—21no—
53C2×C4×D73—21no—
54C2×C4×D73—21no—
55C2×C4×D73—42no—
56C2×C4×D73—41no—
57C2×C4×D73—41no—
58C2×C4×D73C1614yescomplemented
59C2×#10596831—21no—
60C2×#10596831—41no—
61C2×#10596831—41no—
62C2×#10596831C22⋊C412yes—
63C4×Dic73—21no—
64C4×Dic73—21no—
65C4×Dic73—21no—
66C4×Dic73—21no—
67C4×Dic73C22⋊C411yescharacteristic
68C4×Dic73C22⋊C411yescharacteristic
69C4×Dic73M1611yescharacteristic
70C4×Dic73M1611yescharacteristic
71#11924003—42no—
72C22×C4×C73C2×C811yescharacteristic
73C22×C4×C73C2×C811yescharacteristic
74C22×C4×C73C22×C411yescharacteristic
75C2×C8×C73—21no—
76C2×C8×C73—41no—
77C2×C8×C73—41no—
78C2×C8×C73C2×D412yes—
79C42×C73—21no—
80C42×C73—21no—
81C16×C73—42no—
82C23×C73C22×C811yescharacteristic
83C22×D73—21no—
84C22×D73—21no—
85C22×D73—41no—
86C22×D73—41no—
87C22×D73C2×C1612yes—
88C2×C4×C73—21no—
89C2×C4×C73—21no—
90C2×C4×C73—21no—
91C2×C4×C73—21no—
92C2×C4×C73—21no—
93C2×C4×C73—41no—
94C2×C4×C73C2×C1612yes—
95C2×C4×C73C2×C22⋊C411yescharacteristic
96C2×C4×C73C2×M1611yescharacteristic
97C2×Dic73—22no—
98C2×Dic73—21no—
99C2×Dic73—41no—
100C2×Dic73C22⋊C811yescharacteristic
101C2×Dic73C22⋊C811yescharacteristic
102C2×Dic73C22⋊C811yescharacteristic
103C2×Dic73C22⋊C811yescharacteristic
104C2×Dic73C2×C1612yes—
105C2×Dic73M3212yes—
106C4×D73—24no—
107C4×D73—24no—
108C4×D73—42no—
109C4×D73—42no—
110C4×D73—41no—
111C4×D73—41no—
112C4×D73—42no—
113C8×C73—22no—
114C8×C73—41no—
115#10596831—22no—
116#10596831—41no—
117C22×C73—21no—
118C22×C73—21no—
119C22×C73C2×C22⋊C811yescharacteristic
120C22×C73C22×C1611yescharacteristic
121C22×C73C2×M3211yescharacteristic
122C2×D73—22no—
123C2×D73—42no—
124C2×D73—41no—
125C2×D73—41no—
126C2×D73—42no—
127C4×C73—22no—
128C4×C73—21no—
129C4×C73—41no—
130C4×C73#41011yescharacteristic
131C4×C73#41311yescharacteristic
132Dic73—21no—
133Dic73—41no—
134Dic73C22⋊C1612yes—
135Dic73C22⋊C1612yes—
136Dic73#34912yes—
137#11459—731nomaximal, Sylow
138C2×C73—21no—
139C2×C73—21no—
140C2×C73#145811yescharacteristic
141C2×C73C2×C22⋊C1611yescharacteristic
142C2×C73C2×#34911yescharacteristic
143D73—42no—
144C2×#431—731no—
145#2162—732no—
146#1322—731no—
147#1322D7311yescharacteristic, complemented
148#1314—732no—
149C73#1145911yescharacteristic, complemented, Sylow
150C22×C42—731no—
151#431—1462no—
152#431—1462no—
153C2×C4⋊C8—731no—
154C2×C4⋊C8C2×D7311yescharacteristic
155C2×C22⋊C8—731no—
156C2×C22⋊C8—731no—
157C2×C4×C8—731no—
158C2×C4×C8—731no—
159C4⋊C16—1462no—
160C22⋊C16—732no—
161C22⋊C16C2×D7312yes—
162C4×C16—1462no—
163C23×C4—731no—
164C23×C4—731no—
165C23×C4—731no—
166C22×C8—731no—
167C22×C8—731no—
168C22×C8C4×D7311yescharacteristic
169C22×C8C22×D7311yescharacteristic
170C2×C42—731no—
171C2×C42—731no—
172C2×C42—731no—
173C2×C42—1461no—
174C2×C42—1461no—
175C2×C42—1461no—
176C2×C42—1461no—
177C2×C42C4×D7311yescharacteristic
178C2×C16—22no—
179C2×C16—1462no—
180C4⋊C8—22no—
181C4⋊C8—1462no—
182C22⋊C8—1462no—
183C22⋊C8—1462no—
184C4×C8—732no—
185C4×C8—732no—
186C4×C8—1461no—
187C4×C8—1461no—
188C24—731no—
189C22×C4—2921no—
190C22×C4—2921no—
191C22×C4—2922no—
192C22×C4—731no—
193C22×C4—731no—
194C22×C4—731no—
195C22×C4—734no—
196C22×C4—1461no—
197C22×C4—1461no—
198C22×C4—1461no—
199C22×C4—1461no—
200C22×C4—1462no—
201C22×C4—1461no—
202C22×C4—1461no—
203C22×C4C8×D7311yescharacteristic
204C22×C4C8×D7311yescharacteristic
205C22×C4C2×C4×D7311yescharacteristic
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—2921no—
210C2×C8—2921no—
211C2×C8—732no—
212C2×C8—1461no—
213C2×C8D4×D7312yes—
214C42—21no—
215C42—21no—
216C42—731no—
217C42—731no—
218C42—731no—
219C42—731no—
220C42—1461no—
221C42—1461no—
222C42—1461no—
223C42—1461no—
224C16—42no—
225C16—2922no—
226C23—2921no—
227C23—2921no—
228C23—732no—
229C23—1461no—
230C23—1461no—
231C23C2×C8×D7311yescharacteristic
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—2922no—
239C2×C4—2922no—
240C2×C4—2922no—
241C2×C4—2921no—
242C2×C4—2921no—
243C2×C4—2921no—
244C2×C4—732no—
245C2×C4—732no—
246C2×C4—731no—
247C2×C4—731no—
248C2×C4—731no—
249C2×C4—731no—
250C2×C4—1464no—
251C2×C4—1464no—
252C2×C4—1461no—
253C2×C4—1462no—
254C2×C4C16×D7312yes—
255C2×C4C22⋊C4×D7311yescharacteristic
256C2×C4M16×D7311yescharacteristic
257C8—22no—
258C8—41no—
259C8—2921no—
260C8—1462no—
261C22—21no—
262C22—21no—
263C22—2922no—
264C22—2922no—
265C22—2921no—
266C22—2921no—
267C22—1462no—
268C22C22⋊C8×D7311yescharacteristic
269C22C2×C16×D7311yescharacteristic
270C22M32×D7311yescharacteristic
271C4—22no—
272C4—21no—
273C4—41no—
274C4—2921no—
275C4—732no—
276C4—732no—
277C4—732no—
278C4—1461no—
279C4#1798855911yescharacteristic
280C4#1798857011yescharacteristic
281C2—21no—
282C2—21no—
283C2—2922no—
284C2#2260824411yescharacteristic
285C2C22⋊C16×D7311yescharacteristic
286C2#349×D7311yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D73
commutatorC22×C73C22×C16
FrattiniC22×C8C22×D73
FittingC73×#1322C2
socleC22×C73C2×C22⋊C8
prime radicalC23×D73C2×C8
cosocleC22×C8×C73C23
hypercenter#1322D73
hypocenterC73#11459
LCSG ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1322

Complements

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

NH ≅ G/Ntype
2. #22611727C2semidirect
3. #22607871C2semidirect
5. C73×#1322C2semidirect
58. C2×C4×D73C16semidirect
147. #1322D73semidirect
149. C73#11459semidirect

Maximal subgroups

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

subgroupindex
C2×#179886392
#226117272
#226078712
#226078542
C73×#13222
#1145973