Finite Group Databasebeta

← #32995183

Subgroups — #32995183

Order 18688.

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

normal subgroups only

This group has 374 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×#17988637C211yesmaximal, characteristic
2#22611731C212yesmaximal, complemented
3#22607870C212yesmaximal, complemented
4#22607855C211yesmaximal, characteristic
5C73×#1314C211yesmaximal, characteristic
6C22×C4×Dic73C411yescharacteristic
7#17988637—22no—
8#17988637—22no—
9C2×#14556520C411yescharacteristic
10C2×#14556520C2211yescharacteristic
11C2×C22⋊#10596831C411yescharacteristic
12C2×C8×Dic73C411yescharacteristic
13C2×C8×Dic73C2211yescharacteristic
14#17988405—22no—
15C22⋊#11924003C2212yes—
16#17988362—22no—
17C2×C22⋊C8×C73C2211yescharacteristic
18C22⋊C16×C73C2212yes—
19C23×Dic73C811yescharacteristic
20C23×Dic73C811yescharacteristic
21C22×#10596831C2×C411yescharacteristic
22C22×#10596831C2×C411yescharacteristic
23C2×C4×Dic73—21no—
24C2×C4×Dic73—21no—
25C2×C4×Dic73—21no—
26C2×C4×Dic73—21no—
27C2×C4×Dic73C811yescharacteristic
28C2×C4×Dic73C811yescharacteristic
29C2×C4×Dic73C2×C411yescharacteristic
30C2×C4×Dic73C2×C411yescharacteristic
31C2×#11924003—22no—
32#14556520—22no—
33#14556520—22no—
34C22⋊#10596831—22no—
35C8×Dic73—21no—
36C8×Dic73—21no—
37C8×Dic73D412yes—
38C8×Dic73D412yes—
39C23×C4×C73C2×C411yescharacteristic
40C22×C8×C73C2×C411yescharacteristic
41C22×C8×C73C2311yescharacteristic
42C2×C16×C73—22no—
43C22⋊C8×C73—22no—
44C22×Dic73—22no—
45C22×Dic73—21no—
46C22×Dic73—21no—
47C22×Dic73—22no—
48C22×Dic73—21no—
49C22×Dic73—21no—
50C22×Dic73—42no—
51C22×Dic73C2×C811yescharacteristic
52C22×Dic73C2×C811yescharacteristic
53C22×Dic73C2×C811yescharacteristic
54C22×Dic73C2×C811yescharacteristic
55C2×#10596831—21no—
56C2×#10596831—41no—
57C2×#10596831—41no—
58C2×#10596831C22⋊C412yes—
59C4×Dic73—21no—
60C4×Dic73—21no—
61C4×Dic73—21no—
62C4×Dic73—21no—
63C4×Dic73—21no—
64C4×Dic73—21no—
65C4×Dic73C22⋊C411yescharacteristic
66C4×Dic73C22⋊C411yescharacteristic
67C4×Dic73M1611yescharacteristic
68C4×Dic73M1611yescharacteristic
69#11924003—42no—
70C24×C73C2×C811yescharacteristic
71C22×C4×C73—21no—
72C22×C4×C73—21no—
73C22×C4×C73—41no—
74C22×C4×C73—41no—
75C22×C4×C73C2×C811yescharacteristic
76C22×C4×C73C22×C411yescharacteristic
77C2×C8×C73—21no—
78C2×C8×C73—41no—
79C2×C8×C73—41no—
80C2×C8×C73C2×D412yes—
81C16×C73—42no—
82C23×C73—21no—
83C23×C73—21no—
84C23×C73—41no—
85C23×C73—41no—
86C23×C73M3212yes—
87C23×C73C22×C811yescharacteristic
88C2×C4×C73—21no—
89C2×C4×C73—21no—
90C2×C4×C73—21no—
91C2×C4×C73—42no—
92C2×C4×C73—41no—
93C2×C4×C73—41no—
94C2×C4×C73—41no—
95C2×C4×C73C2×C22⋊C411yescharacteristic
96C2×C4×C73C2×M1611yescharacteristic
97C2×Dic73—24no—
98C2×Dic73—24no—
99C2×Dic73—21no—
100C2×Dic73—21no—
101C2×Dic73—21no—
102C2×Dic73—21no—
103C2×Dic73—22no—
104C2×Dic73—21no—
105C2×Dic73—42no—
106C2×Dic73—42no—
107C2×Dic73—41no—
108C2×Dic73C22⋊C811yescharacteristic
109C2×Dic73C22⋊C811yescharacteristic
110C2×Dic73C22⋊C811yescharacteristic
111C2×Dic73C22⋊C811yescharacteristic
112C2×Dic73M3212yes—
113C8×C73—22no—
114C8×C73—41no—
115#10596831—22no—
116#10596831—41no—
117C22×C73—21no—
118C22×C73—21no—
119C22×C73—22no—
120C22×C73—42no—
121C22×C73—42no—
122C22×C73—41no—
123C22×C73—41no—
124C22×C73C2×C22⋊C811yescharacteristic
125C22×C73C2×M3211yescharacteristic
126C22×C73C2×M3211yescharacteristic
127C4×C73—21no—
128C4×C73—41no—
129C4×C73#41011yescharacteristic
130C4×C73#41311yescharacteristic
131Dic73—21no—
132Dic73—21no—
133Dic73—22no—
134Dic73—21no—
135Dic73—41no—
136Dic73#34912yes—
137#11460—731nomaximal, Sylow
138C2×C73—21no—
139C2×C73—21no—
140C2×C73—42no—
141C2×C73#145811yescharacteristic
142C2×C73#211211yescharacteristic
143C2×C73C2×#34911yescharacteristic
144C2×#431—731no—
145#2163—732no—
146#1322—732no—
147#1314—731no—
148#1314D7311yescharacteristic
149C73#1146011yescharacteristic, complemented, Sylow
150C22×C42—731no—
151#431—1462no—
152#431—1462no—
153C2×C4⋊C8—731no—
154C2×C4⋊C8—731no—
155C2×C22⋊C8—731no—
156C2×C22⋊C8C2×D7311yescharacteristic
157C2×C4×C8—731no—
158C2×C4×C8—731no—
159C4⋊C16—1462no—
160C22⋊C16—732no—
161C22⋊C16C2×D7312yes—
162#346—1462no—
163C23×C4—731no—
164C23×C4—731no—
165C23×C4C4×D7311yescharacteristic
166C22×C8—731no—
167C22×C8—731no—
168C22×C8C4×D7311yescharacteristic
169C22×C8C22×D7311yescharacteristic
170C2×C42—731no—
171C2×C42—731no—
172C2×C42—731no—
173C2×C42—731no—
174C2×C42—1461no—
175C2×C42—1461no—
176C2×C42—1461no—
177C2×C42—1461no—
178C2×C16—22no—
179C2×C16—1462no—
180C4⋊C8—1462no—
181C4⋊C8—1462no—
182C22⋊C8—22no—
183C22⋊C8—1462no—
184C4×C8—732no—
185C4×C8—732no—
186C4×C8—1461no—
187C4×C8—1461no—
188C24C8×D7311yescharacteristic
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—2922no—
194C22×C4—731no—
195C22×C4—731no—
196C22×C4—731no—
197C22×C4—731no—
198C22×C4—1461no—
199C22×C4—1461no—
200C22×C4—1461no—
201C22×C4—1461no—
202C22×C4—1462no—
203C22×C4—1462no—
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—731no—
215C42—731no—
216C42—731no—
217C42—731no—
218C42—1461no—
219C42—1461no—
220C42—1461no—
221C42—1461no—
222C42—1461no—
223C42—1461no—
224C16—42no—
225C16—2922no—
226C23—21no—
227C23—21no—
228C23—41no—
229C23—41no—
230C23#1455653012yes—
231C23C2×C8×D7311yescharacteristic
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—2922no—
240C2×C4—2922no—
241C2×C4—2921no—
242C2×C4—731no—
243C2×C4—732no—
244C2×C4—731no—
245C2×C4—731no—
246C2×C4—731no—
247C2×C4—1461no—
248C2×C4—1462no—
249C2×C4—1464no—
250C2×C4—1464no—
251C2×C4—1461no—
252C2×C4—1461no—
253C2×C4—1461no—
254C2×C4—1461no—
255C2×C4C22⋊C4×D7311yescharacteristic
256C2×C4M16×D7311yescharacteristic
257C8—22no—
258C8—41no—
259C8—2921no—
260C8—1462no—
261C22—21no—
262C22—21no—
263C22—22no—
264C22—42no—
265C22—42no—
266C22—41no—
267C22—41no—
268C22C22⋊C8×D7311yescharacteristic
269C22M32×D7311yescharacteristic
270C22C2×#1455653011yescharacteristic
271C4—21no—
272C4—41no—
273C4—2921no—
274C4—732no—
275C4—1462no—
276C4—1461no—
277C4—1461no—
278C4—1461no—
279C4#1798855911yescharacteristic
280C4#1798856811yescharacteristic
281C2—21no—
282C2—21no—
283C2—42no—
284C2#2260824211yescharacteristic
285C2#2261154011yescharacteristic
286C2#349×D7311yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D73
commutatorC23×C73C22×C8
FrattiniC22×C8C22×D73
FittingC73×#1314C2
socleC22×C73C2×C22⋊C8
prime radicalC24×C73C2×C8
cosocleC22×C8×C73C23
hypercenter#1314D73
hypocenterC73#11460
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. #22611731C2semidirect
3. #22607870C2semidirect
149. C73#11460semidirect

Maximal subgroups

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

subgroupindex
C2×#179886372
#226117312
#226078702
#226078552
C73×#13142
#1146073