Finite Group Databasebeta

← #32995426

Subgroups — #32995426

Order 18688.

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

normal subgroups only

This group has 288 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#22618576C211yesmaximal, characteristic
2#22611743C211yesmaximal, characteristic, complemented
3#22607883C212yesmaximal, complemented
4#22607868C212yesmaximal
5C73×#2165C211yesmaximal, characteristic, complemented
6C4×#11924028C411yescharacteristic
7#17988649—21no—
8#17988649—21no—
9#17988638—21no—
10#17988638—21no—
11#17988615C411yescharacteristic
12#17988615C2211yescharacteristic
13#17988555C411yescharacteristic
14#17988555C2211yescharacteristic
15C2×C4×#10596831C411yescharacteristic
16#17988408—21no—
17#17988369C2212yes—
18#17988365—21no—
19C4×M16×C73C2211yescharacteristic
20#364×C73—21no—
21#349×C73C2212yes—
22#347×C73—21no—
23C2×#11924028C2×C411yescharacteristic
24C22×#10596831C2×C411yescharacteristic
25C22×#10596831C2×C411yescharacteristic
26C4×Dic146—21no—
27C4×D292—21no—
28C4×C22⋊D73—22no—
29#14556550C2×C411yescharacteristic
30#14556548—21no—
31C42×D73—21no—
32#14556531—22no—
33#14556520—21no—
34#14556520—21no—
35#14556520—21no—
36#14556520—21no—
37#14556503—21no—
38#14556503—21no—
39#14556503—21no—
40#14556503—21no—
41C4×#10596831—21no—
42C4×#10596831—21no—
43C2×M16×C73C2×C411yescharacteristic
44C2×M16×C73C2311yescharacteristic
45C2×C42×C73C2×C411yescharacteristic
46M32×C73—22no—
47#97×C73D411yescharacteristic
48#97×C73D411yescharacteristic
49C4×C8×C73D411yescharacteristic
50C4×C8×C73D411yescharacteristic
51#11924028—41no—
52#11924028—41no—
53C2×Dic146M1611yescharacteristic
54C2×D292M1611yescharacteristic
55C2×C22⋊D73—21no—
56C2×C4×D73—22no—
57C2×C4×D73—21no—
58C2×#10596831—21no—
59C2×#10596831—21no—
60C2×#10596831—41no—
61C2×#10596831—41no—
62C4⋊Dic73—21no—
63#11924007—22no—
64C22⋊Dic73M1612yes—
65#11924005—22no—
66C4×Dic73—22no—
67C4×Dic73—21no—
68#11924003—42no—
69C22×C4×C73—21no—
70C22×C4×C73C22×C411yescharacteristic
71M16×C73—41no—
72M16×C73—41no—
73C2×C8×C73C22⋊C411yescharacteristic
74C2×C8×C73C22⋊C411yescharacteristic
75C2×C8×C73C2×D411yescharacteristic
76C2×C8×C73C2×D411yescharacteristic
77C42×C73—21no—
78C42×C73C22⋊C411yescharacteristic
79C42×C73C22⋊C411yescharacteristic
80C16×C73—42no—
81C23×C73C2×M1611yescharacteristic
82C22×D73—21no—
83C2×C4×C73—21no—
84C2×C4×C73—21no—
85C2×C4×C73—21no—
86C2×C4×C73—21no—
87C2×C4×C73—42no—
88C2×C4×C73C2×C22⋊C411yescharacteristic
89C2×C4×C73C2×M1611yescharacteristic
90C22⋊D73—42no—
91C2×Dic73—22no—
92C2×Dic73—21no—
93D292—41no—
94C4×D73—44no—
95C4×D73—41no—
96C4×D73—41no—
97Dic146—41no—
98C8×C73—21no—
99C8×C73—21no—
100C8×C73—21no—
101C8×C73—21no—
102#10596831—41no—
103#10596831—41no—
104C22×C73—21no—
105C22×C73—41no—
106C22×C73#40711yescharacteristic
107C2×D73—42no—
108C2×D73—41no—
109C4×C73—22no—
110C4×C73—21no—
111C4×C73—41no—
112C4×C73#41011yescharacteristic
113C4×C73#41311yescharacteristic
114Dic73—42no—
115Dic73—41no—
116#11524—731nomaximal, Sylow
117C2×C73—21no—
118C2×C73—41no—
119C2×C73#146411yescharacteristic
120D73—81no—
121#2922—731no—
122#2165—731no—
123#2165D7311yescharacteristic, complemented
124#1328—732no—
125#1321—732no—
126C73#1152411yescharacteristic, complemented, Sylow
127C4×#41—731no—
128#433—1461no—
129#433—1461no—
130#431—1461no—
131#431—1461no—
132#424—731no—
133#424—731no—
134#408—731no—
135#408—731no—
136C4×M16C2×D7311yescharacteristic
137C2×C4×C8—731no—
138#364—21no—
139#364—1461no—
140#349—732no—
141#349C2×D7312yes—
142#347—21no—
143#347—1461no—
144C2×#41—731no—
145C2×M16C4×D7311yescharacteristic
146C2×M16C22×D7311yescharacteristic
147C22×C8—731no—
148C22×C8—731no—
149C4×Q8—1461no—
150C4×D4—1461no—
151C4×D4—1462no—
152#117—731no—
153#117—1461no—
154C2×C42—1461no—
155C2×C42C4×D7311yescharacteristic
156M32—22no—
157M32—1462no—
158C4⋊C8—1461no—
159C4⋊C8—1461no—
160C4⋊C8—1461no—
161C4⋊C8—1461no—
162C22⋊C8—1461no—
163C22⋊C8—1461no—
164C22⋊C8—1461no—
165C22⋊C8—1461no—
166#97D29211yescharacteristic
167#97C22⋊D7311yescharacteristic
168C4×C8—1461no—
169C4×C8—1461no—
170C4×C8D29211yescharacteristic
171C4×C8C22⋊D7311yescharacteristic
172#41—2921no—
173#41—2921no—
174C2×Q8—731no—
175C2×D4—731no—
176C2×D4—1461no—
177C22×C4—21no—
178C22×C4—1462no—
179C22×C4—1461no—
180C22×C4C2×C4×D7311yescharacteristic
181M16—41no—
182M16—41no—
183C2×C8—2921no—
184C2×C8—2921no—
185C2×C8—1461no—
186C2×C8—1461no—
187C2×C8#1192400511yescharacteristic
188C2×C8#1192400511yescharacteristic
189C2×C8C2×C22⋊D7311yescharacteristic
190C2×C8C2×D29211yescharacteristic
191C4⋊C4—1462no—
192C4⋊C4—1461no—
193C22⋊C4—732no—
194C22⋊C4—1462no—
195C42—21no—
196C42—1462no—
197C42—1461no—
198C42#1192400511yescharacteristic
199C42#1192400511yescharacteristic
200C16—42no—
201C16—2922no—
202C23—1461no—
203C23M16×D7311yescharacteristic
204Q8—2921no—
205D4—2921no—
206D4—2922no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—42no—
212C2×C4—2924no—
213C2×C4—2921no—
214C2×C4—2921no—
215C2×C4—1462no—
216C2×C4—1461no—
217C2×C4C2×#1192400511yescharacteristic
218C2×C4M16×D7311yescharacteristic
219C8—21no—
220C8—21no—
221C8—21no—
222C8—21no—
223C8—2921no—
224C8—2921no—
225C22—21no—
226C22—41no—
227C22—2922no—
228C22—2921no—
229C22#1798855011yescharacteristic
230C4—22no—
231C4—21no—
232C4—41no—
233C4—2922no—
234C4—2921no—
235C4#1798856011yescharacteristic
236C4#1798856911yescharacteristic
237C2—21no—
238C2—41no—
239C2—5841no—
240C2#2260826311yescharacteristic
subgroupG/H
centerC4#17988560
commutatorC22×C4×C73C22×C4
FrattiniC2×M16C22×D73
FittingC73×#2165C2
socleC2×C73#1464
prime radicalC2×#11924028C2×C4
cosocleC2×M16×C73C23
hypercenter#2165D73
hypocenterC73#11524
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. #22607883C2semidirect
5. C73×#2165C2semidirect
123. #2165D73semidirect
126. C73#11524semidirect

Maximal subgroups

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

subgroupindex
#226185762
#226117432
#226078832
#226078682
C73×#21652
#1152473