Finite Group Databasebeta

← #34005253

Subgroups — #34005253

Order 18688.

630 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 132 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#22625788C212yesmaximal
2#22624809C214yesmaximal, complemented
3#384×D73C211yesmaximal, characteristic
4C2×#14556658C2214yes—
5C2×#14556648—22no—
6#17989453—22no—
7C2×C4⋊C4×D73—22no—
8C2×C4⋊C4×D73C412yes—
9C2×C42×D73C2211yescharacteristic
10#17988457C411yescharacteristic
11#17988456—22no—
12#17988456C2212yes—
13#384×C73C411yescharacteristic
14C22×#10596833—24no—
15#14556658—28no—
16#14556648—44no—
17C22×C4×D73—22no—
18C22×C4×D73C2×C411yescharacteristic
19C22×C4×D73D412yes—
20C2×C4⋊Dic73—22no—
21C2×C4⋊Dic73C2×C412yes—
22C2×#11924007—22no—
23C2×#11924007—24no—
24C4⋊C4×D73—44no—
25C4⋊C4×D73—44no—
26C4⋊C4×D73—44no—
27C2×C4×Dic73D412yes—
28C2×C4×Dic73C2311yescharacteristic
29C42×D73—24no—
30C2×C4⋊C4×C73—22no—
31C2×C4⋊C4×C73C2×C412yes—
32C2×C42×C73C2×C411yescharacteristic
33C2×#10596833—44no—
34C2×#10596833—44no—
35C2×#10596833—44no—
36C23×D73C22⋊C411yescharacteristic
37C22×Dic73—22no—
38C22×Dic73C22⋊C411yescharacteristic
39C22×Dic73C2×D412yes—
40C2×C4×D73—22no—
41C2×C4×D73—22no—
42C2×C4×D73—22no—
43C2×C4×D73—24no—
44C2×C4×D73—24no—
45C2×C4×D73—44no—
46C2×C4×D73—48no—
47C2×C4×D73D814yes—
48C2×C4×D73SD1614yes—
49C4⋊Dic73—44no—
50C4⋊Dic73—42no—
51#11924007—44no—
52#11924007—44no—
53#11924007—44no—
54C4×Dic73—24no—
55C4×Dic73#4114yes—
56C22×C4×C73—22no—
57C22×C4×C73C22⋊C412yes—
58C22×C4×C73C22×C411yescharacteristic
59C4⋊C4×C73—42no—
60C4⋊C4×C73—44no—
61C42×C73—22no—
62C23×C73C2×C22⋊C411yescharacteristic
63C22×D73—21no—
64C22×D73—21no—
65C22×D73—22no—
66C22×D73#10012yes—
67C22×D73#10214yes—
68C2×C4×C73—22no—
69C2×C4×C73—21no—
70C2×C4×C73—21no—
71C2×C4×C73—22no—
72C2×C4×C73—42no—
73C2×C4×C73—44no—
74C2×C4×C73#10214yes—
75C2×C4×C73#11712yes—
76C2×Dic73—21no—
77C2×Dic73—21no—
78C2×Dic73—42no—
79C2×Dic73—44no—
80C2×Dic73#10112yes—
81C2×Dic73#12314yes—
82C2×Dic73C2×D812yes—
83C2×Dic73C2×SD1612yes—
84C2×Dic73#13714yes—
85C4×D73—28no—
86C4×D73—28no—
87C4×D73—28no—
88C4×D73—88no—
89#10596833—84no—
90C22×C73#38111yescharacteristic
91C22×C73C2×#10011yescharacteristic
92C22×C73C2×#10111yescharacteristic
93C22×C73C2×#10212yes—
94C22×C73#41912yes—
95C2×D73—24no—
96C2×D73—22no—
97C2×D73—22no—
98C2×D73—24no—
99C2×D73#33114yes—
100C4×C73—22no—
101C4×C73—24no—
102C4×C73—82no—
103Dic73—22no—
104Dic73—82no—
105Dic73#48014yes—
106Dic73#48314yes—
107#8723—731nomaximal, Sylow
108C2×C73C2×#33111yescharacteristic
109C2×C73#154612yes—
110C2×C73#155811yescharacteristic
111C2×C73#182111yescharacteristic
112C2×C73#182312yes—
113D73—24no—
114C2×#384—731no—
115#1797—732no—
116#1296—734no—
117C73#872311yescharacteristic, complemented, Sylow
118C22×C4⋊C4—732no—
119C22×C4⋊C4—1462no—
120C22×C42—731no—
121C2×C4⋊C8—734no—
122C2×C22⋊C8—1462no—
123#384—732no—
124#384—731no—
125#384—1462no—
126#384#6440811yescharacteristic
127#336—1462no—
128C23×C4—732no—
129C23×C4—731no—
130C23×C4—1462no—
131C22×C8—1464no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—2924no—
134C2×C4⋊C4—2924no—
135C2×C4⋊C4—2924no—
136C2×C4⋊C4—732no—
137C2×C4⋊C4—1462no—
138C2×C4⋊C4—1464no—
139C2×C4⋊C4—1462no—
140C2×C4⋊C4C2×#6440812yes—
141C2×C42—731no—
142C2×C42—732no—
143C2×C42—1464no—
144C2×C42C2×#6440811yescharacteristic
145C4⋊C8—1468no—
146C22⋊C8—2924no—
147C24—731no—
148C22×C4—22no—
149C22×C4—2928no—
150C22×C4—2924no—
151C22×C4—732no—
152C22×C4—731no—
153C22×C4—734no—
154C22×C4—734no—
155C22×C4—1464no—
156C22×C4—1462no—
157C22×C4—1464no—
158C22×C4—1462no—
159C22×C4—1462no—
160C22×C4—1462no—
161C22×C4C22⋊#6440812yes—
162C22×C4C22×#6440811yescharacteristic
163C2×C8—2924no—
164C2×C8—2924no—
165C2×C8—2924no—
166C4⋊C4—42no—
167C4⋊C4—44no—
168C4⋊C4—2922no—
169C4⋊C4—2924no—
170C4⋊C4—2924no—
171C4⋊C4—2924no—
172C4⋊C4—2924no—
173C42—22no—
174C42—734no—
175C42—1464no—
176C23—732no—
177C23—734no—
178C23—1461no—
179C23—1461no—
180C23—1462no—
181C23C2×C22⋊#6440811yescharacteristic
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—42no—
187C2×C4—44no—
188C2×C4—2922no—
189C2×C4—2924no—
190C2×C4—5848no—
191C2×C4—734no—
192C2×C4—732no—
193C2×C4—732no—
194C2×C4—734no—
195C2×C4—732no—
196C2×C4—1468no—
197C2×C4—1468no—
198C2×C4—1468no—
199C2×C4—1461no—
200C2×C4—1461no—
201C2×C4#1455665314yes—
202C2×C4#1455667012yes—
203C8—5844no—
204C22—734no—
205C22—1462no—
206C22—1464no—
207C22—1464no—
208C22—1462no—
209C22#1798953011yescharacteristic
210C22D73⋊#10011yescharacteristic
211C22D73⋊#10111yescharacteristic
212C22C2×#1455665312yes—
213C22#1798958012yes—
214C4—22no—
215C4—24no—
216C4—82no—
217C4—5842no—
218C4—734no—
219C4—734no—
220C4—1462no—
221C2—1464no—
222C2D73⋊#33111yescharacteristic
223C2#2262544112yes—
224C2#2262546511yescharacteristic
225C2#2262583111yescharacteristic
226C2#2262583412yes—
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
Fitting#384×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC23×D73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenter#384#64408
hypocenterC73#8723
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
2. #22624809C2semidirect
117. C73#8723semidirect

Maximal subgroups

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

subgroupindex
#226257882
#226248092
#384×D732
#872373