Finite Group Databasebeta

← #34005249

Subgroups — #34005249

Order 18688.

526 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 100 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#22625658C212yesmaximal
2#22624807C214yesmaximal, complemented
3#22612336C211yesmaximal, characteristic
4C2×#14556648—22no—
5C2×#14556646C2214yes—
6#17989453—22no—
7C2×#14556549C412yes—
8C2×C42×D73C2211yescharacteristic
9#17988457C411yescharacteristic
10#17988452C2212yes—
11#17988444—22no—
12C4⋊C4×Dic73—22no—
13#384×C73C411yescharacteristic
14C22×#10596833—24no—
15#14556648—28no—
16#14556646—28no—
17C22×C4×D73C2×C411yescharacteristic
18C22×C4×D73D412yes—
19#14556549—44no—
20C2×C4⋊Dic73—22no—
21C2×C4⋊Dic73C2×C412yes—
22C2×#11924005—22no—
23C2×C4×Dic73—22no—
24C2×C4×Dic73D412yes—
25C2×C4×Dic73C2311yescharacteristic
26C42×D73—24no—
27#14556498—24no—
28C2×C4⋊C4×C73—22no—
29C2×C4⋊C4×C73C2×C412yes—
30C2×C42×C73C2×C411yescharacteristic
31C2×#10596833—28no—
32C2×#10596833—44no—
33C2×#10596833—44no—
34C23×D73C22⋊C411yescharacteristic
35C22×Dic73—22no—
36C22×Dic73C22⋊C411yescharacteristic
37C22×Dic73C2×D412yes—
38C2×C4×D73—22no—
39C2×C4×D73—22no—
40C2×C4×D73—22no—
41C2×C4×D73—24no—
42C2×C4×D73—24no—
43C2×C4×D73—42no—
44C4⋊Dic73—44no—
45C4⋊Dic73—42no—
46#11924005—44no—
47C4×Dic73—24no—
48C4×Dic73—44no—
49C4×Dic73#4114yes—
50C22×C4×C73—22no—
51C22×C4×C73C22⋊C412yes—
52C22×C4×C73C22×C411yescharacteristic
53C4⋊C4×C73—42no—
54C4⋊C4×C73—44no—
55C42×C73—22no—
56C23×C73C2×C22⋊C411yescharacteristic
57C22×D73—21no—
58C22×D73—21no—
59C22×D73—22no—
60C22×D73—22no—
61C22×D73#10112yes—
62C2×C4×C73—22no—
63C2×C4×C73—21no—
64C2×C4×C73—21no—
65C2×C4×C73—22no—
66C2×C4×C73—42no—
67C2×C4×C73—44no—
68C2×C4×C73#10414yes—
69C2×C4×C73#11712yes—
70C2×Dic73—21no—
71C2×Dic73—21no—
72C2×Dic73—22no—
73C2×Dic73—22no—
74C2×Dic73—42no—
75C2×Dic73—44no—
76C2×Dic73#10012yes—
77C2×Dic73#12314yes—
78C4×D73—28no—
79C4×D73—48no—
80#10596833—48no—
81C22×C73#38111yescharacteristic
82C22×C73C2×#10011yescharacteristic
83C22×C73C2×#10111yescharacteristic
84C22×C73C2×#10412yes—
85C22×C73#42112yes—
86C2×D73—22no—
87C2×D73—24no—
88C2×D73—41no—
89C2×D73—42no—
90C2×D73—41no—
91C4×C73—22no—
92C4×C73—24no—
93C4×C73—82no—
94Dic73—22no—
95Dic73—24no—
96Dic73—82no—
97#8721—731nomaximal, Sylow
98C2×C73#152511yescharacteristic
99C2×C73#153112yes—
100C2×C73#153611yescharacteristic
101C2×C73#182111yescharacteristic
102C2×C73#182612yes—
103D73—42no—
104#2285—731no—
105#1749—732no—
106#1295—734no—
107C73#872111yescharacteristic, complemented, Sylow
108C2×#117—732no—
109C22×C42—731no—
110C2×C22⋊C8—1462no—
111C2×#97—734no—
112#384—731no—
113#384#6440811yescharacteristic
114#382—732no—
115#380—1462no—
116C4×C4⋊C4—1462no—
117#336—1462no—
118C23×C4—732no—
119C23×C4—731no—
120C22×C8—1464no—
121#117—2924no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—732no—
124C2×C4⋊C4—1462no—
125C2×C4⋊C4C2×#6440812yes—
126C2×C22⋊C4—1462no—
127C2×C42—731no—
128C2×C42—732no—
129C2×C42—1464no—
130C2×C42—1462no—
131C2×C42C2×#6440811yescharacteristic
132C22⋊C8—1468no—
133#97—1468no—
134#95—1464no—
135C24—731no—
136C22×C4—22no—
137C22×C4—2922no—
138C22×C4—731no—
139C22×C4—732no—
140C22×C4—1464no—
141C22×C4—1462no—
142C22×C4—1462no—
143C22×C4—1462no—
144C22×C4—1464no—
145C22×C4—1462no—
146C22×C4C22⋊#6440812yes—
147C22×C4C22×#6440811yescharacteristic
148C2×C8—2924no—
149C2×C8—2924no—
150C2×C8—1468no—
151C4⋊C4—42no—
152C4⋊C4—44no—
153C4⋊C4—2922no—
154C4⋊C4—2924no—
155C22⋊C4—2924no—
156C42—22no—
157C42—2924no—
158C42—734no—
159C42—1464no—
160C23—732no—
161C23—1461no—
162C23—1461no—
163C23—1462no—
164C23—1462no—
165C23C2×C22⋊#6440811yescharacteristic
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—42no—
171C2×C4—44no—
172C2×C4—2928no—
173C2×C4—2922no—
174C2×C4—2924no—
175C2×C4—734no—
176C2×C4—732no—
177C2×C4—1468no—
178C2×C4—1461no—
179C2×C4—1461no—
180C2×C4—1462no—
181C2×C4—1462no—
182C2×C4#1455665614yes—
183C2×C4#1455667012yes—
184C8—2928no—
185C22—2922no—
186C22—2921no—
187C22—2921no—
188C22—1462no—
189C22—1464no—
190C22#1798953011yescharacteristic
191C22D73⋊#10011yescharacteristic
192C22D73⋊#10111yescharacteristic
193C22C2×#1455665612yes—
194C22#1798958512yes—
195C4—22no—
196C4—24no—
197C4—82no—
198C4—5842no—
199C4—1462no—
200C4—1464no—
201C2—2922no—
202C2#2262538711yescharacteristic
203C2#2262540512yes—
204C2#2262542011yescharacteristic
205C2#2262583111yescharacteristic
206C2#2262583812yes—
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#8721
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. #22624807C2semidirect
107. C73#8721semidirect

Maximal subgroups

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

subgroupindex
#226256582
#226248072
#226123362
#872173