Finite Group Databasebeta

← #32983596

Subgroups — #32983596

Order 18688.

452 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 103 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#22620307C211yesmaximal, characteristic
2#22619033C211yesmaximal, characteristic, complemented
3#22608600C212yesmaximal
4#22607820C212yesmaximal, complemented
5C73×#1561C211yesmaximal, characteristic, complemented
6C2×#14556588C2211yescharacteristic
7#17988953C414yes—
8#17988935—22no—
9#17988718—22no—
10Q8×#10596831—22no—
11#17988648—22no—
12#17988640C2212yes—
13#17988591—22no—
14C4×#11924012C2211yescharacteristic
15#17988330—22no—
16#17988330—22no—
17#533×C73C2211yescharacteristic
18#423×C73C2212yes—
19#332×C73—22no—
20C22×D292—21no—
21C22×D292—22no—
22C2×#11924012—21no—
23#14556588—21no—
24#14556588—22no—
25#14556588C2×C412yes—
26C4⋊#10596831—24no—
27C4⋊#10596831D412yes—
28C4⋊#10596831D412yes—
29#14556511—28no—
30C22⋊#10596831—22no—
31#14556502—22no—
32#14556502C2×C412yes—
33C4×#10596831—22no—
34C4×#10596831—22no—
35C4×#10596831—21no—
36C2×M16×C73—22no—
37C4⋊Q8×C73D412yes—
38#125×C73C2×C412yes—
39C22⋊Q8×C73—22no—
40C4×Q8×C73—22no—
41#117×C73D412yes—
42C2×C42×C73C2311yescharacteristic
43C4⋊C8×C73—22no—
44C4⋊C8×C73—22no—
45C23×D73—41no—
46C2×D292—22no—
47C2×D292—24no—
48C2×D292—44no—
49C2×D292—41no—
50C2×D292—42no—
51C2×D292—82no—
52C2×D292—82no—
53#11924012—42no—
54C2×#10596831—22no—
55C2×#10596831—22no—
56C2×#10596831—22no—
57C2×Q8×C73—22no—
58C22×C4×C73C2×D411yescharacteristic
59C22×C4×C73#4112yes—
60M16×C73—82no—
61C2×C8×C73—42no—
62C4⋊C4×C73—24no—
63C4⋊C4×C73—22no—
64C4⋊C4×C73C22⋊C414yes—
65C22⋊C4×C73—22no—
66C42×C73—22no—
67C42×C73C22×C411yescharacteristic
68C42×C73C2×D412yes—
69C42×C73C2×D411yescharacteristic
70C23×C73#12311yescharacteristic
71C22×D73—42no—
72C22×D73—82no—
73C22×D73—84no—
74Q8×C73—42no—
75Q8×C73—42no—
76C2×C4×C73—22no—
77C2×C4×C73—22no—
78C2×C4×C73—22no—
79C2×C4×C73—24no—
80C2×C4×C73C2×C22⋊C411yescharacteristic
81C2×C4×C73C4×D412yes—
82C2×C4×C73#12011yescharacteristic
83C2×C4×C73#12112yes—
84D292—44no—
85D292—84no—
86D292—82no—
87D292—82no—
88D292—82no—
89C8×C73—82no—
90#10596831—42no—
91#10596831—42no—
92#10596831—42no—
93C22×C73—21no—
94C22×C73—22no—
95C22×C73#38611yescharacteristic
96C2×D73—82no—
97C2×D73—84no—
98C2×D73—164no—
99C4×C73—21no—
100C4×C73—22no—
101C4×C73—44no—
102C4×C73#45412yes—
103C4×C73#47112yes—
104C4×C73#47212yes—
105#9438—731nomaximal, Sylow
106C2×C73—41no—
107C2×C73#188811yescharacteristic
108C2×C73#190712yes—
109D73—162no—
110#3138—731no—
111#2986—731no—
112#1561D7311yescharacteristic, complemented
113#1545—732no—
114#1299—732no—
115C73#943811yescharacteristic, complemented, Sylow
116#533C2×D7311yescharacteristic
117C2×#127—731no—
118#490—734no—
119#486—1462no—
120#446—1462no—
121C8×Q8—1462no—
122#433—1462no—
123#431—732no—
124#423C2×D7312yes—
125#418—1462no—
126C4×M16—731no—
127#332—22no—
128#331—1462no—
129#331—1462no—
130C22×D4—1461no—
131C22×D4—1462no—
132C2×M16—22no—
133C2×M16—1461no—
134C4⋊Q8C22⋊D7312yes—
135#127—732no—
136#127—1462no—
137#127—1461no—
138#125C4×D7312yes—
139C22⋊Q8—22no—
140C4×Q8—22no—
141#117D29212yes—
142C2×C42C22×D7311yescharacteristic
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—732no—
146C4⋊C8—732no—
147C4⋊C8—1464no—
148#102—1468no—
149C22⋊C8—1462no—
150#97—732no—
151#97—1462no—
152C4×C8—1462no—
153C4×C8—1462no—
154C4×C8—1461no—
155C24—2921no—
156C2×Q8—22no—
157C2×D4—2924no—
158C2×D4—2921no—
159C2×D4—2922no—
160C2×D4—5842no—
161C2×D4—5842no—
162C2×D4—1462no—
163C2×D4—1464no—
164C22×C4C2×D29211yescharacteristic
165C22×C4#1192402712yes—
166M16—82no—
167M16—2922no—
168C2×C8—42no—
169C2×C8—1462no—
170C2×C8—1462no—
171C2×C8—1462no—
172C4⋊C4—24no—
173C4⋊C4—22no—
174C4⋊C4#1192400514yes—
175C22⋊C4—22no—
176C42—22no—
177C42C2×C4×D7311yescharacteristic
178C42D4×D7312yes—
179C42C2×C22⋊D7311yescharacteristic
180C23—2922no—
181C23—5842no—
182C23—5844no—
183C23#1455657511yescharacteristic
184Q8—42no—
185Q8—42no—
186D4—2924no—
187D4—5844no—
188D4—5842no—
189D4—5842no—
190D4—5842no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4C2×#1192400511yescharacteristic
196C2×C4#1455655112yes—
197C2×C4#1455656111yescharacteristic
198C2×C4#1455656312yes—
199C8—82no—
200C8—2922no—
201C8—2922no—
202C8—2922no—
203C22—22no—
204C22—21no—
205C22—5844no—
206C22—5842no—
207C22—11684no—
208C22#1798846411yescharacteristic
209C4—22no—
210C4—21no—
211C4—44no—
212C4#1798876212yes—
213C4#1798885212yes—
214C4#1798885512yes—
215C2—41no—
216C2—11682no—
217C2#2261033211yescharacteristic
218C2#2261043712yes—
subgroupG/H
centerC22#17988464
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1561C2
socleC22×C73#386
prime radicalC2×#14556588C22
cosocleC2×C42×C73C23
hypercenter#1561D73
hypocenterC73#9438
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1561

Complements

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

NH ≅ G/Ntype
2. #22619033C2semidirect
4. #22607820C2semidirect
5. C73×#1561C2semidirect
112. #1561D73semidirect
115. C73#9438semidirect

Maximal subgroups

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

subgroupindex
#226203072
#226190332
#226086002
#226078202
C73×#15612
#943873