Finite Group Databasebeta

← #32983586

Subgroups — #32983586

Order 18688.

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

normal subgroups only

This group has 400 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#22620313C211yesmaximal, characteristic
2#22619033C211yesmaximal, characteristic
3#22608617C212yesmaximal
4#22607820C212yesmaximal
5C73×#1561C211yesmaximal, characteristic
6C2×#14556591C2211yescharacteristic
7#17988957C414yes—
8#17988939—22no—
9#17988718—22no—
10Q8×#10596831—22no—
11#17988648—22no—
12#17988640C2212yes—
13#17988595—22no—
14C4×#11924012C2211yescharacteristic
15#17988332—22no—
16#17988332—22no—
17#533×C73C2211yescharacteristic
18#423×C73C2212yes—
19#332×C73—22no—
20C22×Dic146—21no—
21C2×#11924012—21no—
22#14556591—22no—
23#14556591—21no—
24#14556591C2×C412yes—
25C2×C4⋊Dic73—22no—
26C4⋊#10596831—24no—
27C4⋊#10596831D412yes—
28C4⋊#10596831D412yes—
29#14556514—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—
45C2×Dic146—22no—
46C2×Dic146—44no—
47C2×Dic146—41no—
48C22×Dic73—41no—
49#11924012—42no—
50C2×#10596831—22no—
51C2×#10596831—22no—
52C2×#10596831—22no—
53C4⋊Dic73—24no—
54C4⋊Dic73—42no—
55C2×Q8×C73—22no—
56C22×C4×C73C2×D411yescharacteristic
57C22×C4×C73#4112yes—
58M16×C73—82no—
59C2×C8×C73—42no—
60C4⋊C4×C73—24no—
61C4⋊C4×C73—22no—
62C4⋊C4×C73C22⋊C414yes—
63C22⋊C4×C73—22no—
64C42×C73—22no—
65C42×C73C22×C411yescharacteristic
66C42×C73C2×D412yes—
67C42×C73C2×D411yescharacteristic
68C23×C73#12311yescharacteristic
69Q8×C73—42no—
70Q8×C73—42no—
71C2×C4×C73—22no—
72C2×C4×C73—22no—
73C2×C4×C73—22no—
74C2×C4×C73—24no—
75C2×C4×C73C2×C22⋊C411yescharacteristic
76C2×C4×C73C4×D412yes—
77C2×C4×C73#12011yescharacteristic
78C2×C4×C73#12112yes—
79C2×Dic73—42no—
80C2×Dic73—82no—
81Dic146—44no—
82Dic146—82no—
83C8×C73—82no—
84#10596831—42no—
85#10596831—42no—
86#10596831—42no—
87C22×C73—21no—
88C22×C73—22no—
89C22×C73#38611yescharacteristic
90C4×C73—21no—
91C4×C73—22no—
92C4×C73—44no—
93C4×C73#45412yes—
94C4×C73#47112yes—
95C4×C73#47312yes—
96Dic73—82no—
97#9436—731nomaximal, Sylow
98C2×C73—41no—
99C2×C73#188811yescharacteristic
100C2×C73#190812yes—
101#3139—731no—
102#2986—731no—
103#1561D7311yescharacteristic
104#1549—732no—
105#1299—732no—
106C73#943611yescharacteristic, complemented, Sylow
107#533C2×D7311yescharacteristic
108C2×C4⋊Q8—731no—
109#491—734no—
110#487—1462no—
111#446—1462no—
112C8×Q8—1462no—
113#433—1462no—
114#431—732no—
115#423C2×D7312yes—
116#419—1462no—
117C4×M16—731no—
118#332—22no—
119#332—1462no—
120#332—1462no—
121C22×Q8—1461no—
122C2×M16—22no—
123C2×M16—1461no—
124C4⋊Q8—732no—
125C4⋊Q8—1462no—
126C4⋊Q8—1461no—
127C4⋊Q8C22⋊D7312yes—
128#125C4×D7312yes—
129C22⋊Q8—22no—
130C4×Q8—22no—
131#117D29212yes—
132C2×C4⋊C4—1462no—
133C2×C42C22×D7311yescharacteristic
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—732no—
137C4⋊C8—732no—
138C4⋊C8—1464no—
139#103—1468no—
140C22⋊C8—1462no—
141#97—732no—
142#97—1462no—
143C4×C8—1462no—
144C4×C8—1462no—
145C4×C8—1461no—
146C2×Q8—22no—
147C2×Q8—2924no—
148C2×Q8—2921no—
149C2×Q8—1462no—
150C22×C4—2921no—
151C22×C4C2×D29211yescharacteristic
152C22×C4#1192402712yes—
153M16—82no—
154M16—2922no—
155C2×C8—42no—
156C2×C8—1462no—
157C2×C8—1462no—
158C2×C8—1462no—
159C4⋊C4—24no—
160C4⋊C4—22no—
161C4⋊C4—2922no—
162C4⋊C4—1464no—
163C4⋊C4#1192400514yes—
164C22⋊C4—22no—
165C42—22no—
166C42C2×C4×D7311yescharacteristic
167C42D4×D7312yes—
168C42C2×C22⋊D7311yescharacteristic
169C23#1455657511yescharacteristic
170Q8—42no—
171Q8—42no—
172Q8—2924no—
173Q8—5842no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—2922no—
179C2×C4—5842no—
180C2×C4C2×#1192400511yescharacteristic
181C2×C4#1455655112yes—
182C2×C4#1455656111yescharacteristic
183C2×C4#1455656312yes—
184C8—82no—
185C8—2922no—
186C8—2922no—
187C8—2922no—
188C22—22no—
189C22—21no—
190C22#1798846411yescharacteristic
191C4—22no—
192C4—21no—
193C4—44no—
194C4—5842no—
195C4#1798876212yes—
196C4#1798884812yes—
197C4#1798886012yes—
198C2—41no—
199C2#2261033211yescharacteristic
200C2#2261044412yes—
subgroupG/H
centerC22#17988464
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1561C2
socleC22×C73#386
prime radicalC23×C73#123
cosocleC2×C42×C73C23
hypercenter#1561D73
hypocenterC73#9436
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
106. C73#9436semidirect

Maximal subgroups

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

subgroupindex
#226203132
#226190332
#226086172
#226078202
C73×#15612
#943673