Finite Group Databasebeta

← #32990080

Subgroups — #32990080

Order 18688.

344 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 81 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#22610050C211yesmaximal, characteristic
2#22608631C212yesmaximal
3#22608443C212yesmaximal
4#22607801C211yesmaximal, characteristic
5C73×#3249C211yesmaximal, characteristic
6#17988644C2212yes—
7C2×C4⋊#10596831C2211yescharacteristic
8#17988433C2211yescharacteristic
9#17988341—22no—
10#17988333—22no—
11#17988333—22no—
12#17988324—24no—
13#533×C73C2212yes—
14#477×C73—22no—
15#467×C73—22no—
16#462×C73—22no—
17C2×C4⋊C8×C73C2211yescharacteristic
18C22×#10596831—21no—
19C2×C4×Dic73—21no—
20C4⋊#10596831—22no—
21C4⋊#10596831—22no—
22C22⋊#10596831—22no—
23#14556502—22no—
24#14556498—21no—
25#14556498D412yes—
26C2×Q16×C73—22no—
27C22×C8×C73D412yes—
28C4⋊Q8×C73D412yes—
29C4⋊Q8×C73D412yes—
30#125×C73—22no—
31C22⋊Q8×C73—22no—
32C22⋊Q8×C73—22no—
33C4×Q8×C73—22no—
34C4×Q8×C73—22no—
35#117×C73—22no—
36C2×C42×C73C2311yescharacteristic
37#107×C73—22no—
38C4⋊C8×C73—22no—
39#103×C73—24no—
40C22×Dic73—22no—
41C2×#10596831—42no—
42C2×#10596831—41no—
43C2×#10596831—42no—
44C4×Dic73—41no—
45C4×Dic73—41no—
46C2×Q8×C73—22no—
47C2×Q8×C73—22no—
48C22×C4×C73C2×D411yescharacteristic
49C22×C4×C73C2×D411yescharacteristic
50C22×C4×C73#4111yescharacteristic
51Q16×C73—82no—
52C2×C8×C73—22no—
53C2×C8×C73—42no—
54C4⋊C4×C73—22no—
55C4⋊C4×C73—22no—
56C4⋊C4×C73—22no—
57C4⋊C4×C73—22no—
58C4⋊C4×C73—42no—
59C4⋊C4×C73—42no—
60C22⋊C4×C73—42no—
61C42×C73—42no—
62C42×C73C2×D412yes—
63C42×C73#4112yes—
64C23×C73#12111yescharacteristic
65Q8×C73—42no—
66Q8×C73—42no—
67Q8×C73—82no—
68C2×C4×C73—22no—
69C2×C4×C73—21no—
70C2×C4×C73—21no—
71C2×C4×C73—41no—
72C2×C4×C73—42no—
73C2×C4×C73—42no—
74C2×C4×C73#12011yescharacteristic
75C2×C4×C73#12112yes—
76C2×C4×C73#12312yes—
77C2×C4×C73#12411yescharacteristic
78C2×Dic73—42no—
79C2×Dic73—42no—
80C2×Dic73—42no—
81C8×C73—42no—
82#10596831—82no—
83#10596831—81no—
84C22×C73—21no—
85C22×C73—21no—
86C22×C73#39411yescharacteristic
87C22×C73#47312yes—
88C4×C73—22no—
89C4×C73—21no—
90C4×C73—22no—
91C4×C73—82no—
92C4×C73—82no—
93C4×C73#45411yescharacteristic
94C4×C73#45511yescharacteristic
95Dic73—82no—
96#10517—731nomaximal, Sylow
97C2×C73—22no—
98C2×C73#204011yescharacteristic
99C2×C73#204711yescharacteristic
100C2×C73#205311yescharacteristic
101#3249D7311yescharacteristic
102#1832—731no—
103#1552—732no—
104#1506—732no—
105#1290—731no—
106C73#1051711yescharacteristic, complemented, Sylow
107#533C2×D7312yes—
108#477—22no—
109#467—22no—
110#462—22no—
111#432—732no—
112C2×C4⋊C8—731no—
113C2×C4⋊C8C2×D7311yescharacteristic
114#376—731no—
115#336—1462no—
116#332—1462no—
117#332—1462no—
118#328—1464no—
119C2×Q16—22no—
120C22×C8—1461no—
121C22×C8C22⋊D7312yes—
122C4⋊Q8C22⋊D7312yes—
123C4⋊Q8C22⋊D7312yes—
124#125—22no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129#117—22no—
130C2×C42—1461no—
131C2×C42C22×D7311yescharacteristic
132#107—22no—
133C4⋊C8—22no—
134C4⋊C8—1462no—
135C4⋊C8—1462no—
136#103—24no—
137C22⋊C8—1462no—
138#97—1462no—
139#95—732no—
140#95—1461no—
141C2×Q8—22no—
142C2×Q8—22no—
143C22×C4—1462no—
144C22×C4D4×D7311yescharacteristic
145C22×C4C2×C22⋊D7311yescharacteristic
146C22×C4#1192402711yescharacteristic
147Q16—82no—
148C2×C8—22no—
149C2×C8—42no—
150C2×C8—2922no—
151C2×C8—2921no—
152C2×C8—2922no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C22⋊C4—42no—
160C42—42no—
161C42—2921no—
162C42—2921no—
163C42C2×C22⋊D7312yes—
164C42#1192402712yes—
165C23#1455656211yescharacteristic
166Q8—42no—
167Q8—42no—
168Q8—82no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—2922no—
176C2×C4—2922no—
177C2×C4—2922no—
178C2×C4#1455656011yescharacteristic
179C2×C4#1455656312yes—
180C2×C4#1455657212yes—
181C2×C4#1455657811yescharacteristic
182C8—42no—
183C8—5842no—
184C8—5841no—
185C22—21no—
186C22—21no—
187C22#1798850111yescharacteristic
188C22#1798885912yes—
189C4—22no—
190C4—21no—
191C4—22no—
192C4—82no—
193C4—82no—
194C4—5842no—
195C4#1798876011yescharacteristic
196C4#1798876511yescharacteristic
197C2—22no—
198C2#2261115711yescharacteristic
199C2#2261119611yescharacteristic
200C2#2261123411yescharacteristic
subgroupG/H
centerC22#17988501
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#3249C2
socleC22×C73#394
prime radicalC23×C73#121
hypercenter#3249D73
hypocenterC73#10517
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3249

Complements

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

NH ≅ G/Ntype
106. C73#10517semidirect

Maximal subgroups

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

subgroupindex
#226100502
#226086312
#226084432
#226078012
C73×#32492
#1051773