Finite Group Databasebeta

← #32975213

Subgroups — #32975213

Order 18688.

410 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 87 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#22616963C211yesmaximal, characteristic
2#22611408C212yesmaximal
3#22607757C211yesmaximal, characteristic
4#22607755C212yesmaximal
5C73×#2096C211yesmaximal, characteristic
6C2×#14556589—21no—
7C2×#14556522—22no—
8#17988517—21no—
9#17988508—22no—
10#17988485C2212yes—
11#17988472—22no—
12#17988349—22no—
13#17988342C2212yes—
14C2×#106×C73—21no—
15C2×#103×C73—21no—
16#391×C73C2211yescharacteristic
17#384×C73C2211yescharacteristic
18#336×C73C2211yescharacteristic
19C22×Dic146—21no—
20C22×#10596831—22no—
21#14556589—42no—
22#14556589—42no—
23C2×C4⋊Dic73—21no—
24C2×C4⋊Dic73D412yes—
25C2×C4⋊Dic73D412yes—
26C2×#11924007—22no—
27C2×#11924007—22no—
28C2×C4×Dic73—21no—
29#14556522—44no—
30#14556498—21no—
31#14556498—22no—
32#14556498—22no—
33C22×Q8×C73D411yescharacteristic
34C22×C8×C73—21no—
35C2×C4⋊C4×C73—22no—
36C2×C4⋊C4×C73—21no—
37C2×C4⋊C4×C73D411yescharacteristic
38C2×C42×C73C2311yescharacteristic
39#106×C73—42no—
40#103×C73—42no—
41#95×C73—22no—
42C2×Dic146—42no—
43C2×Dic146—81no—
44C2×Dic146—81no—
45C22×Dic73—22no—
46C22×Dic73—22no—
47C2×#10596831—24no—
48C2×#10596831—42no—
49C2×#10596831—42no—
50C4⋊Dic73—22no—
51C4⋊Dic73—42no—
52C4⋊Dic73—81no—
53C4⋊Dic73SD1614yes—
54#11924007—44no—
55#11924007—42no—
56#11924007—42no—
57C4×Dic73—42no—
58C2×Q8×C73—21no—
59C2×Q8×C73—41no—
60C2×Q8×C73—41no—
61C2×Q8×C73SD1612yes—
62C22×C4×C73—21no—
63C22×C4×C73—21no—
64C22×C4×C73—41no—
65C22×C4×C73C2×D412yes—
66C22×C4×C73C2×D411yescharacteristic
67C2×C8×C73—22no—
68C2×C8×C73—41no—
69C2×C8×C73—41no—
70C4⋊C4×C73—41no—
71C4⋊C4×C73—41no—
72C4⋊C4×C73—41no—
73C4⋊C4×C73—42no—
74C4⋊C4×C73—42no—
75C42×C73—41no—
76C23×C73#12011yescharacteristic
77Q8×C73—42no—
78Q8×C73—81no—
79C2×C4×C73—21no—
80C2×C4×C73—22no—
81C2×C4×C73—21no—
82C2×C4×C73—22no—
83C2×C4×C73—22no—
84C2×C4×C73—41no—
85C2×C4×C73—41no—
86C2×C4×C73—41no—
87C2×C4×C73—42no—
88C2×C4×C73—82no—
89C2×C4×C73—81no—
90C2×C4×C73C2×SD1612yes—
91C2×C4×C73C2×SD1611yescharacteristic
92C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
93C2×C4×C73#13712yes—
94C2×Dic73—24no—
95C2×Dic73—42no—
96C2×Dic73—42no—
97C2×Dic73—42no—
98C2×Dic73—42no—
99C2×Dic73—42no—
100Dic146—84no—
101C8×C73—42no—
102#10596831—44no—
103C22×C73#44812yes—
104C22×C73#45011yescharacteristic
105C22×C73#45311yescharacteristic
106C22×C73#45612yes—
107C22×C73#45811yescharacteristic
108C4×C73—22no—
109C4×C73—22no—
110C4×C73—21no—
111C4×C73—21no—
112C4×C73—42no—
113C4×C73—81no—
114C4×C73—161no—
115Dic73—44no—
116Dic73—82no—
117#7817—731nomaximal, Sylow
118C2×C73#161112yes—
119C2×C73#161311yescharacteristic
120C2×C73#162311yescharacteristic
121C2×C73#165112yes—
122C2×C73#166011yescharacteristic
123#2734—731no—
124#2096D7311yescharacteristic
125#2084—732no—
126#1272—731no—
127#1271—732no—
128C73#781711yescharacteristic, complemented, Sylow
129C2×C4⋊Q8—1461no—
130C2×#106—21no—
131C2×#106—1462no—
132C2×#103—21no—
133#398—1461no—
134#395—1462no—
135#391C2×D7311yescharacteristic
136#389—732no—
137#387—1462no—
138#384C2×D7311yescharacteristic
139#340—1462no—
140#336—732no—
141#336C2×D7311yescharacteristic
142C22×Q8—1461no—
143C22×Q8C22⋊D7311yescharacteristic
144C22×C8—21no—
145C22×C8—1462no—
146C4⋊Q8—2922no—
147C4⋊Q8—2922no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—732no—
151C2×C4⋊C4—732no—
152C2×C4⋊C4—1462no—
153C2×C4⋊C4—1462no—
154C2×C4⋊C4—1461no—
155C2×C4⋊C4C22⋊D7311yescharacteristic
156C2×C42—1461no—
157C2×C42C22×D7311yescharacteristic
158#106—42no—
159#106—2924no—
160#103—42no—
161#95—22no—
162#95—1461no—
163#95—1462no—
164#95—1462no—
165C2×Q8—21no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—2922no—
169C2×Q8—5841no—
170C2×Q8—5841no—
171C2×Q8#1192401512yes—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—41no—
175C22×C4—1462no—
176C22×C4—1462no—
177C22×C4D4×D7312yes—
178C22×C4C2×C22⋊D7311yescharacteristic
179C2×C8—22no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—2922no—
183C2×C8—2922no—
184C2×C8—1464no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—2922no—
191C4⋊C4—2924no—
192C4⋊C4—2922no—
193C4⋊C4—2922no—
194C4⋊C4—5841no—
195C4⋊C4—734no—
196C4⋊C4—1462no—
197C42—41no—
198C42—2922no—
199C23#1455655911yescharacteristic
200Q8—42no—
201Q8—81no—
202Q8—5844no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—82no—
213C2×C4—81no—
214C2×C4—2922no—
215C2×C4—2922no—
216C2×C4—2922no—
217C2×C4—2922no—
218C2×C4—2922no—
219C2×C4—1464no—
220C2×C4SD16×D7312yes—
221C2×C4C2×#1192401511yescharacteristic
222C2×C4#1455661611yescharacteristic
223C2×C4#1455662012yes—
224C8—42no—
225C8—2924no—
226C22#1798872412yes—
227C22#1798873711yescharacteristic
228C22#1798875411yescharacteristic
229C22#1798876812yes—
230C22#1798877611yescharacteristic
231C4—22no—
232C4—22no—
233C4—21no—
234C4—21no—
235C4—42no—
236C4—81no—
237C4—161no—
238C4—2924no—
239C4—5842no—
240C2#2260893712yes—
241C2#2260895011yescharacteristic
242C2#2260900711yescharacteristic
243C2#2260919312yes—
244C2#2260924011yescharacteristic
subgroupG/H
centerC23#14556559
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#2096C2
socle, prime radicalC23×C73#120
hypercenter#2096D73
hypocenterC73#7817
LCSG ▷ C2×C42×C73 ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2096

Complements

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

NH ≅ G/Ntype
128. C73#7817semidirect

Maximal subgroups

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

subgroupindex
#226169632
#226114082
#226077572
#226077552
C73×#20962
#781773