Finite Group Databasebeta

← #32978360

Subgroups — #32978360

Order 18688.

620 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 131 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#22610913C211yesmaximal, characteristic
2#22608661C212yesmaximal
3#22608569C211yesmaximal, characteristic
4#22608443C212yesmaximal
5C73×#3556C211yesmaximal, characteristic
6#17988644C2212yes—
7#17988611C2211yescharacteristic
8#17988433C2211yescharacteristic
9#17988346—22no—
10#17988336—22no—
11#17988333—22no—
12#17988329—22no—
13#17988324—24no—
14#564×C73—22no—
15#557×C73—22no—
16#557×C73—22no—
17#557×C73C414yescomplemented
18#541×C73—24no—
19#541×C73—22no—
20#533×C73C2212yes—
21#533×C73C2211yescharacteristic
22C2×#11924012—21no—
23C2×C4×Dic73—21no—
24C4⋊#10596831—22no—
25C4⋊#10596831—22no—
26C22⋊#10596831—22no—
27#14556502—22no—
28#14556498—22no—
29C4⋊Q8×C73—22no—
30C4⋊Q8×C73—24no—
31C4⋊Q8×C73C2×C412yes—
32C4⋊Q8×C73D412yes—
33C4⋊Q8×C73D412yes—
34#126×C73—28no—
35#125×C73—22no—
36#125×C73—28no—
37#125×C73—28no—
38#125×C73—24no—
39#125×C73—28no—
40#125×C73—22no—
41#125×C73D412yes—
42C22⋊Q8×C73—22no—
43C22⋊Q8×C73—22no—
44C22⋊Q8×C73D414yes—
45C4×Q8×C73—24no—
46C4×Q8×C73—24no—
47C4×Q8×C73—24no—
48C4×Q8×C73—22no—
49C4×Q8×C73—22no—
50C4×Q8×C73C2×C414yes—
51#117×C73—22no—
52#117×C73D412yes—
53C2×C42×C73C2311yescharacteristic
54C22×Dic73—41no—
55#11924012—42no—
56C2×#10596831—42no—
57C2×#10596831—41no—
58C4×Dic73—22no—
59C4×Dic73—41no—
60C2×Q8×C73—22no—
61C2×Q8×C73—22no—
62C2×Q8×C73C22⋊C414yes—
63C22×C4×C73C2×D411yescharacteristic
64C22×C4×C73C2×D412yes—
65C4⋊C4×C73—28no—
66C4⋊C4×C73—28no—
67C4⋊C4×C73—28no—
68C4⋊C4×C73—22no—
69C4⋊C4×C73—24no—
70C4⋊C4×C73—22no—
71C4⋊C4×C73—22no—
72C4⋊C4×C73—22no—
73C4⋊C4×C73—24no—
74C4⋊C4×C73—24no—
75C4⋊C4×C73C22⋊C414yes—
76C4⋊C4×C73C22⋊C414yes—
77C22⋊C4×C73—24no—
78C22⋊C4×C73—22no—
79C42×C73—28no—
80C42×C73—24no—
81C42×C73—22no—
82C42×C73C22×C411yescharacteristic
83C42×C73C2×D412yes—
84C42×C73C2×D411yescharacteristic
85C23×C73#12011yescharacteristic
86Q8×C73—44no—
87Q8×C73—44no—
88Q8×C73—42no—
89Q8×C73—42no—
90C2×C4×C73—21no—
91C2×C4×C73—21no—
92C2×C4×C73—24no—
93C2×C4×C73—24no—
94C2×C4×C73—24no—
95C2×C4×C73—42no—
96C2×C4×C73C2×C22⋊C412yes—
97C2×C4×C73C2×C22⋊C411yescharacteristic
98C2×C4×C73#12012yes—
99C2×C4×C73#12011yescharacteristic
100C2×Dic73—42no—
101C2×Dic73—82no—
102#10596831—82no—
103#10596831—81no—
104C22×C73—21no—
105C22×C73—22no—
106C22×C73#37911yescharacteristic
107C4×C73—21no—
108C4×C73—22no—
109C4×C73—22no—
110C4×C73—44no—
111C4×C73—44no—
112C4×C73—44no—
113C4×C73#45411yescharacteristic
114C4×C73#45611yescharacteristic
115Dic73—82no—
116#8495—731nomaximal, Sylow
117C2×C73—41no—
118C2×C73#179311yescharacteristic
119C2×C73#179612yes—
120#3556D7311yescharacteristic
121#1995—731no—
122#1559—732no—
123#1537—731no—
124#1506—732no—
125C73#849511yescharacteristic, complemented, Sylow
126#564—22no—
127#557—22no—
128#557—22no—
129#557Dic7314yescomplemented
130#541—24no—
131#541—22no—
132#533C2×D7312yes—
133#533C2×D7311yescharacteristic
134#432—732no—
135#423—731no—
136#376—731no—
137#339—1462no—
138#333—1462no—
139#332—1462no—
140#330—1462no—
141#328—1464no—
142C2×M16—1461no—
143C4⋊Q8—22no—
144C4⋊Q8—24no—
145C4⋊Q8C2×Dic7312yes—
146C4⋊Q8C22⋊D7312yes—
147C4⋊Q8C22⋊D7312yes—
148#126—28no—
149#125—22no—
150#125—28no—
151#125—28no—
152#125—24no—
153#125—28no—
154#125—22no—
155#125C22⋊D7312yes—
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158C22⋊Q8C22⋊D7314yes—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8C2×Dic7314yes—
165#117—22no—
166#117C22⋊D7312yes—
167C2×C42—1461no—
168C2×C42C22×D7311yescharacteristic
169C4⋊C8—1462no—
170C4⋊C8—1462no—
171C22⋊C8—1462no—
172#97—1462no—
173#95—1462no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8C22⋊Dic7314yes—
177C22×C4—2921no—
178C22×C4C2×C22⋊D7311yescharacteristic
179C22×C4C2×C22⋊D7312yes—
180M16—2922no—
181C2×C8—2922no—
182C2×C8—2921no—
183C4⋊C4—28no—
184C4⋊C4—28no—
185C4⋊C4—28no—
186C4⋊C4—22no—
187C4⋊C4—24no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—24no—
192C4⋊C4—24no—
193C4⋊C4C22⋊Dic7314yes—
194C4⋊C4C22⋊Dic7314yes—
195C22⋊C4—24no—
196C22⋊C4—22no—
197C42—28no—
198C42—24no—
199C42—22no—
200C42—2921no—
201C42—1462no—
202C42C22×Dic7311yescharacteristic
203C42C2×C22⋊D7311yescharacteristic
204C42C2×C22⋊D7312yes—
205C23#1455656011yescharacteristic
206Q8—44no—
207Q8—44no—
208Q8—42no—
209Q8—42no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—24no—
215C2×C4—42no—
216C2×C4—2922no—
217C2×C4—5842no—
218C2×C4C2×C22⋊Dic7312yes—
219C2×C4C2×C22⋊Dic7311yescharacteristic
220C2×C4#1455656012yes—
221C2×C4#1455656011yescharacteristic
222C8—5842no—
223C8—5841no—
224C22—21no—
225C22—22no—
226C22#1798844311yescharacteristic
227C4—21no—
228C4—22no—
229C4—22no—
230C4—44no—
231C4—44no—
232C4—44no—
233C4—5842no—
234C4#1798876011yescharacteristic
235C4#1798877011yescharacteristic
236C2—41no—
237C2#2260989011yescharacteristic
238C2#2260990612yes—
subgroupG/H
centerC22#17988443
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#3556C2
socleC22×C73#379
prime radicalC23×C73#120
cosocleC2×C42×C73C23
hypercenter#3556D73
hypocenterC73#8495
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3556

Complements

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

NH ≅ G/Ntype
17. #557×C73C4semidirect
125. C73#8495semidirect
129. #557Dic73semidirect

Maximal subgroups

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

subgroupindex
#226109132
#226086612
#226085692
#226084432
C73×#35562
#849573