Finite Group Databasebeta

← #34004119

Subgroups — #34004119

Order 18688.

1174 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 1174 conjugacy classes of subgroups — too many to draw. Only its 136 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#22625754C212yesmaximal
2C2×#17989432C214yesmaximal, complemented
3C22×#14556571C211yesmaximal, characteristic
4C2×#14556648—24no—
5C2×C22⋊#10596833C2214yes—
6#17989432—28no—
7C23×Dic146C411yescharacteristic
8C23×C4×D73C2211yescharacteristic
9C2×#14556571—24no—
10C2×#14556571—28no—
11C22×C4⋊Dic73C2212yes—
12C22×#11924005C412yes—
13C22×C4⋊C4×C73C411yescharacteristic
14C22×#10596833—24no—
15#14556648—48no—
16C22⋊#10596833—28no—
17C24×D73C2×C411yescharacteristic
18C22×Dic146—24no—
19C22×Dic146—22no—
20C22×Dic146—44no—
21C23×Dic73C2×C412yes—
22C23×Dic73C2311yescharacteristic
23C22×C4×D73—24no—
24C22×C4×D73—44no—
25C22×C4×D73D414yes—
26#14556571—232no—
27C2×C4⋊Dic73—24no—
28C2×C4⋊Dic73—24no—
29C2×C4⋊Dic73D418yes—
30C2×#11924005—24no—
31C2×#11924005—24no—
32C2×#11924005—24no—
33C23×C4×C73C2×C412yes—
34C23×C4×C73C2×C411yescharacteristic
35C2×C4⋊C4×C73—24no—
36C2×C4⋊C4×C73—22no—
37C2×#10596833—48no—
38C2×#10596833—44no—
39C23×D73—22no—
40C23×D73—24no—
41C23×D73—24no—
42C23×D73—42no—
43C23×D73C22⋊C412yes—
44C2×Dic146—28no—
45C2×Dic146—416no—
46C2×Dic146—44no—
47C2×Dic146—44no—
48C22×Dic73—22no—
49C22×Dic73—22no—
50C22×Dic73—42no—
51C22×Dic73—44no—
52C22×Dic73C22⋊C414yes—
53C22×Dic73C2×D412yes—
54C22×Dic73C2×D414yes—
55C2×C4×D73—28no—
56C2×C4×D73—28no—
57C2×C4×D73—44no—
58C2×C4×D73—416no—
59C4⋊Dic73—216no—
60#11924005—216no—
61C24×C73C22×C411yescharacteristic
62C22×C4×C73—22no—
63C22×C4×C73—24no—
64C22×C4×C73—22no—
65C22×C4×C73—22no—
66C22×C4×C73—28no—
67C22×C4×C73C22⋊C414yes—
68C22×C4×C73C22⋊C412yes—
69C4⋊C4×C73—28no—
70C23×C73—22no—
71C23×C73—22no—
72C23×C73C2×C22⋊C412yes—
73C23×C73C2×C22⋊C411yescharacteristic
74C23×C73#13114yes—
75C22×D73—28no—
76C22×D73—22no—
77C22×D73—24no—
78C22×D73—24no—
79C22×D73—22no—
80C22×D73—416no—
81C22×D73—44no—
82C22×D73—44no—
83C22×D73#10114yes—
84C2×C4×C73—28no—
85C2×C4×C73—22no—
86C2×C4×C73—22no—
87C2×C4×C73—24no—
88C2×C4×C73—28no—
89C2×C4×C73—44no—
90C2×C4×C73—44no—
91C2×C4×C73#9914yes—
92C2×Dic73—24no—
93C2×Dic73—24no—
94C2×Dic73—44no—
95C2×Dic73—44no—
96C2×Dic73—44no—
97C2×Dic73—44no—
98C2×Dic73#12014yes—
99C4×D73—416no—
100Dic146—416no—
101#10596833—84no—
102C22×C73—28no—
103C22×C73—22no—
104C22×C73—22no—
105C22×C73#37911yescharacteristic
106C22×C73#40814yes—
107C22×C73#40812yes—
108C22×C73C2×#9912yes—
109C22×C73C2×#10112yes—
110C2×D73—28no—
111C2×D73—24no—
112C2×D73—416no—
113C2×D73—44no—
114C2×D73—42no—
115C2×D73—42no—
116C4×C73—24no—
117C4×C73—44no—
118Dic73—24no—
119Dic73—44no—
120#8308—731nomaximal, Sylow
121C2×C73—24no—
122C2×C73#147214yes—
123C2×C73#178011yescharacteristic
124C2×C73#178111yescharacteristic
125C2×C73#178411yescharacteristic
126D73—44no—
127C22×C22⋊Q8—731no—
128#1779—732no—
129C2×#324—734no—
130C73#830811yescharacteristic, complemented, Sylow
131C23×Q8—731no—
132C24×C4—731no—
133C2×C22⋊Q8—1468no—
134C2×C22⋊Q8—1464no—
135C22×C4⋊C4—732no—
136C22×C4⋊C4#6440811yescharacteristic
137C22×C22⋊C4—732no—
138C2×C22⋊C8—734no—
139C2×C22⋊C8—1464no—
140#324—1468no—
141C25—731no—
142C22×Q8—2924no—
143C22×Q8—1464no—
144C22×Q8—1462no—
145C23×C4—2924no—
146C23×C4—731no—
147C23×C4—732no—
148C23×C4—734no—
149C23×C4—1464no—
150C23×C4C2×#6440812yes—
151C23×C4C2×#6440811yescharacteristic
152C22×C8—1464no—
153C22⋊Q8—14632no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—738no—
157C2×C4⋊C4—1464no—
158C2×C4⋊C4—1464no—
159C2×C22⋊C4—1464no—
160C2×C22⋊C4—1464no—
161C2×C22⋊C4—1464no—
162C22⋊C8—2928no—
163C22⋊C8—1468no—
164C24—2922no—
165C24—732no—
166C24—1464no—
167C24—1464no—
168C24—1462no—
169C24C22×#6440811yescharacteristic
170C2×Q8—29216no—
171C2×Q8—2924no—
172C2×Q8—2924no—
173C2×Q8—1468no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—28no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—29216no—
180C22×C4—2924no—
181C22×C4—2924no—
182C22×C4—2922no—
183C22×C4—734no—
184C22×C4—734no—
185C22×C4—732no—
186C22×C4—1468no—
187C22×C4—1462no—
188C22×C4—1462no—
189C22×C4—1468no—
190C22×C4C22⋊#6440814yes—
191C22×C4C22⋊#6440812yes—
192C2×C8—2924no—
193C2×C8—2928no—
194C4⋊C4—28no—
195C4⋊C4—14616no—
196C22⋊C4—14616no—
197C23—22no—
198C23—22no—
199C23—2924no—
200C23—29216no—
201C23—2924no—
202C23—734no—
203C23—1468no—
204C23—1462no—
205C23—1464no—
206C23—1464no—
207C23—1462no—
208C23C2×C22⋊#6440812yes—
209C23C2×C22⋊#6440811yescharacteristic
210C23#1455667814yes—
211Q8—29216no—
212C2×C4—28no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—28no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—29216no—
220C2×C4—2924no—
221C2×C4—2924no—
222C2×C4—2924no—
223C2×C4—2924no—
224C2×C4—734no—
225C2×C4—1464no—
226C2×C4—1464no—
227C2×C4#1455665014yes—
228C8—5844no—
229C22—28no—
230C22—22no—
231C22—22no—
232C22—29216no—
233C22—2924no—
234C22—2922no—
235C22—2922no—
236C22—1468no—
237C22—1464no—
238C22#1798952811yescharacteristic
239C22#1798956014yes—
240C22#1798956112yes—
241C22C2×#1455665012yes—
242C22D73⋊#10112yes—
243C4—24no—
244C4—44no—
245C4—2924no—
246C4—1464no—
247C2—24no—
248C2—2924no—
249C2#2262520914yes—
250C2#2262575511yescharacteristic
251C2#2262575911yescharacteristic
252C2#2262576611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC24×C73C22×C4
FrattiniC24C22×#64408
FittingC22×C4⋊C4×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC24×D73C2×C4
cosocleC23×Dic73C23
hypercenterC22×C4⋊C4#64408
hypocenterC73#8308
LCSG ▷ C24×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#17989432C2semidirect
130. C73#8308semidirect

Maximal subgroups

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

subgroupindex
#226257542
C2×#179894322
C22×#145565712
#830873