Finite Group Databasebeta

← #34001418

Subgroups — #34001418

Order 18688.

846 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 124 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#22625821C212yesmaximal, complemented
2#22624748C214yesmaximal, complemented
3C2×#17988488C211yesmaximal, characteristic
4C2×C22⋊#10596833—22no—
5#17989453—22no—
6#17989453C2214yescomplemented
7C23×D292C411yescharacteristic
8C22×#11924005—22no—
9C22×C4×Dic73C2211yescharacteristic
10#17988488—22no—
11#17988488C412yes—
12#17988488C2212yes—
13C22×C4⋊C4×C73C411yescharacteristic
14C22×#10596833—24no—
15C22⋊#10596833—44no—
16C24×D73—21no—
17C22×D292—21no—
18C22×D292—22no—
19C22×D292—42no—
20C22×D292—42no—
21C22×D292C814yescomplemented
22C22×D292C2×C412yes—
23C23×Dic73D412yes—
24C2×#11924005—24no—
25C2×#11924005—24no—
26C2×#11924005—48no—
27C2×C4×Dic73—42no—
28C2×C4×Dic73C2×C411yescharacteristic
29C2×C4×Dic73D412yes—
30C2×C4×Dic73C2311yescharacteristic
31C23×C4×C73—21no—
32C23×C4×C73C2×C411yescharacteristic
33C2×C4⋊C4×C73—21no—
34C2×C4⋊C4×C73—42no—
35C2×C4⋊C4×C73C2×C412yes—
36C2×#10596833—44no—
37C2×#10596833—44no—
38C2×#10596833—44no—
39C23×D73—24no—
40C23×D73—22no—
41C23×D73—44no—
42C23×D73—44no—
43C23×D73—42no—
44C23×D73—42no—
45C2×D292—24no—
46C2×D292—22no—
47C2×D292—48no—
48C2×D292—42no—
49C2×D292—42no—
50C2×D292—48no—
51C2×D292—42no—
52C2×D292—42no—
53C2×D292C2×C814yes—
54C22×Dic73—24no—
55C22×Dic73—24no—
56C22×Dic73—42no—
57C22×Dic73C22⋊C412yes—
58C22×Dic73C2×D412yes—
59#11924005—48no—
60#11924005—48no—
61C4×Dic73—41no—
62C4×Dic73—42no—
63C4×Dic73—41no—
64C24×C73C22⋊C411yescharacteristic
65C22×C4×C73—22no—
66C22×C4×C73—22no—
67C22×C4×C73—22no—
68C22×C4×C73—44no—
69C22×C4×C73—42no—
70C22×C4×C73C22⋊C411yescharacteristic
71C22×C4×C73C2×C812yes—
72C22×C4×C73M1612yes—
73C22×C4×C73C22×C411yescharacteristic
74C4⋊C4×C73—42no—
75C4⋊C4×C73—42no—
76C23×C73—22no—
77C23×C73—22no—
78C23×C73C22⋊C812yes—
79C23×C73#9912yes—
80C23×C73#10012yes—
81C23×C73C2×C22⋊C411yescharacteristic
82C22×D73—24no—
83C22×D73—416no—
84C22×D73—48no—
85C22×D73—42no—
86C22×D73—42no—
87C22×D73—44no—
88C22×D73—44no—
89C22×D73—84no—
90C22×D73—88no—
91C22×D73—84no—
92C2×C4×C73—24no—
93C2×C4×C73—24no—
94C2×C4×C73—21no—
95C2×C4×C73—21no—
96C2×C4×C73—21no—
97C2×C4×C73—21no—
98C2×C4×C73—42no—
99C2×C4×C73—44no—
100C2×C4×C73—84no—
101C2×C4×C73C22⋊C812yes—
102C2×C4×C73C22×C811yescharacteristic
103C2×C4×C73C2×M1611yescharacteristic
104C2×Dic73—22no—
105C2×Dic73—22no—
106C2×Dic73—22no—
107C2×Dic73—22no—
108C2×Dic73—22no—
109C2×Dic73—48no—
110C2×Dic73AGL(1,ℤ/8)14yes—
111D292—44no—
112D292—48no—
113D292—44no—
114#10596833—84no—
115C22×C73—22no—
116C22×C73—24no—
117C22×C73—24no—
118C22×C73—22no—
119C22×C73#32314yes—
120C22×C73C2×C22⋊C811yescharacteristic
121C22×C73C2×#9911yescharacteristic
122C22×C73C2×#10011yescharacteristic
123C22×C73#41812yes—
124C22×C73#42112yes—
125C2×D73—44no—
126C2×D73—48no—
127C2×D73—816no—
128C2×D73—82no—
129C2×D73—84no—
130C2×D73—82no—
131C4×C73—22no—
132C4×C73—21no—
133C4×C73—21no—
134C4×C73—82no—
135Dic73—24no—
136Dic73—42no—
137#7592—731nomaximal, Sylow
138C2×C73—24no—
139C2×C73C2×#32311yescharacteristic
140C2×C73#147812yes—
141C2×C73#151012yes—
142C2×C73#153311yescharacteristic
143C2×C73#155511yescharacteristic
144D73—84no—
145C2×#390—731no—
146#1817—732no—
147#1270—734no—
148C73#759211yescharacteristic, complemented, Sylow
149C23×D4—731no—
150C22×C4⋊C4#6440811yescharacteristic
151C22×C22⋊C4—1462no—
152C22×C42—731no—
153C2×C22⋊C8—1462no—
154#390—732no—
155#390—732no—
156#390—1462no—
157#336—734no—
158#336—1462no—
159C25—1461no—
160C22×D4—2922no—
161C22×D4—2922no—
162C22×D4—734no—
163C22×D4—732no—
164C22×D4—1461no—
165C22×D4—1462no—
166C23×C4—21no—
167C23×C4—732no—
168C23×C4C2×#6440811yescharacteristic
169C22×C8—1464no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4C2×#6440812yes—
173C2×C22⋊C4—2928no—
174C2×C22⋊C4—1464no—
175C2×C22⋊C4—1464no—
176C2×C42—2922no—
177C2×C42—732no—
178C2×C42—731no—
179C2×C42—731no—
180C22⋊C8—2924no—
181C24—2924no—
182C24—2924no—
183C24—2922no—
184C24—2922no—
185C24—1464no—
186C24—1462no—
187C24C22⋊#6440811yescharacteristic
188C2×D4—2928no—
189C2×D4—2922no—
190C2×D4—2922no—
191C2×D4—2928no—
192C2×D4—2922no—
193C2×D4—2922no—
194C2×D4—734no—
195C2×D4—1462no—
196C2×D4—1464no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—44no—
201C22×C4—42no—
202C22×C4—2922no—
203C22×C4—732no—
204C22×C4—732no—
205C22×C4—1464no—
206C22×C4—1464no—
207C22×C4C22⋊#6440811yescharacteristic
208C22×C4#1192403612yes—
209C22×C4#1192403812yes—
210C22×C4C22×#6440811yescharacteristic
211C2×C8—2924no—
212C2×C8—2924no—
213C2×C8—2924no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C22⋊C4—2928no—
217C22⋊C4—2928no—
218C42—2922no—
219C42—2921no—
220C42—2921no—
221C23—22no—
222C23—22no—
223C23—29216no—
224C23—2928no—
225C23—2922no—
226C23—2922no—
227C23—2924no—
228C23—2924no—
229C23—5848no—
230C23—5844no—
231C23—5844no—
232C23—1464no—
233C23#1455664812yes—
234C23#1455665012yes—
235C23#1455665112yes—
236C23C2×C22⋊#6440811yescharacteristic
237D4—2924no—
238D4—2924no—
239D4—2928no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—84no—
249C2×C4—2928no—
250C2×C4—734no—
251C2×C4—1462no—
252C2×C4—1462no—
253C2×C4—1462no—
254C2×C4—1462no—
255C2×C4—1462no—
256C2×C4#1455664812yes—
257C2×C4C2×#1192403611yescharacteristic
258C2×C4C2×#1192403811yescharacteristic
259C8—5844no—
260C22—24no—
261C22—22no—
262C22—22no—
263C22—24no—
264C22—2924no—
265C22—2928no—
266C22—58416no—
267C22—5844no—
268C22—5842no—
269C22—5842no—
270C22#1798943114yes—
271C22C2×#1455664811yescharacteristic
272C22C2×#1455665011yescharacteristic
273C22C2×#1455665111yescharacteristic
274C22#1798957912yes—
275C22#1798958612yes—
276C4—21no—
277C4—21no—
278C4—22no—
279C4—82no—
280C4—2922no—
281C4—1464no—
282C2—24no—
283C2—5844no—
284C2C2×#1798943111yescharacteristic
285C2#2262523212yes—
286C2#2262533812yes—
287C2#2262541211yescharacteristic
288C2#2262545811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC22×C4⋊C4×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC23×D292C4
cosocleC2×C4×Dic73C23
hypercenterC22×C4⋊C4#64408
hypocenterC73#7592
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #22625821C2semidirect
2. #22624748C2semidirect
6. #17989453C22semidirect
21. C22×D292C8semidirect
148. C73#7592semidirect

Maximal subgroups

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

subgroupindex
#226258212
#226247482
C2×#179884882
#759273