Finite Group Databasebeta

← #34001448

Subgroups — #34001448

Order 18688.

492 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 106 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#22625847C212yesmaximal, complemented
2#22624752C212yesmaximal, complemented
3#22624747C212yesmaximal
4C4×#14556578C211yesmaximal, characteristic
5C2×#14556658—22no—
6#17989453—22no—
7#17989453C2214yes—
8C2×#14556578C411yescharacteristic
9C2×C4×Dic146C411yescharacteristic
10#17988492C2211yescharacteristic
11#17988490C2211yescharacteristic
12#17988481—22no—
13#17988452C411yescharacteristic
14C4×C22⋊Dic73—22no—
15C42×Dic73C2211yescharacteristic
16C2×C4×D4×C73C411yescharacteristic
17C22×#10596833—24no—
18#14556658—44no—
19C22×Dic146C2×C411yescharacteristic
20#14556578—22no—
21#14556578C814yescomplemented
22C4×Dic146—42no—
23C2×C4⋊Dic73C2×C411yescharacteristic
24C2×#11924007—21no—
25C2×C22⋊Dic73—22no—
26C2×C22⋊Dic73—22no—
27C2×C4×Dic73—21no—
28C2×C4×Dic73C2×C411yescharacteristic
29C2×C4×Dic73D412yes—
30C2×C4×Dic73D412yes—
31C2×C4×Dic73C2311yescharacteristic
32#14556498—22no—
33#14556498—22no—
34C22×D4×C73C2×C411yescharacteristic
35C23×C4×C73—21no—
36C4×D4×C73—42no—
37C2×C4⋊C4×C73C2×C411yescharacteristic
38C2×C22⋊C4×C73—21no—
39C2×C42×C73C2×C411yescharacteristic
40C2×#10596833—44no—
41C2×#10596833—44no—
42C2×#10596833—44no—
43C2×Dic146—21no—
44C2×Dic146—41no—
45C2×Dic146—41no—
46C2×Dic146C2×C812yes—
47C22×Dic73—21no—
48C22×Dic73—21no—
49C22×Dic73C22⋊C412yes—
50C22×Dic73C2×D412yes—
51C4⋊Dic73—41no—
52#11924007—42no—
53C22⋊Dic73—28no—
54C22⋊Dic73—42no—
55C22⋊Dic73—42no—
56C4×Dic73—22no—
57C4×Dic73—22no—
58C4×Dic73—42no—
59C4×Dic73—41no—
60C4×Dic73—42no—
61C4×Dic73C2×C812yes—
62C4×Dic73M1612yes—
63C24×C73—21no—
64C2×D4×C73—21no—
65C2×D4×C73—41no—
66C2×D4×C73—41no—
67C2×D4×C73C2×C812yes—
68C22×C4×C73—21no—
69C22×C4×C73—44no—
70C22×C4×C73—42no—
71C22×C4×C73C22⋊C411yescharacteristic
72C22×C4×C73C22⋊C411yescharacteristic
73C22×C4×C73C22×C411yescharacteristic
74C4⋊C4×C73—41no—
75C22⋊C4×C73—42no—
76C42×C73—22no—
77C23×C73—22no—
78C23×C73—41no—
79C23×C73—41no—
80C23×C73—42no—
81C23×C73—42no—
82C23×C73C2×C22⋊C411yescharacteristic
83D4×C73—42no—
84D4×C73—42no—
85C2×C4×C73—21no—
86C2×C4×C73—21no—
87C2×C4×C73—22no—
88C2×C4×C73—21no—
89C2×C4×C73—22no—
90C2×C4×C73—41no—
91C2×C4×C73—42no—
92C2×C4×C73—84no—
93C2×C4×C73#10012yes—
94C2×C4×C73C22×C811yescharacteristic
95C2×C4×C73C2×M1611yescharacteristic
96C2×Dic73—22no—
97C2×Dic73—22no—
98C2×Dic73—22no—
99C2×Dic73—22no—
100C2×Dic73—22no—
101C2×Dic73—41no—
102C2×Dic73—41no—
103C2×Dic73—41no—
104C2×Dic73—42no—
105C2×Dic73C22⋊C814yes—
106C2×Dic73AGL(1,ℤ/8)12yes—
107C2×Dic73#13712yes—
108Dic146—42no—
109Dic146—42no—
110#10596833—84no—
111C22×C73—42no—
112C22×C73—44no—
113C22×C73—81no—
114C22×C73—82no—
115C22×C73—81no—
116C22×C73C2×C22⋊C811yescharacteristic
117C22×C73#41011yescharacteristic
118C22×C73C2×#10011yescharacteristic
119C22×C73#41712yes—
120C22×C73#42112yes—
121C4×C73—21no—
122C4×C73—21no—
123C4×C73—22no—
124C4×C73—41no—
125C4×C73—81no—
126Dic73—24no—
127Dic73—22no—
128Dic73—22no—
129Dic73—42no—
130Dic73—81no—
131#7601—731nomaximal, Sylow
132C2×C73—82no—
133C2×C73#146011yescharacteristic
134C2×C73#147712yes—
135C2×C73#151212yes—
136C2×C73#153311yescharacteristic
137C2×C73#156011yescharacteristic
138C4×#124—731no—
139#1831—732no—
140#1272—732no—
141#1270—732no—
142C73#760111yescharacteristic, complemented, Sylow
143C2×#124—731no—
144C2×C4×Q8—731no—
145C2×C4×D4#6440811yescharacteristic
146C2×C4⋊C8—1462no—
147#391—731no—
148#390—731no—
149#388—1462no—
150#382—731no—
151C4×C22⋊C4—1462no—
152C43—731no—
153#336—734no—
154#336—1462no—
155C22×Q8—731no—
156C22×D4C2×#6440811yescharacteristic
157C23×C4—21no—
158C22×C8—1464no—
159#124—734no—
160#124—1462no—
161C4×Q8—2922no—
162C4×D4—42no—
163C2×C4⋊C4—731no—
164C2×C4⋊C4—1461no—
165C2×C4⋊C4C2×#6440811yescharacteristic
166C2×C22⋊C4—21no—
167C2×C22⋊C4—1462no—
168C2×C22⋊C4—1462no—
169C2×C42—732no—
170C2×C42—731no—
171C2×C42—731no—
172C2×C42—732no—
173C2×C42—1461no—
174C2×C42C2×#6440811yescharacteristic
175C4⋊C8—2924no—
176#95—1462no—
177#95—1462no—
178C24—21no—
179C2×Q8—2921no—
180C2×Q8—2921no—
181C2×Q8—732no—
182C2×Q8—1461no—
183C2×D4—21no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4#1192403612yes—
187C22×C4—21no—
188C22×C4—44no—
189C22×C4—42no—
190C22×C4—732no—
191C22×C4—732no—
192C22×C4—1461no—
193C22×C4—1461no—
194C22×C4C22⋊#6440811yescharacteristic
195C22×C4C22⋊#6440811yescharacteristic
196C22×C4C22×#6440811yescharacteristic
197C2×C8—2924no—
198C2×C8—2924no—
199C2×C8—2924no—
200C4⋊C4—41no—
201C4⋊C4—2921no—
202C4⋊C4—2922no—
203C22⋊C4—42no—
204C22⋊C4—2922no—
205C22⋊C4—2922no—
206C22⋊C4—1468no—
207C42—22no—
208C42—2921no—
209C42—2922no—
210C42—2922no—
211C42—732no—
212C42—732no—
213C42—1462no—
214C42—1462no—
215C23—22no—
216C23—41no—
217C23—41no—
218C23—42no—
219C23—42no—
220C23C2×C22⋊#6440811yescharacteristic
221Q8—2922no—
222Q8—2922no—
223D4—42no—
224D4—42no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—84no—
233C2×C4—2921no—
234C2×C4—2921no—
235C2×C4—2921no—
236C2×C4—2922no—
237C2×C4—734no—
238C2×C4—732no—
239C2×C4—732no—
240C2×C4—1462no—
241C2×C4—1462no—
242C2×C4—1462no—
243C2×C4—1462no—
244C2×C4—1462no—
245C2×C4#1455665112yes—
246C2×C4C2×#1192403611yescharacteristic
247C2×C4D73⋊M1611yescharacteristic
248C8—5844no—
249C22—42no—
250C22—44no—
251C22—82no—
252C22—81no—
253C22—81no—
254C22#1798955511yescharacteristic
255C22#1798956611yescharacteristic
256C22C2×#1455665111yescharacteristic
257C22#1798957812yes—
258C22#1798958512yes—
259C4—22no—
260C4—21no—
261C4—21no—
262C4—41no—
263C4—81no—
264C4—2922no—
265C4—5841no—
266C4—1462no—
267C4—1462no—
268C4—1464no—
269C2—82no—
270C2#2262517011yescharacteristic
271C2#2262522712yes—
272C2#2262534212yes—
273C2#2262541011yescharacteristic
274C2#2262546811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC2×C4×D4×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC22×D4×C73C2×C4
cosocleC2×C4×Dic73C23
hypercenterC2×C4×D4#64408
hypocenterC73#7601
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #22625847C2semidirect
2. #22624752C2semidirect
21. #14556578C8semidirect
142. C73#7601semidirect

Maximal subgroups

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

subgroupindex
#226258472
#226247522
#226247472
C4×#145565782
#760173