Finite Group Databasebeta

← #34012535

Subgroups — #34012535

Order 18688.

512 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C2×#17989495C212yesmaximal
2C4×#14556652C212yesmaximal, complemented
3#22625029C212yesmaximal, complemented
4#22612643C211yesmaximal, characteristic
5C2×#14556652C2214yes—
6C4×#11924037—22no—
7#17989495C418yescomplemented
8#17989467—22no—
9C2×#14556578C2211yescharacteristic
10C2×C4×Dic146C2211yescharacteristic
11#17988492C2211yescharacteristic
12#17988490C411yescharacteristic
13#17988475—22no—
14#17988460—22no—
15#17988456C411yescharacteristic
16#17988433C411yescharacteristic
17C2×C4×D4×C73C411yescharacteristic
18C2×#11924037—24no—
19#14556652—24no—
20#14556652C2×C418yescomplemented
21#14556646—42no—
22C4×#10596833—42no—
23C22×Dic146C2311yescharacteristic
24#14556578—22no—
25#14556578C2×C414yes—
26C4×Dic146—42no—
27C2×C4⋊Dic73C2×C411yescharacteristic
28C2×#11924007—22no—
29C2×#11924007D412yes—
30C2×C22⋊Dic73—22no—
31C2×C22⋊Dic73—22no—
32C2×C4×Dic73C2×C411yescharacteristic
33C2×C4×Dic73C2×C411yescharacteristic
34C2×C4×Dic73D412yes—
35#14556498—22no—
36#14556498—21no—
37#14556498—21no—
38C22×D4×C73C2×C411yescharacteristic
39C23×C4×C73—21no—
40C4×D4×C73—42no—
41C2×C4⋊C4×C73C2×C411yescharacteristic
42C2×C22⋊C4×C73—21no—
43C2×C42×C73C2×C411yescharacteristic
44#11924037—28no—
45#11924037—44no—
46C2×#10596833—44no—
47C2×Dic146—41no—
48C2×Dic146—41no—
49C2×Dic146C22×C412yes—
50C2×Dic146#4112yes—
51C22×Dic73—21no—
52C22×Dic73—21no—
53C22×Dic73C22⋊C412yes—
54C22×Dic73C2×D412yes—
55C4⋊Dic73—41no—
56#11924007—42no—
57#11924007—82no—
58C22⋊Dic73—28no—
59C22⋊Dic73—82no—
60C4×Dic73—22no—
61C4×Dic73—22no—
62C4×Dic73—21no—
63C4×Dic73—41no—
64C4×Dic73C4212yes—
65C24×C73—21no—
66C2×D4×C73—21no—
67C2×D4×C73—41no—
68C2×D4×C73—41no—
69C2×D4×C73C4212yes—
70C22×C4×C73—21no—
71C22×C4×C73—42no—
72C22×C4×C73—82no—
73C22×C4×C73C22⋊C411yescharacteristic
74C22×C4×C73C22⋊C411yescharacteristic
75C22×C4×C73C22×C411yescharacteristic
76C4⋊C4×C73—41no—
77C22⋊C4×C73—42no—
78C42×C73—22no—
79C23×C73—22no—
80C23×C73—42no—
81C23×C73—42no—
82C23×C73—41no—
83C23×C73—41no—
84C23×C73C2×C22⋊C411yescharacteristic
85D4×C73—42no—
86D4×C73—42no—
87C2×C4×C73—21no—
88C2×C4×C73—21no—
89C2×C4×C73—21no—
90C2×C4×C73—41no—
91C2×C4×C73—41no—
92C2×C4×C73—42no—
93C2×C4×C73—41no—
94C2×C4×C73—84no—
95C2×C4×C73#9912yes—
96C2×C4×C73C2×C4211yescharacteristic
97C2×C4×C73#11711yescharacteristic
98C2×Dic73—22no—
99C2×Dic73—22no—
100C2×Dic73—22no—
101C2×Dic73—22no—
102C2×Dic73—41no—
103C2×Dic73—41no—
104C2×Dic73—42no—
105C2×Dic73—41no—
106C2×Dic73—41no—
107C2×Dic73—41no—
108C2×Dic73C4×D414yes—
109Dic146—42no—
110Dic146—42no—
111#10596833—48no—
112C22×C73—21no—
113C22×C73—21no—
114C22×C73—44no—
115C22×C73—42no—
116C22×C73—82no—
117C22×C73—81no—
118C22×C73—81no—
119C22×C73C4×C22⋊C411yescharacteristic
120C22×C73C2×#9911yescharacteristic
121C22×C73#41011yescharacteristic
122C4×C73—22no—
123C4×C73—21no—
124C4×C73—21no—
125C4×C73—41no—
126C4×C73—81no—
127Dic73—24no—
128Dic73—42no—
129Dic73—42no—
130Dic73—81no—
131#12632—731nomaximal, Sylow
132C2×C73—21no—
133C2×C73—21no—
134C2×C73—82no—
135C2×C73C4×#9911yescharacteristic
136C2×C73#213311yescharacteristic
137C2×C73#213411yescharacteristic
138#2326—731no—
139C2×#355—732no—
140C4×#101—732no—
141#1394—732no—
142C73#1263211yescharacteristic, complemented, Sylow
143C2×#124—731no—
144C2×C4×Q8—731no—
145C2×C4×D4#6440811yescharacteristic
146C2×#101—734no—
147C4×M16—1462no—
148#391—731no—
149#390—731no—
150#388—1462no—
151#385—1462no—
152#384—731no—
153#376—731no—
154#355—738no—
155#343—1462no—
156C22×Q8—731no—
157C22×D4C2×#6440811yescharacteristic
158C23×C4—21no—
159C2×M16—1464no—
160#124—734no—
161#124—1462no—
162C4×Q8—2922no—
163C4×D4—42no—
164C2×C4⋊C4—732no—
165C2×C4⋊C4—731no—
166C2×C4⋊C4—1462no—
167C2×C4⋊C4C2×#6440811yescharacteristic
168C2×C22⋊C4—21no—
169C2×C22⋊C4—1462no—
170C2×C22⋊C4—1462no—
171C2×C42—731no—
172C2×C42—731no—
173C2×C42—732no—
174C2×C42C2×#6440811yescharacteristic
175#101—738no—
176#101—1464no—
177#97—2922no—
178C4×C8—2922no—
179#95—1462no—
180#95—1461no—
181#95—1461no—
182C24—21no—
183C2×Q8—2921no—
184C2×Q8—2921no—
185C2×Q8—732no—
186C2×Q8—732no—
187C2×D4—21no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4C4×#6440812yes—
191C22×C4—21no—
192C22×C4—42no—
193C22×C4—82no—
194C22×C4—732no—
195C22×C4—732no—
196C22×C4—1461no—
197C22×C4—1461no—
198C22×C4C22⋊#6440811yescharacteristic
199C22×C4C22⋊#6440811yescharacteristic
200C22×C4C22×#6440811yescharacteristic
201M16—2924no—
202M16—1468no—
203C2×C8—2924no—
204C4⋊C4—41no—
205C4⋊C4—2922no—
206C4⋊C4—2921no—
207C4⋊C4—5842no—
208C22⋊C4—42no—
209C22⋊C4—5842no—
210C22⋊C4—1468no—
211C42—22no—
212C42—2921no—
213C42—732no—
214C42—1462no—
215C42—1462no—
216C42—1461no—
217C23—22no—
218C23—42no—
219C23—42no—
220C23—41no—
221C23—41no—
222C23C2×C22⋊#6440811yescharacteristic
223Q8—2922no—
224Q8—2922no—
225D4—42no—
226D4—42no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—84no—
235C2×C4—2921no—
236C2×C4—2921no—
237C2×C4—2922no—
238C2×C4—2921no—
239C2×C4—2921no—
240C2×C4—2921no—
241C2×C4—734no—
242C2×C4—1462no—
243C2×C4—1462no—
244C2×C4—1462no—
245C2×C4—1462no—
246C2×C4#1455664912yes—
247C2×C4C2×C4×#6440811yescharacteristic
248C2×C4#1455667011yescharacteristic
249C8—2928no—
250C22—21no—
251C22—21no—
252C22—44no—
253C22—42no—
254C22—82no—
255C22—81no—
256C22—81no—
257C22C4×C22⋊#6440811yescharacteristic
258C22C2×#1455664911yescharacteristic
259C22#1798956611yescharacteristic
260C4—21no—
261C4—21no—
262C4—22no—
263C4—41no—
264C4—81no—
265C4—2922no—
266C4—2922no—
267C4—5841no—
268C4—1464no—
269C2—21no—
270C2—21no—
271C2—82no—
272C2C4×#1455664911yescharacteristic
273C2#2262622611yescharacteristic
274C2#2262623011yescharacteristic
subgroupG/H
centerC22C2×#14556649
commutatorC2×C4×C73C2×C42
FrattiniC22×C4C22×#64408
FittingC2×C4×D4×C73C4
socleC22×C73C2×#99
prime radicalC22×D4×C73C2×C4
cosocleC22×Dic146C23
hypercenterC2×C4×D4#64408
hypocenterC73#12632
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C4×#14556652C2semidirect
3. #22625029C2semidirect
7. #17989495C4semidirect
20. #14556652C2×C4semidirect
142. C73#12632semidirect

Maximal subgroups

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

subgroupindex
C2×#179894952
C4×#145566522
#226250292
#226126432
#1263273