Finite Group Databasebeta

← #34010172

Subgroups — #34010172

Order 18688.

440 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 74 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#22625187C212yesmaximal
2#22624873C212yesmaximal, complemented
3#22624861C212yesmaximal, complemented
4#22612644C211yesmaximal, characteristic
5#17989557—22no—
6#17989432C2214yes—
7#17989430—22no—
8C2×Q8×Dic73C411yescharacteristic
9C2×D4×Dic73C2211yescharacteristic
10#17988494C2211yescharacteristic
11#17988490C411yescharacteristic
12#17988480—22no—
13#17988462—22no—
14#17988457C2211yescharacteristic
15#17988433C411yescharacteristic
16C2×#124×C73C411yescharacteristic
17C2×#11924037—42no—
18C22⋊#10596833—24no—
19C22⋊#10596833—42no—
20C23×Dic73D412yes—
21Q8×Dic73—42no—
22D4×Dic73—42no—
23C2×C4⋊Dic73—21no—
24C2×C4⋊Dic73—22no—
25C2×C4⋊Dic73C2×C411yescharacteristic
26C2×C4⋊Dic73C2311yescharacteristic
27C2×C22⋊Dic73—22no—
28C2×C22⋊Dic73D412yes—
29C2×C4×Dic73—21no—
30C2×C4×Dic73C2×C411yescharacteristic
31C2×C4×Dic73C2×C411yescharacteristic
32#14556498—21no—
33#14556498—21no—
34#14556498—22no—
35C22×Q8×C73C2×C411yescharacteristic
36C22×D4×C73C2×C411yescharacteristic
37#124×C73—22no—
38#124×C73—22no—
39C2×C22⋊C4×C73—22no—
40C2×C42×C73C2×C411yescharacteristic
41#11924037—84no—
42C2×#10596833—44no—
43C22×Dic73—22no—
44C22×Dic73—22no—
45C22×Dic73—21no—
46C22×Dic73—21no—
47C22×Dic73—82no—
48C22×Dic73C22⋊C411yescharacteristic
49C22×Dic73C22⋊C411yescharacteristic
50C22×Dic73C2×D412yes—
51C4⋊Dic73—41no—
52C4⋊Dic73—42no—
53C4⋊Dic73—41no—
54C4⋊Dic73—82no—
55C22⋊Dic73—42no—
56C22⋊Dic73—82no—
57C4×Dic73—22no—
58C4×Dic73—42no—
59C4×Dic73—41no—
60C24×C73C22⋊C412yes—
61C2×Q8×C73—21no—
62C2×Q8×C73—41no—
63C2×Q8×C73—41no—
64C2×Q8×C73M1612yescomplemented
65C2×D4×C73—21no—
66C2×D4×C73—21no—
67C2×D4×C73—41no—
68C2×D4×C73—41no—
69C22×C4×C73—21no—
70C22×C4×C73—21no—
71C22×C4×C73C22×C411yescharacteristic
72C22⋊C4×C73—28no—
73C42×C73—21no—
74C42×C73M1612yescomplemented
75C23×C73—22no—
76C23×C73—22no—
77C23×C73—22no—
78C23×C73—42no—
79C23×C73—42no—
80C23×C73C2×C22⋊C411yescharacteristic
81Q8×C73—42no—
82Q8×C73—42no—
83D4×C73—42no—
84D4×C73—42no—
85C2×C4×C73—21no—
86C2×C4×C73—21no—
87C2×C4×C73—22no—
88C2×C4×C73—22no—
89C2×C4×C73—41no—
90C2×C4×C73—41no—
91C2×C4×C73—41no—
92C2×C4×C73—41no—
93C2×C4×C73C2×M1611yescharacteristic
94C2×C4×C73C2×M1611yescharacteristic
95C2×Dic73—21no—
96C2×Dic73—22no—
97C2×Dic73—41no—
98C2×Dic73—44no—
99C2×Dic73—42no—
100C2×Dic73—41no—
101C2×Dic73—41no—
102C2×Dic73—42no—
103C2×Dic73—41no—
104C2×Dic73—41no—
105C2×Dic73—41no—
106C2×Dic73—82no—
107C2×Dic73#9912yes—
108#10596833—84no—
109C22×C73—21no—
110C22×C73—21no—
111C22×C73—44no—
112C22×C73—42no—
113C22×C73—42no—
114C22×C73—42no—
115C22×C73—82no—
116C22×C73#40711yescharacteristic
117C22×C73C2×#9911yescharacteristic
118C22×C73#41311yescharacteristic
119C4×C73—21no—
120C4×C73—21no—
121C4×C73—42no—
122C4×C73—42no—
123Dic73—22no—
124Dic73—42no—
125Dic73—41no—
126Dic73—81no—
127Dic73—81no—
128#11535—731nomaximal, Sylow
129C2×C73—21no—
130C2×C73—21no—
131C2×C73—82no—
132C2×C73#146511yescharacteristic
133C2×C73#212711yescharacteristic
134C2×C73#213511yescharacteristic
135#2326—731no—
136#1465—732no—
137#1325—732no—
138#1319—732no—
139C73#1153511yescharacteristic, complemented, Sylow
140C2×#124#6440811yescharacteristic
141C2×C4×Q8—731no—
142C2×C4×D4—731no—
143#407—1462no—
144#391—731no—
145#390—731no—
146#388—1462no—
147#385—1462no—
148#384—731no—
149#376—731no—
150#324—734no—
151#323—1462no—
152C22×Q8C2×#6440811yescharacteristic
153C22×D4C2×#6440811yescharacteristic
154C23×C4—732no—
155C2×M16—2922no—
156#124—22no—
157#124—22no—
158C4×Q8—2922no—
159C4×D4—2922no—
160C2×C4⋊C4—731no—
161C2×C4⋊C4—731no—
162C2×C4⋊C4—1461no—
163C2×C4⋊C4—1462no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—732no—
166C2×C22⋊C4—1462no—
167C2×C42—731no—
168C2×C42—731no—
169C2×C42—1461no—
170C2×C42C2×#6440811yescharacteristic
171C22⋊C8—2922no—
172C22⋊C8—1464no—
173#95—1461no—
174#95—1461no—
175#95—1462no—
176C24C22⋊#6440812yes—
177C2×Q8—21no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8#1192403712yescomplemented
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—41no—
184C2×D4—41no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—5842no—
188C22×C4—731no—
189C22×C4—731no—
190C22×C4—732no—
191C22×C4—1461no—
192C22×C4—1461no—
193C22×C4—1462no—
194C22×C4—1462no—
195C22×C4C22×#6440811yescharacteristic
196M16—5844no—
197C2×C8—2924no—
198C4⋊C4—2921no—
199C4⋊C4—2921no—
200C4⋊C4—2922no—
201C4⋊C4—5842no—
202C22⋊C4—28no—
203C22⋊C4—2922no—
204C22⋊C4—5842no—
205C42—21no—
206C42—2921no—
207C42—2922no—
208C42—1462no—
209C42#1192403712yescomplemented
210C23—22no—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23C2×C22⋊#6440811yescharacteristic
216Q8—42no—
217Q8—42no—
218D4—42no—
219D4—42no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—2924no—
229C2×C4—2922no—
230C2×C4—2921no—
231C2×C4—2921no—
232C2×C4—2922no—
233C2×C4—2921no—
234C2×C4—2921no—
235C2×C4—2921no—
236C2×C4—2921no—
237C2×C4—5842no—
238C2×C4—732no—
239C2×C4—1461no—
240C2×C4—1462no—
241C2×C4C2×#1192403711yescharacteristic
242C2×C4C2×#1192403711yescharacteristic
243C8—5844no—
244C22—21no—
245C22—21no—
246C22—44no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—82no—
251C22#1798955711yescharacteristic
252C22D73⋊#9911yescharacteristic
253C22#1798957411yescharacteristic
254C4—21no—
255C4—21no—
256C4—42no—
257C4—42no—
258C4—2922no—
259C4—2921no—
260C4—5841no—
261C4—5841no—
262C4—1462no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#2262518711yescharacteristic
267C2#2262620911yescharacteristic
268C2#2262623411yescharacteristic
subgroupG/H
centerC22D73⋊#99
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×#124×C73C4
socleC22×C73C2×#99
prime radicalC22×D4×C73C2×C4
cosocleC2×C4⋊Dic73C23
hypercenterC2×#124#64408
hypocenterC73#11535
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. #22624873C2semidirect
3. #22624861C2semidirect
64. C2×Q8×C73M16semidirect
74. C42×C73M16semidirect
139. C73#11535semidirect
180. C2×Q8#11924037semidirect
209. C42#11924037semidirect

Maximal subgroups

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

subgroupindex
#226251872
#226248732
#226248612
#226126442
#1153573