Finite Group Databasebeta

← #34010155

Subgroups — #34010155

Order 18688.

460 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 78 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#22625188C212yesmaximal
2#22624872C212yesmaximal, complemented
3#22624861C212yesmaximal, complemented
4C4×#14556580C211yesmaximal, characteristic
5#17989558—22no—
6#17989432C2214yes—
7#17989431—22no—
8C2×#14556580C411yescharacteristic
9C2×C4×D292C2211yescharacteristic
10#17988494C2211yescharacteristic
11#17988488C411yescharacteristic
12#17988477—22no—
13#17988452C2211yescharacteristic
14C4×#11924005—22no—
15C42×Dic73C411yescharacteristic
16C2×C4×Q8×C73C411yescharacteristic
17C2×#11924038—42no—
18#14556648—42no—
19C22⋊#10596833—24no—
20C22×D292C2×C411yescharacteristic
21C22×C4×D73D412yes—
22#14556580—22no—
23#14556580—22no—
24C4×D292—41no—
25C4×D292—41no—
26C2×C4⋊Dic73C2311yescharacteristic
27C2×#11924005—22no—
28C2×#11924005—22no—
29C2×#11924005D412yes—
30C2×C4×Dic73—21no—
31C2×C4×Dic73—21no—
32C2×C4×Dic73C2×C411yescharacteristic
33C2×C4×Dic73C2×C411yescharacteristic
34#14556498—22no—
35#14556498—22no—
36C22×Q8×C73C2×C411yescharacteristic
37C4×Q8×C73—41no—
38C4×Q8×C73—41no—
39C2×C4⋊C4×C73—21no—
40C2×C4⋊C4×C73C2×C411yescharacteristic
41C2×C42×C73—21no—
42C2×C42×C73C2×C411yescharacteristic
43#11924038—84no—
44C2×#10596833—44no—
45C23×D73C22⋊C412yes—
46C2×D292—21no—
47C2×D292—21no—
48C2×D292—41no—
49C2×D292—41no—
50C22×Dic73—21no—
51C22×Dic73—21no—
52C22×Dic73C2×D412yes—
53C2×C4×D73—22no—
54C2×C4×D73—22no—
55C2×C4×D73—44no—
56C4⋊Dic73—41no—
57#11924005—28no—
58#11924005—42no—
59#11924005—44no—
60C4×Dic73—21no—
61C4×Dic73—42no—
62C4×Dic73—42no—
63C4×Dic73—41no—
64C4×Dic73M1612yescomplemented
65C2×Q8×C73—21no—
66C2×Q8×C73—41no—
67C2×Q8×C73—41no—
68C2×Q8×C73M1612yescomplemented
69C22×C4×C73—21no—
70C22×C4×C73—21no—
71C22×C4×C73C22⋊C411yescharacteristic
72C22×C4×C73C22⋊C411yescharacteristic
73C22×C4×C73C22×C411yescharacteristic
74C4⋊C4×C73—41no—
75C4⋊C4×C73—42no—
76C42×C73—21no—
77C42×C73—21no—
78C42×C73—41no—
79C42×C73—41no—
80C23×C73C2×C22⋊C411yescharacteristic
81C22×D73—22no—
82C22×D73—22no—
83C22×D73—22no—
84C22×D73—42no—
85C22×D73—42no—
86Q8×C73—42no—
87Q8×C73—42no—
88C2×C4×C73—21no—
89C2×C4×C73—21no—
90C2×C4×C73—22no—
91C2×C4×C73—22no—
92C2×C4×C73—21no—
93C2×C4×C73—41no—
94C2×C4×C73—41no—
95C2×C4×C73—41no—
96C2×C4×C73—42no—
97C2×C4×C73—41no—
98C2×C4×C73#9911yescharacteristic
99C2×C4×C73#9911yescharacteristic
100C2×C4×C73C2×M1611yescharacteristic
101C2×C4×C73C2×M1611yescharacteristic
102C2×Dic73—22no—
103C2×Dic73—22no—
104C2×Dic73—22no—
105C2×Dic73—41no—
106C2×Dic73—41no—
107C2×Dic73—41no—
108C2×Dic73—41no—
109C2×Dic73—42no—
110D292—42no—
111D292—42no—
112C4×D73—44no—
113C4×D73—82no—
114#10596833—84no—
115C22×C73—21no—
116C22×C73—21no—
117C22×C73#40711yescharacteristic
118C22×C73C2×#9911yescharacteristic
119C22×C73#41311yescharacteristic
120C2×D73—44no—
121C2×D73—42no—
122C2×D73—42no—
123C2×D73—42no—
124C2×D73—82no—
125C4×C73—21no—
126C4×C73—21no—
127C4×C73—21no—
128C4×C73—41no—
129C4×C73—42no—
130C4×C73—81no—
131C4×C73#35412yes—
132Dic73—42no—
133Dic73—42no—
134Dic73—42no—
135#11531—731nomaximal, Sylow
136C2×C73—21no—
137C2×C73—21no—
138C2×C73#146511yescharacteristic
139C2×C73C2×#35411yescharacteristic
140C2×C73#213211yescharacteristic
141D73—82no—
142C4×#124—731no—
143#1465—732no—
144#1324—732no—
145#1319—732no—
146C73#1153111yescharacteristic, complemented, Sylow
147C2×#124—731no—
148C2×C4×Q8#6440811yescharacteristic
149C2×C4×D4—731no—
150#407—1462no—
151#391—731no—
152#390—731no—
153#388—1462no—
154#382—731no—
155C4×C22⋊C4—1462no—
156C43—731no—
157#324—734no—
158#323—1462no—
159C22×Q8C2×#6440811yescharacteristic
160C22×D4—731no—
161C23×C4—732no—
162C2×M16—2922no—
163#124—1462no—
164#124—1462no—
165C4×Q8—41no—
166C4×Q8—41no—
167C4×D4—2921no—
168C4×D4—2921no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—731no—
171C2×C4⋊C4C2×#6440811yescharacteristic
172C2×C22⋊C4—732no—
173C2×C22⋊C4—1462no—
174C2×C22⋊C4—1462no—
175C2×C42—21no—
176C2×C42—731no—
177C2×C42—731no—
178C2×C42—1461no—
179C2×C42—1461no—
180C2×C42C2×#6440811yescharacteristic
181C22⋊C8—2922no—
182C22⋊C8—1464no—
183#95—1462no—
184#95—1462no—
185C24—732no—
186C2×Q8—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8#1192403812yescomplemented
190C2×D4—2921no—
191C2×D4—2921no—
192C2×D4—1461no—
193C2×D4—1461no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—2924no—
197C22×C4—732no—
198C22×C4—1461no—
199C22×C4—1461no—
200C22×C4—1462no—
201C22×C4—1462no—
202C22×C4C22⋊#6440811yescharacteristic
203C22×C4C22⋊#6440811yescharacteristic
204C22×C4C22×#6440811yescharacteristic
205M16—5844no—
206C2×C8—2924no—
207C4⋊C4—41no—
208C4⋊C4—42no—
209C4⋊C4—2921no—
210C22⋊C4—2922no—
211C22⋊C4—2924no—
212C22⋊C4—1468no—
213C42—21no—
214C42—21no—
215C42—41no—
216C42—41no—
217C42—2921no—
218C42—2922no—
219C42—2922no—
220C42—732no—
221C42—1461no—
222C23—2922no—
223C23—2922no—
224C23—1462no—
225C23—1462no—
226C23—1462no—
227C23C2×C22⋊#6440811yescharacteristic
228Q8—42no—
229Q8—42no—
230D4—2922no—
231D4—2922no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—2921no—
243C2×C4—2921no—
244C2×C4—2921no—
245C2×C4—2921no—
246C2×C4—2924no—
247C2×C4—2922no—
248C2×C4—5842no—
249C2×C4—1462no—
250C2×C4—1462no—
251C2×C4—1462no—
252C2×C4#1455664911yescharacteristic
253C2×C4#1455664911yescharacteristic
254C2×C4D73⋊M1611yescharacteristic
255C2×C4C2×#1192403811yescharacteristic
256C8—5844no—
257C22—21no—
258C22—21no—
259C22—2924no—
260C22—2922no—
261C22—2922no—
262C22—2922no—
263C22—5842no—
264C22#1798955811yescharacteristic
265C22C2×#1455664911yescharacteristic
266C22#1798957311yescharacteristic
267C4—21no—
268C4—21no—
269C4—21no—
270C4—42no—
271C4—41no—
272C4—81no—
273C4—2922no—
274C4—2922no—
275C4—2922no—
276C4#1798949312yes—
277C2—21no—
278C2—21no—
279C2—5842no—
280C2#2262518611yescharacteristic
281C2C2×#1798949311yescharacteristic
282C2#2262622211yescharacteristic
subgroupG/H
centerC22C2×#14556649
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×C4×Q8×C73C4
socleC22×C73C2×#99
prime radicalC22×D292C2×C4
cosocleC2×C4⋊Dic73C23
hypercenterC2×C4×Q8#64408
hypocenterC73#11531
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #22624872C2semidirect
3. #22624861C2semidirect
64. C4×Dic73M16semidirect
68. C2×Q8×C73M16semidirect
146. C73#11531semidirect
189. C2×Q8#11924038semidirect

Maximal subgroups

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

subgroupindex
#226251882
#226248722
#226248612
C4×#145565802
#1153173