Finite Group Databasebeta

← #34033543

Subgroups — #34033543

Order 18688.

526 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 83 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#22627043C212yesmaximal
2#22626957C214yesmaximal
3#22626405C211yesmaximal, characteristic
4#17989700—22no—
5#17989673—22no—
6#17989672C2214yes—
7C2×#14556670C411yescharacteristic
8C2×#14556670C2211yescharacteristic
9(C2×#64408)⋊Q8—22no—
10(C2×#64408)⋊Q8—22no—
11#17989530C412yes—
12#17989530C2212yes—
13C22×Q8×D73C411yescharacteristic
14C4⋊#10596842—42no—
15C22⋊#10596842—24no—
16C4×#10596842—42no—
17#14556670—42no—
18#14556670—42no—
19C2×C4⋊#64408—21no—
20C2×C4⋊#64408D412yes—
21C2×C22⋊#64408C2×C412yes—
22C2×C22⋊#64408C2×C411yescharacteristic
23C2×C22⋊#64408C2311yescharacteristic
24C2×C4×#64408—21no—
25C2×C4×#64408D412yes—
26(C2×#64408)⋊C4—24no—
27(C2×#64408)⋊C4—24no—
28C22×Dic146C812yescomplemented
29C22×Dic146C811yescharacteristic, complemented
30C2×Q8×D73—21no—
31C2×Q8×D73—21no—
32C2×Q8×D73—21no—
33C2×Q8×D73—21no—
34C2×Q8×D73—42no—
35C2×Q8×D73—42no—
36C22×C4×D73C2×C411yescharacteristic
37C22×C4×D73C2×C412yes—
38C22×Q8×C73C811yescharacteristic, complemented
39C2×#10596842—44no—
40C22×#64408—22no—
41C22×#64408C22⋊C412yes—
42C22×#64408C2×D412yes—
43C4⋊#64408—42no—
44C4⋊#64408—42no—
45C22⋊#64408—41no—
46C22⋊#64408—41no—
47C22⋊#64408—42no—
48C4×#64408—22no—
49C4×#64408—22no—
50C4×#64408—42no—
51C23×D73C22×C411yescharacteristic
52C2×Dic146—21no—
53C2×Dic146—21no—
54C2×Dic146—22no—
55C2×Dic146—22no—
56C2×Dic146—44no—
57C2×Dic146—42no—
58Q8×D73—42no—
59Q8×D73—42no—
60Q8×D73—44no—
61Q8×D73—44no—
62Q8×D73—42no—
63Q8×D73—42no—
64C22×Dic73C2×C811yescharacteristic
65C22×Dic73C2×C812yes—
66C2×C4×D73—22no—
67C2×C4×D73—22no—
68C2×C4×D73—22no—
69C2×C4×D73—22no—
70C2×C4×D73—42no—
71C2×C4×D73—42no—
72C2×C4×D73C22⋊C411yescharacteristic
73C2×C4×D73C22⋊C411yescharacteristic
74C2×C4×D73M1611yescharacteristic
75C2×C4×D73M1611yescharacteristic
76C2×Q8×C73—21no—
77C2×Q8×C73—21no—
78C2×Q8×C73—42no—
79C22×C4×C73C2×C812yes—
80C22×C4×C73C2×C811yescharacteristic
81C23×C73C22×C811yescharacteristic
82C22×D73—21no—
83C22×D73—21no—
84C22×D73—42no—
85C22×D73C2×C22⋊C411yescharacteristic
86C22×D73C2×M1611yescharacteristic
87C2×#64408—22no—
88C2×#64408—22no—
89C2×#64408—42no—
90C2×#64408—42no—
91C2×#64408—44no—
92C2×#64408—42no—
93#10596842—84no—
94Q8×C73—42no—
95Q8×C73—42no—
96C2×C4×C73—22no—
97C2×C4×C73—22no—
98C2×C4×C73—42no—
99C2×C4×C73C22⋊C811yescharacteristic
100C2×C4×C73C22⋊C811yescharacteristic
101C2×Dic73—22no—
102C2×Dic73—22no—
103C2×Dic73—42no—
104C2×Dic73C22⋊C811yescharacteristic
105C2×Dic73C22⋊C811yescharacteristic
106C4×D73—21no—
107C4×D73—21no—
108C4×D73—22no—
109C4×D73—22no—
110C4×D73—44no—
111C4×D73—42no—
112C4×D73—42no—
113C4×D73#9912yes—
114C4×D73#10012yes—
115Dic146—44no—
116Dic146—42no—
117Dic146—42no—
118Dic146—44no—
119C22×C73—21no—
120C22×C73—41no—
121C22×C73C2×C22⋊C811yescharacteristic
122C2×D73—22no—
123C2×D73—21no—
124C2×D73—42no—
125C2×D73—42no—
126C2×D73—41no—
127C2×D73C2×#9911yescharacteristic
128C2×D73C2×#10011yescharacteristic
129#64408—42no—
130#64408—42no—
131#64408—82no—
132C4×C73—21no—
133C4×C73—42no—
134C4×C73#32312yes—
135Dic73—21no—
136Dic73—42no—
137Dic73#32312yes—
138#11428—731nomaximal, Sylow
139C2×C73—21no—
140C2×C73—41no—
141C2×C73C2×#32311yescharacteristic
142D73—21no—
143D73—41no—
144D73#212312yes—
145#2310—731no—
146#1458—732no—
147#1317—734no—
148C73#1142811yescharacteristic, complemented, Sylow
149C23×Q8—731no—
150C2×#117—731no—
151C2×#117—731no—
152#431—1462no—
153#391—1462no—
154#391—1462no—
155#381—732no—
156#381—732no—
157#324—1462no—
158#323—734no—
159C22×Q8—2922no—
160C22×Q8—2922no—
161C22×Q8—732no—
162C22×Q8—731no—
163C22×Q8—1461no—
164C22×Q8—1461no—
165C22×Q8—1461no—
166C22×Q8—1461no—
167C22×Q8#1059684211yescharacteristic, complemented
168C23×C4—732no—
169C23×C4—731no—
170#117—2922no—
171#117—2922no—
172C2×C4⋊C4—732no—
173C2×C4⋊C4—1461no—
174C2×C22⋊C4—731no—
175C2×C22⋊C4—731no—
176C2×C22⋊C4—732no—
177C2×C42—732no—
178C2×C42—1461no—
179C4⋊C8—2922no—
180C22⋊C8—1464no—
181C4×C8—2922no—
182#95—1464no—
183#95—1464no—
184C24—731no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—42no—
188C2×Q8—2922no—
189C2×Q8—2924no—
190C2×Q8—2922no—
191C2×Q8—2922no—
192C2×Q8—2922no—
193C2×Q8—2922no—
194C2×Q8—2924no—
195C2×Q8—2924no—
196C2×Q8—1462no—
197C2×Q8—1462no—
198C2×Q8—1461no—
199C2×Q8—1461no—
200C22×C4—2922no—
201C22×C4—2922no—
202C22×C4—732no—
203C22×C4—732no—
204C22×C4—732no—
205C22×C4—731no—
206C22×C4—731no—
207C22×C4—731no—
208C22×C4—731no—
209C22×C4—731no—
210C22×C4—1462no—
211C22×C4—1462no—
212C22×C4—1462no—
213C22×C4—1462no—
214C22×C4—1462no—
215C22×C4C2×#1059684212yes—
216C22×C4C2×#1059684211yescharacteristic
217C2×C8—2924no—
218C4⋊C4—2922no—
219C4⋊C4—2922no—
220C22⋊C4—2922no—
221C22⋊C4—2921no—
222C22⋊C4—2921no—
223C42—2922no—
224C42—1462no—
225C42—1462no—
226C23—2922no—
227C23—731no—
228C23—731no—
229C23—1461no—
230C23—1461no—
231C23C22×#1059684211yescharacteristic
232Q8—42no—
233Q8—42no—
234Q8—2922no—
235Q8—2924no—
236Q8—2924no—
237Q8—2922no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—42no—
241C2×C4—2922no—
242C2×C4—2922no—
243C2×C4—2924no—
244C2×C4—2922no—
245C2×C4—2922no—
246C2×C4—2922no—
247C2×C4—2922no—
248C2×C4—2924no—
249C2×C4—731no—
250C2×C4—732no—
251C2×C4—732no—
252C2×C4—731no—
253C2×C4—1462no—
254C2×C4—1461no—
255C2×C4—1461no—
256C2×C4—1462no—
257C2×C4—1462no—
258C2×C4—1462no—
259C2×C4—1462no—
260C2×C4—1462no—
261C2×C4C22⋊#1059684211yescharacteristic
262C2×C4C22⋊#1059684211yescharacteristic
263C8—5844no—
264C22—21no—
265C22—41no—
266C22—2921no—
267C22—2922no—
268C22—2922no—
269C22—731no—
270C22—731no—
271C22—1462no—
272C22—1461no—
273C22C2×C22⋊#1059684211yescharacteristic
274C4—21no—
275C4—42no—
276C4—2922no—
277C4—2922no—
278C4—2922no—
279C4—5842no—
280C4—732no—
281C4—1461no—
282C4#1798967212yes—
283C2—21no—
284C2—41no—
285C2—2921no—
286C2—732no—
287C2—1461no—
288C2C2×#1798967211yescharacteristic
subgroupG/H
centerC2C2×#17989672
commutatorC23×C73C22×C8
FrattiniC23C22×#10596842
FittingC22×Q8×C73C8
socleC2×C73C2×#323
prime radicalC23×D73C22×C4
cosocleC2×C22⋊#64408C23
hypercenterC22×Q8#10596842
hypocenterC73#11428
LCSG ▷ C23×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
28. C22×Dic146C8semidirect
29. C22×Dic146C8semidirect
38. C22×Q8×C73C8semidirect
148. C73#11428semidirect
167. C22×Q8#10596842semidirect

Maximal subgroups

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

subgroupindex
#226270432
#226269572
#226264052
#1142873