Finite Group Databasebeta

← #34009815

Subgroups — #34009815

Order 18688.

686 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 112 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#22625162C212yesmaximal, complemented
2#22624859C214yesmaximal, complemented
3#22612513C211yesmaximal, characteristic
4#17989603—22no—
5#17989432C2214yescomplemented
6#17989430—22no—
7C2×#14556550C411yescharacteristic
8C2×#14556550C2211yescharacteristic
9#17988490—22no—
10#17988490C412yes—
11#17988490C2212yes—
12#17988443—22no—
13C23×D4×C73C411yescharacteristic
14C4⋊#10596833—42no—
15C22⋊#10596833—24no—
16C4×#10596833—42no—
17#14556550—42no—
18#14556550—42no—
19C2×C4⋊Dic73—21no—
20C2×C4⋊Dic73C2×C411yescharacteristic
21C2×C4⋊Dic73C2311yescharacteristic
22C2×C22⋊Dic73—24no—
23C2×C22⋊Dic73—24no—
24C2×C22⋊Dic73—21no—
25C2×C22⋊Dic73D412yes—
26C2×C4×Dic73C2×C412yes—
27C2×C4×Dic73D412yes—
28C25×C73—21no—
29C22×D4×C73—21no—
30C22×D4×C73—22no—
31C22×D4×C73—42no—
32C22×D4×C73—42no—
33C22×D4×C73C814yescomplemented
34C22×D4×C73C2×C412yes—
35C23×C4×C73C2×C411yescharacteristic
36C2×#10596833—44no—
37C22×Dic73—22no—
38C22×Dic73C22⋊C412yes—
39C22×Dic73C2×D412yes—
40C4⋊Dic73—41no—
41C4⋊Dic73—41no—
42C4⋊Dic73—42no—
43C22⋊Dic73—42no—
44C22⋊Dic73—42no—
45C22⋊Dic73—84no—
46C22⋊Dic73—84no—
47C4×Dic73—22no—
48C4×Dic73—22no—
49C4×Dic73—42no—
50C24×C73—22no—
51C24×C73—24no—
52C24×C73—42no—
53C24×C73—42no—
54C24×C73—82no—
55C24×C73—82no—
56C24×C73C22⋊C411yescharacteristic
57C2×D4×C73—24no—
58C2×D4×C73—22no—
59C2×D4×C73—42no—
60C2×D4×C73—42no—
61C2×D4×C73—48no—
62C2×D4×C73—48no—
63C2×D4×C73—42no—
64C2×D4×C73—42no—
65C2×D4×C73C2×C814yes—
66C22×C4×C73—22no—
67C22×C4×C73—22no—
68C22×C4×C73C22⋊C411yescharacteristic
69C22×C4×C73C2×C812yes—
70C22×C4×C73M1612yes—
71C22×C4×C73C22×C411yescharacteristic
72C23×C73—21no—
73C23×C73—21no—
74C23×C73—24no—
75C23×C73—41no—
76C23×C73—41no—
77C23×C73—48no—
78C23×C73—44no—
79C23×C73—44no—
80C23×C73—42no—
81C23×C73—42no—
82C23×C73—88no—
83C23×C73—88no—
84C23×C73—84no—
85C23×C73—84no—
86C23×C73C22⋊C812yes—
87C23×C73C2×C22⋊C411yescharacteristic
88D4×C73—44no—
89D4×C73—44no—
90D4×C73—48no—
91C2×C4×C73—21no—
92C2×C4×C73—21no—
93C2×C4×C73—24no—
94C2×C4×C73—24no—
95C2×C4×C73C22⋊C812yes—
96C2×C4×C73#9912yes—
97C2×C4×C73#10012yes—
98C2×C4×C73C22×C811yescharacteristic
99C2×C4×C73C2×M1611yescharacteristic
100C2×Dic73—22no—
101C2×Dic73—22no—
102C2×Dic73—42no—
103C2×Dic73—42no—
104C2×Dic73—42no—
105C2×Dic73—42no—
106C2×Dic73—42no—
107#10596833—84no—
108C22×C73—21no—
109C22×C73—21no—
110C22×C73—22no—
111C22×C73—42no—
112C22×C73—42no—
113C22×C73—48no—
114C22×C73—41no—
115C22×C73—41no—
116C22×C73—44no—
117C22×C73—84no—
118C22×C73—816no—
119C22×C73—82no—
120C22×C73—82no—
121C22×C73C2×C22⋊C811yescharacteristic
122C22×C73C2×#9911yescharacteristic
123C22×C73C2×#10011yescharacteristic
124C4×C73—21no—
125C4×C73—21no—
126C4×C73#32314yes—
127Dic73—42no—
128Dic73—42no—
129Dic73—82no—
130#11433—731nomaximal, Sylow
131C2×C73—21no—
132C2×C73—21no—
133C2×C73—42no—
134C2×C73—84no—
135C2×C73C2×#32311yescharacteristic
136C2×C73#212811yescharacteristic
137C2×C73#213411yescharacteristic
138#2309—731no—
139#1458—732no—
140#1318—734no—
141C73#1143311yescharacteristic, complemented, Sylow
142C23×D4#6440811yescharacteristic
143C2×#117—731no—
144C2×#117—731no—
145#431—1462no—
146#390—732no—
147#390—732no—
148#390—1462no—
149#379—1462no—
150#324—734no—
151#323—1462no—
152C25—21no—
153C22×D4—21no—
154C22×D4—22no—
155C22×D4—42no—
156C22×D4—42no—
157C22×D4#1059683314yescomplemented
158C22×D4C2×#6440812yes—
159C23×C4C2×#6440811yescharacteristic
160#117—2922no—
161#117—2922no—
162C2×C4⋊C4—731no—
163C2×C4⋊C4—731no—
164C2×C4⋊C4—1461no—
165C2×C22⋊C4—732no—
166C2×C22⋊C4—1461no—
167C2×C22⋊C4—1464no—
168C2×C22⋊C4—1464no—
169C2×C42—732no—
170C2×C42—732no—
171C4⋊C8—2922no—
172C22⋊C8—1464no—
173C4×C8—2922no—
174C24—22no—
175C24—24no—
176C24—42no—
177C24—42no—
178C24—82no—
179C24—82no—
180C24C22⋊#6440811yescharacteristic
181C2×D4—22no—
182C2×D4—24no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—48no—
186C2×D4—48no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4C2×#1059683314yes—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—732no—
193C22×C4—732no—
194C22×C4—1462no—
195C22×C4C22⋊#6440811yescharacteristic
196C22×C4C2×#1059683312yes—
197C22×C4#1192403712yes—
198C22×C4C22×#6440811yescharacteristic
199C2×C8—2924no—
200C4⋊C4—2921no—
201C4⋊C4—2921no—
202C4⋊C4—2922no—
203C22⋊C4—2922no—
204C22⋊C4—2922no—
205C22⋊C4—5844no—
206C22⋊C4—5844no—
207C42—2922no—
208C42—1462no—
209C42—1462no—
210C23—21no—
211C23—21no—
212C23—24no—
213C23—41no—
214C23—41no—
215C23—44no—
216C23—44no—
217C23—48no—
218C23—42no—
219C23—42no—
220C23—88no—
221C23—88no—
222C23—84no—
223C23—84no—
224C23C22⋊#1059683312yes—
225C23C2×C22⋊#6440811yescharacteristic
226D4—44no—
227D4—44no—
228D4—48no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—2922no—
234C2×C4—2922no—
235C2×C4—2922no—
236C2×C4—2922no—
237C2×C4—2922no—
238C2×C4—1462no—
239C2×C4—1462no—
240C2×C4C22⋊#1059683312yes—
241C2×C4#1455664912yes—
242C2×C4#1455665112yes—
243C2×C4C22×#1059683311yescharacteristic
244C2×C4C2×#1192403711yescharacteristic
245C8—5844no—
246C22—21no—
247C22—21no—
248C22—22no—
249C22—42no—
250C22—42no—
251C22—48no—
252C22—41no—
253C22—41no—
254C22—44no—
255C22—84no—
256C22—816no—
257C22—82no—
258C22—82no—
259C22C2×C22⋊#1059683311yescharacteristic
260C22C2×#1455664911yescharacteristic
261C22C2×#1455665111yescharacteristic
262C4—21no—
263C4—21no—
264C4—2922no—
265C4—2922no—
266C4—5842no—
267C4#1798943014yes—
268C2—21no—
269C2—21no—
270C2—42no—
271C2—84no—
272C2C2×#1798943011yescharacteristic
273C2#2262621011yescharacteristic
274C2#2262623011yescharacteristic
subgroupG/H
centerC22C2×#14556649
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
Fitting, prime radicalC23×D4×C73C4
socleC22×C73C2×#99
cosocleC2×C4⋊Dic73C23
hypercenterC23×D4#64408
hypocenterC73#11433
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. #22625162C2semidirect
2. #22624859C2semidirect
5. #17989432C22semidirect
33. C22×D4×C73C8semidirect
141. C73#11433semidirect
157. C22×D4#10596833semidirect

Maximal subgroups

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

subgroupindex
#226251622
#226248592
#226125132
#1143373