Finite Group Databasebeta

← C73×#12858 #32922262

Subgroups — C73×#12858 #32922262

Order 18688.

786 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 118 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
1C73×#2309C211yesmaximal, characteristic
2C73×#1781C212yesmaximal
3C73×#1393C214yesmaximal, complemented
4C23×D4×C73C2211yescharacteristic
5C2×#117×C73C411yescharacteristic, complemented
6C2×#117×C73C411yescharacteristic, complemented
7C2×#99×C73—22no—
8C2×#99×C73C2214yes—
9#390×C73—22no—
10#390×C73C412yescomplemented
11#390×C73C2212yes—
12#379×C73—22no—
13#379×C73—22no—
14C25×C73D412yes—
15C22×D4×C73—24no—
16C22×D4×C73—42no—
17C22×D4×C73—42no—
18C22×D4×C73C2×C411yescharacteristic
19C22×D4×C73D412yes—
20C22×D4×C73C2311yescharacteristic
21C23×C4×C73C2×C411yescharacteristic
22#117×C73—42no—
23#117×C73—42no—
24C2×C4⋊C4×C73—21no—
25C2×C4⋊C4×C73C2×C412yes—
26C2×C22⋊C4×C73—21no—
27C2×C22⋊C4×C73—21no—
28C2×C22⋊C4×C73—24no—
29C2×C22⋊C4×C73—24no—
30C2×C22⋊C4×C73—24no—
31C2×C22⋊C4×C73—44no—
32C2×C42×C73—21no—
33C2×C42×C73C2×C412yes—
34#99×C73—28no—
35#99×C73—44no—
36C24×C73—24no—
37C24×C73—24no—
38C24×C73—42no—
39C24×C73—84no—
40C24×C73C22⋊C412yes—
41C24×C73C22⋊C411yescharacteristic
42C24×C73C2×D412yes—
43C2×D4×C73—24no—
44C2×D4×C73—24no—
45C2×D4×C73—22no—
46C2×D4×C73—41no—
47C2×D4×C73—41no—
48C2×D4×C73—42no—
49C2×D4×C73—41no—
50C2×D4×C73—41no—
51C2×D4×C73—416no—
52C2×D4×C73#4114yes—
53C22×C4×C73—22no—
54C22×C4×C73—22no—
55C22×C4×C73—22no—
56C22×C4×C73—22no—
57C22×C4×C73—22no—
58C22×C4×C73—44no—
59C22×C4×C73C22⋊C411yescharacteristic
60C22×C4×C73C22×C411yescharacteristic
61C4⋊C4×C73—42no—
62C4⋊C4×C73—42no—
63C22⋊C4×C73—42no—
64C22⋊C4×C73—42no—
65C22⋊C4×C73—44no—
66C22⋊C4×C73—44no—
67C22⋊C4×C73—84no—
68C22⋊C4×C73—84no—
69C22⋊C4×C73—88no—
70C42×C73—42no—
71C42×C73—42no—
72C23×C73—21no—
73C23×C73—21no—
74C23×C73—22no—
75C23×C73—22no—
76C23×C73—22no—
77C23×C73—22no—
78C23×C73—22no—
79C23×C73—42no—
80C23×C73—48no—
81C23×C73—44no—
82C23×C73—44no—
83C23×C73—48no—
84C23×C73—84no—
85C23×C73—88no—
86C23×C73—88no—
87C23×C73#9912yes—
88C23×C73C2×C22⋊C411yescharacteristic
89C23×C73#12314yes—
90D4×C73—44no—
91D4×C73—48no—
92D4×C73—44no—
93C2×C4×C73—28no—
94C2×C4×C73—21no—
95C2×C4×C73—21no—
96C2×C4×C73—21no—
97C2×C4×C73—21no—
98C2×C4×C73—42no—
99C2×C4×C73—42no—
100C2×C4×C73—44no—
101C2×C4×C73—42no—
102C2×C4×C73—42no—
103C2×C4×C73—84no—
104C2×C4×C73—88no—
105C2×C4×C73#9912yes—
106C2×C4×C73#11712yes—
107C22×C73—22no—
108C22×C73—22no—
109C22×C73—24no—
110C22×C73—24no—
111C22×C73—44no—
112C22×C73—42no—
113C22×C73—42no—
114C22×C73—44no—
115C22×C73—44no—
116C22×C73—42no—
117C22×C73—44no—
118C22×C73—82no—
119C22×C73—82no—
120C22×C73—816no—
121C22×C73#38111yescharacteristic
122C22×C73C2×#9911yescharacteristic
123C22×C73C2×#9911yescharacteristic
124C4×C73—22no—
125C4×C73—22no—
126C4×C73—82no—
127C4×C73—82no—
128C4×C73—164no—
129#12858C7311yesmaximal, characteristic, direct product factor, Sylow
130C2×C73—21no—
131C2×C73—21no—
132C2×C73—44no—
133C2×C73—42no—
134C2×C73—82no—
135C2×C73#181911yescharacteristic
136C2×C73#212712yes—
137#2309C2×C7311yescharacteristic
138#1781C2×C7312yes—
139#1393C2×C7314yescomplemented
140C73#1285811yescharacteristic, direct product factor, Sylow
141C23×D4C22×C7311yescharacteristic
142C2×#117C4×C7311yescharacteristic, complemented
143C2×#117C4×C7311yescharacteristic, complemented
144C2×#99—22no—
145C2×#99C22×C7314yes—
146#390—22no—
147#390C4×C7312yescomplemented
148#390C22×C7312yes—
149#379—22no—
150#379—22no—
151C25D4×C7312yes—
152C22×D4—24no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4C2×C4×C7311yescharacteristic
156C22×D4D4×C7312yes—
157C22×D4C23×C7311yescharacteristic
158C23×C4C2×C4×C7311yescharacteristic
159#117—42no—
160#117—42no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4C2×C4×C7312yes—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—44no—
169C2×C42—21no—
170C2×C42C2×C4×C7312yes—
171#99—28no—
172#99—44no—
173C24—24no—
174C24—24no—
175C24—42no—
176C24—84no—
177C24C22⋊C4×C7312yes—
178C24C22⋊C4×C7311yescharacteristic
179C24C2×D4×C7312yes—
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—41no—
184C2×D4—41no—
185C2×D4—42no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—416no—
189C2×D4#41×C7314yes—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—44no—
196C22×C4C22⋊C4×C7311yescharacteristic
197C22×C4C22×C4×C7311yescharacteristic
198C4⋊C4—42no—
199C4⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—44no—
203C22⋊C4—44no—
204C22⋊C4—84no—
205C22⋊C4—84no—
206C22⋊C4—88no—
207C42—42no—
208C42—42no—
209C23—21no—
210C23—21no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—42no—
217C23—48no—
218C23—44no—
219C23—44no—
220C23—48no—
221C23—84no—
222C23—88no—
223C23—88no—
224C23#99×C7312yes—
225C23C2×C22⋊C4×C7311yescharacteristic
226C23#123×C7314yes—
227D4—44no—
228D4—48no—
229D4—44no—
230C2×C4—28no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—44no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—84no—
241C2×C4—88no—
242C2×C4#99×C7312yes—
243C2×C4#117×C7312yes—
244C22—22no—
245C22—22no—
246C22—24no—
247C22—24no—
248C22—44no—
249C22—42no—
250C22—42no—
251C22—44no—
252C22—44no—
253C22—42no—
254C22—44no—
255C22—82no—
256C22—82no—
257C22—816no—
258C22#381×C7311yescharacteristic
259C22C2×#99×C7311yescharacteristic
260C22C2×#99×C7311yescharacteristic
261C4—22no—
262C4—22no—
263C4—82no—
264C4—82no—
265C4—164no—
266C2—21no—
267C2—21no—
268C2—44no—
269C2—42no—
270C2—82no—
271C2C73×#181911yescharacteristic
272C2C73×#212712yes—
subgroupG/H
center, socleC22×C73C2×#99
commutatorC22×C4C22×C4×C73
Frattini, cosocleC22×D4C23×C73
prime radicalC23×D4×C73C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C24×C73 ◁ C23×D4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
3. C73×#1393C2semidirect
5. C2×#117×C73C4semidirect
6. C2×#117×C73C4semidirect
10. #390×C73C4semidirect
129. #12858C73direct (N × H)
139. #1393C2×C73semidirect
140. C73#12858direct (N × H)
142. C2×#117C4×C73semidirect
143. C2×#117C4×C73semidirect
147. #390C4×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#23092
C73×#17812
C73×#13932
#1285873