Finite Group Databasebeta

← C73×#11512 #32920916

Subgroups — C73×#11512 #32920916

Order 18688.

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

normal subgroups only

This group has 578 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
1C2×#384×C73C211yesmaximal, characteristic
2C73×#1463C212yesmaximal
3C73×#1325C214yesmaximal, complemented
4C22×C4⋊C4×C73—22no—
5C22×C4⋊C4×C73C411yescharacteristic
6C22×C4⋊C4×C73C2211yescharacteristic
7C22×C42×C73C411yescharacteristic
8#407×C73—22no—
9#384×C73—22no—
10#384×C73C412yes—
11#384×C73C2212yes—
12#324×C73—22no—
13#324×C73C2214yes—
14C23×C4×C73—21no—
15C23×C4×C73—21no—
16C23×C4×C73C2×C411yescharacteristic
17C23×C4×C73D412yes—
18C2×M16×C73—42no—
19C2×C4⋊C4×C73—21no—
20C2×C4⋊C4×C73—24no—
21C2×C4⋊C4×C73—24no—
22C2×C4⋊C4×C73—41no—
23C2×C4⋊C4×C73—41no—
24C2×C4⋊C4×C73—48no—
25C2×C4⋊C4×C73—41no—
26C2×C4⋊C4×C73—41no—
27C2×C4⋊C4×C73C2×C412yes—
28C2×C4⋊C4×C73C2×C411yescharacteristic
29C2×C4⋊C4×C73D412yes—
30C2×C4⋊C4×C73C2311yescharacteristic
31C2×C42×C73—21no—
32C2×C42×C73—24no—
33C2×C42×C73C2×C412yes—
34C22⋊C8×C73—24no—
35C22⋊C8×C73—42no—
36C24×C73C22⋊C411yescharacteristic
37C22×C4×C73—24no—
38C22×C4×C73—24no—
39C22×C4×C73—24no—
40C22×C4×C73—21no—
41C22×C4×C73—21no—
42C22×C4×C73—22no—
43C22×C4×C73—22no—
44C22×C4×C73—22no—
45C22×C4×C73—22no—
46C22×C4×C73—22no—
47C22×C4×C73—22no—
48C22×C4×C73—44no—
49C22×C4×C73—44no—
50C22×C4×C73—41no—
51C22×C4×C73—41no—
52C22×C4×C73C22⋊C411yescharacteristic
53C22×C4×C73C22⋊C412yes—
54C22×C4×C73C22×C411yescharacteristic
55C22×C4×C73C2×D412yes—
56M16×C73—84no—
57C2×C8×C73—44no—
58C4⋊C4×C73—41no—
59C4⋊C4×C73—42no—
60C4⋊C4×C73—41no—
61C4⋊C4×C73—42no—
62C4⋊C4×C73—48no—
63C4⋊C4×C73—48no—
64C4⋊C4×C73—42no—
65C42×C73—24no—
66C42×C73—22no—
67C42×C73M1614yescomplemented
68C23×C73—22no—
69C23×C73—22no—
70C23×C73—21no—
71C23×C73—21no—
72C23×C73#9911yescharacteristic
73C23×C73#9911yescharacteristic
74C23×C73C2×C22⋊C411yescharacteristic
75C2×C4×C73—24no—
76C2×C4×C73—216no—
77C2×C4×C73—24no—
78C2×C4×C73—24no—
79C2×C4×C73—24no—
80C2×C4×C73—21no—
81C2×C4×C73—24no—
82C2×C4×C73—21no—
83C2×C4×C73—21no—
84C2×C4×C73—21no—
85C2×C4×C73—22no—
86C2×C4×C73—22no—
87C2×C4×C73—44no—
88C2×C4×C73—42no—
89C2×C4×C73—42no—
90C2×C4×C73—44no—
91C2×C4×C73—42no—
92C2×C4×C73—82no—
93C2×C4×C73—82no—
94C2×C4×C73—82no—
95C2×C4×C73#10112yes—
96C2×C4×C73C2×M1612yes—
97C8×C73—84no—
98C22×C73—24no—
99C22×C73—24no—
100C22×C73—21no—
101C22×C73—21no—
102C22×C73—21no—
103C22×C73—21no—
104C22×C73—21no—
105C22×C73—21no—
106C22×C73#35312yes—
107C22×C73#35612yes—
108C22×C73#40711yescharacteristic
109C22×C73C2×#9911yescharacteristic
110C22×C73C2×#10111yescharacteristic
111C4×C73—28no—
112C4×C73—22no—
113C4×C73—22no—
114C4×C73—42no—
115C4×C73—42no—
116C4×C73—82no—
117#11512C7311yesmaximal, characteristic, direct product factor, Sylow
118C2×C73—21no—
119C2×C73—22no—
120C2×C73—22no—
121C2×C73—21no—
122C2×C73#146311yescharacteristic
123C2×C73C2×#35311yescharacteristic
124C2×C73C2×#35611yescharacteristic
125C2×#384C2×C7311yescharacteristic
126#1463C2×C7312yes—
127#1325C2×C7314yescomplemented
128C73#1151211yescharacteristic, direct product factor, Sylow
129C22×C4⋊C4—22no—
130C22×C4⋊C4C4×C7311yescharacteristic
131C22×C4⋊C4C22×C7311yescharacteristic
132C22×C42C4×C7311yescharacteristic
133#407—22no—
134#384—22no—
135#384C4×C7312yes—
136#384C22×C7312yes—
137#324—22no—
138#324C22×C7314yes—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4C2×C4×C7311yescharacteristic
142C23×C4D4×C7312yes—
143C2×M16—42no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—48no—
152C2×C4⋊C4C2×C4×C7312yes—
153C2×C4⋊C4C2×C4×C7311yescharacteristic
154C2×C4⋊C4D4×C7312yes—
155C2×C4⋊C4C23×C7311yescharacteristic
156C2×C42—21no—
157C2×C42—24no—
158C2×C42C2×C4×C7312yes—
159C22⋊C8—24no—
160C22⋊C8—42no—
161C24C22⋊C4×C7311yescharacteristic
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4C22⋊C4×C7311yescharacteristic
178C22×C4C22⋊C4×C7312yes—
179C22×C4C22×C4×C7311yescharacteristic
180C22×C4C2×D4×C7312yes—
181M16—84no—
182C2×C8—44no—
183C4⋊C4—41no—
184C4⋊C4—42no—
185C4⋊C4—41no—
186C4⋊C4—42no—
187C4⋊C4—48no—
188C4⋊C4—48no—
189C4⋊C4—42no—
190C42—24no—
191C42—22no—
192C42M16×C7314yescomplemented
193C23—21no—
194C23—21no—
195C23—22no—
196C23—22no—
197C23#99×C7311yescharacteristic
198C23#99×C7311yescharacteristic
199C23C2×C22⋊C4×C7311yescharacteristic
200C2×C4—24no—
201C2×C4—216no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—21no—
206C2×C4—24no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—44no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—44no—
216C2×C4—42no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4#101×C7312yes—
221C2×C4C2×M16×C7312yes—
222C8—84no—
223C22—24no—
224C22—24no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22#353×C7312yes—
232C22#356×C7312yes—
233C22#407×C7311yescharacteristic
234C22C2×#99×C7311yescharacteristic
235C22C2×#101×C7311yescharacteristic
236C4—28no—
237C4—22no—
238C4—22no—
239C4—42no—
240C4—42no—
241C4—82no—
242C2—21no—
243C2—21no—
244C2—22no—
245C2—22no—
246C2C73×#146311yescharacteristic
247C2C2×#353×C7311yescharacteristic
248C2C2×#356×C7311yescharacteristic
subgroupG/H
center, socleC22×C73C2×#99
commutatorC22×C4C22×C4×C73
Frattini, cosocleC2×C4⋊C4C23×C73
prime radicalC24×C73C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C24×C73 ◁ C22×C4⋊C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
3. C73×#1325C2semidirect
67. C42×C73M16semidirect
117. #11512C73direct (N × H)
127. #1325C2×C73semidirect
128. C73#11512direct (N × H)
192. C42M16×C73semidirect

Maximal subgroups

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

subgroupindex
C2×#384×C732
C73×#14632
C73×#13252
#1151273