Finite Group Databasebeta

← C73×#12246 #32921650

Subgroups — C73×#12246 #32921650

Order 18688.

558 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2537C211yesmaximal, characteristic
2C73×#2060C211yesmaximal, characteristic, complemented
3C73×#1599C212yesmaximal, complemented
4C73×#1345C212yesmaximal, complemented
5C73×#1325C211yesmaximal, characteristic, complemented
6#544×C73C2211yescharacteristic
7C2×#121×C73—22no—
8C2×C4×D4×C73—21no—
9#450×C73—22no—
10#428×C73—21no—
11#394×C73—21no—
12#394×C73C2212yes—
13#384×C73C2211yescharacteristic
14#380×C73—22no—
15#343×C73C2211yescharacteristic
16#339×C73—21no—
17#328×C73—22no—
18#324×C73C2212yes—
19C22×D4×C73—21no—
20C22×D4×C73—21no—
21C22×D4×C73D412yes—
22C23×C4×C73—22no—
23C2×SD16×C73—42no—
24C2×M16×C73—21no—
25C4⋊Q8×C73—41no—
26#125×C73—41no—
27C22⋊Q8×C73—22no—
28#121×C73—48no—
29C4×D4×C73—42no—
30C4×D4×C73—42no—
31#117×C73—21no—
32C2×C4⋊C4×C73—21no—
33C2×C4⋊C4×C73—22no—
34C2×C4⋊C4×C73D411yescharacteristic
35C2×C4⋊C4×C73C2311yescharacteristic
36C2×C22⋊C4×C73—22no—
37C2×C22⋊C4×C73—22no—
38C2×C22⋊C4×C73—22no—
39C2×C42×C73D411yescharacteristic
40#107×C73—41no—
41#106×C73—41no—
42#102×C73—42no—
43C22⋊C8×C73—22no—
44#95×C73—21no—
45#95×C73D412yes—
46C24×C73—22no—
47C2×Q8×C73—22no—
48C2×D4×C73—24no—
49C2×D4×C73—42no—
50C2×D4×C73—42no—
51C2×D4×C73—42no—
52C2×D4×C73—44no—
53C2×D4×C73—44no—
54C2×D4×C73—81no—
55C22×C4×C73—21no—
56C22×C4×C73—22no—
57C22×C4×C73—21no—
58C22×C4×C73—44no—
59C22×C4×C73—44no—
60C22×C4×C73—44no—
61C22×C4×C73C2×D411yescharacteristic
62C22×C4×C73C2×D412yes—
63SD16×C73—84no—
64M16×C73—22no—
65M16×C73—41no—
66C2×C8×C73—41no—
67C2×C8×C73—42no—
68C4⋊C4×C73—22no—
69C4⋊C4×C73—21no—
70C4⋊C4×C73—21no—
71C4⋊C4×C73—41no—
72C4⋊C4×C73—41no—
73C4⋊C4×C73—42no—
74C4⋊C4×C73—44no—
75C4⋊C4×C73—42no—
76C22⋊C4×C73—44no—
77C22⋊C4×C73—44no—
78C22⋊C4×C73—41no—
79C22⋊C4×C73—42no—
80C22⋊C4×C73—42no—
81C42×C73—21no—
82C42×C73—41no—
83C42×C73SD1612yescomplemented
84C23×C73—24no—
85C23×C73—42no—
86C23×C73—42no—
87C23×C73—42no—
88C23×C73—42no—
89C23×C73—42no—
90C23×C73—42no—
91C23×C73#12011yescharacteristic
92Q8×C73—82no—
93D4×C73—48no—
94D4×C73—44no—
95D4×C73—82no—
96D4×C73—84no—
97C2×C4×C73—22no—
98C2×C4×C73—22no—
99C2×C4×C73—22no—
100C2×C4×C73—21no—
101C2×C4×C73—21no—
102C2×C4×C73—22no—
103C2×C4×C73—42no—
104C2×C4×C73—42no—
105C2×C4×C73—42no—
106C2×C4×C73—42no—
107C2×C4×C73—42no—
108C2×C4×C73—41no—
109C2×C4×C73—41no—
110C2×C4×C73—41no—
111C2×C4×C73—88no—
112C2×C4×C73C2×SD1611yescharacteristic
113C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
114C8×C73—42no—
115C8×C73—82no—
116C22×C73—21no—
117C22×C73—21no—
118C22×C73—44no—
119C22×C73—44no—
120C22×C73—44no—
121C22×C73—82no—
122C22×C73—82no—
123C22×C73—82no—
124C22×C73—82no—
125C22×C73#45011yescharacteristic
126C22×C73#45611yescharacteristic
127C22×C73B(4,𝔽2)11yescharacteristic
128C4×C73—24no—
129C4×C73—21no—
130C4×C73—21no—
131C4×C73—42no—
132C4×C73—82no—
133C4×C73—81no—
134C4×C73—81no—
135C4×C73—81no—
136#12246C7311yesmaximal, characteristic, direct product factor, Sylow
137C2×C73—21no—
138C2×C73—21no—
139C2×C73—84no—
140C2×C73#160411yescharacteristic
141C2×C73#219611yescharacteristic
142C2×C73#220411yescharacteristic
143#2537C2×C7311yescharacteristic
144#2060C2×C7311yescharacteristic, complemented
145#1599C2×C7312yescomplemented
146#1345C2×C7312yescomplemented
147#1325C2×C7311yescharacteristic, complemented
148C73#1224611yescharacteristic, direct product factor, Sylow
149#544C22×C7311yescharacteristic
150C2×#121—22no—
151C2×C4×D4—21no—
152#450—22no—
153#428—21no—
154#394—21no—
155#394C22×C7312yes—
156#384C22×C7311yescharacteristic
157#380—22no—
158#343C22×C7311yescharacteristic
159#339—21no—
160#328—22no—
161#324C22×C7312yes—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4D4×C7312yes—
165C23×C4—22no—
166C2×SD16—42no—
167C2×M16—21no—
168C4⋊Q8—41no—
169#125—41no—
170C22⋊Q8—22no—
171#121—48no—
172C4×D4—42no—
173C4×D4—42no—
174#117—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4D4×C7311yescharacteristic
178C2×C4⋊C4C23×C7311yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C42D4×C7311yescharacteristic
183#107—41no—
184#106—41no—
185#102—42no—
186C22⋊C8—22no—
187#95—21no—
188#95D4×C7312yes—
189C24—22no—
190C2×Q8—22no—
191C2×D4—24no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—44no—
196C2×D4—44no—
197C2×D4—81no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4C2×D4×C7311yescharacteristic
205C22×C4C2×D4×C7312yes—
206SD16—84no—
207M16—22no—
208M16—41no—
209C2×C8—41no—
210C2×C8—42no—
211C4⋊C4—22no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—42no—
217C4⋊C4—44no—
218C4⋊C4—42no—
219C22⋊C4—44no—
220C22⋊C4—44no—
221C22⋊C4—41no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C42—21no—
225C42—41no—
226C42SD16×C7312yescomplemented
227C23—24no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23#120×C7311yescharacteristic
235Q8—82no—
236D4—48no—
237D4—44no—
238D4—82no—
239D4—84no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—88no—
255C2×C4C2×SD16×C7311yescharacteristic
256C2×C4AGL(1,ℤ/8)×C7311yescharacteristic
257C8—42no—
258C8—82no—
259C22—21no—
260C22—21no—
261C22—44no—
262C22—44no—
263C22—44no—
264C22—82no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22#450×C7311yescharacteristic
269C22#456×C7311yescharacteristic
270C22B(4,𝔽2)×C7311yescharacteristic
271C4—24no—
272C4—21no—
273C4—21no—
274C4—42no—
275C4—82no—
276C4—81no—
277C4—81no—
278C4—81no—
279C2—21no—
280C2—21no—
281C2—84no—
282C2C73×#160411yescharacteristic
283C2C73×#219611yescharacteristic
284C2C73×#220411yescharacteristic
subgroupG/H
center, socleC22×C73B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C73
prime radicalC73×#2537C2
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C23×C73 ◁ C2×C4⋊C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C73×#2060C2semidirect
3. C73×#1599C2semidirect
4. C73×#1345C2semidirect
5. C73×#1325C2semidirect
83. C42×C73SD16semidirect
136. #12246C73direct (N × H)
144. #2060C2×C73semidirect
145. #1599C2×C73semidirect
146. #1345C2×C73semidirect
147. #1325C2×C73semidirect
148. C73#12246direct (N × H)
226. C42SD16×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#25372
C73×#20602
C73×#15992
C73×#13452
C73×#13252
#1224673