Finite Group Databasebeta

← C73×#9032 #32918436

Subgroups — C73×#9032 #32918436

Order 18688.

516 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#2983C211yesmaximal, characteristic, complemented
2C73×#2398C211yesmaximal, characteristic
3C4×#101×C73C211yesmaximal, characteristic, complemented
4C73×#1490C212yesmaximal
5C73×#1300C212yesmaximal, complemented
6C2×C4×Q8×C73C2211yescharacteristic
7C2×C4×D4×C73C2211yescharacteristic
8#446×C73—24no—
9#432×C73—21no—
10#432×C73C2212yes—
11#431×C73—21no—
12C2×#101×C73—21no—
13C4×M16×C73C2212yes—
14#400×C73C411yescharacteristic
15#400×C73C411yescharacteristic
16#397×C73—21no—
17#394×C73C411yescharacteristic
18#394×C73C411yescharacteristic
19#393×C73—21no—
20C4×C22⋊C4×C73—21no—
21C4×C22⋊C4×C73—21no—
22#376×C73C2211yescharacteristic
23#343×C73—21no—
24D4⋊C8×C73—24no—
25C22×Q8×C73D411yescharacteristic
26C22×D4×C73C2×C411yescharacteristic
27C23×C4×C73—21no—
28C2×M16×C73—22no—
29C4×Q8×C73—22no—
30C4×Q8×C73—22no—
31C4×D4×C73—22no—
32C4×D4×C73—22no—
33C2×C4⋊C4×C73C2×C411yescharacteristic
34C2×C4⋊C4×C73C2×C411yescharacteristic
35C2×C4⋊C4×C73C2×C411yescharacteristic
36C2×C4⋊C4×C73D411yescharacteristic
37C2×C22⋊C4×C73—21no—
38C2×C22⋊C4×C73—21no—
39C2×C22⋊C4×C73—21no—
40C2×C22⋊C4×C73—21no—
41C2×C22⋊C4×C73—21no—
42C2×C42×C73—21no—
43C2×C42×C73D411yescharacteristic
44C2×C42×C73D411yescharacteristic
45C2×C42×C73C2311yescharacteristic
46C4⋊C8×C73—22no—
47C4⋊C8×C73—22no—
48C4⋊C8×C73D414yes—
49#101×C73—24no—
50C22⋊C8×C73—22no—
51#97×C73—22no—
52#97×C73—22no—
53#97×C73—22no—
54C4×C8×C73—22no—
55C4×C8×C73—22no—
56#95×C73—21no—
57#95×C73—21no—
58#95×C73—21no—
59#95×C73—21no—
60#95×C73—21no—
61#95×C73C2×C411yescharacteristic
62#95×C73C2×C411yescharacteristic
63C24×C73—21no—
64C2×Q8×C73—22no—
65C2×Q8×C73—22no—
66C2×Q8×C73—41no—
67C2×D4×C73—22no—
68C2×D4×C73—42no—
69C22×C4×C73—21no—
70C22×C4×C73—21no—
71C22×C4×C73—21no—
72C22×C4×C73—42no—
73C22×C4×C73—42no—
74C22×C4×C73—41no—
75C22×C4×C73—41no—
76C22×C4×C73C22⋊C411yescharacteristic
77C22×C4×C73C22⋊C411yescharacteristic
78C22×C4×C73C22⋊C411yescharacteristic
79C22×C4×C73C22⋊C411yescharacteristic
80C22×C4×C73C22×C411yescharacteristic
81C22×C4×C73C2×D411yescharacteristic
82C22×C4×C73C2×D411yescharacteristic
83M16×C73—24no—
84M16×C73—42no—
85C2×C8×C73—24no—
86C2×C8×C73—42no—
87C4⋊C4×C73—22no—
88C4⋊C4×C73—22no—
89C4⋊C4×C73—22no—
90C4⋊C4×C73—22no—
91C22⋊C4×C73—42no—
92C22⋊C4×C73—42no—
93C22⋊C4×C73—42no—
94C22⋊C4×C73—42no—
95C22⋊C4×C73—42no—
96C42×C73—21no—
97C42×C73—21no—
98C42×C73—21no—
99C42×C73—21no—
100C42×C73—42no—
101C42×C73C2×D412yes—
102C42×C73#4112yes—
103C23×C73—22no—
104C23×C73—41no—
105C23×C73—41no—
106C23×C73—41no—
107C23×C73—41no—
108C23×C73—41no—
109C23×C73—41no—
110C23×C73C2×C22⋊C411yescharacteristic
111Q8×C73—44no—
112D4×C73—24no—
113D4×C73—42no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—21no—
117C2×C4×C73—21no—
118C2×C4×C73—22no—
119C2×C4×C73—21no—
120C2×C4×C73—21no—
121C2×C4×C73—21no—
122C2×C4×C73—21no—
123C2×C4×C73—21no—
124C2×C4×C73—41no—
125C2×C4×C73—41no—
126C2×C4×C73—41no—
127C2×C4×C73—41no—
128C2×C4×C73—41no—
129C2×C4×C73—41no—
130C2×C4×C73—41no—
131C2×C4×C73—41no—
132C2×C4×C73—41no—
133C2×C4×C73—41no—
134C2×C4×C73—41no—
135C2×C4×C73—41no—
136C2×C4×C73—82no—
137C2×C4×C73—82no—
138C2×C4×C73#10412yes—
139C2×C4×C73C4×D412yes—
140C2×C4×C73#12011yescharacteristic
141C2×C4×C73#12112yes—
142C2×C4×C73#12311yescharacteristic
143C8×C73—44no—
144C8×C73—44no—
145C22×C73—21no—
146C22×C73—21no—
147C22×C73—42no—
148C22×C73—42no—
149C22×C73—42no—
150C22×C73—81no—
151C22×C73—81no—
152C22×C73—81no—
153C22×C73—81no—
154C22×C73#38611yescharacteristic
155C22×C73C2×#10411yescharacteristic
156C22×C73#42111yescharacteristic
157C4×C73—22no—
158C4×C73—22no—
159C4×C73—21no—
160C4×C73—22no—
161C4×C73—41no—
162C4×C73—41no—
163C4×C73—41no—
164C4×C73—81no—
165C4×C73—81no—
166C4×C73—81no—
167C4×C73#45412yes—
168#9032C7311yesmaximal, characteristic, direct product factor, Sylow
169C2×C73—21no—
170C2×C73—21no—
171C2×C73—82no—
172C2×C73#187111yescharacteristic
173C2×C73#188811yescharacteristic
174C2×C73#189111yescharacteristic
175#2983C2×C7311yescharacteristic, complemented
176#2398C2×C7311yescharacteristic
177C4×#101C2×C7311yescharacteristic, complemented
178#1490C2×C7312yes—
179#1300C2×C7312yescomplemented
180C73#903211yescharacteristic, direct product factor, Sylow
181C2×C4×Q8C22×C7311yescharacteristic
182C2×C4×D4C22×C7311yescharacteristic
183#446—24no—
184#432—21no—
185#432C22×C7312yes—
186#431—21no—
187C2×#101—21no—
188C4×M16C22×C7312yes—
189#400C4×C7311yescharacteristic
190#400C4×C7311yescharacteristic
191#397—21no—
192#394C4×C7311yescharacteristic
193#394C4×C7311yescharacteristic
194#393—21no—
195C4×C22⋊C4—21no—
196C4×C22⋊C4—21no—
197#376C22×C7311yescharacteristic
198#343—21no—
199D4⋊C8—24no—
200C22×Q8D4×C7311yescharacteristic
201C22×D4C2×C4×C7311yescharacteristic
202C23×C4—21no—
203C2×M16—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×D4—22no—
207C4×D4—22no—
208C2×C4⋊C4C2×C4×C7311yescharacteristic
209C2×C4⋊C4C2×C4×C7311yescharacteristic
210C2×C4⋊C4C2×C4×C7311yescharacteristic
211C2×C4⋊C4D4×C7311yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C42—21no—
218C2×C42D4×C7311yescharacteristic
219C2×C42D4×C7311yescharacteristic
220C2×C42C23×C7311yescharacteristic
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8D4×C7314yes—
224#101—24no—
225C22⋊C8—22no—
226#97—22no—
227#97—22no—
228#97—22no—
229C4×C8—22no—
230C4×C8—22no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95C2×C4×C7311yescharacteristic
237#95C2×C4×C7311yescharacteristic
238C24—21no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—41no—
242C2×D4—22no—
243C2×D4—42no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4C22⋊C4×C7311yescharacteristic
252C22×C4C22⋊C4×C7311yescharacteristic
253C22×C4C22⋊C4×C7311yescharacteristic
254C22×C4C22⋊C4×C7311yescharacteristic
255C22×C4C22×C4×C7311yescharacteristic
256C22×C4C2×D4×C7311yescharacteristic
257C22×C4C2×D4×C7311yescharacteristic
258M16—24no—
259M16—42no—
260C2×C8—24no—
261C2×C8—42no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—42no—
276C42C2×D4×C7312yes—
277C42#41×C7312yes—
278C23—22no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23C2×C22⋊C4×C7311yescharacteristic
286Q8—44no—
287D4—24no—
288D4—42no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4#104×C7312yes—
314C2×C4C4×D4×C7312yes—
315C2×C4#120×C7311yescharacteristic
316C2×C4#121×C7312yes—
317C2×C4#123×C7311yescharacteristic
318C8—44no—
319C8—44no—
320C22—21no—
321C22—21no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22#386×C7311yescharacteristic
330C22C2×#104×C7311yescharacteristic
331C22#421×C7311yescharacteristic
332C4—22no—
333C4—22no—
334C4—21no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4#454×C7312yes—
343C2—21no—
344C2—21no—
345C2—82no—
346C2C73×#187111yescharacteristic
347C2C73×#188811yescharacteristic
348C2C73×#189111yescharacteristic
subgroupG/H
centerC2×C4×C73#120
commutatorC22×C4C22×C4×C73
Frattini, cosocleC2×C42C23×C73
socleC22×C73#386
prime radicalC22×D4×C73C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C73 ◁ C2×C42×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C73×#2983C2semidirect
3. C4×#101×C73C2semidirect
5. C73×#1300C2semidirect
168. #9032C73direct (N × H)
175. #2983C2×C73semidirect
177. C4×#101C2×C73semidirect
179. #1300C2×C73semidirect
180. C73#9032direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#29832
C73×#23982
C4×#101×C732
C73×#14902
C73×#13002
#903273