Finite Group Databasebeta

← C73×#9200 #32918604

Subgroups — C73×#9200 #32918604

Order 18688.

560 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#2627C211yesmaximal, characteristic
2C73×#2316C211yesmaximal, characteristic
3C73×#1520C212yesmaximal
4C2×#324×C73C211yesmaximal, characteristic
5C73×#1289C212yesmaximal
6C2×C22⋊Q8×C73C2211yescharacteristic
7C2×#117×C73—21no—
8C22×C4⋊C4×C73C2211yescharacteristic
9C2×C22⋊C8×C73C2212yes—
10#400×C73C412yescomplemented
11#398×C73—22no—
12#395×C73C412yescomplemented
13#389×C73—24no—
14#385×C73—21no—
15#381×C73—21no—
16#380×C73—21no—
17#380×C73C2211yescharacteristic
18#380×C73C2212yes—
19#328×C73—24no—
20#324×C73—22no—
21C22×Q8×C73C2×C411yescharacteristic
22C23×C4×C73D411yescharacteristic
23C23×C4×C73D411yescharacteristic
24C23×C4×C73C2311yescharacteristic
25C22×C8×C73—22no—
26C22⋊Q8×C73—22no—
27C22⋊Q8×C73—22no—
28#117×C73—42no—
29#117×C73—42no—
30C2×C4⋊C4×C73—22no—
31C2×C4⋊C4×C73—22no—
32C2×C4⋊C4×C73—22no—
33C2×C4⋊C4×C73—21no—
34C2×C4⋊C4×C73—22no—
35C2×C4⋊C4×C73—22no—
36C2×C4⋊C4×C73—22no—
37C2×C4⋊C4×C73—21no—
38C2×C4⋊C4×C73C2×C411yescharacteristic
39C2×C4⋊C4×C73C2×C412yes—
40C2×C4⋊C4×C73D412yes—
41C2×C22⋊C4×C73—22no—
42C2×C22⋊C4×C73—21no—
43C2×C22⋊C4×C73—21no—
44C2×C42×C73—22no—
45C22⋊C8×C73—22no—
46C22⋊C8×C73—22no—
47#95×C73—22no—
48#95×C73—22no—
49#95×C73—22no—
50#95×C73—22no—
51#95×C73C2×C412yes—
52#95×C73D414yes—
53C24×C73C2×D411yescharacteristic
54C2×Q8×C73—22no—
55C2×Q8×C73—81no—
56C22×C4×C73—21no—
57C22×C4×C73—21no—
58C22×C4×C73—21no—
59C22×C4×C73—21no—
60C22×C4×C73—21no—
61C22×C4×C73—21no—
62C22×C4×C73—21no—
63C22×C4×C73—22no—
64C22×C4×C73—21no—
65C22×C4×C73—24no—
66C22×C4×C73—81no—
67C22×C4×C73—81no—
68C22×C4×C73C22⋊C412yes—
69C22×C4×C73C22⋊C412yes—
70C22×C4×C73SD1612yes—
71C22×C4×C73Q1612yes—
72C22×C4×C73C22×C411yescharacteristic
73C22×C4×C73C2×D412yes—
74C22×C4×C73C2×D411yescharacteristic
75C22×C4×C73#4112yes—
76C2×C8×C73—42no—
77C2×C8×C73—82no—
78C4⋊C4×C73—24no—
79C4⋊C4×C73—24no—
80C4⋊C4×C73—22no—
81C4⋊C4×C73—44no—
82C4⋊C4×C73—42no—
83C4⋊C4×C73—82no—
84C4⋊C4×C73—81no—
85C4⋊C4×C73—82no—
86C4⋊C4×C73—82no—
87C22⋊C4×C73—41no—
88C22⋊C4×C73—41no—
89C22⋊C4×C73—42no—
90C22⋊C4×C73—41no—
91C22⋊C4×C73—42no—
92C22⋊C4×C73—41no—
93C42×C73—44no—
94C23×C73—41no—
95C23×C73—41no—
96C23×C73C2×C22⋊C411yescharacteristic
97C23×C73#12011yescharacteristic
98C23×C73#12311yescharacteristic
99C23×C73C2×SD1611yescharacteristic
100C23×C73C2×Q1611yescharacteristic
101C23×C73AGL(1,ℤ/8)11yescharacteristic
102C23×C73#13711yescharacteristic
103Q8×C73—82no—
104C2×C4×C73—22no—
105C2×C4×C73—22no—
106C2×C4×C73—22no—
107C2×C4×C73—22no—
108C2×C4×C73—22no—
109C2×C4×C73—22no—
110C2×C4×C73—22no—
111C2×C4×C73—41no—
112C2×C4×C73—42no—
113C2×C4×C73—42no—
114C2×C4×C73—41no—
115C2×C4×C73—42no—
116C2×C4×C73—42no—
117C2×C4×C73—42no—
118C2×C4×C73—44no—
119C2×C4×C73—42no—
120C2×C4×C73—42no—
121C2×C4×C73—44no—
122C2×C4×C73—42no—
123C2×C4×C73—41no—
124C2×C4×C73—41no—
125C2×C4×C73—42no—
126C2×C4×C73—44no—
127C2×C4×C73—81no—
128C2×C4×C73#10314yes—
129C2×C4×C73C4×D412yes—
130C2×C4×C73#12112yes—
131C8×C73—82no—
132C22×C73—21no—
133C22×C73—21no—
134C22×C73—21no—
135C22×C73—21no—
136C22×C73—21no—
137C22×C73—21no—
138C22×C73—41no—
139C22×C73—41no—
140C22×C73—41no—
141C22×C73—41no—
142C22×C73#38611yescharacteristic
143C22×C73C2×#10311yescharacteristic
144C22×C73#41711yescharacteristic
145C22×C73#45011yescharacteristic
146C22×C73#45111yescharacteristic
147C22×C73#48311yescharacteristic
148C22×C73#48411yescharacteristic
149C4×C73—24no—
150C4×C73—22no—
151C4×C73—22no—
152C4×C73—42no—
153C4×C73—84no—
154C4×C73—82no—
155C4×C73—81no—
156C4×C73—81no—
157#9200C7311yesmaximal, characteristic, direct product factor, Sylow
158C2×C73—21no—
159C2×C73—21no—
160C2×C73—21no—
161C2×C73—21no—
162C2×C73—21no—
163C2×C73—21no—
164C2×C73#187211yescharacteristic
165C2×C73#191211yescharacteristic
166C2×C73#191311yescharacteristic
167#2627C2×C7311yescharacteristic
168#2316C2×C7311yescharacteristic
169#1520C2×C7312yes—
170C2×#324C2×C7311yescharacteristic
171#1289C2×C7312yes—
172C73#920011yescharacteristic, direct product factor, Sylow
173C2×C22⋊Q8C22×C7311yescharacteristic
174C2×#117—21no—
175C22×C4⋊C4C22×C7311yescharacteristic
176C2×C22⋊C8C22×C7312yes—
177#400C4×C7312yescomplemented
178#398—22no—
179#395C4×C7312yescomplemented
180#389—24no—
181#385—21no—
182#381—21no—
183#380—21no—
184#380C22×C7311yescharacteristic
185#380C22×C7312yes—
186#328—24no—
187#324—22no—
188C22×Q8C2×C4×C7311yescharacteristic
189C23×C4D4×C7311yescharacteristic
190C23×C4D4×C7311yescharacteristic
191C23×C4C23×C7311yescharacteristic
192C22×C8—22no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195#117—42no—
196#117—42no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C2×C4×C7311yescharacteristic
206C2×C4⋊C4C2×C4×C7312yes—
207C2×C4⋊C4D4×C7312yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C42—22no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95C2×C4×C7312yes—
219#95D4×C7314yes—
220C24C2×D4×C7311yescharacteristic
221C2×Q8—22no—
222C2×Q8—81no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—81no—
234C22×C4—81no—
235C22×C4C22⋊C4×C7312yes—
236C22×C4C22⋊C4×C7312yes—
237C22×C4SD16×C7312yes—
238C22×C4Q16×C7312yes—
239C22×C4C22×C4×C7311yescharacteristic
240C22×C4C2×D4×C7312yes—
241C22×C4C2×D4×C7311yescharacteristic
242C22×C4#41×C7312yes—
243C2×C8—42no—
244C2×C8—82no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—22no—
248C4⋊C4—44no—
249C4⋊C4—42no—
250C4⋊C4—82no—
251C4⋊C4—81no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—42no—
257C22⋊C4—41no—
258C22⋊C4—42no—
259C22⋊C4—41no—
260C42—44no—
261C23—41no—
262C23—41no—
263C23C2×C22⋊C4×C7311yescharacteristic
264C23#120×C7311yescharacteristic
265C23#123×C7311yescharacteristic
266C23C2×SD16×C7311yescharacteristic
267C23C2×Q16×C7311yescharacteristic
268C23AGL(1,ℤ/8)×C7311yescharacteristic
269C23#137×C7311yescharacteristic
270Q8—82no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—81no—
295C2×C4#103×C7314yes—
296C2×C4C4×D4×C7312yes—
297C2×C4#121×C7312yes—
298C8—82no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22#386×C7311yescharacteristic
310C22C2×#103×C7311yescharacteristic
311C22#417×C7311yescharacteristic
312C22#450×C7311yescharacteristic
313C22#451×C7311yescharacteristic
314C22#483×C7311yescharacteristic
315C22#484×C7311yescharacteristic
316C4—24no—
317C4—22no—
318C4—22no—
319C4—42no—
320C4—84no—
321C4—82no—
322C4—81no—
323C4—81no—
324C2—21no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2—21no—
329C2—21no—
330C2C73×#187211yescharacteristic
331C2C73×#191211yescharacteristic
332C2C73×#191311yescharacteristic
subgroupG/H
center, socleC22×C73#386
commutatorC22×C4C22×C4×C73
Frattini, cosocleC23×C4C23×C73
prime radicalC24×C73C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C73 ◁ C23×C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
10. #400×C73C4semidirect
12. #395×C73C4semidirect
157. #9200C73direct (N × H)
172. C73#9200direct (N × H)
177. #400C4×C73semidirect
179. #395C4×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#26272
C73×#23162
C73×#15202
C2×#324×C732
C73×#12892
#920073