Finite Group Databasebeta

← C73×#9304 #32918708

Subgroups — C73×#9304 #32918708

Order 18688.

436 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3078C211yesmaximal, characteristic
2C73×#2982C211yesmaximal, characteristic
3C73×#1805C211yesmaximal, characteristic
4C73×#1518C211yesmaximal, characteristic
5C73×#1481C211yesmaximal, characteristic
6C73×#1301C211yesmaximal, characteristic
7C73×#1286C211yesmaximal, characteristic
8#533×C73C2211yescharacteristic
9C2×C4×Q8×C73C2211yescharacteristic
10#479×C73—22no—
11#477×C73C412yescomplemented
12#475×C73C412yescomplemented
13C8×Q8×C73—22no—
14C8×Q8×C73—22no—
15#431×C73—22no—
16#431×C73C2211yescharacteristic
17#431×C73C2211yescharacteristic
18#428×C73—21no—
19#423×C73C2211yescharacteristic
20C2×C4⋊C8×C73C2211yescharacteristic
21#419×C73—21no—
22C2×#103×C73—21no—
23#382×C73C2211yescharacteristic
24#340×C73—21no—
25#328×C73—22no—
26Q8⋊C8×C73—22no—
27C22×Q8×C73D411yescharacteristic
28C2×M16×C73—21no—
29C22×C8×C73—21no—
30C4⋊Q8×C73C2×C411yescharacteristic
31C4⋊Q8×C73C2×C411yescharacteristic
32#125×C73—21no—
33C22⋊Q8×C73—21no—
34C22⋊Q8×C73—21no—
35C4×Q8×C73—21no—
36C4×Q8×C73—21no—
37C4×Q8×C73—21no—
38C4×Q8×C73—21no—
39C4×Q8×C73C2×C412yes—
40C4×Q8×C73D412yes—
41#117×C73—21no—
42C2×C4⋊C4×C73—21no—
43C2×C4⋊C4×C73—21no—
44C2×C4⋊C4×C73D411yescharacteristic
45C2×C42×C73—21no—
46C2×C42×C73C2311yescharacteristic
47#107×C73—22no—
48#106×C73—22no—
49C4⋊C8×C73—24no—
50C4⋊C8×C73—21no—
51C4⋊C8×C73—21no—
52C4⋊C8×C73—21no—
53C4⋊C8×C73—21no—
54C4⋊C8×C73—21no—
55C4⋊C8×C73C2×C412yes—
56#103×C73—22no—
57#103×C73—22no—
58#103×C73—42no—
59C22⋊C8×C73—21no—
60C22⋊C8×C73D412yes—
61C4×C8×C73—24no—
62C4×C8×C73—21no—
63C4×C8×C73D412yes—
64#95×C73—21no—
65#95×C73—21no—
66C2×Q8×C73—21no—
67C2×Q8×C73—21no—
68C2×Q8×C73—21no—
69C2×Q8×C73—21no—
70C2×Q8×C73—21no—
71C2×Q8×C73C22⋊C412yes—
72C2×Q8×C73SD1612yes—
73C2×Q8×C73Q1612yes—
74C22×C4×C73—21no—
75C22×C4×C73—21no—
76C22×C4×C73—41no—
77C22×C4×C73C2×D411yescharacteristic
78C22×C4×C73C2×D411yescharacteristic
79C22×C4×C73#4111yescharacteristic
80M16×C73—41no—
81M16×C73—41no—
82C2×C8×C73—22no—
83C2×C8×C73—22no—
84C2×C8×C73—22no—
85C2×C8×C73—41no—
86C2×C8×C73—41no—
87C2×C8×C73—41no—
88C4⋊C4×C73—22no—
89C4⋊C4×C73—21no—
90C4⋊C4×C73—22no—
91C4⋊C4×C73—21no—
92C4⋊C4×C73—21no—
93C4⋊C4×C73—21no—
94C4⋊C4×C73—21no—
95C4⋊C4×C73—41no—
96C4⋊C4×C73—41no—
97C4⋊C4×C73—41no—
98C4⋊C4×C73—41no—
99C4⋊C4×C73C22⋊C412yes—
100C22⋊C4×C73—41no—
101C42×C73—22no—
102C42×C73—22no—
103C42×C73—41no—
104C42×C73C22×C411yescharacteristic
105C42×C73C2×D411yescharacteristic
106C42×C73C2×D411yescharacteristic
107C42×C73#4111yescharacteristic
108C23×C73#12111yescharacteristic
109Q8×C73—22no—
110Q8×C73—22no—
111Q8×C73—22no—
112Q8×C73—42no—
113Q8×C73—81no—
114Q8×C73#10314yes—
115C2×C4×C73—22no—
116C2×C4×C73—22no—
117C2×C4×C73—22no—
118C2×C4×C73—22no—
119C2×C4×C73—41no—
120C2×C4×C73—41no—
121C2×C4×C73—41no—
122C2×C4×C73—41no—
123C2×C4×C73—81no—
124C2×C4×C73—81no—
125C2×C4×C73—81no—
126C2×C4×C73—81no—
127C2×C4×C73C2×C22⋊C411yescharacteristic
128C2×C4×C73C4×D411yescharacteristic
129C2×C4×C73C4×D411yescharacteristic
130C2×C4×C73#12011yescharacteristic
131C2×C4×C73#12111yescharacteristic
132C2×C4×C73#12311yescharacteristic
133C2×C4×C73C2×SD1611yescharacteristic
134C2×C4×C73C2×Q1611yescharacteristic
135C2×C4×C73#13511yescharacteristic
136C2×C4×C73#13511yescharacteristic
137C8×C73—42no—
138C8×C73—42no—
139C8×C73—42no—
140C8×C73—81no—
141C22×C73—21no—
142C22×C73—21no—
143C22×C73#38611yescharacteristic
144C22×C73#46511yescharacteristic
145C22×C73#46711yescharacteristic
146C4×C73—21no—
147C4×C73—21no—
148C4×C73—24no—
149C4×C73—21no—
150C4×C73—21no—
151C4×C73—42no—
152C4×C73—81no—
153C4×C73—81no—
154C4×C73—161no—
155C4×C73C2×#10311yescharacteristic
156C4×C73#41611yescharacteristic
157C4×C73#46511yescharacteristic
158C4×C73#46711yescharacteristic
159#9304C7311yesmaximal, characteristic, direct product factor, Sylow
160C2×C73—21no—
161C2×C73—21no—
162C2×C73#188111yescharacteristic
163C2×C73#189211yescharacteristic
164C2×C73#190511yescharacteristic
165#3078C2×C7311yescharacteristic
166#2982C2×C7311yescharacteristic
167#1805C2×C7311yescharacteristic
168#1518C2×C7311yescharacteristic
169#1481C2×C7311yescharacteristic
170#1301C2×C7311yescharacteristic
171#1286C2×C7311yescharacteristic
172C73#930411yescharacteristic, direct product factor, Sylow
173#533C22×C7311yescharacteristic
174C2×C4×Q8C22×C7311yescharacteristic
175#479—22no—
176#477C4×C7312yescomplemented
177#475C4×C7312yescomplemented
178C8×Q8—22no—
179C8×Q8—22no—
180#431—22no—
181#431C22×C7311yescharacteristic
182#431C22×C7311yescharacteristic
183#428—21no—
184#423C22×C7311yescharacteristic
185C2×C4⋊C8C22×C7311yescharacteristic
186#419—21no—
187C2×#103—21no—
188#382C22×C7311yescharacteristic
189#340—21no—
190#328—22no—
191Q8⋊C8—22no—
192C22×Q8D4×C7311yescharacteristic
193C2×M16—21no—
194C22×C8—21no—
195C4⋊Q8C2×C4×C7311yescharacteristic
196C4⋊Q8C2×C4×C7311yescharacteristic
197#125—21no—
198C22⋊Q8—21no—
199C22⋊Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×Q8—21no—
204C4×Q8C2×C4×C7312yes—
205C4×Q8D4×C7312yes—
206#117—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4D4×C7311yescharacteristic
210C2×C42—21no—
211C2×C42C23×C7311yescharacteristic
212#107—22no—
213#106—22no—
214C4⋊C8—24no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8C2×C4×C7312yes—
221#103—22no—
222#103—22no—
223#103—42no—
224C22⋊C8—21no—
225C22⋊C8D4×C7312yes—
226C4×C8—24no—
227C4×C8—21no—
228C4×C8D4×C7312yes—
229#95—21no—
230#95—21no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8C22⋊C4×C7312yes—
237C2×Q8SD16×C7312yes—
238C2×Q8Q16×C7312yes—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—41no—
242C22×C4C2×D4×C7311yescharacteristic
243C22×C4C2×D4×C7311yescharacteristic
244C22×C4#41×C7311yescharacteristic
245M16—41no—
246M16—41no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—41no—
253C4⋊C4—22no—
254C4⋊C4—21no—
255C4⋊C4—22no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4C22⋊C4×C7312yes—
265C22⋊C4—41no—
266C42—22no—
267C42—22no—
268C42—41no—
269C42C22×C4×C7311yescharacteristic
270C42C2×D4×C7311yescharacteristic
271C42C2×D4×C7311yescharacteristic
272C42#41×C7311yescharacteristic
273C23#121×C7311yescharacteristic
274Q8—22no—
275Q8—22no—
276Q8—22no—
277Q8—42no—
278Q8—81no—
279Q8#103×C7314yes—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4C2×C22⋊C4×C7311yescharacteristic
293C2×C4C4×D4×C7311yescharacteristic
294C2×C4C4×D4×C7311yescharacteristic
295C2×C4#120×C7311yescharacteristic
296C2×C4#121×C7311yescharacteristic
297C2×C4#123×C7311yescharacteristic
298C2×C4C2×SD16×C7311yescharacteristic
299C2×C4C2×Q16×C7311yescharacteristic
300C2×C4#135×C7311yescharacteristic
301C2×C4#135×C7311yescharacteristic
302C8—42no—
303C8—42no—
304C8—42no—
305C8—81no—
306C22—21no—
307C22—21no—
308C22#386×C7311yescharacteristic
309C22#465×C7311yescharacteristic
310C22#467×C7311yescharacteristic
311C4—21no—
312C4—21no—
313C4—24no—
314C4—21no—
315C4—21no—
316C4—42no—
317C4—81no—
318C4—81no—
319C4—161no—
320C4C2×#103×C7311yescharacteristic
321C4#416×C7311yescharacteristic
322C4#465×C7311yescharacteristic
323C4#467×C7311yescharacteristic
324C2—21no—
325C2—21no—
326C2C73×#188111yescharacteristic
327C2C73×#189211yescharacteristic
328C2C73×#190511yescharacteristic
subgroupG/H
center, socleC22×C73#386
commutatorC42C22×C4×C73
Frattini, cosocleC2×C42C23×C73
prime radicalC23×C73#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C73 ◁ C2×C42×C73 ◁ G

Complements

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

NH ≅ G/Ntype
11. #477×C73C4semidirect
12. #475×C73C4semidirect
159. #9304C73direct (N × H)
172. C73#9304direct (N × H)
176. #477C4×C73semidirect
177. #475C4×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#30782
C73×#29822
C73×#18052
C73×#15182
C73×#14812
C73×#13012
C73×#12862
#930473