Finite Group Databasebeta

← C73×#8202 #32917606

Subgroups — C73×#8202 #32917606

Order 18688.

570 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C8×D4×C73C211yesmaximal, characteristic
2C73×#1770C211yesmaximal, characteristic, complemented
3C4×#102×C73C211yesmaximal, characteristic
4C2×D4⋊C8×C73C211yesmaximal, characteristic
5C2×#322×C73C211yesmaximal, characteristic, complemented
6C73×#1279C211yesmaximal, characteristic, complemented
7C73×#1277C211yesmaximal, characteristic, complemented
8C23×C8×C73—21no—
9C2×C4×D4×C73C2211yescharacteristic
10C8×D4×C73—22no—
11C8×D4×C73C414yescomplemented
12C2×C4⋊C8×C73C2211yescharacteristic
13C2×C4⋊C8×C73C2211yescharacteristic
14C2×#102×C73C412yes—
15C2×C22⋊C8×C73—21no—
16C2×#97×C73—21no—
17C2×C4×C8×C73C2211yescharacteristic
18C2×C4×C8×C73C2211yescharacteristic
19C2×C4×C8×C73C2211yescharacteristic
20C4×C4⋊C4×C73C2211yescharacteristic
21#340×C73C412yes—
22#336×C73—21no—
23D4⋊C8×C73C414yescomplemented
24#322×C73—22no—
25C22×D4×C73C2×C411yescharacteristic
26C23×C4×C73—21no—
27C22×C8×C73—21no—
28C22×C8×C73—21no—
29C22×C8×C73—21no—
30C22×C8×C73—24no—
31C22×C8×C73—42no—
32C22×C8×C73—42no—
33C22×C8×C73C2×C412yes—
34C22×C8×C73D411yescharacteristic
35C22×C8×C73D411yescharacteristic
36C4×D4×C73—21no—
37C4×D4×C73—21no—
38C4×D4×C73C2×C412yes—
39C4×D4×C73C2×C412yes—
40C2×C4⋊C4×C73C2×C412yes—
41C2×C4⋊C4×C73C2×C411yescharacteristic
42C2×C22⋊C4×C73—21no—
43C2×C42×C73—21no—
44C2×C42×C73C2311yescharacteristic
45C4⋊C8×C73—21no—
46C4⋊C8×C73—21no—
47C4⋊C8×C73C2×C412yes—
48C4⋊C8×C73C2×C412yes—
49#102×C73—24no—
50C22⋊C8×C73—24no—
51#97×C73—42no—
52C4×C8×C73—21no—
53C4×C8×C73—21no—
54C4×C8×C73—21no—
55C4×C8×C73C2×C412yes—
56C4×C8×C73C2×C412yes—
57C4×C8×C73D412yes—
58#95×C73—21no—
59C24×C73—21no—
60C2×D4×C73—21no—
61C2×D4×C73—22no—
62C2×D4×C73C4212yes—
63C2×D4×C73M1614yes—
64C22×C4×C73—21no—
65C22×C4×C73—22no—
66C22×C4×C73—22no—
67C22×C4×C73—22no—
68C22×C4×C73—41no—
69C22×C4×C73—41no—
70C22×C4×C73—41no—
71C22×C4×C73—41no—
72C22×C4×C73C22×C411yescharacteristic
73C22×C4×C73C2×D411yescharacteristic
74C22×C4×C73#4111yescharacteristic
75C2×C8×C73—21no—
76C2×C8×C73—21no—
77C2×C8×C73—22no—
78C2×C8×C73—22no—
79C2×C8×C73—22no—
80C2×C8×C73—22no—
81C2×C8×C73—22no—
82C2×C8×C73—22no—
83C2×C8×C73—42no—
84C2×C8×C73—41no—
85C2×C8×C73—41no—
86C2×C8×C73—41no—
87C2×C8×C73—41no—
88C2×C8×C73—48no—
89C2×C8×C73C22⋊C412yes—
90C2×C8×C73C22⋊C412yes—
91C2×C8×C73D812yes—
92C2×C8×C73SD1612yes—
93C4⋊C4×C73—21no—
94C4⋊C4×C73—22no—
95C4⋊C4×C73—22no—
96C4⋊C4×C73C4212yes—
97C22⋊C4×C73—22no—
98C22⋊C4×C73—22no—
99C42×C73—41no—
100C42×C73—41no—
101C42×C73C22×C411yescharacteristic
102C42×C73C22×C411yescharacteristic
103C42×C73C2×D411yescharacteristic
104C42×C73#4111yescharacteristic
105C23×C73—22no—
106C23×C73—24no—
107C23×C73—41no—
108C23×C73—41no—
109C23×C73—42no—
110C23×C73C4×D411yescharacteristic
111D4×C73—22no—
112D4×C73—24no—
113D4×C73#9714yescomplemented
114C2×C4×C73—22no—
115C2×C4×C73—22no—
116C2×C4×C73—41no—
117C2×C4×C73—41no—
118C2×C4×C73—42no—
119C2×C4×C73—42no—
120C2×C4×C73—42no—
121C2×C4×C73—42no—
122C2×C4×C73—42no—
123C2×C4×C73—42no—
124C2×C4×C73—42no—
125C2×C4×C73C2×C4211yescharacteristic
126C2×C4×C73C2×C22⋊C411yescharacteristic
127C2×C4×C73#11711yescharacteristic
128C2×C4×C73C4×D411yescharacteristic
129C2×C4×C73C4×D411yescharacteristic
130C2×C4×C73C4×D411yescharacteristic
131C2×C4×C73C2×M1612yes—
132C2×C4×C73#13112yes—
133C2×C4×C73C2×D811yescharacteristic
134C2×C4×C73C2×SD1611yescharacteristic
135C2×C4×C73#13511yescharacteristic
136C2×C4×C73#13511yescharacteristic
137C8×C73—22no—
138C8×C73—42no—
139C8×C73—42no—
140C8×C73—42no—
141C8×C73—42no—
142C8×C73#10214yescomplemented
143C22×C73—24no—
144C22×C73—44no—
145C22×C73—44no—
146C22×C73—42no—
147C22×C73—42no—
148C22×C73C4×C22⋊C411yescharacteristic
149C22×C73#43512yes—
150C22×C73C4×D811yescharacteristic
151C22×C73C4×SD1611yescharacteristic
152C22×C73#44412yes—
153C4×C73—21no—
154C4×C73—21no—
155C4×C73—42no—
156C4×C73—82no—
157C4×C73C2×#9711yescharacteristic
158C4×C73#40511yescharacteristic
159C4×C73C2×#10211yescharacteristic
160C4×C73#41611yescharacteristic
161C4×C73#43512yes—
162C4×C73C4×D811yescharacteristic
163C4×C73C4×SD1611yescharacteristic
164#8202C7311yesmaximal, characteristic, direct product factor, Sylow
165C2×C73—44no—
166C2×C73#158112yes—
167C2×C73#158212yes—
168C2×C73#175211yescharacteristic
169C2×C73C4×#10211yescharacteristic
170C2×C73#176511yescharacteristic
171C2×C8×D4C2×C7311yescharacteristic
172#1770C2×C7311yescharacteristic, complemented
173C4×#102C2×C7311yescharacteristic
174C2×D4⋊C8C2×C7311yescharacteristic
175C2×#322C2×C7311yescharacteristic, complemented
176#1279C2×C7311yescharacteristic, complemented
177#1277C2×C7311yescharacteristic, complemented
178C73#820211yescharacteristic, direct product factor, Sylow
179C23×C8—21no—
180C2×C4×D4C22×C7311yescharacteristic
181C8×D4—22no—
182C8×D4C4×C7314yescomplemented
183C2×C4⋊C8C22×C7311yescharacteristic
184C2×C4⋊C8C22×C7311yescharacteristic
185C2×#102C4×C7312yes—
186C2×C22⋊C8—21no—
187C2×#97—21no—
188C2×C4×C8C22×C7311yescharacteristic
189C2×C4×C8C22×C7311yescharacteristic
190C2×C4×C8C22×C7311yescharacteristic
191C4×C4⋊C4C22×C7311yescharacteristic
192#340C4×C7312yes—
193#336—21no—
194D4⋊C8C4×C7314yescomplemented
195#322—22no—
196C22×D4C2×C4×C7311yescharacteristic
197C23×C4—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—24no—
202C22×C8—42no—
203C22×C8—42no—
204C22×C8C2×C4×C7312yes—
205C22×C8D4×C7311yescharacteristic
206C22×C8D4×C7311yescharacteristic
207C4×D4—21no—
208C4×D4—21no—
209C4×D4C2×C4×C7312yes—
210C4×D4C2×C4×C7312yes—
211C2×C4⋊C4C2×C4×C7312yes—
212C2×C4⋊C4C2×C4×C7311yescharacteristic
213C2×C22⋊C4—21no—
214C2×C42—21no—
215C2×C42C23×C7311yescharacteristic
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8C2×C4×C7312yes—
219C4⋊C8C2×C4×C7312yes—
220#102—24no—
221C22⋊C8—24no—
222#97—42no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8C2×C4×C7312yes—
227C4×C8C2×C4×C7312yes—
228C4×C8D4×C7312yes—
229#95—21no—
230C24—21no—
231C2×D4—21no—
232C2×D4—22no—
233C2×D4C42×C7312yes—
234C2×D4M16×C7314yes—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4C22×C4×C7311yescharacteristic
244C22×C4C2×D4×C7311yescharacteristic
245C22×C4#41×C7311yescharacteristic
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—42no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—48no—
260C2×C8C22⋊C4×C7312yes—
261C2×C8C22⋊C4×C7312yes—
262C2×C8D8×C7312yes—
263C2×C8SD16×C7312yes—
264C4⋊C4—21no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4C42×C7312yes—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C42—41no—
271C42—41no—
272C42C22×C4×C7311yescharacteristic
273C42C22×C4×C7311yescharacteristic
274C42C2×D4×C7311yescharacteristic
275C42#41×C7311yescharacteristic
276C23—22no—
277C23—24no—
278C23—41no—
279C23—41no—
280C23—42no—
281C23C4×D4×C7311yescharacteristic
282D4—22no—
283D4—24no—
284D4#97×C7314yescomplemented
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4C2×C42×C7311yescharacteristic
297C2×C4C2×C22⋊C4×C7311yescharacteristic
298C2×C4#117×C7311yescharacteristic
299C2×C4C4×D4×C7311yescharacteristic
300C2×C4C4×D4×C7311yescharacteristic
301C2×C4C4×D4×C7311yescharacteristic
302C2×C4C2×M16×C7312yes—
303C2×C4#131×C7312yes—
304C2×C4C2×D8×C7311yescharacteristic
305C2×C4C2×SD16×C7311yescharacteristic
306C2×C4#135×C7311yescharacteristic
307C2×C4#135×C7311yescharacteristic
308C8—22no—
309C8—42no—
310C8—42no—
311C8—42no—
312C8—42no—
313C8#102×C7314yescomplemented
314C22—24no—
315C22—42no—
316C22—42no—
317C22—44no—
318C22—44no—
319C22C4×C22⋊C4×C7311yescharacteristic
320C22#435×C7312yes—
321C22C4×D8×C7311yescharacteristic
322C22C4×SD16×C7311yescharacteristic
323C22#444×C7312yes—
324C4—21no—
325C4—21no—
326C4—42no—
327C4—82no—
328C4C2×#97×C7311yescharacteristic
329C4#405×C7311yescharacteristic
330C4C2×#102×C7311yescharacteristic
331C4#416×C7311yescharacteristic
332C4#435×C7312yes—
333C4C4×D8×C7311yescharacteristic
334C4C4×SD16×C7311yescharacteristic
335C2—44no—
336C2C73×#158112yes—
337C2C73×#158212yes—
338C2C73×#175211yescharacteristic
339C2C4×#102×C7311yescharacteristic
340C2C73×#176511yescharacteristic
subgroupG/H
centerC22×C4×C73C2×D4
commutatorC2×C4C2×C42×C73
Frattini, cosocleC2×C42C23×C73
socleC23×C73C4×D4
prime radicalC22×D4×C73C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C73 ◁ C2×C4×C8×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C73×#1770C2semidirect
5. C2×#322×C73C2semidirect
6. C73×#1279C2semidirect
7. C73×#1277C2semidirect
11. C8×D4×C73C4semidirect
23. D4⋊C8×C73C4semidirect
113. D4×C73#97semidirect
142. C8×C73#102semidirect
164. #8202C73direct (N × H)
172. #1770C2×C73semidirect
175. C2×#322C2×C73semidirect
176. #1279C2×C73semidirect
177. #1277C2×C73semidirect
178. C73#8202direct (N × H)
182. C8×D4C4×C73semidirect
194. D4⋊C8C4×C73semidirect
284. D4#97×C73semidirect
313. C8#102×C73semidirect

Maximal subgroups

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

subgroupindex
C2×C8×D4×C732
C73×#17702
C4×#102×C732
C2×D4⋊C8×C732
C2×#322×C732
C73×#12792
C73×#12772
#820273