Finite Group Databasebeta

← C73×#7716 #32917120

Subgroups — C73×#7716 #32917120

Order 18688.

414 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#2861C211yesmaximal, characteristic
2C73×#1839C211yesmaximal, characteristic, complemented
3C73×#1831C211yesmaximal, characteristic, complemented
4C73×#1272C212yesmaximal, complemented
5C73×#1270C212yesmaximal
6C2×#125×C73C411yescharacteristic
7C2×#124×C73C411yescharacteristic
8C2×C4⋊C8×C73—21no—
9#391×C73C411yescharacteristic
10#391×C73C2211yescharacteristic
11#390×C73C411yescharacteristic
12#390×C73C2211yescharacteristic
13#388×C73—21no—
14#388×C73—21no—
15#388×C73—21no—
16#388×C73—21no—
17C43×C73C2211yescharacteristic
18#336×C73—21no—
19#336×C73—21no—
20#336×C73—21no—
21#336×C73C2212yes—
22#336×C73C2212yes—
23C22×Q8×C73C2×C411yescharacteristic
24C22×D4×C73C2×C411yescharacteristic
25C22×C8×C73—22no—
26C22×C8×C73—22no—
27#125×C73—42no—
28#124×C73—42no—
29C2×C4⋊C4×C73—21no—
30C2×C4⋊C4×C73—21no—
31C2×C4⋊C4×C73C2×C411yescharacteristic
32C2×C4⋊C4×C73C2×C411yescharacteristic
33C2×C22⋊C4×C73—21no—
34C2×C22⋊C4×C73—21no—
35C2×C22⋊C4×C73—21no—
36C2×C22⋊C4×C73—21no—
37C2×C22⋊C4×C73—21no—
38C2×C22⋊C4×C73—21no—
39C2×C42×C73C2×C411yescharacteristic
40C2×C42×C73C2×C411yescharacteristic
41C2×C42×C73D411yescharacteristic
42C2×C42×C73D411yescharacteristic
43C2×C42×C73D411yescharacteristic
44C2×C42×C73D411yescharacteristic
45C2×C42×C73C2311yescharacteristic
46C4⋊C8×C73—42no—
47#95×C73—21no—
48#95×C73—21no—
49#95×C73—21no—
50#95×C73—21no—
51C24×C73—21no—
52C2×Q8×C73—22no—
53C2×Q8×C73—41no—
54C2×Q8×C73—41no—
55C2×D4×C73—22no—
56C2×D4×C73—41no—
57C2×D4×C73—41no—
58C22×C4×C73—21no—
59C22×C4×C73—21no—
60C22×C4×C73—21no—
61C22×C4×C73C22⋊C411yescharacteristic
62C22×C4×C73C22⋊C411yescharacteristic
63C22×C4×C73C22⋊C411yescharacteristic
64C22×C4×C73C22⋊C411yescharacteristic
65C22×C4×C73C22×C411yescharacteristic
66C22×C4×C73C2×D411yescharacteristic
67C22×C4×C73C2×D411yescharacteristic
68C2×C8×C73—42no—
69C2×C8×C73—42no—
70C2×C8×C73—42no—
71C2×C8×C73—42no—
72C2×C8×C73—42no—
73C2×C8×C73—42no—
74C4⋊C4×C73—41no—
75C4⋊C4×C73—41no—
76C4⋊C4×C73—42no—
77C4⋊C4×C73—42no—
78C22⋊C4×C73—42no—
79C22⋊C4×C73—42no—
80C22⋊C4×C73—42no—
81C22⋊C4×C73—42no—
82C22⋊C4×C73—42no—
83C22⋊C4×C73—41no—
84C22⋊C4×C73—41no—
85C42×C73—22no—
86C42×C73—22no—
87C42×C73—22no—
88C42×C73—21no—
89C42×C73—21no—
90C42×C73—41no—
91C42×C73—41no—
92C42×C73—41no—
93C23×C73—41no—
94C23×C73—41no—
95C23×C73—41no—
96C23×C73—41no—
97C23×C73—41no—
98C23×C73—41no—
99C23×C73—41no—
100C23×C73C2×C22⋊C411yescharacteristic
101Q8×C73—82no—
102D4×C73—82no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—21no—
106C2×C4×C73—21no—
107C2×C4×C73—21no—
108C2×C4×C73—21no—
109C2×C4×C73—21no—
110C2×C4×C73—21no—
111C2×C4×C73—21no—
112C2×C4×C73—21no—
113C2×C4×C73—21no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—21no—
117C2×C4×C73—21no—
118C2×C4×C73—21no—
119C2×C4×C73—21no—
120C2×C4×C73—41no—
121C2×C4×C73—41no—
122C2×C4×C73—41no—
123C2×C4×C73—41no—
124C2×C4×C73—41no—
125C2×C4×C73—41no—
126C2×C4×C73—41no—
127C2×C4×C73—41no—
128C2×C4×C73—41no—
129C2×C4×C73#10012yes—
130C2×C4×C73#13112yes—
131C2×C4×C73#13512yes—
132C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C73#13711yescharacteristic
134C8×C73—82no—
135C8×C73—82no—
136C22×C73—81no—
137C22×C73—81no—
138C22×C73—81no—
139C22×C73—81no—
140C22×C73—81no—
141C22×C73—81no—
142C22×C73—81no—
143C22×C73#40811yescharacteristic
144C22×C73#41011yescharacteristic
145C22×C73C2×#10011yescharacteristic
146C22×C73#41611yescharacteristic
147C22×C73#41711yescharacteristic
148C22×C73#42111yescharacteristic
149C22×C73#42111yescharacteristic
150C4×C73—22no—
151C4×C73—22no—
152C4×C73—21no—
153C4×C73—21no—
154C4×C73—22no—
155C4×C73—41no—
156C4×C73—41no—
157C4×C73—41no—
158C4×C73—81no—
159C4×C73—81no—
160C4×C73—81no—
161#7716C7311yesmaximal, characteristic, direct product factor, Sylow
162C2×C73—161no—
163C2×C73#146911yescharacteristic
164C2×C73#148211yescharacteristic
165C2×C73#149511yescharacteristic
166C2×C73#151211yescharacteristic
167C2×C73#152111yescharacteristic
168C2×C73#153611yescharacteristic
169C2×C73#155111yescharacteristic
170#2861C2×C7311yescharacteristic
171#1839C2×C7311yescharacteristic, complemented
172#1831C2×C7311yescharacteristic, complemented
173#1272C2×C7312yescomplemented
174#1270C2×C7312yes—
175C73#771611yescharacteristic, direct product factor, Sylow
176C2×#125C4×C7311yescharacteristic
177C2×#124C4×C7311yescharacteristic
178C2×C4⋊C8—21no—
179#391C4×C7311yescharacteristic
180#391C22×C7311yescharacteristic
181#390C4×C7311yescharacteristic
182#390C22×C7311yescharacteristic
183#388—21no—
184#388—21no—
185#388—21no—
186#388—21no—
187C43C22×C7311yescharacteristic
188#336—21no—
189#336—21no—
190#336—21no—
191#336C22×C7312yes—
192#336C22×C7312yes—
193C22×Q8C2×C4×C7311yescharacteristic
194C22×D4C2×C4×C7311yescharacteristic
195C22×C8—22no—
196C22×C8—22no—
197#125—42no—
198#124—42no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4C2×C4×C7311yescharacteristic
202C2×C4⋊C4C2×C4×C7311yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C42C2×C4×C7311yescharacteristic
210C2×C42C2×C4×C7311yescharacteristic
211C2×C42D4×C7311yescharacteristic
212C2×C42D4×C7311yescharacteristic
213C2×C42D4×C7311yescharacteristic
214C2×C42D4×C7311yescharacteristic
215C2×C42C23×C7311yescharacteristic
216C4⋊C8—42no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221C24—21no—
222C2×Q8—22no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×D4—22no—
226C2×D4—41no—
227C2×D4—41no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4C22⋊C4×C7311yescharacteristic
232C22×C4C22⋊C4×C7311yescharacteristic
233C22×C4C22⋊C4×C7311yescharacteristic
234C22×C4C22⋊C4×C7311yescharacteristic
235C22×C4C22×C4×C7311yescharacteristic
236C22×C4C2×D4×C7311yescharacteristic
237C22×C4C2×D4×C7311yescharacteristic
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C42—22no—
256C42—22no—
257C42—22no—
258C42—21no—
259C42—21no—
260C42—41no—
261C42—41no—
262C42—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23C2×C22⋊C4×C7311yescharacteristic
271Q8—82no—
272D4—82no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4#100×C7312yes—
300C2×C4#131×C7312yes—
301C2×C4#135×C7312yes—
302C2×C4AGL(1,ℤ/8)×C7311yescharacteristic
303C2×C4#137×C7311yescharacteristic
304C8—82no—
305C8—82no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22#408×C7311yescharacteristic
314C22#410×C7311yescharacteristic
315C22C2×#100×C7311yescharacteristic
316C22#416×C7311yescharacteristic
317C22#417×C7311yescharacteristic
318C22#421×C7311yescharacteristic
319C22#421×C7311yescharacteristic
320C4—22no—
321C4—21no—
322C4—21no—
323C4—22no—
324C4—22no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—81no—
329C4—81no—
330C4—81no—
331C2—161no—
332C2C73×#146911yescharacteristic
333C2C73×#148211yescharacteristic
334C2C73×#149511yescharacteristic
335C2C73×#151211yescharacteristic
336C2C73×#152111yescharacteristic
337C2C73×#153611yescharacteristic
338C2C73×#155111yescharacteristic
subgroupG/H
center, socleC23×C73C2×C22⋊C4
commutatorC22×C4C22×C4×C73
Frattini, cosocleC2×C42C23×C73
prime radicalC22×D4×C73C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C73 ◁ C43×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C73×#1839C2semidirect
3. C73×#1831C2semidirect
4. C73×#1272C2semidirect
161. #7716C73direct (N × H)
171. #1839C2×C73semidirect
172. #1831C2×C73semidirect
173. #1272C2×C73semidirect
175. C73#7716direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#28612
C73×#18392
C73×#18312
C73×#12722
C73×#12702
#771673