Finite Group Databasebeta

← C73×#10205 #32919609

Subgroups — C73×#10205 #32919609

Order 18688.

512 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3246C211yesmaximal, characteristic
2C73×#3226C211yesmaximal, characteristic
3C73×#3139C211yesmaximal, characteristic, complemented
4C73×#1988C211yesmaximal, characteristic, complemented
5C73×#1980C211yesmaximal, characteristic
6C73×#1503C211yesmaximal, characteristic, complemented
7C73×#1283C211yesmaximal, characteristic, complemented
8#533×C73C2211yescharacteristic
9#532×C73C2211yescharacteristic
10C2×C4⋊Q8×C73C2211yescharacteristic
11#500×C73—21no—
12#491×C73—22no—
13#487×C73—21no—
14#487×C73—21no—
15#486×C73—21no—
16#480×C73—22no—
17#476×C73—21no—
18#475×C73—21no—
19#474×C73—22no—
20#465×C73—22no—
21#461×C73—21no—
22#460×C73—21no—
23#431×C73C2211yescharacteristic
24C2×C4⋊C8×C73C2211yescharacteristic
25C2×#104×C73—22no—
26#419×C73—22no—
27C4×M16×C73C2211yescharacteristic
28C4×C4⋊C4×C73C2211yescharacteristic
29#340×C73—22no—
30#328×C73—22no—
31C2×#41×C73—21no—
32C22×Q8×C73—21no—
33C2×SD16×C73—22no—
34C2×M16×C73D412yes—
35C22×C8×C73D412yes—
36C4⋊Q8×C73—21no—
37C4⋊Q8×C73—21no—
38C4⋊Q8×C73—21no—
39C4⋊Q8×C73D411yescharacteristic
40C4⋊Q8×C73D412yes—
41C4⋊Q8×C73D411yescharacteristic
42C4⋊Q8×C73D411yescharacteristic
43#127×C73D411yescharacteristic
44#125×C73—21no—
45#124×C73—21no—
46C22⋊Q8×C73—21no—
47C22⋊Q8×C73—21no—
48#121×C73—21no—
49#121×C73—21no—
50C4×Q8×C73—21no—
51C4×Q8×C73—21no—
52C4×D4×C73—21no—
53C4×D4×C73—21no—
54#117×C73—21no—
55C2×C4⋊C4×C73—21no—
56C2×C4⋊C4×C73—21no—
57C2×C4⋊C4×C73—21no—
58C2×C42×C73—21no—
59C2×C42×C73C2311yescharacteristic
60#107×C73—24no—
61#106×C73—22no—
62C4⋊C8×C73—21no—
63C4⋊C8×C73—21no—
64C4⋊C8×C73—21no—
65#104×C73—44no—
66#103×C73—24no—
67#103×C73—24no—
68#103×C73—22no—
69#102×C73—22no—
70#102×C73—24no—
71C22⋊C8×C73D412yes—
72#97×C73—21no—
73C4×C8×C73—21no—
74C4×C8×C73—21no—
75#95×C73—21no—
76#41×C73—42no—
77#41×C73—42no—
78C2×Q8×C73—22no—
79C2×Q8×C73—21no—
80C2×Q8×C73—21no—
81C2×Q8×C73—21no—
82C2×Q8×C73—41no—
83C2×Q8×C73—42no—
84C2×Q8×C73—42no—
85C2×D4×C73—21no—
86C2×D4×C73—21no—
87C2×D4×C73—21no—
88C2×D4×C73—41no—
89C22×C4×C73—41no—
90C22×C4×C73—41no—
91C22×C4×C73—41no—
92C22×C4×C73C2×D411yescharacteristic
93C22×C4×C73C2×D411yescharacteristic
94C22×C4×C73C2×D411yescharacteristic
95SD16×C73—82no—
96M16×C73—22no—
97M16×C73—22no—
98C2×C8×C73—22no—
99C2×C8×C73—22no—
100C2×C8×C73—22no—
101C2×C8×C73—42no—
102C4⋊C4×C73—22no—
103C4⋊C4×C73—21no—
104C4⋊C4×C73—22no—
105C4⋊C4×C73—21no—
106C4⋊C4×C73—21no—
107C4⋊C4×C73—21no—
108C4⋊C4×C73—21no—
109C4⋊C4×C73—22no—
110C4⋊C4×C73—21no—
111C4⋊C4×C73—41no—
112C4⋊C4×C73—41no—
113C4⋊C4×C73—41no—
114C4⋊C4×C73—41no—
115C4⋊C4×C73—41no—
116C22⋊C4×C73—41no—
117C22⋊C4×C73—41no—
118C22⋊C4×C73—41no—
119C42×C73—41no—
120C42×C73—42no—
121C42×C73C2×D411yescharacteristic
122C42×C73C2×D411yescharacteristic
123C42×C73C2×D411yescharacteristic
124C42×C73#4111yescharacteristic
125C23×C73—41no—
126C23×C73#12711yescharacteristic
127Q8×C73—44no—
128Q8×C73—41no—
129Q8×C73—41no—
130Q8×C73—42no—
131Q8×C73—81no—
132Q8×C73—82no—
133D4×C73—42no—
134D4×C73—41no—
135D4×C73—41no—
136D4×C73—42no—
137D4×C73—81no—
138D4×C73—81no—
139C2×C4×C73—21no—
140C2×C4×C73—21no—
141C2×C4×C73—41no—
142C2×C4×C73—42no—
143C2×C4×C73—41no—
144C2×C4×C73—41no—
145C2×C4×C73—41no—
146C2×C4×C73—42no—
147C2×C4×C73—81no—
148C2×C4×C73—81no—
149C2×C4×C73—81no—
150C2×C4×C73—81no—
151C2×C4×C73—82no—
152C2×C4×C73#12011yescharacteristic
153C2×C4×C73#12011yescharacteristic
154C2×C4×C73#12011yescharacteristic
155C2×C4×C73#12111yescharacteristic
156C2×C4×C73#12111yescharacteristic
157C2×C4×C73#12111yescharacteristic
158C2×C4×C73#13512yes—
159C2×C4×C73AGL(1,ℤ/8)12yes—
160C8×C73—42no—
161C8×C73—42no—
162C8×C73—42no—
163C22×C73—21no—
164C22×C73—21no—
165C22×C73—81no—
166C22×C73—81no—
167C22×C73—81no—
168C22×C73#39211yescharacteristic
169C22×C73#49511yescharacteristic
170C22×C73#49611yescharacteristic
171C4×C73—21no—
172C4×C73—21no—
173C4×C73—21no—
174C4×C73—82no—
175C4×C73—81no—
176C4×C73—81no—
177C4×C73—81no—
178C4×C73—82no—
179C4×C73#44912yes—
180C4×C73#45511yescharacteristic
181C4×C73#45611yescharacteristic
182C4×C73#46312yes—
183#10205C7311yesmaximal, characteristic, direct product factor, Sylow
184C2×C73—21no—
185C2×C73—21no—
186C2×C73—161no—
187C2×C73#200711yescharacteristic
188C2×C73#200911yescharacteristic
189C2×C73#201211yescharacteristic
190#3246C2×C7311yescharacteristic
191#3226C2×C7311yescharacteristic
192#3139C2×C7311yescharacteristic, complemented
193#1988C2×C7311yescharacteristic, complemented
194#1980C2×C7311yescharacteristic
195#1503C2×C7311yescharacteristic, complemented
196#1283C2×C7311yescharacteristic, complemented
197C73#1020511yescharacteristic, direct product factor, Sylow
198#533C22×C7311yescharacteristic
199#532C22×C7311yescharacteristic
200C2×C4⋊Q8C22×C7311yescharacteristic
201#500—21no—
202#491—22no—
203#487—21no—
204#487—21no—
205#486—21no—
206#480—22no—
207#476—21no—
208#475—21no—
209#474—22no—
210#465—22no—
211#461—21no—
212#460—21no—
213#431C22×C7311yescharacteristic
214C2×C4⋊C8C22×C7311yescharacteristic
215C2×#104—22no—
216#419—22no—
217C4×M16C22×C7311yescharacteristic
218C4×C4⋊C4C22×C7311yescharacteristic
219#340—22no—
220#328—22no—
221C2×#41—21no—
222C22×Q8—21no—
223C2×SD16—22no—
224C2×M16D4×C7312yes—
225C22×C8D4×C7312yes—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8D4×C7312yes—
230C4⋊Q8D4×C7311yescharacteristic
231C4⋊Q8D4×C7311yescharacteristic
232C4⋊Q8D4×C7311yescharacteristic
233#127D4×C7311yescharacteristic
234#125—21no—
235#124—21no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238#121—21no—
239#121—21no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×D4—21no—
243C4×D4—21no—
244#117—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C42—21no—
249C2×C42C23×C7311yescharacteristic
250#107—24no—
251#106—22no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255#104—44no—
256#103—24no—
257#103—24no—
258#103—22no—
259#102—22no—
260#102—24no—
261C22⋊C8D4×C7312yes—
262#97—21no—
263C4×C8—21no—
264C4×C8—21no—
265#95—21no—
266#41—42no—
267#41—42no—
268C2×Q8—22no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—41no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4C2×D4×C7311yescharacteristic
283C22×C4C2×D4×C7311yescharacteristic
284C22×C4C2×D4×C7311yescharacteristic
285SD16—82no—
286M16—22no—
287M16—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—42no—
292C4⋊C4—22no—
293C4⋊C4—21no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—22no—
300C4⋊C4—21no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C42—41no—
310C42—42no—
311C42C2×D4×C7311yescharacteristic
312C42C2×D4×C7311yescharacteristic
313C42C2×D4×C7311yescharacteristic
314C42#41×C7311yescharacteristic
315C23—41no—
316C23#127×C7311yescharacteristic
317Q8—44no—
318Q8—41no—
319Q8—41no—
320Q8—42no—
321Q8—81no—
322Q8—82no—
323D4—42no—
324D4—41no—
325D4—41no—
326D4—42no—
327D4—81no—
328D4—81no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—82no—
342C2×C4#120×C7311yescharacteristic
343C2×C4#120×C7311yescharacteristic
344C2×C4#120×C7311yescharacteristic
345C2×C4#121×C7311yescharacteristic
346C2×C4#121×C7311yescharacteristic
347C2×C4#121×C7311yescharacteristic
348C2×C4#135×C7312yes—
349C2×C4AGL(1,ℤ/8)×C7312yes—
350C8—42no—
351C8—42no—
352C8—42no—
353C22—21no—
354C22—21no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22#392×C7311yescharacteristic
359C22#495×C7311yescharacteristic
360C22#496×C7311yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—82no—
365C4—81no—
366C4—81no—
367C4—81no—
368C4—82no—
369C4#449×C7312yes—
370C4#455×C7311yescharacteristic
371C4#456×C7311yescharacteristic
372C4#463×C7312yes—
373C2—21no—
374C2—21no—
375C2—161no—
376C2C73×#200711yescharacteristic
377C2C73×#200911yescharacteristic
378C2C73×#201211yescharacteristic
subgroupG/H
center, socleC22×C73#392
commutator, Frattini, cosocleC2×C42C23×C73
prime radical#532×C73C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C73 ◁ C2×C42×C73 ◁ G

Complements

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

NH ≅ G/Ntype
3. C73×#3139C2semidirect
4. C73×#1988C2semidirect
6. C73×#1503C2semidirect
7. C73×#1283C2semidirect
183. #10205C73direct (N × H)
192. #3139C2×C73semidirect
193. #1988C2×C73semidirect
195. #1503C2×C73semidirect
196. #1283C2×C73semidirect
197. C73#10205direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#32462
C73×#32262
C73×#31392
C73×#19882
C73×#19802
C73×#15032
C73×#12832
#1020573