Finite Group Databasebeta

← C73×#10204 #32919608

Subgroups — C73×#10204 #32919608

Order 18688.

598 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3245C211yesmaximal, characteristic
2C73×#3226C211yesmaximal, characteristic, complemented
3C73×#3152C211yesmaximal, characteristic, complemented
4C73×#1988C211yesmaximal, characteristic, complemented
5C73×#1980C211yesmaximal, characteristic, complemented
6C73×#1502C211yesmaximal, characteristic, complemented
7C73×#1283C211yesmaximal, characteristic, complemented
8C2×#137×C73—22no—
9#532×C73C2211yescharacteristic
10#532×C73C2211yescharacteristic
11C2×C4⋊Q8×C73C2211yescharacteristic
12#500×C73—21no—
13#497×C73—22no—
14#494×C73—21no—
15#492×C73—21no—
16#486×C73—21no—
17#480×C73—22no—
18#474×C73—22no—
19#474×C73—21no—
20#474×C73—21no—
21#466×C73—22no—
22#459×C73—21no—
23#459×C73—21no—
24#431×C73C2211yescharacteristic
25C2×C4⋊C8×C73C2211yescharacteristic
26C2×#104×C73—22no—
27C4×M16×C73C2211yescharacteristic
28C4×C4⋊C4×C73C2211yescharacteristic
29#340×C73—22no—
30#327×C73—22no—
31C2×#41×C73—21no—
32C2×#41×C73—21no—
33C22×Q8×C73—21no—
34#137×C73—44no—
35#137×C73—44no—
36C2×Q16×C73—24no—
37C2×SD16×C73—24no—
38C2×D8×C73—22no—
39C2×M16×C73D412yes—
40C22×C8×C73D412yes—
41C4⋊Q8×C73—21no—
42C4⋊Q8×C73—21no—
43C4⋊Q8×C73—21no—
44C4⋊Q8×C73D411yescharacteristic
45C4⋊Q8×C73D412yes—
46C4⋊Q8×C73D411yescharacteristic
47#127×C73D411yescharacteristic
48#127×C73D411yescharacteristic
49#124×C73—21no—
50#124×C73—21no—
51#121×C73—21no—
52#121×C73—21no—
53#121×C73—21no—
54#121×C73—21no—
55C4×D4×C73—21no—
56C4×D4×C73—21no—
57C4×D4×C73—21no—
58C4×D4×C73—21no—
59C2×C4⋊C4×C73—21no—
60C2×C4⋊C4×C73—21no—
61C2×C4⋊C4×C73—21no—
62C2×C42×C73—21no—
63C2×C42×C73C2311yescharacteristic
64#107×C73—22no—
65#107×C73—24no—
66C4⋊C8×C73—21no—
67C4⋊C8×C73—21no—
68C4⋊C8×C73—21no—
69#104×C73—44no—
70#102×C73—22no—
71#102×C73—22no—
72#102×C73—24no—
73C22⋊C8×C73D412yes—
74#97×C73—21no—
75C4×C8×C73—21no—
76C4×C8×C73—21no—
77#95×C73—21no—
78#41×C73—42no—
79#41×C73—42no—
80#41×C73—42no—
81#41×C73—42no—
82C2×Q8×C73—21no—
83C2×Q8×C73—22no—
84C2×Q8×C73—21no—
85C2×Q8×C73—41no—
86C2×Q8×C73—42no—
87C2×Q8×C73—42no—
88C2×D4×C73—21no—
89C2×D4×C73—21no—
90C2×D4×C73—21no—
91C2×D4×C73—21no—
92C2×D4×C73—21no—
93C2×D4×C73—21no—
94C2×D4×C73—41no—
95C2×D4×C73—41no—
96C22×C4×C73—41no—
97C22×C4×C73—41no—
98C22×C4×C73—41no—
99C22×C4×C73—41no—
100C22×C4×C73C2×D411yescharacteristic
101C22×C4×C73C2×D411yescharacteristic
102C22×C4×C73C2×D411yescharacteristic
103Q16×C73—48no—
104SD16×C73—48no—
105D8×C73—82no—
106M16×C73—22no—
107M16×C73D814yescomplemented
108C2×C8×C73—22no—
109C2×C8×C73—22no—
110C2×C8×C73—22no—
111C2×C8×C73—42no—
112C4⋊C4×C73—22no—
113C4⋊C4×C73—22no—
114C4⋊C4×C73—21no—
115C4⋊C4×C73—21no—
116C4⋊C4×C73—21no—
117C4⋊C4×C73—22no—
118C4⋊C4×C73—21no—
119C4⋊C4×C73—41no—
120C4⋊C4×C73—41no—
121C4⋊C4×C73—41no—
122C22⋊C4×C73—41no—
123C22⋊C4×C73—41no—
124C22⋊C4×C73—41no—
125C22⋊C4×C73—41no—
126C42×C73—42no—
127C42×C73C2×D411yescharacteristic
128C42×C73C2×D411yescharacteristic
129C42×C73C2×D411yescharacteristic
130C42×C73#4111yescharacteristic
131C23×C73—41no—
132C23×C73—41no—
133C23×C73#12711yescharacteristic
134Q8×C73—44no—
135Q8×C73—42no—
136Q8×C73—42no—
137Q8×C73—82no—
138D4×C73—42no—
139D4×C73—42no—
140D4×C73—41no—
141D4×C73—41no—
142D4×C73—41no—
143D4×C73—41no—
144D4×C73—42no—
145D4×C73—81no—
146D4×C73—81no—
147D4×C73—81no—
148D4×C73—81no—
149D4×C73—81no—
150C2×C4×C73—21no—
151C2×C4×C73—21no—
152C2×C4×C73—41no—
153C2×C4×C73—42no—
154C2×C4×C73—41no—
155C2×C4×C73—41no—
156C2×C4×C73—42no—
157C2×C4×C73—81no—
158C2×C4×C73—81no—
159C2×C4×C73—81no—
160C2×C4×C73—81no—
161C2×C4×C73—82no—
162C2×C4×C73—82no—
163C2×C4×C73#12011yescharacteristic
164C2×C4×C73#12011yescharacteristic
165C2×C4×C73#12011yescharacteristic
166C2×C4×C73#12111yescharacteristic
167C2×C4×C73#12111yescharacteristic
168C2×C4×C73#12111yescharacteristic
169C2×C4×C73C2×D812yes—
170C2×C4×C73AGL(1,ℤ/8)12yes—
171C8×C73—24no—
172C8×C73—42no—
173C8×C73—42no—
174C22×C73—21no—
175C22×C73—21no—
176C22×C73—81no—
177C22×C73—81no—
178C22×C73—81no—
179C22×C73—81no—
180C22×C73—81no—
181C22×C73—81no—
182C22×C73#39211yescharacteristic
183C22×C73#49311yescharacteristic
184C22×C73#49611yescharacteristic
185C4×C73—21no—
186C4×C73—21no—
187C4×C73—21no—
188C4×C73—82no—
189C4×C73—81no—
190C4×C73—81no—
191C4×C73—82no—
192C4×C73#44712yes—
193C4×C73#45511yescharacteristic
194C4×C73#45611yescharacteristic
195C4×C73#45912yes—
196#10204C7311yesmaximal, characteristic, direct product factor, Sylow
197C2×C73—21no—
198C2×C73—21no—
199C2×C73—161no—
200C2×C73—161no—
201C2×C73#200711yescharacteristic
202C2×C73#200911yescharacteristic
203C2×C73#201111yescharacteristic
204#3245C2×C7311yescharacteristic
205#3226C2×C7311yescharacteristic, complemented
206#3152C2×C7311yescharacteristic, complemented
207#1988C2×C7311yescharacteristic, complemented
208#1980C2×C7311yescharacteristic, complemented
209#1502C2×C7311yescharacteristic, complemented
210#1283C2×C7311yescharacteristic, complemented
211C73#1020411yescharacteristic, direct product factor, Sylow
212C2×#137—22no—
213#532C22×C7311yescharacteristic
214#532C22×C7311yescharacteristic
215C2×C4⋊Q8C22×C7311yescharacteristic
216#500—21no—
217#497—22no—
218#494—21no—
219#492—21no—
220#486—21no—
221#480—22no—
222#474—22no—
223#474—21no—
224#474—21no—
225#466—22no—
226#459—21no—
227#459—21no—
228#431C22×C7311yescharacteristic
229C2×C4⋊C8C22×C7311yescharacteristic
230C2×#104—22no—
231C4×M16C22×C7311yescharacteristic
232C4×C4⋊C4C22×C7311yescharacteristic
233#340—22no—
234#327—22no—
235C2×#41—21no—
236C2×#41—21no—
237C22×Q8—21no—
238#137—44no—
239#137—44no—
240C2×Q16—24no—
241C2×SD16—24no—
242C2×D8—22no—
243C2×M16D4×C7312yes—
244C22×C8D4×C7312yes—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8D4×C7312yes—
249C4⋊Q8D4×C7311yescharacteristic
250C4⋊Q8D4×C7311yescharacteristic
251#127D4×C7311yescharacteristic
252#127D4×C7311yescharacteristic
253#124—21no—
254#124—21no—
255#121—21no—
256#121—21no—
257#121—21no—
258#121—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C42—21no—
267C2×C42C23×C7311yescharacteristic
268#107—22no—
269#107—24no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273#104—44no—
274#102—22no—
275#102—22no—
276#102—24no—
277C22⋊C8D4×C7312yes—
278#97—21no—
279C4×C8—21no—
280C4×C8—21no—
281#95—21no—
282#41—42no—
283#41—42no—
284#41—42no—
285#41—42no—
286C2×Q8—21no—
287C2×Q8—22no—
288C2×Q8—21no—
289C2×Q8—41no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—41no—
299C2×D4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4C2×D4×C7311yescharacteristic
305C22×C4C2×D4×C7311yescharacteristic
306C22×C4C2×D4×C7311yescharacteristic
307Q16—48no—
308SD16—48no—
309D8—82no—
310M16—22no—
311M16D8×C7314yescomplemented
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—42no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—22no—
322C4⋊C4—21no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C42—42no—
331C42C2×D4×C7311yescharacteristic
332C42C2×D4×C7311yescharacteristic
333C42C2×D4×C7311yescharacteristic
334C42#41×C7311yescharacteristic
335C23—41no—
336C23—41no—
337C23#127×C7311yescharacteristic
338Q8—44no—
339Q8—42no—
340Q8—42no—
341Q8—82no—
342D4—42no—
343D4—42no—
344D4—41no—
345D4—41no—
346D4—41no—
347D4—41no—
348D4—42no—
349D4—81no—
350D4—81no—
351D4—81no—
352D4—81no—
353D4—81no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4#120×C7311yescharacteristic
368C2×C4#120×C7311yescharacteristic
369C2×C4#120×C7311yescharacteristic
370C2×C4#121×C7311yescharacteristic
371C2×C4#121×C7311yescharacteristic
372C2×C4#121×C7311yescharacteristic
373C2×C4C2×D8×C7312yes—
374C2×C4AGL(1,ℤ/8)×C7312yes—
375C8—24no—
376C8—42no—
377C8—42no—
378C22—21no—
379C22—21no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#392×C7311yescharacteristic
387C22#493×C7311yescharacteristic
388C22#496×C7311yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—82no—
393C4—81no—
394C4—81no—
395C4—82no—
396C4#447×C7312yes—
397C4#455×C7311yescharacteristic
398C4#456×C7311yescharacteristic
399C4#459×C7312yes—
400C2—21no—
401C2—21no—
402C2—161no—
403C2—161no—
404C2C73×#200711yescharacteristic
405C2C73×#200911yescharacteristic
406C2C73×#201111yescharacteristic
subgroupG/H
center, socleC22×C73#392
commutator, Frattini, cosocleC2×C42C23×C73
prime radicalC73×#3245C2
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
2. C73×#3226C2semidirect
3. C73×#3152C2semidirect
4. C73×#1988C2semidirect
5. C73×#1980C2semidirect
6. C73×#1502C2semidirect
7. C73×#1283C2semidirect
107. M16×C73D8semidirect
196. #10204C73direct (N × H)
205. #3226C2×C73semidirect
206. #3152C2×C73semidirect
207. #1988C2×C73semidirect
208. #1980C2×C73semidirect
209. #1502C2×C73semidirect
210. #1283C2×C73semidirect
211. C73#10204direct (N × H)
311. M16D8×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#32452
C73×#32262
C73×#31522
C73×#19882
C73×#19802
C73×#15022
C73×#12832
#1020473