Finite Group Databasebeta

← C73×#33778 #32943182

Subgroups — C73×#33778 #32943182

Order 18688.

598 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3168C211yesmaximal, characteristic, complemented
2C73×#3147C211yesmaximal, characteristic, complemented
3C73×#2249C212yesmaximal, complemented
4C73×#2242C212yesmaximal, complemented
5C73×#2209C211yesmaximal, characteristic, complemented
6C73×#2208C211yesmaximal, characteristic, complemented
7C73×#2207C211yesmaximal, characteristic, complemented
8C73×#2206C211yesmaximal, characteristic, complemented
9C73×#2189C212yesmaximal, complemented
10C73×#2186C212yesmaximal, complemented
11C73×#2162C211yesmaximal, characteristic, complemented
12C2×#135×C73—21no—
13C2×#135×C73C2212yes—
14#532×C73—21no—
15#529×C73—21no—
16C2×Q32×C73C2212yes—
17C2×SD32×C73C2212yes—
18C2×SD32×C73C2212yes—
19C2×D16×C73C2212yes—
20#499×C73C2211yescharacteristic
21#495×C73—21no—
22#495×C73C2212yes—
23#495×C73C2211yescharacteristic
24#494×C73C2211yescharacteristic
25#493×C73C2211yescharacteristic
26#492×C73—21no—
27#488×C73—21no—
28#467×C73C2211yescharacteristic
29#466×C73C2211yescharacteristic
30#465×C73—21no—
31C4×Q16×C73C2211yescharacteristic
32C4×SD16×C73—21no—
33C4×D8×C73C2211yescharacteristic
34C2×C4×C8×C73C2211yescharacteristic
35C4⋊C16×C73C2211yescharacteristic
36C4⋊C16×C73C2211yescharacteristic
37#358×C73C2212yes—
38#358×C73C2212yes—
39#357×C73C2212yes—
40#357×C73C2212yes—
41C22⋊C16×C73C2212yes—
42C4×C16×C73C2212yes—
43C2×#41×C73—22no—
44#135×C73—42no—
45#135×C73—42no—
46#135×C73—44no—
47C2×Q16×C73—21no—
48C2×Q16×C73C2311yescharacteristic
49C2×Q16×C73C2312yes—
50C2×SD16×C73—21no—
51C2×SD16×C73—22no—
52C2×SD16×C73—22no—
53C2×D8×C73—21no—
54C2×D8×C73C2311yescharacteristic
55C2×D8×C73C2312yes—
56C22×C8×C73D411yescharacteristic
57C22×C8×C73C2311yescharacteristic
58C4⋊Q8×C73—21no—
59#127×C73—21no—
60#126×C73—41no—
61#125×C73—21no—
62#124×C73—21no—
63#124×C73—22no—
64#123×C73—41no—
65C22⋊Q8×C73—21no—
66#121×C73—21no—
67#121×C73—41no—
68#121×C73—41no—
69C4×Q8×C73—21no—
70C4×D4×C73—21no—
71C4×D4×C73—41no—
72C4×D4×C73—41no—
73C4×D4×C73—41no—
74#117×C73—41no—
75C2×C42×C73D411yescharacteristic
76Q32×C73—42no—
77SD32×C73—42no—
78SD32×C73—42no—
79D16×C73—42no—
80C2×C16×C73C2312yes—
81C2×C16×C73C2312yes—
82#107×C73C2311yescharacteristic
83#107×C73C2311yescharacteristic
84#106×C73—21no—
85#103×C73—21no—
86#103×C73—21no—
87#102×C73—21no—
88#102×C73—21no—
89C4×C8×C73D411yescharacteristic
90C4×C8×C73D411yescharacteristic
91C4×C8×C73C2311yescharacteristic
92C4×C8×C73C2311yescharacteristic
93#41×C73—44no—
94#41×C73—44no—
95C2×Q8×C73—21no—
96C2×Q8×C73—22no—
97C2×D4×C73—21no—
98C2×D4×C73—22no—
99C2×D4×C73—41no—
100C2×D4×C73—42no—
101C2×D4×C73—41no—
102C2×D4×C73—41no—
103C22×C4×C73—41no—
104C22×C4×C73—42no—
105C22×C4×C73D811yescharacteristic
106C22×C4×C73D811yescharacteristic
107C22×C4×C73C2×D411yescharacteristic
108Q16×C73—21no—
109Q16×C73—22no—
110Q16×C73—21no—
111Q16×C73—22no—
112Q16×C73—42no—
113SD16×C73—44no—
114SD16×C73—42no—
115SD16×C73—44no—
116D8×C73—21no—
117D8×C73—22no—
118D8×C73—21no—
119D8×C73—22no—
120D8×C73—42no—
121C2×C8×C73—22no—
122C2×C8×C73—21no—
123C2×C8×C73—21no—
124C2×C8×C73C2×D411yescharacteristic
125C2×C8×C73C2×D411yescharacteristic
126C2×C8×C73C2×D411yescharacteristic
127C2×C8×C73C2411yescharacteristic
128C4⋊C4×C73—21no—
129C4⋊C4×C73—21no—
130C4⋊C4×C73—41no—
131C4⋊C4×C73—41no—
132C4⋊C4×C73—41no—
133C4⋊C4×C73—41no—
134C4⋊C4×C73—41no—
135C4⋊C4×C73—41no—
136C22⋊C4×C73—41no—
137C22⋊C4×C73—41no—
138C22⋊C4×C73—41no—
139C22⋊C4×C73—41no—
140C22⋊C4×C73—42no—
141C22⋊C4×C73—42no—
142C22⋊C4×C73—41no—
143C42×C73—41no—
144C42×C73D811yescharacteristic
145C42×C73D811yescharacteristic
146C42×C73C2×D411yescharacteristic
147C42×C73C2×D411yescharacteristic
148C16×C73—22no—
149C16×C73—22no—
150C23×C73—41no—
151C23×C73—42no—
152C23×C73C2×D811yescharacteristic
153Q8×C73—42no—
154Q8×C73—41no—
155Q8×C73—42no—
156Q8×C73—41no—
157D4×C73—42no—
158D4×C73—41no—
159D4×C73—42no—
160D4×C73—41no—
161D4×C73—82no—
162D4×C73—82no—
163D4×C73—84no—
164D4×C73—82no—
165C2×C4×C73—21no—
166C2×C4×C73—21no—
167C2×C4×C73—22no—
168C2×C4×C73—41no—
169C2×C4×C73—41no—
170C2×C4×C73—41no—
171C2×C4×C73—41no—
172C2×C4×C73—42no—
173C2×C4×C73—84no—
174C2×C4×C73—82no—
175C2×C4×C73C2×D811yescharacteristic
176C2×C4×C73C2×D811yescharacteristic
177C2×C4×C73C2×D811yescharacteristic
178C2×C4×C73C2×D811yescharacteristic
179C2×C4×C73C2×D811yescharacteristic
180C2×C4×C73C22×D411yescharacteristic
181C8×C73—21no—
182C8×C73—21no—
183C8×C73AGL(1,ℤ/8)12yes—
184C8×C73H(5,𝔽2)11yescharacteristic
185C8×C73H(5,𝔽2)11yescharacteristic
186C22×C73—21no—
187C22×C73—21no—
188C22×C73—21no—
189C22×C73—81no—
190C22×C73—82no—
191C22×C73—82no—
192C22×C73—81no—
193C22×C73—82no—
194C22×C73—81no—
195C22×C73C22×D811yescharacteristic
196C4×C73—21no—
197C4×C73—21no—
198C4×C73—21no—
199C4×C73—21no—
200C4×C73—81no—
201C4×C73—81no—
202C4×C73—82no—
203C4×C73—81no—
204C4×C73#50812yes—
205C4×C73#53511yescharacteristic
206C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
207#33778C7311yesmaximal, characteristic, direct product factor, Sylow
208C2×C73—41no—
209C2×C73—161no—
210C2×C73—162no—
211C2×C73#320511yescharacteristic
212C2×C73C2×#50811yescharacteristic
213C2×C73#341411yescharacteristic
214#3168C2×C7311yescharacteristic, complemented
215#3147C2×C7311yescharacteristic, complemented
216#2249C2×C7312yescomplemented
217#2242C2×C7312yescomplemented
218#2209C2×C7311yescharacteristic, complemented
219#2208C2×C7311yescharacteristic, complemented
220#2207C2×C7311yescharacteristic, complemented
221#2206C2×C7311yescharacteristic, complemented
222#2189C2×C7312yescomplemented
223#2186C2×C7312yescomplemented
224#2162C2×C7311yescharacteristic, complemented
225C73#3377811yescharacteristic, direct product factor, Sylow
226C2×#135—21no—
227C2×#135C22×C7312yes—
228#532—21no—
229#529—21no—
230C2×Q32C22×C7312yes—
231C2×SD32C22×C7312yes—
232C2×SD32C22×C7312yes—
233C2×D16C22×C7312yes—
234#499C22×C7311yescharacteristic
235#495—21no—
236#495C22×C7312yes—
237#495C22×C7311yescharacteristic
238#494C22×C7311yescharacteristic
239#493C22×C7311yescharacteristic
240#492—21no—
241#488—21no—
242#467C22×C7311yescharacteristic
243#466C22×C7311yescharacteristic
244#465—21no—
245C4×Q16C22×C7311yescharacteristic
246C4×SD16—21no—
247C4×D8C22×C7311yescharacteristic
248C2×C4×C8C22×C7311yescharacteristic
249C4⋊C16C22×C7311yescharacteristic
250C4⋊C16C22×C7311yescharacteristic
251#358C22×C7312yes—
252#358C22×C7312yes—
253#357C22×C7312yes—
254#357C22×C7312yes—
255C22⋊C16C22×C7312yes—
256C4×C16C22×C7312yes—
257C2×#41—22no—
258#135—42no—
259#135—42no—
260#135—44no—
261C2×Q16—21no—
262C2×Q16C23×C7312yes—
263C2×Q16C23×C7311yescharacteristic
264C2×SD16—22no—
265C2×SD16—22no—
266C2×SD16—21no—
267C2×D8—21no—
268C2×D8C23×C7312yes—
269C2×D8C23×C7311yescharacteristic
270C22×C8D4×C7311yescharacteristic
271C22×C8C23×C7311yescharacteristic
272C4⋊Q8—21no—
273#127—21no—
274#126—41no—
275#125—21no—
276#124—21no—
277#124—22no—
278#123—41no—
279C22⋊Q8—21no—
280#121—21no—
281#121—41no—
282#121—41no—
283C4×Q8—21no—
284C4×D4—21no—
285C4×D4—41no—
286C4×D4—41no—
287C4×D4—41no—
288#117—41no—
289C2×C42D4×C7311yescharacteristic
290Q32—42no—
291SD32—42no—
292SD32—42no—
293D16—42no—
294C2×C16C23×C7312yes—
295C2×C16C23×C7312yes—
296#107C23×C7311yescharacteristic
297#107C23×C7311yescharacteristic
298#106—21no—
299#103—21no—
300#103—21no—
301#102—21no—
302#102—21no—
303C4×C8D4×C7311yescharacteristic
304C4×C8D4×C7311yescharacteristic
305C4×C8C23×C7311yescharacteristic
306C4×C8C23×C7311yescharacteristic
307#41—44no—
308#41—44no—
309C2×Q8—21no—
310C2×Q8—22no—
311C2×D4—21no—
312C2×D4—22no—
313C2×D4—41no—
314C2×D4—42no—
315C2×D4—41no—
316C2×D4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4D8×C7311yescharacteristic
320C22×C4D8×C7311yescharacteristic
321C22×C4C2×D4×C7311yescharacteristic
322Q16—21no—
323Q16—22no—
324Q16—21no—
325Q16—22no—
326Q16—42no—
327SD16—44no—
328SD16—42no—
329SD16—44no—
330D8—21no—
331D8—22no—
332D8—21no—
333D8—22no—
334D8—42no—
335C2×C8—22no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8C2×D4×C7311yescharacteristic
339C2×C8C2×D4×C7311yescharacteristic
340C2×C8C2×D4×C7311yescharacteristic
341C2×C8C24×C7311yescharacteristic
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4—41no—
357C42—41no—
358C42D8×C7311yescharacteristic
359C42D8×C7311yescharacteristic
360C42C2×D4×C7311yescharacteristic
361C42C2×D4×C7311yescharacteristic
362C16—22no—
363C16—22no—
364C23—41no—
365C23—42no—
366C23C2×D8×C7311yescharacteristic
367Q8—42no—
368Q8—41no—
369Q8—42no—
370Q8—41no—
371D4—42no—
372D4—41no—
373D4—42no—
374D4—41no—
375D4—82no—
376D4—84no—
377D4—82no—
378D4—82no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—84no—
388C2×C4—82no—
389C2×C4C2×D8×C7311yescharacteristic
390C2×C4C2×D8×C7311yescharacteristic
391C2×C4C2×D8×C7311yescharacteristic
392C2×C4C2×D8×C7311yescharacteristic
393C2×C4C2×D8×C7311yescharacteristic
394C2×C4C22×D4×C7311yescharacteristic
395C8—21no—
396C8—21no—
397C8AGL(1,ℤ/8)×C7312yes—
398C8H(5,𝔽2)×C7311yescharacteristic
399C8H(5,𝔽2)×C7311yescharacteristic
400C22—21no—
401C22—21no—
402C22—21no—
403C22—81no—
404C22—82no—
405C22—82no—
406C22—81no—
407C22—82no—
408C22—81no—
409C22C22×D8×C7311yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—81no—
415C4—81no—
416C4—82no—
417C4—81no—
418C4#508×C7312yes—
419C4#535×C7311yescharacteristic
420C4C2×AGL(1,ℤ/8)×C7311yescharacteristic
421C2—41no—
422C2—161no—
423C2—162no—
424C2C73×#320511yescharacteristic
425C2C2×#508×C7311yescharacteristic
426C2C73×#341411yescharacteristic
subgroupG/H
center, socleC22×C73C22×D8
commutator, Frattini, cosocleC2×C8C24×C73
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C2×C42×C73 ◁ C2×C4×C8×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C73×#3168C2semidirect
2. C73×#3147C2semidirect
3. C73×#2249C2semidirect
4. C73×#2242C2semidirect
5. C73×#2209C2semidirect
6. C73×#2208C2semidirect
7. C73×#2207C2semidirect
8. C73×#2206C2semidirect
9. C73×#2189C2semidirect
10. C73×#2186C2semidirect
11. C73×#2162C2semidirect
207. #33778C73direct (N × H)
214. #3168C2×C73semidirect
215. #3147C2×C73semidirect
216. #2249C2×C73semidirect
217. #2242C2×C73semidirect
218. #2209C2×C73semidirect
219. #2208C2×C73semidirect
220. #2207C2×C73semidirect
221. #2206C2×C73semidirect
222. #2189C2×C73semidirect
223. #2186C2×C73semidirect
224. #2162C2×C73semidirect
225. C73#33778direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#31682
C73×#31472
C73×#22492
C73×#22422
C73×#22092
C73×#22082
C73×#22072
C73×#22062
C73×#21892
C73×#21862
C73×#21622
#3377873