Finite Group Databasebeta

← C73×#9251 #32918655

Subgroups — C73×#9251 #32918655

Order 18688.

528 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#474×C73C211yesmaximal, characteristic
2D4×M16×C73C211yesmaximal, characteristic
3C73×#1966C211yesmaximal, characteristic, complemented
4C73×#1546C211yesmaximal, characteristic, complemented
5C73×#1486C211yesmaximal, characteristic
6C73×#1292C211yesmaximal, characteristic, complemented
7C73×#1283C211yesmaximal, characteristic, complemented
8C22×M16×C73—21no—
9C2×C4⋊Q8×C73C2211yescharacteristic
10C2×C4×D4×C73C2211yescharacteristic
11#474×C73—21no—
12#474×C73—21no—
13#474×C73—21no—
14#474×C73—21no—
15#436×C73—21no—
16#435×C73—22no—
17C8×D4×C73—21no—
18#431×C73C2211yescharacteristic
19C2×#107×C73C412yes—
20#423×C73C2211yescharacteristic
21#423×C73C2211yescharacteristic
22C2×C4⋊C8×C73C2211yescharacteristic
23C2×#102×C73C412yes—
24C2×#101×C73—21no—
25C2×#101×C73—21no—
26#407×C73—21no—
27C4×M16×C73C2211yescharacteristic
28#332×C73—22no—
29D4⋊C8×C73—21no—
30D4⋊C8×C73—21no—
31C22×Q8×C73—21no—
32C22×D4×C73C2×C411yescharacteristic
33C23×C4×C73—21no—
34C2×M16×C73—22no—
35C2×M16×C73—21no—
36C2×M16×C73—21no—
37C2×M16×C73—42no—
38C2×M16×C73—42no—
39C2×M16×C73D411yescharacteristic
40C2×M16×C73D411yescharacteristic
41C22×C8×C73—22no—
42C22×C8×C73C2×C412yes—
43C4⋊Q8×C73—21no—
44C4⋊Q8×C73—21no—
45C4⋊Q8×C73—21no—
46C4⋊Q8×C73—21no—
47C4×D4×C73—21no—
48C4×D4×C73—21no—
49C4×D4×C73D411yescharacteristic
50C4×D4×C73D411yescharacteristic
51C4×D4×C73D411yescharacteristic
52C4×D4×C73D411yescharacteristic
53C2×C4⋊C4×C73—21no—
54C2×C4⋊C4×C73C2×C412yes—
55C2×C4⋊C4×C73C2×C411yescharacteristic
56C2×C22⋊C4×C73—21no—
57C2×C42×C73C2311yescharacteristic
58#107×C73—22no—
59#107×C73—22no—
60C4⋊C8×C73—21no—
61C4⋊C8×C73—21no—
62C4⋊C8×C73—21no—
63C4⋊C8×C73—21no—
64C4⋊C8×C73—21no—
65C4⋊C8×C73—21no—
66C4⋊C8×C73D412yes—
67#102×C73—22no—
68#102×C73—22no—
69#101×C73—42no—
70#101×C73—42no—
71C22⋊C8×C73—22no—
72C22⋊C8×C73—22no—
73C22⋊C8×C73—21no—
74#97×C73—21no—
75C4×C8×C73—21no—
76C4×C8×C73—21no—
77C24×C73—21no—
78C2×Q8×C73—41no—
79C2×Q8×C73—41no—
80C2×Q8×C73—41no—
81C2×Q8×C73—41no—
82C2×Q8×C73—41no—
83C2×Q8×C73—41no—
84C2×D4×C73—22no—
85C2×D4×C73—21no—
86C2×D4×C73—41no—
87C2×D4×C73C22⋊C411yescharacteristic
88C2×D4×C73C22⋊C411yescharacteristic
89C22×C4×C73—21no—
90C22×C4×C73—21no—
91C22×C4×C73—21no—
92C22×C4×C73—21no—
93C22×C4×C73—21no—
94C22×C4×C73—22no—
95C22×C4×C73—41no—
96C22×C4×C73—41no—
97C22×C4×C73—41no—
98C22×C4×C73—41no—
99C22×C4×C73C22×C411yescharacteristic
100C22×C4×C73C2×D411yescharacteristic
101C22×C4×C73#4111yescharacteristic
102M16×C73—21no—
103M16×C73—21no—
104M16×C73—21no—
105M16×C73—44no—
106M16×C73—41no—
107M16×C73—41no—
108M16×C73—41no—
109M16×C73—41no—
110M16×C73D812yes—
111C2×C8×C73—21no—
112C2×C8×C73—22no—
113C2×C8×C73—21no—
114C2×C8×C73—22no—
115C2×C8×C73—44no—
116C2×C8×C73—42no—
117C2×C8×C73—41no—
118C2×C8×C73—41no—
119C4⋊C4×C73—21no—
120C4⋊C4×C73—22no—
121C4⋊C4×C73—22no—
122C4⋊C4×C73—41no—
123C4⋊C4×C73—41no—
124C4⋊C4×C73C22⋊C411yescharacteristic
125C4⋊C4×C73C22⋊C411yescharacteristic
126C22⋊C4×C73—21no—
127C22⋊C4×C73—21no—
128C22⋊C4×C73—21no—
129C22⋊C4×C73—21no—
130C42×C73C2×D411yescharacteristic
131C42×C73C2×D411yescharacteristic
132C42×C73C2×D411yescharacteristic
133C42×C73#4111yescharacteristic
134C23×C73—21no—
135C23×C73—21no—
136C23×C73—42no—
137C23×C73—41no—
138C23×C73—41no—
139C23×C73—81no—
140C23×C73C4×D411yescharacteristic
141Q8×C73—81no—
142Q8×C73—81no—
143Q8×C73—81no—
144Q8×C73—81no—
145D4×C73—21no—
146D4×C73—21no—
147D4×C73—21no—
148D4×C73—42no—
149D4×C73#10112yescomplemented
150C2×C4×C73—21no—
151C2×C4×C73—21no—
152C2×C4×C73—21no—
153C2×C4×C73—21no—
154C2×C4×C73—41no—
155C2×C4×C73—41no—
156C2×C4×C73—41no—
157C2×C4×C73—41no—
158C2×C4×C73—41no—
159C2×C4×C73—41no—
160C2×C4×C73—41no—
161C2×C4×C73—41no—
162C2×C4×C73—41no—
163C2×C4×C73—41no—
164C2×C4×C73—41no—
165C2×C4×C73—42no—
166C2×C4×C73—42no—
167C2×C4×C73—42no—
168C2×C4×C73C2×C22⋊C411yescharacteristic
169C2×C4×C73C4×D411yescharacteristic
170C2×C4×C73#12011yescharacteristic
171C2×C4×C73#12111yescharacteristic
172C2×C4×C73#12111yescharacteristic
173C2×C4×C73#12311yescharacteristic
174C2×C4×C73C2×D811yescharacteristic
175C2×C4×C73#13511yescharacteristic
176C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
178C8×C73—22no—
179C8×C73—42no—
180C8×C73—42no—
181C8×C73—41no—
182C8×C73—81no—
183C8×C73—81no—
184C22×C73—21no—
185C22×C73—21no—
186C22×C73—22no—
187C22×C73—41no—
188C22×C73—41no—
189C22×C73—41no—
190C22×C73—41no—
191C22×C73—41no—
192C22×C73—82no—
193C22×C73—82no—
194C22×C73#38611yescharacteristic
195C22×C73C4×D811yescharacteristic
196C22×C73#44211yescharacteristic
197C4×C73—21no—
198C4×C73—21no—
199C4×C73—21no—
200C4×C73—41no—
201C4×C73—41no—
202C4×C73—82no—
203C4×C73C2×#10111yescharacteristic
204C4×C73#41311yescharacteristic
205C4×C73#44711yescharacteristic
206C4×C73#44911yescharacteristic
207C4×C73#45911yescharacteristic
208C4×C73#46311yescharacteristic
209#9251C7311yesmaximal, characteristic, direct product factor, Sylow
210C2×C73—21no—
211C2×C73—21no—
212C2×C73—42no—
213C2×C73—81no—
214C2×C73#187911yescharacteristic
215C2×C73#190111yescharacteristic
216C2×C73#190211yescharacteristic
217C2×#474C2×C7311yescharacteristic
218D4×M16C2×C7311yescharacteristic
219#1966C2×C7311yescharacteristic, complemented
220#1546C2×C7311yescharacteristic, complemented
221#1486C2×C7311yescharacteristic
222#1292C2×C7311yescharacteristic, complemented
223#1283C2×C7311yescharacteristic, complemented
224C73#925111yescharacteristic, direct product factor, Sylow
225C22×M16—21no—
226C2×C4⋊Q8C22×C7311yescharacteristic
227C2×C4×D4C22×C7311yescharacteristic
228#474—21no—
229#474—21no—
230#474—21no—
231#474—21no—
232#436—21no—
233#435—22no—
234C8×D4—21no—
235#431C22×C7311yescharacteristic
236C2×#107C4×C7312yes—
237#423C22×C7311yescharacteristic
238#423C22×C7311yescharacteristic
239C2×C4⋊C8C22×C7311yescharacteristic
240C2×#102C4×C7312yes—
241C2×#101—21no—
242C2×#101—21no—
243#407—21no—
244C4×M16C22×C7311yescharacteristic
245#332—22no—
246D4⋊C8—21no—
247D4⋊C8—21no—
248C22×Q8—21no—
249C22×D4C2×C4×C7311yescharacteristic
250C23×C4—21no—
251C2×M16—22no—
252C2×M16—21no—
253C2×M16—21no—
254C2×M16—42no—
255C2×M16—42no—
256C2×M16D4×C7311yescharacteristic
257C2×M16D4×C7311yescharacteristic
258C22×C8—22no—
259C22×C8C2×C4×C7312yes—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4D4×C7311yescharacteristic
267C4×D4D4×C7311yescharacteristic
268C4×D4D4×C7311yescharacteristic
269C4×D4D4×C7311yescharacteristic
270C2×C4⋊C4—21no—
271C2×C4⋊C4C2×C4×C7312yes—
272C2×C4⋊C4C2×C4×C7311yescharacteristic
273C2×C22⋊C4—21no—
274C2×C42C23×C7311yescharacteristic
275#107—22no—
276#107—22no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8D4×C7312yes—
284#102—22no—
285#102—22no—
286#101—42no—
287#101—42no—
288C22⋊C8—22no—
289C22⋊C8—22no—
290C22⋊C8—21no—
291#97—21no—
292C4×C8—21no—
293C4×C8—21no—
294C24—21no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×D4—22no—
302C2×D4—21no—
303C2×D4—41no—
304C2×D4C22⋊C4×C7311yescharacteristic
305C2×D4C22⋊C4×C7311yescharacteristic
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4C22×C4×C7311yescharacteristic
317C22×C4C2×D4×C7311yescharacteristic
318C22×C4#41×C7311yescharacteristic
319M16—21no—
320M16—21no—
321M16—21no—
322M16—44no—
323M16—41no—
324M16—41no—
325M16—41no—
326M16—41no—
327M16D8×C7312yes—
328C2×C8—21no—
329C2×C8—22no—
330C2×C8—21no—
331C2×C8—22no—
332C2×C8—44no—
333C2×C8—42no—
334C2×C8—41no—
335C2×C8—41no—
336C4⋊C4—21no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4C22⋊C4×C7311yescharacteristic
342C4⋊C4C22⋊C4×C7311yescharacteristic
343C22⋊C4—21no—
344C22⋊C4—21no—
345C22⋊C4—21no—
346C22⋊C4—21no—
347C42C2×D4×C7311yescharacteristic
348C42C2×D4×C7311yescharacteristic
349C42C2×D4×C7311yescharacteristic
350C42#41×C7311yescharacteristic
351C23—21no—
352C23—21no—
353C23—42no—
354C23—41no—
355C23—41no—
356C23—81no—
357C23C4×D4×C7311yescharacteristic
358Q8—81no—
359Q8—81no—
360Q8—81no—
361Q8—81no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—42no—
366D4#101×C7312yescomplemented
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4C2×C22⋊C4×C7311yescharacteristic
386C2×C4C4×D4×C7311yescharacteristic
387C2×C4#120×C7311yescharacteristic
388C2×C4#121×C7311yescharacteristic
389C2×C4#121×C7311yescharacteristic
390C2×C4#123×C7311yescharacteristic
391C2×C4C2×D8×C7311yescharacteristic
392C2×C4#135×C7311yescharacteristic
393C2×C4AGL(1,ℤ/8)×C7311yescharacteristic
394C2×C4AGL(1,ℤ/8)×C7311yescharacteristic
395C8—22no—
396C8—42no—
397C8—42no—
398C8—41no—
399C8—81no—
400C8—81no—
401C22—21no—
402C22—21no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—82no—
410C22—82no—
411C22#386×C7311yescharacteristic
412C22C4×D8×C7311yescharacteristic
413C22#442×C7311yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—41no—
418C4—41no—
419C4—82no—
420C4C2×#101×C7311yescharacteristic
421C4#413×C7311yescharacteristic
422C4#447×C7311yescharacteristic
423C4#449×C7311yescharacteristic
424C4#459×C7311yescharacteristic
425C4#463×C7311yescharacteristic
426C2—21no—
427C2—21no—
428C2—42no—
429C2—81no—
430C2C73×#187911yescharacteristic
431C2C73×#190111yescharacteristic
432C2C73×#190211yescharacteristic
subgroupG/H
center, socleC22×C73#386
commutatorC22×C4C22×C4×C73
Frattini, cosocleC2×C42C23×C73
prime radicalC22×D4×C73C2×C4
LCSG ▷ C22×C4 ▷ 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×#1966C2semidirect
4. C73×#1546C2semidirect
6. C73×#1292C2semidirect
7. C73×#1283C2semidirect
149. D4×C73#101semidirect
209. #9251C73direct (N × H)
219. #1966C2×C73semidirect
220. #1546C2×C73semidirect
222. #1292C2×C73semidirect
223. #1283C2×C73semidirect
224. C73#9251direct (N × H)
366. D4#101×C73semidirect

Maximal subgroups

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

subgroupindex
C2×#474×C732
D4×M16×C732
C73×#19662
C73×#15462
C73×#14862
C73×#12922
C73×#12832
#925173