Finite Group Databasebeta

← #33002949

Subgroups — #33002949

Order 18688.

512 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988857C211yesmaximal, characteristic
2#22611646C211yesmaximal, characteristic, complemented
3C2×#17988387C211yesmaximal, characteristic, complemented
4C2×#17988371C211yesmaximal, characteristic, complemented
5#22608039C211yesmaximal, characteristic, complemented
6#22608026C211yesmaximal, characteristic, complemented
7C73×#2913C211yesmaximal, characteristic, complemented
8C2×#14556616—21no—
9C22×D584C2211yescharacteristic
10C2×#14556531C2211yescharacteristic
11C22×#11924003C2211yescharacteristic
12#17988857—22no—
13#17988857C414yescomplemented
14C2×#14556527C2211yescharacteristic
15C2×#14556507—21no—
16#17988387—22no—
17#17988387—21no—
18#17988387—21no—
19#17988371—22no—
20C2×#131×C73C2211yescharacteristic
21#520×C73—21no—
22#428×C73—21no—
23#428×C73—22no—
24#427×C73—21no—
25#427×C73C2211yescharacteristic
26C2×#107×C73—21no—
27C2×#107×C73C2211yescharacteristic
28C22×D292—21no—
29#14556616—24no—
30#14556616—42no—
31C2×#11924015—41no—
32C2×D584—21no—
33C2×D584—21no—
34C2×D584—21no—
35C2×D584—21no—
36C2×D584—21no—
37C2×D584C2×C412yescomplemented
38C2×#11924013—41no—
39C2×#11924012—21no—
40#14556531—21no—
41#14556531—21no—
42C2×#11924003—21no—
43C2×#11924003—21no—
44C2×#11924003—21no—
45C2×#11924003—21no—
46#14556527—21no—
47#14556527C2×C412yescomplemented
48#14556507—24no—
49C2×#41×C73D411yescharacteristic
50#131×C73—21no—
51#131×C73—21no—
52#131×C73C2×C412yes—
53#131×C73D412yes—
54C2×M16×C73—21no—
55C2×M16×C73D411yescharacteristic
56C22×C8×C73—21no—
57C22×C8×C73C2311yescharacteristic
58C4×Q8×C73—41no—
59C4×D4×C73—42no—
60C4×D4×C73—41no—
61#117×C73—22no—
62#117×C73—21no—
63C2×C4⋊C4×C73—22no—
64C2×C4⋊C4×C73—21no—
65#107×C73—22no—
66#107×C73—21no—
67#107×C73D411yescharacteristic
68#107×C73D412yes—
69#107×C73D411yescharacteristic
70#106×C73—22no—
71#106×C73—21no—
72C23×D73—41no—
73C2×D292—22no—
74C2×D292—41no—
75C2×D292—41no—
76C2×D292—41no—
77C2×D292—41no—
78C2×D292—41no—
79#11924015—44no—
80D584—22no—
81D584—22no—
82D584—41no—
83D584—41no—
84#11924013—44no—
85#11924012—22no—
86#11924012—41no—
87C2×#10596831—41no—
88#11924003—41no—
89#11924003—41no—
90#41×C73—21no—
91#41×C73—21no—
92#41×C73C22⋊C412yes—
93#41×C73SD1612yes—
94C2×Q8×C73D811yescharacteristic
95C2×D4×C73—21no—
96C2×D4×C73D811yescharacteristic
97C22×C4×C73—21no—
98C22×C4×C73—42no—
99C22×C4×C73—41no—
100C22×C4×C73C2×D411yescharacteristic
101M16×C73—21no—
102M16×C73—22no—
103M16×C73—22no—
104M16×C73C22⋊C412yes—
105C2×C8×C73—21no—
106C2×C8×C73—22no—
107C2×C8×C73—22no—
108C2×C8×C73C22×C411yescharacteristic
109C2×C8×C73C2×D411yescharacteristic
110C2×C8×C73C2×D411yescharacteristic
111C2×C8×C73C2×D411yescharacteristic
112C2×C8×C73#4111yescharacteristic
113C2×C8×C73#4111yescharacteristic
114C4⋊C4×C73—21no—
115C4⋊C4×C73—22no—
116C4⋊C4×C73—21no—
117C4⋊C4×C73—42no—
118C4⋊C4×C73—41no—
119C22⋊C4×C73—42no—
120C22⋊C4×C73—41no—
121C42×C73—42no—
122C42×C73—41no—
123C23×C73—21no—
124C23×C73AGL(1,ℤ/8)11yescharacteristic
125C22×D73—42no—
126C22×D73—81no—
127C22×D73—81no—
128C22×D73—161no—
129C22×D73—161no—
130Q8×C73—21no—
131Q8×C73#10212yes—
132D4×C73—22no—
133D4×C73—22no—
134D4×C73—21no—
135D4×C73#10212yes—
136C2×C4×C73—21no—
137C2×C4×C73—22no—
138C2×C4×C73—22no—
139C2×C4×C73—41no—
140C2×C4×C73—42no—
141C2×C4×C73—41no—
142C2×C4×C73—161no—
143C2×C4×C73—162no—
144C2×C4×C73C2×C22⋊C411yescharacteristic
145C2×C4×C73#12011yescharacteristic
146C2×C4×C73#12311yescharacteristic
147C2×C4×C73C2×D811yescharacteristic
148C2×C4×C73C2×SD1611yescharacteristic
149C2×C4×C73#13711yescharacteristic
150D292—42no—
151D292—42no—
152D292—81no—
153D292—81no—
154C8×C73—22no—
155C8×C73C4×D411yescharacteristic
156C8×C73C4×D411yescharacteristic
157C8×C73#12111yescharacteristic
158C8×C73#12111yescharacteristic
159#10596831—42no—
160C22×C73—21no—
161C22×C73—21no—
162C22×C73—22no—
163C22×C73—41no—
164C22×C73—41no—
165C22×C73#41711yescharacteristic
166C22×C73#44711yescharacteristic
167C22×C73#48111yescharacteristic
168C2×D73—82no—
169C2×D73—162no—
170C2×D73—161no—
171C2×D73—161no—
172C2×D73—162no—
173C4×C73—22no—
174C4×C73—82no—
175C4×C73—81no—
176C4×C73—81no—
177C4×C73#38611yescharacteristic
178C4×C73C2×#10211yescharacteristic
179C4×C73#44811yescharacteristic
180C4×C73#48011yescharacteristic
181#12996—731nomaximal, Sylow
182C2×C73—21no—
183C2×C73—21no—
184C2×C73—42no—
185C2×C73#187411yescharacteristic
186C2×C73#219211yescharacteristic
187C2×C73#219311yescharacteristic
188D73—162no—
189C2×#472—731no—
190#2913D7311yescharacteristic, complemented
191#2140—731no—
192C2×#357—731no—
193C2×#350—731no—
194#1387—731no—
195#1382—731no—
196C73#1299611yescharacteristic, complemented, Sylow
197C2×AGL(1,ℤ/8)—1461no—
198C22×D8—731no—
199C2×#131C2×D7311yescharacteristic
200#520—21no—
201C2×M32—731no—
202C22×C16—731no—
203#472—734no—
204#472—1462no—
205C2×#108—731no—
206#428—21no—
207#428—22no—
208#427—21no—
209#427C2×D7311yescharacteristic
210C2×#107—21no—
211C2×#107C2×D7311yescharacteristic
212C2×#100—1461no—
213#357—1462no—
214#357—1461no—
215#357—1461no—
216#350—1462no—
217C2×#41C22⋊D7311yescharacteristic
218C22×D4—1461no—
219AGL(1,ℤ/8)—2922no—
220AGL(1,ℤ/8)—1464no—
221C2×SD16—2921no—
222C2×D8—2921no—
223C2×D8—732no—
224C2×D8—1461no—
225C2×D8—1461no—
226C2×D8—1461no—
227C2×D8—1461no—
228C2×D8—1461no—
229#131—21no—
230#131—21no—
231#131C4×D7312yes—
232#131D29212yes—
233C2×M16—21no—
234C2×M16—1461no—
235C2×M16C22⋊D7311yescharacteristic
236C22×C8—21no—
237C22×C8C22×D7311yescharacteristic
238C4×Q8—41no—
239C4×D4—42no—
240C4×D4—41no—
241#117—22no—
242#117—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245M32—1461no—
246M32—1461no—
247C2×C16—1461no—
248C2×C16—1461no—
249C2×C16—1461no—
250C2×C16—1461no—
251#108—732no—
252#108—1461no—
253#107—22no—
254#107—21no—
255#107C22⋊D7311yescharacteristic
256#107C22⋊D7312yes—
257#107C22⋊D7311yescharacteristic
258#106—22no—
259#106—21no—
260#100—1464no—
261C24—2921no—
262#41—21no—
263#41—21no—
264#41#1192400512yes—
265#41#1192401512yes—
266C2×Q8#1192401311yescharacteristic
267C2×D4—21no—
268C2×D4—2921no—
269C2×D4—2921no—
270C2×D4—2921no—
271C2×D4—2921no—
272C2×D4—2921no—
273C2×D4—1462no—
274C2×D4#1192401311yescharacteristic
275C22×C4—21no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4C2×C22⋊D7311yescharacteristic
279SD16—2924no—
280D8—2924no—
281D8—2921no—
282D8—2921no—
283D8—1462no—
284D8—1462no—
285M16—21no—
286M16—22no—
287M16—22no—
288M16—2921no—
289M16—1462no—
290M16#1192400512yes—
291C2×C8—21no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—2921no—
295C2×C8C2×C4×D7311yescharacteristic
296C2×C8C2×C22⋊D7311yescharacteristic
297C2×C8C2×C22⋊D7311yescharacteristic
298C2×C8C2×D29211yescharacteristic
299C2×C8#1192402811yescharacteristic
300C2×C8#1192402811yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C22⋊C4—42no—
307C22⋊C4—41no—
308C42—42no—
309C42—41no—
310C16—2921no—
311C16—2921no—
312C23—21no—
313C23—2922no—
314C23—5841no—
315C23—5841no—
316C23—11681no—
317C23—11681no—
318C23#1455661711yescharacteristic
319Q8—21no—
320Q8#1455651112yes—
321D4—22no—
322D4—22no—
323D4—21no—
324D4—2922no—
325D4—2922no—
326D4—5841no—
327D4—5841no—
328D4#1455651112yes—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—161no—
336C2×C4—162no—
337C2×C4C2×#1192400511yescharacteristic
338C2×C4#1455656011yescharacteristic
339C2×C4#1455657411yescharacteristic
340C2×C4C2×#1192401311yescharacteristic
341C2×C4C2×#1192401511yescharacteristic
342C2×C4#1455662211yescharacteristic
343C8—22no—
344C8—2922no—
345C8C4×C22⋊D7311yescharacteristic
346C8C4×C22⋊D7311yescharacteristic
347C8#1455656511yescharacteristic
348C8#1455656511yescharacteristic
349C22—21no—
350C22—21no—
351C22—22no—
352C22—41no—
353C22—41no—
354C22—5842no—
355C22—11682no—
356C22—11682no—
357C22—11681no—
358C22—11681no—
359C22#1798858611yescharacteristic
360C22#1798872111yescharacteristic
361C22#1798890511yescharacteristic
362C4—22no—
363C4—82no—
364C4—81no—
365C4—81no—
366C4#1798846611yescharacteristic
367C4C2×#1455651111yescharacteristic
368C4#1798872611yescharacteristic
369C4#1798890011yescharacteristic
370C2—21no—
371C2—21no—
372C2—42no—
373C2—11682no—
374C2#2261024811yescharacteristic
375C2#2261188811yescharacteristic
376C2#2261189511yescharacteristic
subgroupG/H
centerC22#17988721
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC73×#2913C2
socleC22×C73#447
prime radicalC2×#17988857C2
cosocleC22×C8×C73C23
hypercenter#2913D73
hypocenterC73#12996
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ #2913

Complements

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

NH ≅ G/Ntype
2. #22611646C2semidirect
3. C2×#17988387C2semidirect
4. C2×#17988371C2semidirect
5. #22608039C2semidirect
6. #22608026C2semidirect
7. C73×#2913C2semidirect
13. #17988857C4semidirect
37. C2×D584C2×C4semidirect
47. #14556527C2×C4semidirect
190. #2913D73semidirect
196. C73#12996semidirect

Maximal subgroups

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

subgroupindex
C2×#179888572
#226116462
C2×#179883872
C2×#179883712
#226080392
#226080262
C73×#29132
#1299673