Finite Group Databasebeta

← #33239337

Subgroups — #33239337

Order 18688.

660 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989028C211yesmaximal, characteristic
2#22618808C211yesmaximal, characteristic
3#22611800C212yesmaximal, complemented
4#22611793C212yesmaximal, complemented
5C4×#14556537C212yesmaximal, complemented
6C4×#14556534C212yesmaximal, complemented
7#22611720C211yesmaximal, characteristic, complemented
8#22611636C211yesmaximal, characteristic
9C2×#17988391C211yesmaximal, characteristic, complemented
10#22611528C211yesmaximal, characteristic, complemented
11C73×#2950C211yesmaximal, characteristic, complemented
12C2×#14556613C2211yescharacteristic
13#17989067—21no—
14#17989028C418yescomplemented
15C2×#14556537C2212yes—
16C2×#14556534C2212yes—
17C2×#14556531C2211yescharacteristic
18C22×#11924003C2211yescharacteristic
19#17988689—22no—
20C4×Dic292C2212yes—
21C4×D584C2212yes—
22C2×#14556525C2211yescharacteristic
23#17988589—21no—
24#17988414C2212yes—
25#17988412C2212yes—
26#17988391C2212yes—
27#17988391C2212yes—
28#17988390C2211yescharacteristic
29#17988390C2211yescharacteristic
30#17988387C2211yescharacteristic
31#17988387C2211yescharacteristic
32#17988363C2212yes—
33C4×#11924003C2212yes—
34C22×Q16×C73C2211yescharacteristic
35#519×C73—21no—
36#441×C73—22no—
37C4×Q16×C73C2212yes—
38C4×Q16×C73C2212yes—
39#427×C73C2211yescharacteristic
40#419×C73—21no—
41#405×C73C2211yescharacteristic
42C2×#11924028—21no—
43#14556613—22no—
44#14556613C2×C414yes—
45C2×Dic292C2311yescharacteristic
46C2×#11924017—21no—
47C2×D584C2311yescharacteristic
48C4×Dic146—22no—
49C4×D292—22no—
50#14556551—42no—
51#14556549—41no—
52C2×C4⋊Dic73—21no—
53C4⋊C4×D73—41no—
54#14556537C2×C418yescomplemented
55#14556534C2×C418yescomplemented
56#14556531C2×C414yescomplemented
57C2×#11924003—22no—
58C2×#11924003C2311yescharacteristic
59C2×#11924003C2311yescharacteristic
60C2×#11924003C2312yes—
61#14556525—21no—
62#14556525C2312yes—
63#14556516—22no—
64#14556513—22no—
65C22×Q8×C73—21no—
66C2×Q16×C73—21no—
67C2×Q16×C73—22no—
68C2×Q16×C73C2×C414yescomplemented
69C2×Q16×C73C2312yes—
70C2×M16×C73D411yescharacteristic
71C22×C8×C73C2311yescharacteristic
72C4×Q8×C73—22no—
73C4×Q8×C73—22no—
74C4×Q8×C73—41no—
75C4×Q8×C73—41no—
76#117×C73—21no—
77#117×C73—42no—
78#117×C73D411yescharacteristic
79#107×C73C2311yescharacteristic
80#107×C73C2311yescharacteristic
81#106×C73—21no—
82#103×C73—22no—
83#103×C73—22no—
84#97×C73D412yes—
85C4×C8×C73C2312yes—
86#11924028—24no—
87#11924028—42no—
88C2×Dic146—21no—
89C2×D292—21no—
90C2×C22⋊D73—41no—
91C22×Dic73—41no—
92C2×C4×D73—41no—
93C2×C4×D73—42no—
94Dic292—21no—
95Dic292C22×C412yes—
96#11924017—24no—
97D584—21no—
98D584C22×C412yes—
99C4⋊Dic73—22no—
100C4⋊Dic73—41no—
101#11924007—42no—
102#11924005—42no—
103C4×Dic73—42no—
104#11924003—21no—
105#11924003—21no—
106#11924003C22×C414yes—
107C2×Q8×C73—24no—
108C2×Q8×C73—22no—
109C2×Q8×C73—41no—
110C2×Q8×C73—42no—
111C22×C4×C73—41no—
112C22×C4×C73C2×D411yescharacteristic
113Q16×C73—24no—
114Q16×C73—22no—
115Q16×C73C22×C414yes—
116M16×C73—22no—
117C2×C8×C73C22×C412yes—
118C2×C8×C73C2×D412yes—
119C2×C8×C73C2×D411yescharacteristic
120C2×C8×C73#4112yes—
121C2×C8×C73C2411yescharacteristic
122C4⋊C4×C73—21no—
123C4⋊C4×C73—21no—
124C4⋊C4×C73—42no—
125C4⋊C4×C73—42no—
126C4⋊C4×C73D811yescharacteristic
127C4⋊C4×C73D811yescharacteristic
128C22⋊C4×C73—41no—
129C22⋊C4×C73D812yes—
130C42×C73—41no—
131C42×C73—42no—
132C42×C73—42no—
133C42×C73C2×D412yes—
134C23×C73C2×D811yescharacteristic
135C22×D73—41no—
136Q8×C73—24no—
137Q8×C73—42no—
138Q8×C73—44no—
139C2×C4×C73—42no—
140C2×C4×C73—44no—
141C2×C4×C73—41no—
142C2×C4×C73—41no—
143C2×C4×C73C4×D412yes—
144C2×C4×C73C2×D812yes—
145C2×C4×C73C2×D812yes—
146C2×C4×C73C2×D811yescharacteristic
147C2×C4×C73#13512yes—
148C2×C4×C73C22×D411yescharacteristic
149C22⋊D73—44no—
150C2×Dic73—41no—
151C2×Dic73—42no—
152C2×Dic73—161no—
153D292—22no—
154D292—41no—
155C4×D73—44no—
156C4×D73—162no—
157Dic146—22no—
158Dic146—41no—
159C8×C73—22no—
160C8×C73C4×D412yes—
161C8×C73C23×C411yescharacteristic
162C8×C73C2×#4111yescharacteristic
163C22×C73—21no—
164C22×C73—21no—
165C22×C73C4×D812yes—
166C22×C73C22×D811yescharacteristic
167C2×D73—42no—
168C2×D73—81no—
169C2×D73—81no—
170C4×C73—22no—
171C4×C73—22no—
172C4×C73—44no—
173C4×C73—81no—
174C4×C73—81no—
175C4×C73C4×D812yes—
176C4×C73C2×C4×D411yescharacteristic
177C4×C73C2×#13511yescharacteristic
178Dic73—42no—
179Dic73—82no—
180#33455—731nomaximal, Sylow
181C2×C73—22no—
182C2×C73C2×C4×D811yescharacteristic
183C2×C73#341611yescharacteristic
184C2×C73#341711yescharacteristic
185D73—82no—
186C2×#510—731no—
187#2951—731no—
188#2950D7311yescharacteristic, complemented
189#2176—732no—
190#2175—732no—
191C4×Q32—732no—
192C4×SD32—732no—
193#2159—731no—
194#2138—731no—
195C2×#358—731no—
196#2108—731no—
197C73#3345511yescharacteristic, complemented, Sylow
198C2×#135—731no—
199C22×Q16C2×D7311yescharacteristic
200#520—1461no—
201#519—21no—
202#510—738no—
203C2×Q32—732no—
204C2×SD32—732no—
205C2×M32—731no—
206C22×C16—731no—
207#441—22no—
208#440—1462no—
209C4×Q16—732no—
210C4×Q16C2×D7312yes—
211C4×Q16C2×D7312yes—
212C4×D8—732no—
213#427C2×D7311yescharacteristic
214C2×#107—731no—
215#419—21no—
216#417—1461no—
217#405C2×D7311yescharacteristic
218#367—732no—
219#366—732no—
220#358—732no—
221#358—732no—
222#358—731no—
223#358—731no—
224#357—731no—
225#357—731no—
226#346—732no—
227C4×C16—732no—
228C2×#41—1461no—
229C22×Q8—21no—
230#135—734no—
231#135—1462no—
232C2×Q16—21no—
233C2×Q16—22no—
234C2×Q16—731no—
235C2×Q16C4×D7314yescomplemented
236C2×Q16C22×D7312yes—
237C2×SD16—1461no—
238C2×D8—731no—
239C2×M16C22⋊D7311yescharacteristic
240C22×C8C22×D7311yescharacteristic
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8—41no—
244C4×Q8—41no—
245C4×Q8—1462no—
246C4×D4—2922no—
247C4×D4—1462no—
248#117—21no—
249#117—42no—
250#117—2921no—
251#117C22⋊D7311yescharacteristic
252C2×C4⋊C4—2921no—
253C2×C4⋊C4—1461no—
254Q32—738no—
255SD32—738no—
256M32—734no—
257C2×C16—732no—
258C2×C16—731no—
259C2×C16—731no—
260C2×C16—1462no—
261#107—732no—
262#107—1461no—
263#107C22×D7311yescharacteristic
264#107C22×D7311yescharacteristic
265#106—21no—
266#103—22no—
267#103—22no—
268#103—1462no—
269#102—1462no—
270#97C22⋊D7312yes—
271C4×C8C22×D7312yes—
272#41—2922no—
273#41—1464no—
274C2×Q8—24no—
275C2×Q8—22no—
276C2×Q8—41no—
277C2×Q8—42no—
278C2×Q8—1461no—
279C2×D4—2921no—
280C2×D4—1461no—
281C22×C4—41no—
282C22×C4—2921no—
283C22×C4—2921no—
284C22×C4—2922no—
285C22×C4C2×C22⋊D7311yescharacteristic
286Q16—24no—
287Q16—22no—
288Q16—732no—
289Q16—1461no—
290Q16C2×C4×D7314yes—
291SD16—1464no—
292D8—732no—
293D8—1461no—
294M16—22no—
295C2×C8C2×C4×D7312yes—
296C2×C8C2×C22⋊D7312yes—
297C2×C8C2×C22⋊D7311yescharacteristic
298C2×C8#1192402812yes—
299C2×C8C23×D7311yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—2922no—
305C4⋊C4—2921no—
306C4⋊C4—1462no—
307C4⋊C4#1192401311yescharacteristic
308C4⋊C4#1192401311yescharacteristic
309C22⋊C4—41no—
310C22⋊C4—2922no—
311C22⋊C4#1192401312yes—
312C42—41no—
313C42—42no—
314C42—42no—
315C42—2922no—
316C42C2×C22⋊D7312yes—
317C16—734no—
318C16—1461no—
319C16—1461no—
320C23—2921no—
321C23C2×#1192401311yescharacteristic
322Q8—24no—
323Q8—44no—
324Q8—42no—
325Q8—2921no—
326Q8—1462no—
327D4—2924no—
328D4—2921no—
329D4—1462no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—2921no—
335C2×C4—2924no—
336C2×C4—2922no—
337C2×C4—11682no—
338C2×C4—11681no—
339C2×C4C4×C22⋊D7312yes—
340C2×C4C2×#1192401312yes—
341C2×C4C2×#1192401312yes—
342C2×C4C2×#1192401311yescharacteristic
343C2×C4#1455661112yes—
344C2×C4C22×C22⋊D7311yescharacteristic
345C8—22no—
346C8C4×C22⋊D7312yes—
347C8C22×C4×D7311yescharacteristic
348C8C2×#1192402811yescharacteristic
349C22—21no—
350C22—21no—
351C22—2922no—
352C22—5841no—
353C22—5841no—
354C22C4×#1192401312yes—
355C22C22×#1192401311yescharacteristic
356C4—22no—
357C4—22no—
358C4—44no—
359C4—81no—
360C4—81no—
361C4—2922no—
362C4—5842no—
363C4C4×#1192401312yes—
364C4C2×C4×C22⋊D7311yescharacteristic
365C4C2×#1455661111yescharacteristic
366C2—22no—
367C2—5842no—
368C2C2×C4×#1192401311yescharacteristic
369C2#2262307511yescharacteristic
370C2#2262308211yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC73×#2950C2
socleC22×C73C22×D8
prime radicalC2×#14556613C22
cosocleC2×C8×C73C24
hypercenter#2950D73
hypocenterC73#33455
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2950

Complements

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

NH ≅ G/Ntype
3. #22611800C2semidirect
4. #22611793C2semidirect
5. C4×#14556537C2semidirect
6. C4×#14556534C2semidirect
7. #22611720C2semidirect
9. C2×#17988391C2semidirect
10. #22611528C2semidirect
11. C73×#2950C2semidirect
14. #17989028C4semidirect
54. #14556537C2×C4semidirect
55. #14556534C2×C4semidirect
56. #14556531C2×C4semidirect
68. C2×Q16×C73C2×C4semidirect
188. #2950D73semidirect
197. C73#33455semidirect
235. C2×Q16C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179890282
#226188082
#226118002
#226117932
C4×#145565372
C4×#145565342
#226117202
#226116362
C2×#179883912
#226115282
C73×#29502
#3345573