Finite Group Databasebeta

← #33240848

Subgroups — #33240848

Order 18688.

568 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
1#22619601C211yesmaximal, characteristic
2#22612139C212yesmaximal, complemented
3#22612133C212yesmaximal, complemented
4#22612122C212yesmaximal, complemented
5#22612111C212yesmaximal, complemented
6#22611720C211yesmaximal, characteristic, complemented
7C2×#17988412C211yesmaximal, characteristic, complemented
8#22611641C211yesmaximal, characteristic, complemented
9#22611636C211yesmaximal, characteristic, complemented
10#22611545C211yesmaximal, characteristic, complemented
11C73×#2913C211yesmaximal, characteristic, complemented
12C2×#14556613C2211yescharacteristic
13#17989165—21no—
14C2×#14556531C2211yescharacteristic
15C22×#11924003C2211yescharacteristic
16#17988833—22no—
17#17988828C2212yes—
18#17988823C2212yes—
19C2×#14556525C2211yescharacteristic
20#17988589—21no—
21#17988414C2212yes—
22#17988412C2212yes—
23#17988412C2212yes—
24#17988412C2212yes—
25#17988390C2211yescharacteristic
26#17988390C2212yes—
27#17988390C2211yescharacteristic
28#17988387C2211yescharacteristic
29#17988387C2212yes—
30#17988387C2211yescharacteristic
31C22⋊#11924003C2212yes—
32C22⋊#11924003C2212yes—
33C2×#131×C73C2211yescharacteristic
34#520×C73—21no—
35#428×C73—22no—
36#428×C73—21no—
37#427×C73C2211yescharacteristic
38#427×C73C2212yes—
39C2×#107×C73C2212yes—
40C2×#107×C73C2211yescharacteristic
41C2×#11924028—21no—
42#14556613—22no—
43#14556613—22no—
44C2×Dic292C2311yescharacteristic
45C2×#11924017—21no—
46C2×D584C2311yescharacteristic
47#14556570—22no—
48#14556567—41no—
49#14556565—22no—
50#14556563—42no—
51#14556562—41no—
52C2×C4⋊Dic73—21no—
53#14556531—22no—
54C2×#11924003—22no—
55C2×#11924003C2311yescharacteristic
56C2×#11924003C2311yescharacteristic
57C2×#11924003C2312yes—
58#14556525—21no—
59#14556525C2312yes—
60#14556516—22no—
61#14556513—22no—
62C2×#41×C73D411yescharacteristic
63#131×C73—22no—
64#131×C73—22no—
65C2×M16×C73D412yes—
66C2×M16×C73D411yescharacteristic
67C22×C8×C73C2312yes—
68C22×C8×C73C2311yescharacteristic
69C4×Q8×C73—41no—
70C4×D4×C73—41no—
71C4×D4×C73—42no—
72#117×C73—22no—
73#117×C73—21no—
74C2×C4⋊C4×C73—22no—
75C2×C4⋊C4×C73—21no—
76#107×C73—22no—
77#107×C73—21no—
78#107×C73C2311yescharacteristic
79#107×C73C2312yes—
80#107×C73C2311yescharacteristic
81#106×C73—22no—
82#106×C73—21no—
83#11924028—42no—
84#11924028—42no—
85C2×Dic146—21no—
86C2×D292—21no—
87C2×C22⋊D73—41no—
88C2×C22⋊D73—42no—
89C22×Dic73—41no—
90C2×C4×D73—41no—
91Dic292—21no—
92Dic292—21no—
93#11924017—42no—
94D584—21no—
95D584—21no—
96C4⋊Dic73—22no—
97C4⋊Dic73—41no—
98#11924007—42no—
99C22⋊Dic73—42no—
100#11924005—42no—
101#11924003—21no—
102#11924003—21no—
103#11924003—22no—
104#41×C73—22no—
105#41×C73—22no—
106C2×Q8×C73D811yescharacteristic
107C2×D4×C73D811yescharacteristic
108C2×D4×C73D812yes—
109C22×C4×C73—42no—
110C22×C4×C73—41no—
111C22×C4×C73C2×D412yes—
112C22×C4×C73C2×D411yescharacteristic
113M16×C73—22no—
114M16×C73—24no—
115C2×C8×C73—22no—
116C2×C8×C73—22no—
117C2×C8×C73C2×D412yes—
118C2×C8×C73C2×D411yescharacteristic
119C2×C8×C73#4112yes—
120C2×C8×C73#4112yes—
121C2×C8×C73C2411yescharacteristic
122C4⋊C4×C73—21no—
123C4⋊C4×C73—22no—
124C4⋊C4×C73—21no—
125C4⋊C4×C73—42no—
126C4⋊C4×C73—41no—
127C22⋊C4×C73—42no—
128C22⋊C4×C73—41no—
129C42×C73—42no—
130C42×C73—41no—
131C23×C73C2×D812yes—
132C23×C73C2×D811yescharacteristic
133C22×D73—41no—
134Q8×C73—22no—
135D4×C73—24no—
136D4×C73—22no—
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×C73#12312yes—
145C2×C4×C73C2×D812yes—
146C2×C4×C73C2×D811yescharacteristic
147C2×C4×C73#13712yes—
148C2×C4×C73C22×D411yescharacteristic
149C22⋊D73—82no—
150C22⋊D73—162no—
151C2×Dic73—42no—
152C2×Dic73—41no—
153C2×Dic73—161no—
154D292—41no—
155D292—41no—
156C4×D73—82no—
157Dic146—41no—
158Dic146—41no—
159C8×C73—22no—
160C8×C73#12312yes—
161C8×C73C2×#4111yescharacteristic
162C8×C73C2×#4111yescharacteristic
163C22×C73—21no—
164C22×C73—21no—
165C22×C73—22no—
166C22×C73—22no—
167C22×C73—22no—
168C22×C73#48012yes—
169C22×C73C22×D811yescharacteristic
170C2×D73—81no—
171C2×D73—81no—
172C2×D73—81no—
173C4×C73—22no—
174C4×C73—82no—
175C4×C73—81no—
176C4×C73—81no—
177C4×C73#48012yes—
178C4×C73C2×#12311yescharacteristic
179C4×C73C2×#13711yescharacteristic
180Dic73—81no—
181Dic73—82no—
182#33652—731nomaximal, Sylow
183C2×C73—22no—
184C2×C73—42no—
185C2×C73C2×#48011yescharacteristic
186C2×C73#341511yescharacteristic
187C2×C73#341711yescharacteristic
188D73—161no—
189#3061—731no—
190#2913D7311yescharacteristic, complemented
191#2237—732no—
192#2236—732no—
193#2234—732no—
194#2232—732no—
195#2159—731no—
196C2×#366—731no—
197#2139—731no—
198#2138—731no—
199#2113—731no—
200C73#3365211yescharacteristic, complemented, Sylow
201C2×#135—731no—
202C2×#131C2×D7311yescharacteristic
203#537—1461no—
204#520—21no—
205C2×M32—731no—
206C22×C16—731no—
207#468—1462no—
208#467—732no—
209#466—732no—
210#428—21no—
211#428—22no—
212#427C2×D7311yescharacteristic
213#427C2×D7312yes—
214C2×#107—731no—
215C2×#107C2×D7311yescharacteristic
216C2×#107C2×D7312yes—
217#417—1461no—
218#367—732no—
219#366—732no—
220#366—732no—
221#366—732no—
222#358—731no—
223#358—731no—
224#358—732no—
225#357—731no—
226#357—731no—
227#357—732no—
228C22⋊C16—732no—
229C22⋊C16—732no—
230C2×#41—1461no—
231C2×#41C22⋊D7311yescharacteristic
232#135—1462no—
233#135—1462no—
234C2×Q16—731no—
235C2×SD16—1461no—
236C2×D8—731no—
237#131—22no—
238#131—22no—
239C2×M16C22⋊D7311yescharacteristic
240C2×M16C22⋊D7312yes—
241C22×C8C22×D7312yes—
242C22×C8C22×D7311yescharacteristic
243C22⋊Q8—2921no—
244C22⋊Q8—1462no—
245#121—2921no—
246#121—2922no—
247#121—1462no—
248C4×Q8—41no—
249C4×D4—42no—
250C4×D4—41no—
251#117—22no—
252#117—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—1461no—
256M32—1462no—
257C2×C16—731no—
258C2×C16—731no—
259C2×C16—732no—
260C2×C16—1462no—
261#107—22no—
262#107—21no—
263#107—732no—
264#107—1461no—
265#107C22×D7311yescharacteristic
266#107C22×D7312yes—
267#107C22×D7311yescharacteristic
268#106—22no—
269#106—21no—
270#103—1462no—
271#102—1462no—
272#41—22no—
273#41—22no—
274#41—2922no—
275#41—2922no—
276C2×Q8—1461no—
277C2×Q8#1192401311yescharacteristic
278C2×D4—2921no—
279C2×D4—2922no—
280C2×D4—1461no—
281C2×D4#1192401312yes—
282C2×D4#1192401311yescharacteristic
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—2921no—
286C22×C4—2921no—
287C22×C4C2×C22⋊D7312yes—
288C22×C4C2×C22⋊D7311yescharacteristic
289Q16—1461no—
290Q16—1461no—
291SD16—2922no—
292D8—1461no—
293D8—1461no—
294M16—22no—
295M16—24no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8C2×C22⋊D7312yes—
299C2×C8C2×C22⋊D7311yescharacteristic
300C2×C8#1192402812yes—
301C2×C8#1192402812yes—
302C2×C8C23×D7311yescharacteristic
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—2922no—
309C4⋊C4—2921no—
310C4⋊C4—1462no—
311C22⋊C4—42no—
312C22⋊C4—41no—
313C22⋊C4—2922no—
314C22⋊C4—2922no—
315C42—42no—
316C42—41no—
317C16—1461no—
318C16—1461no—
319C16—1462no—
320C23—2921no—
321C23C2×#1192401312yes—
322C23C2×#1192401311yescharacteristic
323Q8—22no—
324Q8—2921no—
325Q8—2921no—
326D4—24no—
327D4—22no—
328D4—2921no—
329D4—2921no—
330D4—5842no—
331D4—11682no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—161no—
338C2×C4—162no—
339C2×C4—2922no—
340C2×C4—2921no—
341C2×C4—5842no—
342C2×C4—11681no—
343C2×C4#1455657412yes—
344C2×C4C2×#1192401312yes—
345C2×C4C2×#1192401311yescharacteristic
346C2×C4#1455662212yes—
347C2×C4C22×C22⋊D7311yescharacteristic
348C8—22no—
349C8#1455657412yes—
350C8C2×#1192402811yescharacteristic
351C8C2×#1192402811yescharacteristic
352C22—21no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—5841no—
358C22—5841no—
359C22—5841no—
360C22#1798890012yes—
361C22C22×#1192401311yescharacteristic
362C4—22no—
363C4—82no—
364C4—81no—
365C4—81no—
366C4—5842no—
367C4—5841no—
368C4#1798890012yes—
369C4C2×#1455657411yescharacteristic
370C4C2×#1455662211yescharacteristic
371C2—22no—
372C2—42no—
373C2—11681no—
374C2C2×#1798890011yescharacteristic
375C2#2262307011yescharacteristic
376C2#2262308211yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2913C2
socleC22×C73C22×D8
prime radical#22619601C2
hypercenter#2913D73
hypocenterC73#33652
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ #2913

Complements

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

NH ≅ G/Ntype
2. #22612139C2semidirect
3. #22612133C2semidirect
4. #22612122C2semidirect
5. #22612111C2semidirect
6. #22611720C2semidirect
7. C2×#17988412C2semidirect
8. #22611641C2semidirect
9. #22611636C2semidirect
10. #22611545C2semidirect
11. C73×#2913C2semidirect
190. #2913D73semidirect
200. C73#33652semidirect

Maximal subgroups

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

subgroupindex
#226196012
#226121392
#226121332
#226121222
#226121112
#226117202
C2×#179884122
#226116412
#226116362
#226115452
C73×#29132
#3365273