Finite Group Databasebeta

← #33240844

Subgroups — #33240844

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#22618531C211yesmaximal, characteristic, complemented
2#22612136C212yesmaximal, complemented
3#22612131C212yesmaximal, complemented
4#22612119C212yesmaximal, complemented
5#22612109C212yesmaximal, complemented
6#22611717C211yesmaximal, characteristic, complemented
7C2×#17988412C211yesmaximal, characteristic, complemented
8#22611643C211yesmaximal, characteristic, complemented
9#22611638C211yesmaximal, characteristic, complemented
10#22611545C211yesmaximal, characteristic, complemented
11C73×#3061C211yesmaximal, characteristic
12#17989070—21no—
13C2×#14556531C2211yescharacteristic
14C22×#11924003C2211yescharacteristic
15#17988629—21no—
16#17988629—22no—
17#17988625C2211yescharacteristic
18#17988625C2212yes—
19C2×#14556525C2212yes—
20C2×#14556525C2211yescharacteristic
21#17988414C2212yes—
22#17988412C2212yes—
23#17988412C2212yes—
24#17988412C2212yes—
25#17988391C2211yescharacteristic
26#17988391C2212yes—
27#17988391C2211yescharacteristic
28#17988388C2211yescharacteristic
29#17988388C2212yes—
30#17988388C2211yescharacteristic
31C22⋊#11924003C2212yes—
32C22⋊#11924003C2212yes—
33C2×#135×C73C2211yescharacteristic
34C2×#131×C73C2211yescharacteristic
35#537×C73—21no—
36#468×C73—22no—
37#467×C73C2212yes—
38#466×C73C2212yes—
39C2×#107×C73C2211yescharacteristic
40#417×C73—21no—
41Q8×Dic73—41no—
42D4×Dic73—42no—
43D4×Dic73—41no—
44#14556550—22no—
45#14556550—21no—
46C2×C4⋊Dic73—22no—
47C2×C4⋊Dic73—21no—
48#14556531—22no—
49C2×#11924003—22no—
50C2×#11924003C2311yescharacteristic
51C2×#11924003C2311yescharacteristic
52C2×#11924003C2312yes—
53#14556525—22no—
54#14556525—21no—
55#14556525C2311yescharacteristic
56#14556525C2312yes—
57#14556525C2311yescharacteristic
58#14556523—22no—
59#14556523—21no—
60C2×#41×C73—21no—
61C2×#41×C73D411yescharacteristic
62#135×C73—22no—
63#135×C73—22no—
64C2×Q16×C73C2311yescharacteristic
65C2×SD16×C73—21no—
66C2×D8×C73C2311yescharacteristic
67#131×C73—22no—
68#131×C73—22no—
69C2×M16×C73D411yescharacteristic
70C2×M16×C73D412yes—
71C22×C8×C73C2312yes—
72C22×C8×C73C2311yescharacteristic
73C22⋊Q8×C73—22no—
74C22⋊Q8×C73—41no—
75#121×C73—22no—
76#121×C73—41no—
77#121×C73—42no—
78C2×C4⋊C4×C73—21no—
79#107×C73—21no—
80#107×C73C2312yes—
81#103×C73—22no—
82#102×C73—22no—
83C22×Dic73—42no—
84C22×Dic73—41no—
85C4⋊Dic73—21no—
86C4⋊Dic73—22no—
87C4⋊Dic73—21no—
88C4⋊Dic73—42no—
89C4⋊Dic73—41no—
90C22⋊Dic73—42no—
91C22⋊Dic73—41no—
92C4×Dic73—42no—
93C4×Dic73—41no—
94#11924003—21no—
95#11924003—21no—
96#11924003—22no—
97#41×C73—22no—
98#41×C73—22no—
99#41×C73—42no—
100#41×C73—42no—
101C2×Q8×C73—21no—
102C2×Q8×C73D811yescharacteristic
103C2×D4×C73—21no—
104C2×D4×C73—41no—
105C2×D4×C73—42no—
106C2×D4×C73D812yes—
107C2×D4×C73D811yescharacteristic
108C22×C4×C73—41no—
109C22×C4×C73—41no—
110C22×C4×C73C2×D412yes—
111C22×C4×C73C2×D411yescharacteristic
112Q16×C73—21no—
113Q16×C73—21no—
114SD16×C73—42no—
115D8×C73—21no—
116D8×C73—21no—
117M16×C73—22no—
118M16×C73—24no—
119C2×C8×C73—22no—
120C2×C8×C73—22no—
121C2×C8×C73C2×D412yes—
122C2×C8×C73C2×D411yescharacteristic
123C2×C8×C73#4112yes—
124C2×C8×C73#4112yes—
125C2×C8×C73C2411yescharacteristic
126C4⋊C4×C73—22no—
127C4⋊C4×C73—41no—
128C4⋊C4×C73—42no—
129C22⋊C4×C73—42no—
130C22⋊C4×C73—42no—
131C23×C73—41no—
132C23×C73C2×D812yes—
133C23×C73C2×D811yescharacteristic
134Q8×C73—22no—
135Q8×C73—41no—
136Q8×C73—41no—
137D4×C73—24no—
138D4×C73—22no—
139D4×C73—41no—
140D4×C73—41no—
141D4×C73—82no—
142D4×C73—162no—
143C2×C4×C73—22no—
144C2×C4×C73—22no—
145C2×C4×C73—42no—
146C2×C4×C73—41no—
147C2×C4×C73—82no—
148C2×C4×C73—161no—
149C2×C4×C73#12312yes—
150C2×C4×C73C2×D812yes—
151C2×C4×C73C2×D811yescharacteristic
152C2×C4×C73#13712yes—
153C2×C4×C73C22×D411yescharacteristic
154C2×Dic73—41no—
155C2×Dic73—42no—
156C2×Dic73—41no—
157C2×Dic73—161no—
158C2×Dic73—162no—
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—81no—
169C22×C73—81no—
170C22×C73—81no—
171C22×C73#48012yes—
172C22×C73C22×D811yescharacteristic
173C4×C73—22no—
174C4×C73—81no—
175C4×C73—82no—
176C4×C73#48012yes—
177C4×C73C2×#12311yescharacteristic
178C4×C73C2×#13711yescharacteristic
179Dic73—82no—
180Dic73—81no—
181Dic73—81no—
182#33652—731nomaximal, Sylow
183C2×C73—22no—
184C2×C73—42no—
185C2×C73—161no—
186C2×C73C2×#48011yescharacteristic
187C2×C73#341511yescharacteristic
188C2×C73#341711yescharacteristic
189#3061D7311yescharacteristic
190#2913—731no—
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×#135C2×D7311yescharacteristic
202C2×#131C2×D7311yescharacteristic
203#537—21no—
204#520—1461no—
205C2×M32—731no—
206C22×C16—731no—
207#468—22no—
208#467C2×D7312yes—
209#466C2×D7312yes—
210#428—1461no—
211#428—1462no—
212#427—731no—
213#427—732no—
214C2×#107—731no—
215C2×#107—732no—
216C2×#107C2×D7311yescharacteristic
217#417—21no—
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—21no—
231C2×#41C22⋊D7311yescharacteristic
232#135—22no—
233#135—22no—
234C2×Q16C22×D7311yescharacteristic
235C2×SD16—21no—
236C2×D8C22×D7311yescharacteristic
237#131—22no—
238#131—22no—
239C2×M16C22⋊D7311yescharacteristic
240C2×M16C22⋊D7312yes—
241C22×C8C22×D7312yes—
242C22×C8C22×D7311yescharacteristic
243C22⋊Q8—22no—
244C22⋊Q8—41no—
245#121—22no—
246#121—41no—
247#121—42no—
248C4×Q8—2921no—
249C4×D4—2922no—
250C4×D4—2921no—
251#117—1462no—
252#117—1461no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—1462no—
255C2×C4⋊C4—1461no—
256M32—1462no—
257C2×C16—731no—
258C2×C16—731no—
259C2×C16—732no—
260C2×C16—1462no—
261#107—21no—
262#107—731no—
263#107—732no—
264#107—731no—
265#107—1462no—
266#107—1461no—
267#107C22×D7312yes—
268#106—1462no—
269#106—1461no—
270#103—22no—
271#102—22no—
272#41—22no—
273#41—22no—
274#41—42no—
275#41—42no—
276C2×Q8—21no—
277C2×Q8#1192401311yescharacteristic
278C2×D4—21no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4#1192401312yes—
282C2×D4#1192401311yescharacteristic
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—2922no—
286C22×C4—2921no—
287C22×C4C2×C22⋊D7312yes—
288C22×C4C2×C22⋊D7311yescharacteristic
289Q16—21no—
290Q16—21no—
291SD16—42no—
292D8—21no—
293D8—21no—
294M16—22no—
295M16—24no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8C2×C22⋊D7312yes—
299C2×C8C2×C22⋊D7311yescharacteristic
300C2×C8#1192402712yes—
301C2×C8#1192402712yes—
302C2×C8C23×D7311yescharacteristic
303C4⋊C4—22no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—2922no—
307C4⋊C4—2921no—
308C4⋊C4—1461no—
309C4⋊C4—1462no—
310C4⋊C4—1461no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—2922no—
314C22⋊C4—2921no—
315C42—2922no—
316C42—2921no—
317C16—1461no—
318C16—1461no—
319C16—1462no—
320C23—41no—
321C23C2×#1192401312yes—
322C23C2×#1192401311yescharacteristic
323Q8—22no—
324Q8—41no—
325Q8—41no—
326D4—24no—
327D4—22no—
328D4—41no—
329D4—41no—
330D4—82no—
331D4—162no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—82no—
337C2×C4—161no—
338C2×C4—2921no—
339C2×C4—2922no—
340C2×C4—2921no—
341C2×C4—11681no—
342C2×C4—11682no—
343C2×C4#1455657212yes—
344C2×C4C2×#1192401312yes—
345C2×C4C2×#1192401311yescharacteristic
346C2×C4#1455662112yes—
347C2×C4C22×C22⋊D7311yescharacteristic
348C8—22no—
349C8#1455657212yes—
350C8C2×#1192402711yescharacteristic
351C8C2×#1192402711yescharacteristic
352C22—21no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22#1798889812yes—
361C22C22×#1192401311yescharacteristic
362C4—22no—
363C4—82no—
364C4—81no—
365C4—5842no—
366C4—5841no—
367C4—5841no—
368C4#1798889812yes—
369C4C2×#1455657211yescharacteristic
370C4C2×#1455662111yescharacteristic
371C2—22no—
372C2—42no—
373C2—161no—
374C2C2×#1798889811yescharacteristic
375C2#2262307011yescharacteristic
376C2#2262308211yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
Fitting, prime radicalC73×#3061C2
socleC22×C73C22×D8
hypercenter#3061D73
hypocenterC73#33652
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ #3061

Complements

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

NH ≅ G/Ntype
1. #22618531C2semidirect
2. #22612136C2semidirect
3. #22612131C2semidirect
4. #22612119C2semidirect
5. #22612109C2semidirect
6. #22611717C2semidirect
7. C2×#17988412C2semidirect
8. #22611643C2semidirect
9. #22611638C2semidirect
10. #22611545C2semidirect
200. C73#33652semidirect

Maximal subgroups

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

subgroupindex
#226185312
#226121362
#226121312
#226121192
#226121092
#226117172
C2×#179884122
#226116432
#226116382
#226115452
C73×#30612
#3365273