Finite Group Databasebeta

← #33240418

Subgroups — #33240418

Order 18688.

710 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989016C211yesmaximal, characteristic, complemented
2C22×C16×D73C211yesmaximal, characteristic, complemented
3#22619510C212yesmaximal, complemented
4#22612026C212yesmaximal, complemented
5#22612025C212yesmaximal, complemented
6#22612023C211yesmaximal, characteristic, complemented
7#22612022C211yesmaximal, characteristic, complemented
8#22612014C211yesmaximal, characteristic, complemented
9#22611712C211yesmaximal, characteristic, complemented
10#22611635C212yesmaximal, complemented
11C73×#2215C211yesmaximal, characteristic, complemented
12C2×#14556610C2212yes—
13C22×C8×D73C2211yescharacteristic
14#17989097—22no—
15#17989016—24no—
16C2×Dic584C2211yescharacteristic, complemented
17C2×#14556535C2212yescomplemented
18C22×#11924003C2211yescharacteristic
19C2×C16×D73—24no—
20C2×C16×D73C2211yescharacteristic
21C2×C16×D73C2211yescharacteristic
22C2×C16×D73C2211yescharacteristic
23C2×C16×D73C2211yescharacteristic
24#17988829C2212yes—
25#17988828C2212yes—
26#17988820C2212yes—
27#17988815—22no—
28#17988814—22no—
29#17988624C2212yes—
30#17988580—22no—
31#17988414C2212yescomplemented
32#17988413C2211yescharacteristic, complemented
33#17988392C2212yescomplemented
34#17988390C2212yescomplemented
35#17988388C2212yescomplemented
36C2×D16×C73C2211yescharacteristic, complemented
37C22×C16×C73C2211yescharacteristic
38#466×C73C2212yes—
39#366×C73C2211yescharacteristic, complemented
40#357×C73C2212yescomplemented
41C2×#11924027—22no—
42C22×C4×D73D411yescharacteristic
43#14556610—24no—
44#14556610—24no—
45C2×Dic292C2312yes—
46C2×#11924016—22no—
47C22×#10596831D411yescharacteristic
48C2×C8×D73—22no—
49C2×C8×D73—22no—
50C2×C8×D73D411yescharacteristic
51C2×C8×D73D411yescharacteristic
52C2×C8×D73C2311yescharacteristic
53C2×C8×D73C2311yescharacteristic
54#14556573—42no—
55#14556571—22no—
56#14556570—22no—
57#14556562—22no—
58#14556559—42no—
59D4×Dic73—42no—
60#14556551—42no—
61#14556549—22no—
62Dic584—22no—
63#14556535—24no—
64C2×#11924003—22no—
65C2×#11924003C2311yescharacteristic
66C2×#11924003C2311yescharacteristic
67C16×D73—22no—
68C16×D73—22no—
69C16×D73—22no—
70C16×D73D414yescomplemented
71#14556525C2312yes—
72#14556522—22no—
73#14556516—22no—
74#14556514—22no—
75#14556512—22no—
76C2×D8×C73C2312yes—
77C22×C8×C73C2311yescharacteristic
78#121×C73—22no—
79D16×C73—22no—
80C2×C16×C73—22no—
81C2×C16×C73C2311yescharacteristic
82C2×C16×C73C2311yescharacteristic
83#107×C73C2312yes—
84#102×C73—22no—
85C23×D73D811yescharacteristic
86#11924027—44no—
87#11924027—44no—
88C2×Dic146—22no—
89C2×C22⋊D73—42no—
90C2×C22⋊D73—42no—
91C22×Dic73—42no—
92C22×Dic73D811yescharacteristic
93C2×C4×D73—22no—
94C2×C4×D73—22no—
95C2×C4×D73D811yescharacteristic
96C2×C4×D73D811yescharacteristic
97C2×C4×D73C2×D411yescharacteristic
98C2×C4×D73C2×D411yescharacteristic
99Dic292—22no—
100Dic292—22no—
101#11924016—44no—
102C2×#10596831—22no—
103C2×#10596831C2×D411yescharacteristic
104C2×#10596831C2×D411yescharacteristic
105C8×D73—22no—
106C8×D73—21no—
107C8×D73—21no—
108C8×D73—22no—
109C8×D73C2×D412yes—
110C8×D73#4112yes—
111C4⋊Dic73—22no—
112C4⋊Dic73—42no—
113#11924007—42no—
114C22⋊Dic73—42no—
115C22⋊Dic73—42no—
116#11924005—42no—
117#11924005—42no—
118C4×Dic73—42no—
119#11924003—21no—
120#11924003C2×D412yes—
121C2×D4×C73—22no—
122C2×D4×C73—42no—
123C22×C4×C73C2×D411yescharacteristic
124D8×C73—22no—
125D8×C73—22no—
126C2×C8×C73—21no—
127C2×C8×C73—21no—
128C2×C8×C73C2×D411yescharacteristic
129C2×C8×C73C2411yescharacteristic
130C4⋊C4×C73—22no—
131C22⋊C4×C73—42no—
132C16×C73—21no—
133C16×C73C2×D412yes—
134C23×C73—42no—
135C23×C73C2×D811yescharacteristic
136C22×D73—22no—
137C22×D73—22no—
138C22×D73—22no—
139C22×D73C2×D811yescharacteristic
140C22×D73C2×D811yescharacteristic
141D4×C73—42no—
142D4×C73—42no—
143D4×C73—84no—
144C2×C4×C73—21no—
145C2×C4×C73—21no—
146C2×C4×C73—42no—
147C2×C4×C73C2×D811yescharacteristic
148C2×C4×C73C22×D411yescharacteristic
149C22⋊D73—84no—
150C22⋊D73—84no—
151C2×Dic73—22no—
152C2×Dic73—42no—
153C2×Dic73—42no—
154C2×Dic73—84no—
155C2×Dic73C2×D811yescharacteristic
156C2×Dic73C2×D811yescharacteristic
157C4×D73—22no—
158C4×D73—21no—
159C4×D73—21no—
160C4×D73—22no—
161C4×D73#12112yes—
162C4×D73#13512yes—
163Dic146—42no—
164Dic146—42no—
165C8×C73—21no—
166C8×C73C22×D411yescharacteristic
167C8×C73C2×#4111yescharacteristic
168#10596831—21no—
169#10596831#12112yes—
170C22×C73—21no—
171C22×C73—21no—
172C22×C73—82no—
173C22×C73—82no—
174C22×C73—82no—
175C22×C73#50812yescomplemented
176C22×C73C22×D811yescharacteristic
177C2×D73—28no—
178C2×D73—21no—
179C2×D73—21no—
180C2×D73—21no—
181C2×D73—21no—
182C2×D73#46612yes—
183C2×D73#50812yescomplemented
184C4×C73—21no—
185C4×C73—82no—
186C4×C73C2×#12111yescharacteristic
187C4×C73C2×#13511yescharacteristic
188Dic73—21no—
189Dic73—82no—
190Dic73—82no—
191Dic73#46612yes—
192#33600—731nomaximal, Sylow
193C2×C73—22no—
194C2×C73—162no—
195C2×C73C2×#46611yescharacteristic
196C2×C73C2×#50811yescharacteristic
197C2×C73C2×#50811yescharacteristic
198D73—22no—
199D73—22no—
200C2×#508—731no—
201C23×C16—731no—
202#3050—732no—
203#2217—732no—
204#2217—732no—
205#2216—731no—
206#2216—731no—
207#2215—731no—
208#2215D7311yescharacteristic, complemented
209#2158—731no—
210#2138—732no—
211C73#3360011yescharacteristic, complemented, Sylow
212C2×#135—732no—
213C23×C8—731no—
214#525—1462no—
215#508—1464no—
216C2×Q32—731no—
217C2×SD32—732no—
218C2×D16C2×D7311yescharacteristic, complemented
219C22×C16—731no—
220C22×C16—731no—
221C22×C16—731no—
222C22×C16—731no—
223C22×C16—731no—
224C22×C16—1464no—
225C22×C16C2×D7311yescharacteristic
226#467—732no—
227#467—732no—
228#466—732no—
229#466C2×D7312yes—
230#465—1462no—
231#465—1462no—
232#427—732no—
233#416—1462no—
234#367—732no—
235#366—731no—
236#366C2×D7311yescharacteristic, complemented
237#358—732no—
238#358—732no—
239#357—732no—
240#357C2×D7312yescomplemented
241C2×#41—1462no—
242C23×C4—731no—
243#135—1464no—
244#135—1464no—
245C2×Q16—732no—
246C2×SD16—1462no—
247C2×D8C22×D7312yes—
248C22×C8—731no—
249C22×C8—731no—
250C22×C8—731no—
251C22×C8—731no—
252C22×C8—731no—
253C22×C8—1462no—
254C22×C8—1462no—
255C22×C8C22×D7311yescharacteristic
256#123—2922no—
257C22⋊Q8—1462no—
258C22⋊Q8—1462no—
259#121—22no—
260#121—1462no—
261#120—2922no—
262C4×D4—2922no—
263C4×D4—2922no—
264#117—1462no—
265Q32—1462no—
266SD32—1464no—
267D16—22no—
268C2×C16—22no—
269C2×C16—731no—
270C2×C16—734no—
271C2×C16—731no—
272C2×C16—1462no—
273C2×C16—1462no—
274C2×C16—1462no—
275C2×C16—1462no—
276C2×C16C22×D7311yescharacteristic
277C2×C16C22×D7311yescharacteristic
278#107—732no—
279#107C22×D7312yes—
280#106—1462no—
281#103—1462no—
282#103—1462no—
283#102—22no—
284#102—1462no—
285C24—731no—
286#41—2924no—
287#41—2924no—
288C2×Q8—1462no—
289C2×D4—22no—
290C2×D4—42no—
291C2×D4—2922no—
292C2×D4—2922no—
293C22×C4—2922no—
294C22×C4—731no—
295C22×C4—731no—
296C22×C4—731no—
297C22×C4—731no—
298C22×C4—731no—
299C22×C4—1462no—
300C22×C4—1462no—
301C22×C4D4×D7311yescharacteristic
302Q16—1462no—
303Q16—1462no—
304SD16—2924no—
305D8—22no—
306D8—22no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—731no—
310C2×C8—732no—
311C2×C8—732no—
312C2×C8—731no—
313C2×C8—1462no—
314C2×C8—1462no—
315C2×C8—1462no—
316C2×C8—1461no—
317C2×C8—1461no—
318C2×C8D4×D7311yescharacteristic
319C2×C8C23×D7311yescharacteristic
320C4⋊C4—22no—
321C4⋊C4—2922no—
322C4⋊C4—2922no—
323C4⋊C4—1462no—
324C22⋊C4—42no—
325C22⋊C4—2922no—
326C22⋊C4—2922no—
327C22⋊C4—2922no—
328C22⋊C4—2922no—
329C42—2922no—
330C16—21no—
331C16—732no—
332C16—1461no—
333C16D4×D7312yes—
334C23—42no—
335C23—731no—
336C23—731no—
337C23—1462no—
338C23—1462no—
339C23—1462no—
340C23D8×D7311yescharacteristic
341Q8—2922no—
342Q8—2922no—
343D4—42no—
344D4—42no—
345D4—84no—
346D4—5844no—
347D4—5844no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—42no—
351C2×C4—2922no—
352C2×C4—2922no—
353C2×C4—5844no—
354C2×C4—731no—
355C2×C4—732no—
356C2×C4—732no—
357C2×C4—731no—
358C2×C4—1462no—
359C2×C4—1462no—
360C2×C4—1462no—
361C2×C4—1461no—
362C2×C4—1461no—
363C2×C4D8×D7311yescharacteristic
364C2×C4C2×D4×D7311yescharacteristic
365C8—21no—
366C8—732no—
367C8—1461no—
368C8C2×D4×D7311yescharacteristic
369C8#41×D7311yescharacteristic
370C22—21no—
371C22—21no—
372C22—82no—
373C22—82no—
374C22—82no—
375C22—732no—
376C22—732no—
377C22—1468no—
378C22—1461no—
379C22—1461no—
380C22—1461no—
381C22—1461no—
382C22#1798901612yescomplemented
383C22C2×D8×D7311yescharacteristic
384C4—21no—
385C4—82no—
386C4—5842no—
387C4—5842no—
388C4—732no—
389C4—1461no—
390C4#121×D7311yescharacteristic
391C4#135×D7311yescharacteristic
392C2—22no—
393C2—162no—
394C2—1462no—
395C2—1462no—
396C2#466×D7311yescharacteristic
397C2#508×D7311yescharacteristic
398C2C2×#1798901611yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2215C2
socleC22×C73C22×D8
hypercenter#2215D73
hypocenterC73#33600
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2215

Complements

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

NH ≅ G/Ntype
1. C2×#17989016C2semidirect
2. C22×C16×D73C2semidirect
3. #22619510C2semidirect
4. #22612026C2semidirect
5. #22612025C2semidirect
6. #22612023C2semidirect
7. #22612022C2semidirect
8. #22612014C2semidirect
9. #22611712C2semidirect
10. #22611635C2semidirect
11. C73×#2215C2semidirect
16. C2×Dic584C22semidirect
17. C2×#14556535C22semidirect
31. #17988414C22semidirect
32. #17988413C22semidirect
33. #17988392C22semidirect
34. #17988390C22semidirect
35. #17988388C22semidirect
36. C2×D16×C73C22semidirect
39. #366×C73C22semidirect
40. #357×C73C22semidirect
70. C16×D73D4semidirect
175. C22×C73#508semidirect
183. C2×D73#508semidirect
208. #2215D73semidirect
211. C73#33600semidirect
218. C2×D16C2×D73semidirect
236. #366C2×D73semidirect
240. #357C2×D73semidirect
382. C22#17989016semidirect

Maximal subgroups

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

subgroupindex
C2×#179890162
C22×C16×D732
#226195102
#226120262
#226120252
#226120232
#226120222
#226120142
#226117122
#226116352
C73×#22152
#3360073