Finite Group Databasebeta

← #33292094

Subgroups — #33292094

Order 18688.

2502 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989142C212yesmaximal
2#22618329C211yesmaximal, characteristic, complemented
3#22618308C212yesmaximal, complemented
4#22618292C214yesmaximal, complemented
5#22618284C214yesmaximal
6#22618278C214yesmaximal
7C4⋊Q8×Dic73C214yesmaximal, complemented
8#22612498C214yesmaximal, complemented
9#22612336C214yesmaximal
10C43×D73C211yesmaximal, characteristic
11C73×#2868C211yesmaximal, characteristic, complemented
12#17989142—216no—
13C2×#14556591C2214yes—
14C2×#14556591C2212yes—
15C2×#14556582C2214yes—
16C2×#14556571C2214yes—
17C2×#14556567C2214yes—
18C2×Q8×Dic73C2214yes—
19C2×#14556556C2214yes—
20C2×#14556549C2214yes—
21C2×C42×D73C2212yes—
22C2×C42×D73C2212yes—
23C2×C42×D73C2211yescharacteristic
24C2×C42×D73C2212yes—
25#17988494C2218yes—
26#17988492C2218yes—
27#17988486C22116yes—
28#17988483C22116yes—
29#17988457C2214yes—
30#17988457C2214yes—
31#17988452C2218yes—
32#17988444C22116yes—
33C4⋊C4×Dic73C22116yes—
34C42×Dic73C2212yes—
35C42×Dic73C2212yes—
36C42×Dic73C2212yes—
37C42×Dic73C2211yescharacteristic
38C2×C4⋊Q8×C73C2214yes—
39C2×C4⋊Q8×C73C2212yes—
40#384×C73C2214yes—
41#384×C73C2214yes—
42C43×C73C2211yescharacteristic
43C22×Dic146C2314yes—
44C22×C4×D73C2311yescharacteristic
45C22×C4×D73C2311yescharacteristic
46C22×C4×D73C2314yes—
47C22×C4×D73C2311yescharacteristic
48#14556591—28no—
49#14556591—216no—
50#14556582—216no—
51#14556571—216no—
52#14556567—216no—
53Q8×Dic73—216no—
54#14556556—216no—
55#14556549—216no—
56C2×C4⋊Dic73C23116yes—
57C2×C4⋊Dic73C2314yes—
58C2×C4⋊Dic73C2314yes—
59C2×C4⋊Dic73C2318yes—
60C2×#11924007C2318yes—
61C2×#11924005C2318yes—
62C2×C4×Dic73C2314yes—
63C2×C4×Dic73C2314yes—
64C2×C4×Dic73C2312yes—
65C2×C4×Dic73C2312yes—
66C2×C4×Dic73C2318yes—
67C2×C4×Dic73C2311yescharacteristic
68C2×C4×Dic73C2311yescharacteristic
69C2×C4×Dic73C2311yescharacteristic
70C2×C4×Dic73C2314yes—
71C2×C4×Dic73C2312yes—
72C42×D73—28no—
73C42×D73—28no—
74C42×D73D418yes—
75C42×D73Q8116yescomplemented
76#14556498C23132yes—
77C22×Q8×C73C2314yes—
78C4⋊Q8×C73—28no—
79C4⋊Q8×C73—216no—
80C2×C4⋊C4×C73C23116yes—
81C2×C4⋊C4×C73C2314yes—
82C2×C4⋊C4×C73C2314yes—
83C2×C42×C73C2311yescharacteristic
84C2×C42×C73C2312yes—
85C2×C42×C73C2312yes—
86C2×C42×C73C2312yes—
87C23×D73C2411yescharacteristic
88C2×Dic146—28no—
89C2×Dic146—216no—
90C22×Dic73C2418yes—
91C22×Dic73C2411yescharacteristic
92C22×Dic73C2411yescharacteristic
93C22×Dic73C2414yes—
94C22×Dic73C2411yescharacteristic
95C2×C4×D73—24no—
96C2×C4×D73—28no—
97C2×C4×D73—24no—
98C2×C4×D73—28no—
99C2×C4×D73C2×D414yes—
100C2×C4×D73C2×D414yes—
101C2×C4×D73C2×D414yes—
102C2×C4×D73C2×Q8116yes—
103C2×C4×D73C2×Q818yes—
104C4⋊Dic73—28no—
105C4⋊Dic73—28no—
106C4⋊Dic73—232no—
107C4⋊Dic73—216no—
108#11924007—216no—
109#11924005—216no—
110C4×Dic73—24no—
111C4×Dic73—28no—
112C4×Dic73—24no—
113C4×Dic73—28no—
114C4×Dic73—216no—
115C4×Dic73C2×D414yes—
116C4×Dic73C2×D414yes—
117C4×Dic73C2×D414yes—
118C4×Dic73C2×Q818yes—
119C4×Dic73C2×Q8116yes—
120C2×Q8×C73—28no—
121C2×Q8×C73—216no—
122C22×C4×C73C2418yes—
123C22×C4×C73C2411yescharacteristic
124C22×C4×C73C2411yescharacteristic
125C22×C4×C73C2411yescharacteristic
126C22×C4×C73C2414yes—
127C4⋊C4×C73—28no—
128C4⋊C4×C73—28no—
129C4⋊C4×C73—232no—
130C42×C73C2×D414yes—
131C42×C73C2×Q818yes—
132C42×C73#4118yes—
133C42×C73#4118yes—
134C23×C73C2511yescharacteristic
135C22×D73—22no—
136C22×D73—22no—
137C22×D73C22×D412yes—
138C22×D73C22×Q814yes—
139Q8×C73—416no—
140C2×C4×C73—28no—
141C2×C4×C73—216no—
142C2×C4×C73C22×D412yes—
143C2×C4×C73C22×D412yes—
144C2×C4×C73C22×D412yes—
145C2×C4×C73C22×Q814yes—
146C2×C4×C73C22×Q818yes—
147C2×C4×C73C2×#4114yes—
148C2×C4×C73C2×#4118yes—
149C2×C4×C73C2×#4114yes—
150C2×C4×C73C2×#4118yes—
151C2×Dic73—22no—
152C2×Dic73—24no—
153C2×Dic73—22no—
154C2×Dic73—24no—
155C2×Dic73—28no—
156C2×Dic73—216no—
157C2×Dic73C22×D412yes—
158C2×Dic73C22×D412yes—
159C2×Dic73C22×D412yes—
160C2×Dic73C22×Q818yes—
161C2×Dic73C22×Q814yes—
162C4×D73—28no—
163C4×D73—28no—
164C4×D73—232no—
165C4×D73C4⋊Q8116yescomplemented
166Dic146—416no—
167C22×C73C23×D411yescharacteristic
168C22×C73C23×Q812yes—
169C22×C73C22×#4112yes—
170C22×C73C22×#4112yes—
171C2×D73—22no—
172C2×D73—22no—
173C2×D73—28no—
174C2×D73C2×C4⋊Q814yes—
175C4×C73—48no—
176C4×C73C2×C4⋊Q814yes—
177C4×C73#53214yes—
178C4×C73#53214yes—
179C4×C73#533116yescomplemented
180Dic73—22no—
181Dic73—22no—
182Dic73—28no—
183Dic73—48no—
184Dic73C2×C4⋊Q814yes—
185#36645—731nomaximal, Sylow
186C2×C73C22×C4⋊Q811yescharacteristic
187C2×C73C2×#53211yescharacteristic
188C2×C73C2×#53211yescharacteristic
189C2×C73C2×#53314yes—
190D73—24no—
191C2×#533—732no—
192#2868—731no—
193#2868D7311yescharacteristic, complemented
194#2862—732no—
195#2859—734no—
196#2857—734no—
197#2857—734no—
198C4×C4⋊Q8—734no—
199C4×C4⋊Q8—734no—
200#2285—734no—
201C2×C43—731no—
202C73#3664511yescharacteristic, complemented, Sylow
203#533—14616no—
204C2×C4⋊Q8—734no—
205C2×C4⋊Q8—732no—
206C2×C4⋊Q8C2×D7314yes—
207C2×C4⋊Q8C2×D7312yes—
208C2×#125—734no—
209C2×C22⋊Q8—734no—
210C2×C22⋊Q8—734no—
211C2×C4×Q8—734no—
212C2×C4×Q8—734no—
213C2×#117—734no—
214C22×C42—731no—
215C22×C42—732no—
216C22×C42—732no—
217C22×C42—732no—
218#391—738no—
219#391—738no—
220#389—7316no—
221#389—7316no—
222#384—734no—
223#384—734no—
224#384C2×D7314yes—
225#384C2×D7314yes—
226#382—738no—
227#380—7316no—
228C4×C4⋊C4—7316no—
229C43—732no—
230C43—732no—
231C43—732no—
232C43—731no—
233C43C2×D7311yescharacteristic
234C22×Q8—734no—
235C22×Q8C22×D7314yes—
236C23×C4—731no—
237C23×C4—731no—
238C23×C4—731no—
239C23×C4—734no—
240C4⋊Q8—28no—
241C4⋊Q8—216no—
242C4⋊Q8—1468no—
243C4⋊Q8—14616no—
244#125—14616no—
245C22⋊Q8—14616no—
246C22⋊Q8—14616no—
247C4×Q8—14616no—
248C4×Q8—14616no—
249#117—14616no—
250C2×C4⋊C4—734no—
251C2×C4⋊C4—738no—
252C2×C4⋊C4—738no—
253C2×C4⋊C4—734no—
254C2×C4⋊C4—7316no—
255C2×C4⋊C4C22×D7314yes—
256C2×C4⋊C4C22×D7314yes—
257C2×C4⋊C4C22×D73116yes—
258C2×C22⋊C4—738no—
259C2×C42—731no—
260C2×C42—731no—
261C2×C42—731no—
262C2×C42—734no—
263C2×C42—732no—
264C2×C42—738no—
265C2×C42—7316no—
266C2×C42—738no—
267C2×C42—734no—
268C2×C42—734no—
269C2×C42—732no—
270C2×C42—732no—
271C2×C42—1468no—
272C2×C42—1468no—
273C2×C42C22×D7311yescharacteristic
274C2×C42C22×D7312yes—
275C2×C42C22×D7312yes—
276C2×C42C22×D7312yes—
277#95—7332no—
278C24—731no—
279C2×Q8—28no—
280C2×Q8—216no—
281C2×Q8—1468no—
282C2×Q8—14616no—
283C22×C4—731no—
284C22×C4—734no—
285C22×C4—738no—
286C22×C4—734no—
287C22×C4—731no—
288C22×C4—731no—
289C22×C4—734no—
290C22×C4—734no—
291C22×C4—7316no—
292C22×C4—738no—
293C22×C4—1464no—
294C22×C4—1468no—
295C22×C4—1464no—
296C22×C4—1468no—
297C22×C4C23×D7318yes—
298C22×C4C23×D7311yescharacteristic
299C22×C4C23×D7311yescharacteristic
300C22×C4C23×D7311yescharacteristic
301C22×C4C23×D7314yes—
302C4⋊C4—28no—
303C4⋊C4—28no—
304C4⋊C4—232no—
305C4⋊C4—14616no—
306C4⋊C4—14616no—
307C4⋊C4—1468no—
308C4⋊C4—1468no—
309C4⋊C4—14632no—
310C22⋊C4—14616no—
311C42—734no—
312C42—734no—
313C42—734no—
314C42—7316no—
315C42—738no—
316C42—14616no—
317C42—1464no—
318C42—1468no—
319C42—1464no—
320C42—1468no—
321C42D4×D7314yes—
322C42Q8×D7318yes—
323C42#1192402718yes—
324C42#1192402918yes—
325C23—732no—
326C23—734no—
327C23—1462no—
328C23—1462no—
329C23C24×D7311yescharacteristic
330Q8—416no—
331Q8—29216no—
332C2×C4—28no—
333C2×C4—216no—
334C2×C4—732no—
335C2×C4—734no—
336C2×C4—7316no—
337C2×C4—732no—
338C2×C4—732no—
339C2×C4—738no—
340C2×C4—1468no—
341C2×C4—14616no—
342C2×C4—1462no—
343C2×C4—1464no—
344C2×C4—1462no—
345C2×C4—1464no—
346C2×C4—1468no—
347C2×C4—1468no—
348C2×C4—14632no—
349C2×C4C2×D4×D7312yes—
350C2×C4C2×D4×D7312yes—
351C2×C4C2×D4×D7312yes—
352C2×C4C2×Q8×D7314yes—
353C2×C4C2×Q8×D7318yes—
354C2×C4C2×#1192402714yes—
355C2×C4C2×#1192402718yes—
356C2×C4C2×#1192402914yes—
357C2×C4C2×#1192402918yes—
358C22—734no—
359C22—1462no—
360C22—1462no—
361C22—1468no—
362C22C22×D4×D7311yescharacteristic
363C22C22×Q8×D7312yes—
364C22C22×#1192402712yes—
365C22C22×#1192402912yes—
366C4—48no—
367C4—2928no—
368C4—734no—
369C4—1462no—
370C4—1462no—
371C4—1468no—
372C4C4⋊Q8×D7314yes—
373C4#1798913514yes—
374C4#1798913814yes—
375C4#17989142116yescomplemented
376C2—1464no—
377C2C2×C4⋊Q8×D7311yescharacteristic
378C2C2×#1798913511yescharacteristic
379C2C2×#1798913811yescharacteristic
380C2C2×#1798914214yes—
subgroupG/H
center, FrattiniC23C24×D73
commutator, socle, cosocleC23×C73C25
FittingC73×#2868C2
prime radicalC23×D73C24
hypercenter#2868D73
hypocenterC73#36645
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ #2868

Complements

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

NH ≅ G/Ntype
2. #22618329C2semidirect
3. #22618308C2semidirect
4. #22618292C2semidirect
7. C4⋊Q8×Dic73C2semidirect
8. #22612498C2semidirect
11. C73×#2868C2semidirect
75. C42×D73Q8semidirect
165. C4×D73C4⋊Q8semidirect
179. C4×C73#533semidirect
193. #2868D73semidirect
202. C73#36645semidirect
375. C4#17989142semidirect

Maximal subgroups

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

subgroupindex
C2×#179891422
#226183292
#226183082
#226182922
#226182842
#226182782
C4⋊Q8×Dic732
#226124982
#226123362
C43×D732
C73×#28682
#3664573