Finite Group Databasebeta

← #33242911

Subgroups — #33242911

Order 18688.

546 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22622362C211yesmaximal, characteristic, complemented
2#22612158C211yesmaximal, characteristic, complemented
3#22612136C212yesmaximal, complemented
4#22612068C211yesmaximal, characteristic, complemented
5#22612020C212yesmaximal, complemented
6#22611978C211yesmaximal, characteristic, complemented
7#22611871C212yesmaximal, complemented
8C4×#14556535C211yesmaximal, characteristic, complemented
9D4×#11924003C211yesmaximal, characteristic, complemented
10#22611638C212yesmaximal, complemented
11C73×#3283C211yesmaximal, characteristic
12#17989307—21no—
13C2×#14556537C2212yes—
14C2×#14556535C2211yescharacteristic
15C22×#11924003C2212yes—
16#17988992C2211yescharacteristic
17#17988929—22no—
18#17988873—21no—
19#17988828C2212yes—
20C4×Dic292C2211yescharacteristic
21#17988625C2212yes—
22#17988414C2211yescharacteristic
23#17988412C2212yes—
24C4⋊#11924003C2211yescharacteristic
25#17988391C2212yes—
26#17988391C2212yes—
27#17988391C2212yes—
28#17988390C2211yescharacteristic
29#17988388C2211yescharacteristic
30#17988388C2212yes—
31C22⋊#11924003C2212yes—
32C4×#11924003C2211yescharacteristic
33C2×#135×C73C2212yes—
34#547×C73—21no—
35#494×C73C2211yescharacteristic
36#467×C73C2212yes—
37#461×C73—21no—
38#452×C73—22no—
39C4×D8×C73C2211yescharacteristic
40C8×D4×C73C2211yescharacteristic
41C2×Dic292C2311yescharacteristic
42#14556591—21no—
43#14556584—42no—
44#14556578—41no—
45#14556570—22no—
46C4×Dic146—21no—
47#14556556—41no—
48#14556550—22no—
49#14556537—42no—
50#14556535—22no—
51C2×#11924003—42no—
52C2×#11924003C2311yescharacteristic
53C2×#11924003C2311yescharacteristic
54C2×#11924003C2312yes—
55#14556525—21no—
56#14556525C2311yescharacteristic
57#14556525C2312yes—
58#14556523—22no—
59#14556516—22no—
60#14556516—21no—
61C2×#41×C73—22no—
62#135×C73—24no—
63#135×C73—42no—
64C2×Q16×C73—21no—
65C2×Q16×C73C2312yes—
66C2×SD16×C73—22no—
67C2×D8×C73C2311yescharacteristic
68C22×C8×C73C2312yes—
69C4⋊Q8×C73—21no—
70#123×C73—42no—
71C22⋊Q8×C73—22no—
72#121×C73—41no—
73C4×D4×C73—21no—
74C4×D4×C73D411yescharacteristic
75C2×C4⋊C4×C73—41no—
76#107×C73C2311yescharacteristic
77C4⋊C8×C73D411yescharacteristic
78#103×C73—22no—
79#102×C73—21no—
80C22⋊C8×C73D412yes—
81C4×C8×C73C2311yescharacteristic
82C2×Dic146—21no—
83C2×Dic146—41no—
84Dic292—21no—
85Dic292—21no—
86C4⋊Dic73—21no—
87C4⋊Dic73—22no—
88C4⋊Dic73—41no—
89#11924007—42no—
90#11924007—41no—
91C22⋊Dic73—42no—
92C22⋊Dic73—42no—
93C4×Dic73—42no—
94C4×Dic73—41no—
95#11924003—21no—
96#11924003—21no—
97#11924003—22no—
98#41×C73—44no—
99#41×C73—82no—
100C2×Q8×C73—22no—
101C2×D4×C73—21no—
102C2×D4×C73—42no—
103C2×D4×C73D811yescharacteristic
104C22×C4×C73—41no—
105C22×C4×C73—42no—
106C22×C4×C73C2×D412yes—
107Q16×C73—22no—
108Q16×C73—22no—
109Q16×C73—81no—
110SD16×C73—44no—
111D8×C73—21no—
112D8×C73#4112yes—
113C2×C8×C73—22no—
114C2×C8×C73—22no—
115C2×C8×C73C2×D411yescharacteristic
116C2×C8×C73C2×D412yes—
117C2×C8×C73C2411yescharacteristic
118C4⋊C4×C73—21no—
119C4⋊C4×C73—41no—
120C4⋊C4×C73—42no—
121C4⋊C4×C73—41no—
122C4⋊C4×C73D811yescharacteristic
123C22⋊C4×C73—42no—
124C22⋊C4×C73—41no—
125C22⋊C4×C73D812yes—
126C42×C73C2×D411yescharacteristic
127C23×C73—41no—
128C23×C73C2×D812yes—
129Q8×C73—42no—
130Q8×C73—42no—
131D4×C73—22no—
132D4×C73—22no—
133D4×C73—41no—
134D4×C73—84no—
135C2×C4×C73—22no—
136C2×C4×C73—22no—
137C2×C4×C73—41no—
138C2×C4×C73—42no—
139C2×C4×C73—84no—
140C2×C4×C73—82no—
141C2×C4×C73C2×D811yescharacteristic
142C2×C4×C73C2×D812yes—
143C2×C4×C73C2×D811yescharacteristic
144C2×C4×C73C22×D411yescharacteristic
145C2×Dic73—41no—
146C2×Dic73—41no—
147C2×Dic73—42no—
148Dic146—41no—
149Dic146—41no—
150Dic146—82no—
151C8×C73—22no—
152C8×C73—21no—
153C8×C73C2×#4111yescharacteristic
154C8×C73H(5,𝔽2)11yescharacteristic
155C22×C73—22no—
156C22×C73—22no—
157C22×C73—22no—
158C22×C73—42no—
159C22×C73—81no—
160C22×C73—81no—
161C22×C73C22×D811yescharacteristic
162C4×C73—21no—
163C4×C73—22no—
164C4×C73—81no—
165C4×C73—82no—
166C4×C73#50812yes—
167C4×C73#54611yescharacteristic
168C4×C73#57711yescharacteristic
169Dic73—82no—
170Dic73—81no—
171Dic73—81no—
172#33869—731nomaximal, Sylow
173C2×C73—42no—
174C2×C73—82no—
175C2×C73#329411yescharacteristic
176C2×C73C2×#50811yescharacteristic
177C2×C73#341711yescharacteristic
178#3342—731no—
179#3283D7311yescharacteristic
180#2241—731no—
181#2237—732no—
182#2224—731no—
183#2216—732no—
184#2209—731no—
185#2189—732no—
186C4×SD32—731no—
187D4×C16—731no—
188#2138—732no—
189C73#3386911yescharacteristic, complemented, Sylow
190C2×#135C2×D7312yes—
191#556—1461no—
192#547—21no—
193C2×Q32—732no—
194C2×SD32—731no—
195C22×C16—732no—
196#500—731no—
197#494C2×D7311yescharacteristic
198#485—1462no—
199#475—1461no—
200#467—732no—
201#467C2×D7312yes—
202#461—21no—
203#452—22no—
204C4×Q16—731no—
205C4×D8C2×D7311yescharacteristic
206C8×D4C2×D7311yescharacteristic
207#427—732no—
208#367—731no—
209#366—732no—
210C4⋊C16—731no—
211#358—732no—
212#358—732no—
213#358—732no—
214#358—731no—
215#357—731no—
216#357—732no—
217C22⋊C16—732no—
218C4×C16—731no—
219C2×#41—22no—
220#135—24no—
221#135—42no—
222C2×Q16—21no—
223C2×Q16—731no—
224C2×Q16C22×D7312yes—
225C2×SD16—22no—
226C2×D8C22×D7311yescharacteristic
227C22×C8C22×D7312yes—
228C4⋊Q8—21no—
229C4⋊Q8—1461no—
230#126—2922no—
231#124—2921no—
232#123—42no—
233C22⋊Q8—22no—
234C22⋊Q8—1462no—
235#121—41no—
236C4×Q8—2921no—
237C4×Q8—1461no—
238C4×D4—21no—
239C4×D4C22⋊D7311yescharacteristic
240#117—1462no—
241C2×C4⋊C4—41no—
242Q32—2922no—
243SD32—1462no—
244C2×C16—2922no—
245C2×C16—731no—
246C2×C16—731no—
247C2×C16—732no—
248#107—732no—
249#107—731no—
250#107—1461no—
251#107C22×D7311yescharacteristic
252#106—1462no—
253C4⋊C8C22⋊D7311yescharacteristic
254#103—22no—
255#103—1461no—
256#103—1462no—
257#102—21no—
258C22⋊C8C22⋊D7312yes—
259C4×C8C22×D7311yescharacteristic
260#41—44no—
261#41—82no—
262C2×Q8—22no—
263C2×Q8—2921no—
264C2×Q8—1461no—
265C2×D4—21no—
266C2×D4—42no—
267C2×D4#1192401311yescharacteristic
268C22×C4—41no—
269C22×C4—42no—
270C22×C4C2×C22⋊D7312yes—
271Q16—22no—
272Q16—22no—
273Q16—81no—
274Q16—1461no—
275Q16—1461no—
276SD16—44no—
277D8—21no—
278D8#1192402712yes—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8C2×C22⋊D7311yescharacteristic
282C2×C8C2×C22⋊D7312yes—
283C2×C8C23×D7311yescharacteristic
284C4⋊C4—21no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C4⋊C4—41no—
288C4⋊C4—2922no—
289C4⋊C4—2921no—
290C4⋊C4—2921no—
291C4⋊C4—1461no—
292C4⋊C4—1462no—
293C4⋊C4#1192401311yescharacteristic
294C22⋊C4—42no—
295C22⋊C4—41no—
296C22⋊C4—2922no—
297C22⋊C4—2922no—
298C22⋊C4#1192401312yes—
299C42—2922no—
300C42—2921no—
301C42C2×C22⋊D7311yescharacteristic
302C16—1461no—
303C16—1461no—
304C16—1462no—
305C23—41no—
306C23C2×#1192401312yes—
307Q8—42no—
308Q8—42no—
309Q8—2921no—
310Q8—2921no—
311Q8—5842no—
312D4—22no—
313D4—22no—
314D4—41no—
315D4—84no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—84no—
321C2×C4—82no—
322C2×C4—2921no—
323C2×C4—2921no—
324C2×C4—2922no—
325C2×C4C2×#1192401312yes—
326C2×C4C2×#1192401311yescharacteristic
327C2×C4C2×#1192401311yescharacteristic
328C2×C4C22×C22⋊D7311yescharacteristic
329C8—21no—
330C8—22no—
331C8C2×#1192402711yescharacteristic
332C8#1455663611yescharacteristic
333C22—22no—
334C22—22no—
335C22—22no—
336C22—42no—
337C22—81no—
338C22—81no—
339C22C22×#1192401311yescharacteristic
340C4—22no—
341C4—21no—
342C4—82no—
343C4—81no—
344C4—5841no—
345C4—5842no—
346C4—5841no—
347C4#1798901712yes—
348C4#1798923111yescharacteristic
349C4#1798939811yescharacteristic
350C2—42no—
351C2—82no—
352C2#2262174411yescharacteristic
353C2C2×#1798901711yescharacteristic
354C2#2262308211yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
Fitting, prime radicalC73×#3283C2
socleC22×C73C22×D8
hypercenter#3283D73
hypocenterC73#33869
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3283

Complements

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

NH ≅ G/Ntype
1. #22622362C2semidirect
2. #22612158C2semidirect
3. #22612136C2semidirect
4. #22612068C2semidirect
5. #22612020C2semidirect
6. #22611978C2semidirect
7. #22611871C2semidirect
8. C4×#14556535C2semidirect
9. D4×#11924003C2semidirect
10. #22611638C2semidirect
189. C73#33869semidirect

Maximal subgroups

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

subgroupindex
#226223622
#226121582
#226121362
#226120682
#226120202
#226119782
#226118712
C4×#145565352
D4×#119240032
#226116382
C73×#32832
#3386973