Finite Group Databasebeta

← #33069129

Subgroups — #33069129

Order 18688.

662 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
1#22621052C211yesmaximal, characteristic
2#22620856C211yesmaximal, characteristic
3#22618644C211yesmaximal, characteristic
4#22609475C214yesmaximal, complemented
5#22609351C214yesmaximal, complemented
6#22608679C211yesmaximal, characteristic, complemented
7#22608616C211yesmaximal, characteristic, complemented
8#22608603C211yesmaximal, characteristic, complemented
9C2×#487×C73C211yesmaximal, characteristic, complemented
10C2×#14556597—21no—
11#17989137C2211yescharacteristic
12#17988990—21no—
13#17988961—21no—
14#17988937—21no—
15#17988935—21no—
16#17988900—22no—
17#17988864C2214yes—
18#17988789C2214yes—
19#17988726—22no—
20#17988666—21no—
21#17988666C2212yes—
22C8×D292—21no—
23C8×D292C2212yes—
24#17988640C2211yescharacteristic
25#17988640C2211yescharacteristic
26#17988611C2211yescharacteristic
27#17988553—21no—
28#17988340C2212yes—
29#17988338C2212yes—
30#17988333C2214yes—
31#17988333C2214yes—
32C2×C4⋊Q8×C73C2211yescharacteristic
33C2×C4⋊Q8×C73C2211yescharacteristic
34#487×C73—21no—
35#487×C73—21no—
36#487×C73C2212yes—
37#487×C73C2212yes—
38C2×#103×C73—21no—
39C2×#103×C73—21no—
40C2×C4×C8×C73C2211yescharacteristic
41C2×#11924028—21no—
42C2×#11924015—24no—
43#14556597—44no—
44C2×#11924012—21no—
45C2×#11924011—22no—
46C2×C8×D73—22no—
47#14556591D411yescharacteristic
48#14556588D411yescharacteristic
49#14556579—21no—
50#14556565—21no—
51#14556565D412yes—
52C4×D292—21no—
53C4×D292C2312yes—
54#14556524—24no—
55C4⋊#10596831—21no—
56C4⋊#10596831C2312yes—
57C4⋊#10596831C2312yes—
58C4⋊#10596831C2312yes—
59#14556515—24no—
60#14556511—24no—
61C22⋊#10596831—21no—
62C22⋊#10596831—21no—
63#14556503—22no—
64#14556503—22no—
65C4×#10596831—21no—
66C4×#10596831—21no—
67C22×Q8×C73—21no—
68C22×Q8×C73—21no—
69C22×C8×C73D411yescharacteristic
70C22×C8×C73D411yescharacteristic
71C4⋊Q8×C73—21no—
72C4⋊Q8×C73—21no—
73C4⋊Q8×C73—21no—
74C4⋊Q8×C73—21no—
75C4⋊Q8×C73—21no—
76C4⋊Q8×C73—21no—
77C4⋊Q8×C73C2312yes—
78C4⋊Q8×C73C2312yes—
79C2×C4⋊C4×C73—21no—
80C2×C4⋊C4×C73—21no—
81C2×C4⋊C4×C73—21no—
82C2×C4⋊C4×C73—21no—
83C2×C42×C73C2311yescharacteristic
84#103×C73—22no—
85#103×C73—22no—
86#103×C73—22no—
87#103×C73—22no—
88C4×C8×C73D411yescharacteristic
89C4×C8×C73D411yescharacteristic
90C4×C8×C73C2311yescharacteristic
91C4×C8×C73C2311yescharacteristic
92#11924028—42no—
93#11924028—42no—
94C2×Dic146—21no—
95C2×D292—21no—
96C2×D292—21no—
97C2×D292C2×D412yes—
98C2×C22⋊D73—22no—
99C2×C4×D73—22no—
100#11924015—84no—
101#11924012—42no—
102#11924011—44no—
103C2×#10596831—22no—
104C2×#10596831—22no—
105C2×#10596831—22no—
106C8×D73—44no—
107C4⋊Dic73—21no—
108C4⋊Dic73C2×D412yes—
109#11924005—22no—
110#11924005—22no—
111C2×Q8×C73—22no—
112C2×Q8×C73—22no—
113C2×Q8×C73—41no—
114C2×Q8×C73—41no—
115C2×Q8×C73—42no—
116C2×Q8×C73—42no—
117C2×Q8×C73—82no—
118C22×C4×C73—41no—
119C22×C4×C73—41no—
120C22×C4×C73C2×D411yescharacteristic
121C22×C4×C73C2×D411yescharacteristic
122C22×C4×C73#4111yescharacteristic
123C2×C8×C73—22no—
124C2×C8×C73—22no—
125C2×C8×C73C2×D411yescharacteristic
126C2×C8×C73C2×D411yescharacteristic
127C2×C8×C73C2×D411yescharacteristic
128C2×C8×C73C2×D411yescharacteristic
129C4⋊C4×C73—22no—
130C4⋊C4×C73—22no—
131C4⋊C4×C73—22no—
132C4⋊C4×C73—22no—
133C4⋊C4×C73—41no—
134C4⋊C4×C73—41no—
135C4⋊C4×C73—41no—
136C4⋊C4×C73—41no—
137C42×C73C2×D411yescharacteristic
138C42×C73C2×D411yescharacteristic
139C42×C73#4111yescharacteristic
140C42×C73C2411yescharacteristic
141C23×C73#12111yescharacteristic
142C22×D73—22no—
143Q8×C73—42no—
144Q8×C73—42no—
145Q8×C73—44no—
146Q8×C73—82no—
147Q8×C73—82no—
148C2×C4×C73—22no—
149C2×C4×C73—22no—
150C2×C4×C73—42no—
151C2×C4×C73—42no—
152C2×C4×C73—82no—
153C2×C4×C73—82no—
154C2×C4×C73#12111yescharacteristic
155C2×C4×C73#12111yescharacteristic
156C2×C4×C73#12111yescharacteristic
157C2×C4×C73#13512yes—
158C2×C4×C73AGL(1,ℤ/8)12yes—
159C2×C4×C73C22×D411yescharacteristic
160C2×C4×C73C22×D411yescharacteristic
161C2×C4×C73C2×#4111yescharacteristic
162C22⋊D73—44no—
163C2×Dic73—22no—
164D292—24no—
165D292—42no—
166D292—42no—
167C4×D73—42no—
168C4×D73—42no—
169Dic146—42no—
170C8×C73—21no—
171C8×C73—21no—
172C8×C73—21no—
173C8×C73—21no—
174#10596831—42no—
175#10596831—42no—
176#10596831—42no—
177C22×C73—21no—
178C22×C73—21no—
179C22×C73#46312yes—
180C22×C73C2×#12111yescharacteristic
181C2×D73—42no—
182C2×D73—42no—
183C2×D73—42no—
184C4×C73—21no—
185C4×C73—21no—
186C4×C73—82no—
187C4×C73—82no—
188C4×C73#46312yes—
189C4×C73#47312yes—
190C4×C73C2×#13511yescharacteristic
191C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
192C4×C73#57712yes—
193Dic73—42no—
194#19197—731nomaximal, Sylow
195C2×C73—22no—
196C2×C73C2×#46311yescharacteristic
197C2×C73#305711yescharacteristic
198C2×C73C2×#47311yescharacteristic
199D73—82no—
200#3226—731no—
201#3206—731no—
202C2×#487D7311yescharacteristic, complemented
203#2931—731no—
204#1699—734no—
205#1679—734no—
206#1564—731no—
207#1549—731no—
208#1546—731no—
209C73#1919711yescharacteristic, complemented, Sylow
210C2×#131—1461no—
211#532—731no—
212C2×C4⋊Q8C2×D7311yescharacteristic
213C2×C4⋊Q8C2×D7311yescharacteristic
214#500—1461no—
215#492—1461no—
216#487—21no—
217#487—21no—
218#487—1461no—
219#487C2×D7312yes—
220#487C2×D7312yes—
221#486—1461no—
222#480—1462no—
223#474—734no—
224#461—734no—
225#448—1462no—
226#436—732no—
227#436—1461no—
228C8×D4—732no—
229C8×D4—1461no—
230#431—731no—
231#431—731no—
232#423—731no—
233C2×#103—21no—
234C2×#103—21no—
235#408—1461no—
236C2×C4×C8C2×D7311yescharacteristic
237#335—732no—
238#334—732no—
239#332—734no—
240#332—734no—
241C2×#41—1461no—
242C22×Q8—21no—
243C22×Q8—21no—
244C2×SD16—1464no—
245#131—2924no—
246C2×M16—1461no—
247C2×M16—1462no—
248C22×C8—1462no—
249C22×C8C22⋊D7311yescharacteristic
250C22×C8C22⋊D7311yescharacteristic
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—731no—
258C4⋊Q8C22×D7312yes—
259C4⋊Q8C22×D7312yes—
260#127—731no—
261#124—1461no—
262#121—732no—
263#121—1461no—
264C4×D4—732no—
265C4×D4—1461no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C42C22×D7311yescharacteristic
271#107—1464no—
272C4⋊C8—732no—
273C4⋊C8—732no—
274C4⋊C8—732no—
275C4⋊C8—1461no—
276#103—22no—
277#103—22no—
278#103—22no—
279#103—22no—
280#103—1464no—
281#102—1464no—
282C22⋊C8—1461no—
283C22⋊C8—1461no—
284C22⋊C8—1462no—
285C22⋊C8—1462no—
286C4×C8—1461no—
287C4×C8—1461no—
288C4×C8C22⋊D7311yescharacteristic
289C4×C8C22⋊D7311yescharacteristic
290C4×C8C22×D7311yescharacteristic
291C4×C8C22×D7311yescharacteristic
292#41—2922no—
293#41—2922no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—82no—
301C2×Q8—1461no—
302C2×D4—732no—
303C2×D4—1462no—
304C2×D4—1461no—
305C2×D4—1461no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—1462no—
309C22×C4D4×D7311yescharacteristic
310C22×C4C2×C22⋊D7311yescharacteristic
311C22×C4#1192402711yescharacteristic
312SD16—5844no—
313M16—2924no—
314M16—2922no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—2924no—
318C2×C8—1462no—
319C2×C8—1462no—
320C2×C8—1462no—
321C2×C8C2×C22⋊D7311yescharacteristic
322C2×C8C2×C22⋊D7311yescharacteristic
323C2×C8C2×C22⋊D7311yescharacteristic
324C2×C8C2×C22⋊D7311yescharacteristic
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—732no—
334C4⋊C4—1461no—
335C22⋊C4—1462no—
336C22⋊C4—1462no—
337C42D4×D7311yescharacteristic
338C42C2×C22⋊D7311yescharacteristic
339C42#1192402711yescharacteristic
340C42C23×D7311yescharacteristic
341C23—1462no—
342C23#1455656211yescharacteristic
343Q8—44no—
344Q8—42no—
345Q8—42no—
346Q8—82no—
347Q8—82no—
348Q8—2922no—
349D4—2924no—
350D4—2922no—
351D4—2922no—
352D4—1464no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—2922no—
360C2×C4—2922no—
361C2×C4—1462no—
362C2×C4#1455656211yescharacteristic
363C2×C4#1455656211yescharacteristic
364C2×C4#1455656211yescharacteristic
365C2×C4#1455661112yes—
366C2×C4#1455661612yes—
367C2×C4C2×D4×D7311yescharacteristic
368C2×C4C22×C22⋊D7311yescharacteristic
369C2×C4C2×#1192402711yescharacteristic
370C8—21no—
371C8—21no—
372C8—21no—
373C8—21no—
374C8—2922no—
375C8—2922no—
376C8—2922no—
377C22—21no—
378C22—21no—
379C22—2922no—
380C22—2922no—
381C22—2922no—
382C22#1798879912yes—
383C22C2×#1455656211yescharacteristic
384C4—21no—
385C4—21no—
386C4—82no—
387C4—82no—
388C4—2922no—
389C4#1798879912yes—
390C4#1798885912yes—
391C4C2×#1455661111yescharacteristic
392C4C2×#1455661611yescharacteristic
393C4#1798939312yes—
394C2—22no—
395C2—5842no—
396C2C2×#1798879911yescharacteristic
397C2#2261956611yescharacteristic
398C2C2×#1798885911yescharacteristic
subgroupG/H
centerC22C2×#14556562
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC2×#487×C73C2
socleC22×C73C2×#121
prime radical#17989137C22
hypercenterC2×#487D73
hypocenterC73#19197
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
4. #22609475C2semidirect
5. #22609351C2semidirect
6. #22608679C2semidirect
7. #22608616C2semidirect
8. #22608603C2semidirect
9. C2×#487×C73C2semidirect
202. C2×#487D73semidirect
209. C73#19197semidirect

Maximal subgroups

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

subgroupindex
#226210522
#226208562
#226186442
#226094752
#226093512
#226086792
#226086162
#226086032
C2×#487×C732
#1919773