Finite Group Databasebeta

← #33060111

Subgroups — #33060111

Order 18688.

1058 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 183 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620752C212yesmaximal, complemented
2#22615089C211yesmaximal, characteristic
3#22608922C214yesmaximal, complemented
4#22608888C214yesmaximal, complemented
5#22608451C212yesmaximal, complemented
6#22608258C211yesmaximal, characteristic
7C2×#543×C73C211yesmaximal, characteristic
8C2×#14556570C2211yescharacteristic
9#17988917—22no—
10#17988915C2214yescomplemented
11#17988843—24no—
12#17988743—22no—
13#17988742C2214yescomplemented
14#17988551C2212yes—
15#17988512C2212yes—
16#17988499C2212yes—
17#17988481—22no—
18#17988450—21no—
19#17988443—21no—
20C4×C22⋊Dic73C2211yescharacteristic
21#17988324C2218yescomplemented
22#17988311C2214yes—
23#543×C73—24no—
24#543×C73—22no—
25#543×C73C2214yes—
26C2×C22⋊Q8×C73—22no—
27C2×C22⋊Q8×C73—21no—
28C2×C22⋊Q8×C73—22no—
29C2×C22⋊Q8×C73C2212yes—
30C22×C22⋊C4×C73—21no—
31C22×C22⋊C4×C73C2211yescharacteristic
32C22×Dic146C2311yescharacteristic
33C2×#11924018—24no—
34C2×#11924012—22no—
35#14556570—22no—
36#14556570—22no—
37C2×C4⋊Dic73C2311yescharacteristic
38C2×#11924007—21no—
39C2×C22⋊Dic73—21no—
40C2×C22⋊Dic73—21no—
41C2×C22⋊Dic73—22no—
42C2×C22⋊Dic73—22no—
43C2×C4×Dic73—21no—
44#14556522—24no—
45#14556515—24no—
46#14556514—24no—
47C22⋊#10596831—22no—
48C22⋊#10596831C2314yes—
49#14556498—22no—
50#14556498—22no—
51#14556498C2312yes—
52C25×C73D411yescharacteristic
53C22×Q8×C73—21no—
54C22×Q8×C73D412yes—
55C23×C4×C73—21no—
56C23×C4×C73—21no—
57C23×C4×C73D411yescharacteristic
58C23×C4×C73C2311yescharacteristic
59C22⋊Q8×C73—28no—
60C22⋊Q8×C73—28no—
61C22⋊Q8×C73—24no—
62C22⋊Q8×C73—28no—
63C22⋊Q8×C73—24no—
64C22⋊Q8×C73—22no—
65C22⋊Q8×C73—22no—
66C22⋊Q8×C73—28no—
67C22⋊Q8×C73D414yes—
68C22⋊Q8×C73C2314yes—
69C2×C4⋊C4×C73—21no—
70C2×C4⋊C4×C73—22no—
71C2×C4⋊C4×C73—22no—
72C2×C4⋊C4×C73D412yes—
73C2×C22⋊C4×C73—22no—
74C2×C22⋊C4×C73—24no—
75C2×C22⋊C4×C73—22no—
76C2×C22⋊C4×C73—24no—
77C2×C22⋊C4×C73—22no—
78C2×C22⋊C4×C73—22no—
79C2×C22⋊C4×C73D412yes—
80C2×C22⋊C4×C73C2312yes—
81C2×Dic146—22no—
82C2×Dic146—81no—
83C22×Dic73—22no—
84C22×Dic73—21no—
85C22×Dic73—21no—
86#11924018—84no—
87#11924012—82no—
88C2×#10596831—24no—
89C4⋊Dic73—22no—
90#11924007—42no—
91C22⋊Dic73—42no—
92C22⋊Dic73—41no—
93C22⋊Dic73—41no—
94C22⋊Dic73—44no—
95C22⋊Dic73—82no—
96C4×Dic73—81no—
97C24×C73—21no—
98C24×C73—21no—
99C24×C73—24no—
100C24×C73—42no—
101C24×C73—81no—
102C24×C73C2×D411yescharacteristic
103C24×C73C2×D412yes—
104C2×Q8×C73—24no—
105C2×Q8×C73—22no—
106C2×Q8×C73—82no—
107C2×Q8×C73—82no—
108C2×Q8×C73C2×D414yes—
109C22×C4×C73—22no—
110C22×C4×C73—21no—
111C22×C4×C73—22no—
112C22×C4×C73—24no—
113C22×C4×C73—24no—
114C22×C4×C73—21no—
115C22×C4×C73—42no—
116C22×C4×C73—81no—
117C22×C4×C73—82no—
118C22×C4×C73—82no—
119C22×C4×C73C2×D412yes—
120C22×C4×C73C2×D411yescharacteristic
121C22×C4×C73C2×D412yes—
122C22×C4×C73C2×D412yes—
123C22×C4×C73C2411yescharacteristic
124C4⋊C4×C73—24no—
125C4⋊C4×C73—28no—
126C4⋊C4×C73—24no—
127C4⋊C4×C73—22no—
128C4⋊C4×C73—24no—
129C4⋊C4×C73C2×D414yes—
130C22⋊C4×C73—28no—
131C22⋊C4×C73—24no—
132C22⋊C4×C73—24no—
133C22⋊C4×C73—24no—
134C22⋊C4×C73—24no—
135C23×C73—24no—
136C23×C73—24no—
137C23×C73—22no—
138C23×C73—22no—
139C23×C73—22no—
140C23×C73—22no—
141C23×C73—41no—
142C23×C73—42no—
143C23×C73—42no—
144C23×C73—42no—
145C23×C73—42no—
146C23×C73—81no—
147C23×C73—81no—
148C23×C73—88no—
149C23×C73#12011yescharacteristic
150C23×C73#12011yescharacteristic
151C23×C73#13714yes—
152C23×C73C22×D411yescharacteristic
153Q8×C73—44no—
154Q8×C73—82no—
155Q8×C73—84no—
156C2×C4×C73—24no—
157C2×C4×C73—22no—
158C2×C4×C73—24no—
159C2×C4×C73—42no—
160C2×C4×C73—41no—
161C2×C4×C73—42no—
162C2×C4×C73—41no—
163C2×C4×C73—44no—
164C2×C4×C73—42no—
165C2×C4×C73—44no—
166C2×C4×C73—42no—
167C2×C4×C73—81no—
168C2×C4×C73—84no—
169C2×C4×C73—84no—
170C2×C4×C73#12012yes—
171C2×C4×C73C22×D412yes—
172C2×Dic73—42no—
173C2×Dic73—42no—
174C2×Dic73—42no—
175C2×Dic73—41no—
176C2×Dic73—41no—
177C2×Dic73—42no—
178C2×Dic73—42no—
179Dic146—44no—
180#10596831—44no—
181C22×C73—21no—
182C22×C73—21no—
183C22×C73—21no—
184C22×C73—21no—
185C22×C73—21no—
186C22×C73—24no—
187C22×C73—21no—
188C22×C73—21no—
189C22×C73—44no—
190C22×C73—44no—
191C22×C73—48no—
192C22×C73—42no—
193C22×C73—42no—
194C22×C73—82no—
195C22×C73—84no—
196C22×C73#45512yes—
197C22×C73C2×#12011yescharacteristic
198C22×C73C2×#13712yes—
199C4×C73—22no—
200C4×C73—22no—
201C4×C73—42no—
202C4×C73—44no—
203C4×C73—44no—
204Dic73—82no—
205Dic73—81no—
206Dic73—81no—
207#18319—731nomaximal, Sylow
208C2×C73—22no—
209C2×C73—22no—
210C2×C73—41no—
211C2×C73—44no—
212C2×C73#300411yescharacteristic
213C2×C73C2×#45511yescharacteristic
214C2×C73#302611yescharacteristic
215C2×#543D7311yescharacteristic
216#3195—732no—
217#2561—731no—
218#1608—734no—
219#1602—734no—
220#1508—732no—
221#1462—731no—
222C73#1831911yescharacteristic, complemented, Sylow
223#543—24no—
224#543—22no—
225#543C2×D7314yes—
226C2×C22⋊Q8—22no—
227C2×C22⋊Q8—21no—
228C2×C22⋊Q8—22no—
229C2×C22⋊Q8—731no—
230C2×C22⋊Q8C2×D7312yes—
231C22×C22⋊C4—21no—
232C22×C22⋊C4C2×D7311yescharacteristic
233#483—734no—
234#483—1462no—
235#470—1464no—
236#451—734no—
237#451—1462no—
238#407—732no—
239#397—732no—
240#393—732no—
241#388—1462no—
242#381—1461no—
243#379—1461no—
244C4×C22⋊C4—731no—
245#328—738no—
246#323—734no—
247C25C22⋊D7311yescharacteristic
248C22×Q8—21no—
249C22×Q8—731no—
250C22×Q8C22⋊D7312yes—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4C22⋊D7311yescharacteristic
254C23×C4C22×D7311yescharacteristic
255C2×Q16—1464no—
256C2×M16—1462no—
257C22⋊Q8—28no—
258C22⋊Q8—28no—
259C22⋊Q8—24no—
260C22⋊Q8—28no—
261C22⋊Q8—24no—
262C22⋊Q8—22no—
263C22⋊Q8—22no—
264C22⋊Q8—28no—
265C22⋊Q8—1462no—
266C22⋊Q8—1462no—
267C22⋊Q8C22⋊D7314yes—
268C22⋊Q8C22×D7314yes—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—731no—
273C2×C4⋊C4—1461no—
274C2×C4⋊C4C22⋊D7312yes—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—1461no—
282C2×C22⋊C4—1461no—
283C2×C22⋊C4—1462no—
284C2×C22⋊C4—1462no—
285C2×C22⋊C4C22⋊D7312yes—
286C2×C22⋊C4C22×D7312yes—
287C2×C42—1461no—
288#106—1464no—
289#103—1464no—
290#103—1464no—
291C22⋊C8—734no—
292C22⋊C8—1462no—
293#95—732no—
294#95—1462no—
295#95—1462no—
296C24—21no—
297C24—21no—
298C24—24no—
299C24—42no—
300C24—81no—
301C24C2×C22⋊D7311yescharacteristic
302C24C2×C22⋊D7312yes—
303C2×Q8—24no—
304C2×Q8—22no—
305C2×Q8—82no—
306C2×Q8—82no—
307C2×Q8—5841no—
308C2×Q8—1462no—
309C2×Q8C2×C22⋊D7314yes—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—21no—
316C22×C4—42no—
317C22×C4—81no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—1462no—
321C22×C4—1461no—
322C22×C4—1461no—
323C22×C4C2×C22⋊D7312yes—
324C22×C4C2×C22⋊D7311yescharacteristic
325C22×C4C2×C22⋊D7312yes—
326C22×C4C2×C22⋊D7312yes—
327C22×C4C23×D7311yescharacteristic
328Q16—5844no—
329M16—5842no—
330C2×C8—1464no—
331C4⋊C4—24no—
332C4⋊C4—28no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—2922no—
337C4⋊C4—1462no—
338C4⋊C4C2×C22⋊D7314yes—
339C22⋊C4—28no—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—2922no—
345C22⋊C4—2921no—
346C22⋊C4—2921no—
347C22⋊C4—2924no—
348C22⋊C4—5842no—
349C42—5841no—
350C23—24no—
351C23—24no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—41no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—81no—
362C23—81no—
363C23—88no—
364C23#1455656011yescharacteristic
365C23#1455656011yescharacteristic
366C23#1455662214yes—
367C23C22×C22⋊D7311yescharacteristic
368Q8—44no—
369Q8—82no—
370Q8—84no—
371Q8—2924no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—81no—
384C2×C4—84no—
385C2×C4—84no—
386C2×C4—2922no—
387C2×C4—2922no—
388C2×C4—2922no—
389C2×C4—2922no—
390C2×C4—2922no—
391C2×C4—2921no—
392C2×C4—2921no—
393C2×C4#1455656012yes—
394C2×C4C22×C22⋊D7312yes—
395C8—2924no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—24no—
402C22—21no—
403C22—21no—
404C22—44no—
405C22—44no—
406C22—48no—
407C22—42no—
408C22—42no—
409C22—82no—
410C22—84no—
411C22#1798876512yes—
412C22C2×#1455656011yescharacteristic
413C22C2×#1455662212yes—
414C4—22no—
415C4—22no—
416C4—42no—
417C4—44no—
418C4—44no—
419C4—5842no—
420C4—5841no—
421C4—5841no—
422C2—22no—
423C2—22no—
424C2—41no—
425C2—44no—
426C2#2261916511yescharacteristic
427C2C2×#1798876511yescharacteristic
428C2#2261932811yescharacteristic
subgroupG/H
centerC22C2×#14556560
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#543×C73C2
socleC22×C73C2×#120
prime radicalC25×C73D4
hypercenterC2×#543D73
hypocenterC73#18319
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
1. #22620752C2semidirect
3. #22608922C2semidirect
4. #22608888C2semidirect
5. #22608451C2semidirect
10. #17988915C22semidirect
13. #17988742C22semidirect
21. #17988324C22semidirect
222. C73#18319semidirect

Maximal subgroups

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

subgroupindex
#226207522
#226150892
#226089222
#226088882
#226084512
#226082582
C2×#543×C732
#1831973