Finite Group Databasebeta

← #34003727

Subgroups — #34003727

Order 18688.

720 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 132 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#22625658C211yesmaximal, characteristic
2C2×#17989441C212yesmaximal
3C4×(C2×#64408)⋊C4C211yesmaximal, characteristic
4#22624804C212yesmaximal
5C4×#14556567C211yesmaximal, characteristic
6C22×C4×#64408—21no—
7C2×#14556648C2212yes—
8C2×#14556646—21no—
9C2×(C2×#64408)⋊C4—21no—
10C2×(C2×#64408)⋊C4C2212yes—
11#17989453—21no—
12#17989441C418yescomplemented
13C2×#14556567C2211yescharacteristic
14C2×C42×D73C2211yescharacteristic
15#17988494C411yescharacteristic, complemented
16#17988483C411yescharacteristic, complemented
17#17988468—21no—
18#17988458—21no—
19#17988444C2211yescharacteristic
20C4×#11924007—21no—
21C4×C4⋊Dic73C411yescharacteristic, complemented
22C4×#11924005—21no—
23C2×C4×Q8×C73C411yescharacteristic, complemented
24C23×#64408—22no—
25C22×#10596833—22no—
26C2×C4×#64408—24no—
27C2×C4×#64408—42no—
28C2×C4×#64408—42no—
29#14556648—22no—
30#14556648C2×C414yes—
31#14556646—42no—
32(C2×#64408)⋊C4—24no—
33(C2×#64408)⋊C4—22no—
34(C2×#64408)⋊C4C2×C414yes—
35C22×C4×D73D411yescharacteristic
36C22×C4×D73D411yescharacteristic
37C22×C4×D73C2311yescharacteristic
38#14556567—22no—
39#14556567C2×C414yes—
40C2×C4⋊Dic73C2×C411yescharacteristic
41C2×C4⋊Dic73C2×C411yescharacteristic
42C2×#11924007—21no—
43C2×#11924007—21no—
44C2×#11924005—21no—
45C2×#11924005—21no—
46C2×C4×Dic73—21no—
47C2×C4×Dic73C2×C411yescharacteristic
48C2×C4×Dic73D411yescharacteristic
49C2×C4×Dic73D411yescharacteristic
50C42×D73—42no—
51#14556498—21no—
52#14556498—21no—
53#14556498—21no—
54#14556498—21no—
55#14556498—21no—
56C22×Q8×C73C2×C411yescharacteristic
57C4×Q8×C73—42no—
58C2×C4⋊C4×C73—21no—
59C2×C4⋊C4×C73C2×C411yescharacteristic
60C2×C42×C73—21no—
61C2×C42×C73C2×C411yescharacteristic
62C22×#64408—24no—
63C22×#64408—24no—
64C22×#64408—24no—
65C22×#64408—42no—
66C22×#64408—42no—
67C22×#64408—42no—
68C22×#64408—42no—
69C2×#10596833—24no—
70C2×#10596833—42no—
71C2×#10596833—42no—
72C4×#64408—48no—
73C23×D73C2×D411yescharacteristic
74C22×Dic73—21no—
75C22×Dic73—21no—
76C22×Dic73C22⋊C411yescharacteristic
77C22×Dic73C22⋊C411yescharacteristic
78C22×Dic73C2×D411yescharacteristic
79C2×C4×D73—21no—
80C2×C4×D73—21no—
81C2×C4×D73—22no—
82C2×C4×D73—22no—
83C2×C4×D73—22no—
84C2×C4×D73—22no—
85C2×C4×D73—41no—
86C2×C4×D73—41no—
87C2×C4×D73—41no—
88C2×C4×D73C22×C412yes—
89C2×C4×D73#4112yes—
90C4⋊Dic73—21no—
91C4⋊Dic73—41no—
92C4⋊Dic73C4212yescomplemented
93#11924007—24no—
94#11924007—42no—
95#11924005—24no—
96#11924005—41no—
97#11924005—41no—
98C4×Dic73—22no—
99C4×Dic73—41no—
100C4×Dic73—41no—
101C4×Dic73—42no—
102C2×Q8×C73—21no—
103C2×Q8×C73—41no—
104C2×Q8×C73—41no—
105C2×Q8×C73C4212yescomplemented
106C22×C4×C73—21no—
107C22×C4×C73—21no—
108C22×C4×C73C22⋊C411yescharacteristic
109C22×C4×C73C22⋊C411yescharacteristic
110C22×C4×C73C22×C411yescharacteristic
111C4⋊C4×C73—41no—
112C4⋊C4×C73—42no—
113C42×C73—22no—
114C42×C73—42no—
115C23×C73C2×C22⋊C411yescharacteristic
116C22×D73—21no—
117C22×D73—21no—
118C22×D73—21no—
119C22×D73—21no—
120C22×D73C4×D412yes—
121C22×D73AGL(1,ℤ/8)11yescharacteristic
122C22×D73#13712yes—
123C22×D73#13711yescharacteristic
124C2×#64408—28no—
125C2×#64408—44no—
126C2×#64408—44no—
127C2×#64408—44no—
128C2×#64408—44no—
129C2×#64408—44no—
130C2×#64408—44no—
131Q8×C73—42no—
132Q8×C73—42no—
133C2×C4×C73—21no—
134C2×C4×C73—21no—
135C2×C4×C73—21no—
136C2×C4×C73—21no—
137C2×C4×C73—21no—
138C2×C4×C73—21no—
139C2×C4×C73—22no—
140C2×C4×C73—41no—
141C2×C4×C73—41no—
142C2×C4×C73—41no—
143C2×C4×C73—41no—
144C2×C4×C73—41no—
145C2×C4×C73#9912yes—
146C2×C4×C73#10412yes—
147C2×C4×C73C2×C4211yescharacteristic
148C2×C4×C73#11711yescharacteristic
149C2×Dic73—21no—
150C2×Dic73—21no—
151C2×Dic73—21no—
152C2×Dic73—21no—
153C2×Dic73—22no—
154C2×Dic73—21no—
155C2×Dic73—21no—
156C2×Dic73—21no—
157C2×Dic73—41no—
158C2×Dic73—41no—
159C2×Dic73—41no—
160C2×Dic73—41no—
161C2×Dic73—41no—
162C2×Dic73#10412yes—
163C2×Dic73C4×D412yes—
164C4×D73—24no—
165C4×D73—24no—
166C4×D73—42no—
167C4×D73—42no—
168C4×D73—44no—
169#10596833—44no—
170C22×C73C4×C22⋊C411yescharacteristic
171C22×C73C2×#9911yescharacteristic
172C22×C73#41011yescharacteristic
173C22×C73#41711yescharacteristic
174C22×C73#41911yescharacteristic
175C22×C73C2×#10411yescharacteristic
176C22×C73C2×#10411yescharacteristic
177C2×D73—22no—
178C2×D73—22no—
179C2×D73—21no—
180C2×D73—21no—
181C2×D73—22no—
182C2×D73—22no—
183C2×D73—21no—
184C2×D73—21no—
185C2×D73#44012yes—
186C2×D73#44112yes—
187#64408—48no—
188C4×C73—21no—
189C4×C73—21no—
190C4×C73—22no—
191C4×C73—22no—
192C4×C73—42no—
193C4×C73—81no—
194Dic73—22no—
195Dic73—22no—
196Dic73—42no—
197Dic73—41no—
198Dic73—81no—
199#8219—731nomaximal, Sylow
200C2×C73#150411yescharacteristic
201C2×C73#150611yescharacteristic
202C2×C73#150811yescharacteristic
203C2×C73#150911yescharacteristic
204C2×C73C4×#9911yescharacteristic
205C2×C73C4×#10411yescharacteristic
206C2×C73#176311yescharacteristic
207D73—22no—
208D73—22no—
209C4×C22⋊Q8—731no—
210#1749—731no—
211C2×#328—732no—
212C4×#95—731no—
213#1294—732no—
214C73#821911yescharacteristic, complemented, Sylow
215C2×C22⋊Q8—731no—
216C2×C4×Q8#6440811yescharacteristic, complemented
217C22×C42—731no—
218C22×C42—1461no—
219C2×C22⋊C8—732no—
220C2×#97—1461no—
221#391—731no—
222#389—731no—
223#387—1461no—
224#385—1461no—
225#380—731no—
226C4×C4⋊C4—731no—
227C4×C4⋊C4—1461no—
228C4×C22⋊C4—1461no—
229C2×#95—732no—
230C2×#95—1461no—
231#336—1461no—
232#328—738no—
233C22×Q8C2×#6440811yescharacteristic
234C23×C4—731no—
235C23×C4—731no—
236C23×C4—731no—
237C23×C4—1462no—
238C22×C8—1462no—
239C22⋊Q8—734no—
240C22⋊Q8—1462no—
241C4×Q8—42no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—731no—
244C2×C4⋊C4—731no—
245C2×C4⋊C4—1461no—
246C2×C4⋊C4—1461no—
247C2×C4⋊C4C2×#6440811yescharacteristic
248C2×C22⋊C4—1461no—
249C2×C22⋊C4—1461no—
250C2×C42—21no—
251C2×C42—2922no—
252C2×C42—2922no—
253C2×C42—2922no—
254C2×C42—731no—
255C2×C42—731no—
256C2×C42—731no—
257C2×C42—1464no—
258C2×C42—1461no—
259C2×C42C2×#6440811yescharacteristic
260C22⋊C8—734no—
261C22⋊C8—1462no—
262#97—2922no—
263#95—734no—
264#95—1461no—
265#95—1461no—
266#95—1461no—
267#95—1461no—
268#95—1462no—
269#95—1464no—
270#95—1461no—
271C24—731no—
272C2×Q8—21no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8C4×#6440812yescomplemented
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—2921no—
279C22×C4—2921no—
280C22×C4—2921no—
281C22×C4—2922no—
282C22×C4—2922no—
283C22×C4—2922no—
284C22×C4—2922no—
285C22×C4—732no—
286C22×C4—732no—
287C22×C4—731no—
288C22×C4—731no—
289C22×C4—731no—
290C22×C4—1462no—
291C22×C4—1462no—
292C22×C4—1461no—
293C22×C4—1461no—
294C22×C4—1461no—
295C22×C4—1464no—
296C22×C4—1464no—
297C22×C4—1464no—
298C22×C4—1461no—
299C22×C4—1462no—
300C22×C4—1462no—
301C22×C4C22⋊#6440811yescharacteristic
302C22×C4C22⋊#6440811yescharacteristic
303C22×C4C22×#6440811yescharacteristic
304C2×C8—2922no—
305C2×C8—2922no—
306C2×C8—1464no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—2921no—
310C4⋊C4—2922no—
311C4⋊C4—732no—
312C4⋊C4—1464no—
313C4⋊C4—1461no—
314C22⋊C4—2921no—
315C22⋊C4—2921no—
316C22⋊C4—1464no—
317C42—22no—
318C42—42no—
319C42—2921no—
320C42—2921no—
321C42—2928no—
322C42—2922no—
323C42—1462no—
324C23—732no—
325C23—731no—
326C23—731no—
327C23—732no—
328C23—1461no—
329C23—1461no—
330C23—1461no—
331C23—1461no—
332C23C2×C22⋊#6440811yescharacteristic
333Q8—42no—
334Q8—42no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—2924no—
348C2×C4—2922no—
349C2×C4—2922no—
350C2×C4—2924no—
351C2×C4—2924no—
352C2×C4—2924no—
353C2×C4—2924no—
354C2×C4—2924no—
355C2×C4—2924no—
356C2×C4—2921no—
357C2×C4—2921no—
358C2×C4—2921no—
359C2×C4—2921no—
360C2×C4—2921no—
361C2×C4—732no—
362C2×C4—732no—
363C2×C4—1464no—
364C2×C4—1464no—
365C2×C4—1468no—
366C2×C4—1462no—
367C2×C4—1461no—
368C2×C4—1461no—
369C2×C4—1461no—
370C2×C4—1461no—
371C2×C4—1461no—
372C2×C4—1461no—
373C2×C4—1461no—
374C2×C4#1455664912yes—
375C2×C4#1455665612yes—
376C2×C4C2×C4×#6440811yescharacteristic
377C2×C4#1455667011yescharacteristic
378C8—2924no—
379C22—732no—
380C22—732no—
381C22—1461no—
382C22—1461no—
383C22—1462no—
384C22—1462no—
385C22—1461no—
386C22—1461no—
387C22—1462no—
388C22—1462no—
389C22C4×C22⋊#6440811yescharacteristic
390C22C2×#1455664911yescharacteristic
391C22#1798956611yescharacteristic
392C22#1798957811yescharacteristic
393C22#1798958011yescharacteristic
394C22D73⋊#10411yescharacteristic
395C22C2×#1455665611yescharacteristic
396C4—22no—
397C4—22no—
398C4—21no—
399C4—21no—
400C4—42no—
401C4—81no—
402C4—2928no—
403C4—2921no—
404C4—2922no—
405C4—5841no—
406C4—1462no—
407C4—1462no—
408C2—1462no—
409C2—1462no—
410C2#2262531711yescharacteristic
411C2#2262532411yescharacteristic
412C2#2262533111yescharacteristic
413C2#2262533511yescharacteristic
414C2C4×#1455664911yescharacteristic
415C2C4×#1455665611yescharacteristic
416C2#2262570311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC2×C4×C73C2×C42
FrattiniC22×C4C22×#64408
FittingC2×C4×Q8×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC23×D73C2×D4
cosocleC22×C4×D73C23
hypercenterC2×C4×Q8#64408
hypocenterC73#8219
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
12. #17989441C4semidirect
15. #17988494C4semidirect
16. #17988483C4semidirect
21. C4×C4⋊Dic73C4semidirect
23. C2×C4×Q8×C73C4semidirect
92. C4⋊Dic73C42semidirect
105. C2×Q8×C73C42semidirect
214. C73#8219semidirect
216. C2×C4×Q8#64408semidirect
275. C2×Q8C4×#64408semidirect

Maximal subgroups

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

subgroupindex
#226256582
C2×#179894412
C4×(C2×#64408)⋊C42
#226248042
C4×#145565672
#821973