Finite Group Databasebeta

← #32980705

Subgroups — #32980705

Order 18688.

478 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619056C211yesmaximal, characteristic
2#22610909C211yesmaximal, characteristic, complemented
3C4×#14556518C211yesmaximal, characteristic
4#22608352C211yesmaximal, characteristic, complemented
5#22607822C211yesmaximal, characteristic, complemented
6#22607763C211yesmaximal, characteristic, complemented
7C73×#3107C211yesmaximal, characteristic
8#17988718—21no—
9#17988659—22no—
10D4×#10596831—21no—
11#17988644—21no—
12#17988640C2211yescharacteristic
13#17988611C2211yescharacteristic
14C2×C4⋊#10596831—21no—
15C2×#14556518—21no—
16C4×#11924012C2211yescharacteristic
17C42×Dic73C2211yescharacteristic
18#17988346—22no—
19#17988346—21no—
20Q8⋊#10596831—21no—
21Q8⋊#10596831—21no—
22#533×C73C2211yescharacteristic
23C4×#41×C73C2211yescharacteristic
24#485×C73—21no—
25#475×C73C411yescharacteristic, complemented
26#474×C73C411yescharacteristic, complemented
27#462×C73C411yescharacteristic, complemented
28#461×C73C411yescharacteristic, complemented
29#452×C73—21no—
30C4×Q16×C73—21no—
31C4×SD16×C73—21no—
32#431×C73C2211yescharacteristic
33C2×#11924012—21no—
34C2×#11924012D412yes—
35C22×#10596831—22no—
36C2×C4×Dic73—21no—
37C2×C4×Dic73—21no—
38C2×C4×Dic73—21no—
39C4⋊#10596831—22no—
40C4⋊#10596831—21no—
41C4⋊#10596831—21no—
42C4⋊#10596831D411yescharacteristic
43C4⋊#10596831D411yescharacteristic
44#14556518—24no—
45C22⋊#10596831—22no—
46C22⋊#10596831—21no—
47#14556502—22no—
48#14556502—21no—
49C4×#10596831—21no—
50C4×#10596831—21no—
51C2×#41×C73D411yescharacteristic
52C2×Q16×C73—21no—
53C2×SD16×C73—21no—
54C4⋊Q8×C73C2×C411yescharacteristic
55C4⋊Q8×C73C2×C411yescharacteristic
56#125×C73—21no—
57C22⋊Q8×C73—21no—
58C22⋊Q8×C73—21no—
59C4×Q8×C73—21no—
60C4×Q8×C73—21no—
61C4×Q8×C73C2×C411yescharacteristic
62C4×Q8×C73D411yescharacteristic
63C4×D4×C73—21no—
64C4×D4×C73—21no—
65C4×D4×C73C2×C411yescharacteristic
66C4×D4×C73D411yescharacteristic
67#117×C73—21no—
68#117×C73—21no—
69#117×C73D411yescharacteristic
70C2×C42×C73—21no—
71C2×C42×C73C2311yescharacteristic
72#107×C73—21no—
73#106×C73—21no—
74C4⋊C8×C73C2×C411yescharacteristic
75C4⋊C8×C73C2×C411yescharacteristic
76#103×C73—21no—
77#103×C73—21no—
78#103×C73—21no—
79#102×C73—21no—
80C22⋊C8×C73—21no—
81C4×C8×C73—21no—
82C22×Dic73—41no—
83#11924012—22no—
84#11924012—41no—
85#11924012—42no—
86C2×#10596831—21no—
87C2×#10596831—21no—
88C2×#10596831—22no—
89C2×#10596831—41no—
90C2×#10596831—82no—
91C4×Dic73—22no—
92C4×Dic73—22no—
93C4×Dic73—22no—
94C4×Dic73—41no—
95C4×Dic73—41no—
96C4×Dic73—41no—
97#41×C73—22no—
98#41×C73—41no—
99C2×Q8×C73—21no—
100C2×Q8×C73—21no—
101C2×Q8×C73C22⋊C411yescharacteristic
102C2×D4×C73—21no—
103C2×D4×C73C22⋊C411yescharacteristic
104C22×C4×C73—21no—
105C22×C4×C73—21no—
106C22×C4×C73—21no—
107C22×C4×C73C2×D411yescharacteristic
108C22×C4×C73C2×D411yescharacteristic
109C22×C4×C73#4111yescharacteristic
110Q16×C73—42no—
111SD16×C73—42no—
112C2×C8×C73—21no—
113C2×C8×C73—21no—
114C4⋊C4×C73—21no—
115C4⋊C4×C73—21no—
116C4⋊C4×C73—21no—
117C4⋊C4×C73—21no—
118C4⋊C4×C73—21no—
119C4⋊C4×C73—21no—
120C4⋊C4×C73—41no—
121C4⋊C4×C73—41no—
122C4⋊C4×C73C22⋊C411yescharacteristic
123C4⋊C4×C73C22⋊C411yescharacteristic
124C22⋊C4×C73—21no—
125C22⋊C4×C73—21no—
126C22⋊C4×C73—21no—
127C22⋊C4×C73—41no—
128C42×C73—21no—
129C42×C73—21no—
130C42×C73—21no—
131C42×C73—41no—
132C42×C73C22×C411yescharacteristic
133C42×C73C2×D411yescharacteristic
134C42×C73C2×D411yescharacteristic
135C42×C73#4111yescharacteristic
136C23×C73—21no—
137C23×C73#12111yescharacteristic
138Q8×C73—21no—
139Q8×C73—21no—
140Q8×C73—41no—
141Q8×C73—41no—
142Q8×C73—41no—
143Q8×C73—41no—
144D4×C73—21no—
145D4×C73—42no—
146D4×C73#10412yes—
147C2×C4×C73—21no—
148C2×C4×C73—21no—
149C2×C4×C73—21no—
150C2×C4×C73—21no—
151C2×C4×C73—21no—
152C2×C4×C73—21no—
153C2×C4×C73—21no—
154C2×C4×C73—21no—
155C2×C4×C73—21no—
156C2×C4×C73—42no—
157C2×C4×C73—41no—
158C2×C4×C73—41no—
159C2×C4×C73—41no—
160C2×C4×C73—41no—
161C2×C4×C73—41no—
162C2×C4×C73—41no—
163C2×C4×C73C2×C22⋊C411yescharacteristic
164C2×C4×C73C4×D411yescharacteristic
165C2×C4×C73C4×D411yescharacteristic
166C2×C4×C73#12011yescharacteristic
167C2×C4×C73#12111yescharacteristic
168C2×C4×C73#12311yescharacteristic
169C2×Dic73—42no—
170C2×Dic73—42no—
171C2×Dic73—42no—
172C8×C73—41no—
173C8×C73—41no—
174#10596831—42no—
175#10596831—41no—
176#10596831—42no—
177#10596831—41no—
178C22×C73—21no—
179C22×C73—22no—
180C22×C73—21no—
181C22×C73—21no—
182C22×C73—41no—
183C22×C73—41no—
184C22×C73#38611yescharacteristic
185C4×C73—21no—
186C4×C73—21no—
187C4×C73—21no—
188C4×C73—41no—
189C4×C73—41no—
190C4×C73—41no—
191C4×C73—81no—
192C4×C73—81no—
193C4×C73C2×#10411yescharacteristic
194C4×C73#42111yescharacteristic
195C4×C73#44311yescharacteristic
196C4×C73#44411yescharacteristic
197C4×C73#45511yescharacteristic
198C4×C73#45611yescharacteristic
199Dic73—44no—
200#8971—731nomaximal, Sylow
201C2×C73—41no—
202C2×C73—42no—
203C2×C73#187111yescharacteristic
204C2×C73#187611yescharacteristic
205C2×C73#188211yescharacteristic
206#3107D7311yescharacteristic
207#2990—731no—
208#1994—731no—
209C4×#104—731no—
210#1485—731no—
211#1299—731no—
212#1276—731no—
213C73#897111yescharacteristic, complemented, Sylow
214#533C2×D7311yescharacteristic
215C4×#41C2×D7311yescharacteristic
216#485—21no—
217#475Dic7311yescharacteristic, complemented
218#474Dic7311yescharacteristic, complemented
219#462Dic7311yescharacteristic, complemented
220#461Dic7311yescharacteristic, complemented
221#452—21no—
222#446—1461no—
223C4×Q16—21no—
224C4×SD16—21no—
225#435—1462no—
226C8×D4—1461no—
227#432—1461no—
228#431—731no—
229#431C2×D7311yescharacteristic
230#423—731no—
231C2×C4⋊C8—1461no—
232C2×#104—1461no—
233C4×M16—731no—
234C43—731no—
235#339—1462no—
236#339—1461no—
237Q8⋊C8—1461no—
238Q8⋊C8—1461no—
239C2×#41C22⋊D7311yescharacteristic
240C2×Q16—21no—
241C2×SD16—21no—
242C2×M16—732no—
243C2×M16—1461no—
244C22×C8—1462no—
245C4⋊Q8C2×Dic7311yescharacteristic
246C4⋊Q8C2×Dic7311yescharacteristic
247#125—21no—
248C22⋊Q8—21no—
249C22⋊Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8C2×Dic7311yescharacteristic
253C4×Q8C22⋊D7311yescharacteristic
254C4×D4—21no—
255C4×D4—21no—
256C4×D4C2×Dic7311yescharacteristic
257C4×D4C22⋊D7311yescharacteristic
258#117—21no—
259#117—21no—
260#117C22⋊D7311yescharacteristic
261C2×C42—21no—
262C2×C42—1461no—
263C2×C42—1461no—
264C2×C42—1461no—
265C2×C42C22×D7311yescharacteristic
266#107—21no—
267#106—21no—
268C4⋊C8—731no—
269C4⋊C8—731no—
270C4⋊C8—1462no—
271C4⋊C8—1461no—
272C4⋊C8—1461no—
273C4⋊C8C2×Dic7311yescharacteristic
274C4⋊C8C2×Dic7311yescharacteristic
275#104—1464no—
276#103—21no—
277#103—21no—
278#103—21no—
279#102—21no—
280C22⋊C8—21no—
281C22⋊C8—1462no—
282C22⋊C8—1461no—
283#97—1462no—
284#97—1461no—
285C4×C8—21no—
286C4×C8—1461no—
287C4×C8—1461no—
288#41—22no—
289#41—41no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8C22⋊Dic7311yescharacteristic
293C2×D4—21no—
294C2×D4C22⋊Dic7311yescharacteristic
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—2921no—
299C22×C4D4×D7311yescharacteristic
300C22×C4C2×C22⋊D7311yescharacteristic
301C22×C4#1192402711yescharacteristic
302Q16—42no—
303SD16—42no—
304M16—2921no—
305M16—2922no—
306M16—1462no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—2921no—
310C2×C8—5842no—
311C2×C8—1461no—
312C2×C8—1461no—
313C2×C8—1462no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4C22⋊Dic7311yescharacteristic
323C4⋊C4C22⋊Dic7311yescharacteristic
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—41no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—41no—
332C42—2921no—
333C42—2921no—
334C42—2921no—
335C42—1462no—
336C42—1462no—
337C42—1462no—
338C42C22×Dic7311yescharacteristic
339C42D4×D7311yescharacteristic
340C42C2×C22⋊D7311yescharacteristic
341C42#1192402711yescharacteristic
342C23—21no—
343C23#1455656311yescharacteristic
344Q8—21no—
345Q8—21no—
346Q8—41no—
347Q8—41no—
348Q8—41no—
349Q8—41no—
350D4—21no—
351D4—42no—
352D4#1455651812yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—2922no—
370C2×C4—2922no—
371C2×C4—2922no—
372C2×C4C2×C22⋊Dic7311yescharacteristic
373C2×C4D4×Dic7311yescharacteristic
374C2×C4D4×Dic7311yescharacteristic
375C2×C4#1455655911yescharacteristic
376C2×C4#1455656311yescharacteristic
377C2×C4#1455657211yescharacteristic
378C8—41no—
379C8—41no—
380C8—2922no—
381C8—2921no—
382C8—2922no—
383C8—2921no—
384C22—21no—
385C22—22no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22#1798846711yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—81no—
398C4—81no—
399C4—2924no—
400C4C2×#1455651811yescharacteristic
401C4#1798860411yescharacteristic
402C4#1798870411yescharacteristic
403C4#1798870911yescharacteristic
404C4#1798876311yescharacteristic
405C4#1798876811yescharacteristic
406C2—41no—
407C2—42no—
408C2#2261023411yescharacteristic
409C2#2261026011yescharacteristic
410C2#2261030211yescharacteristic
subgroupG/H
centerC2×C4#14556559
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#3107C2
socleC22×C73#386
prime radicalC2×#41×C73D4
cosocleC2×C42×C73C23
hypercenter#3107D73
hypocenterC73#8971
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3107

Complements

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

NH ≅ G/Ntype
2. #22610909C2semidirect
4. #22608352C2semidirect
5. #22607822C2semidirect
6. #22607763C2semidirect
25. #475×C73C4semidirect
26. #474×C73C4semidirect
27. #462×C73C4semidirect
28. #461×C73C4semidirect
213. C73#8971semidirect
217. #475Dic73semidirect
218. #474Dic73semidirect
219. #462Dic73semidirect
220. #461Dic73semidirect

Maximal subgroups

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

subgroupindex
#226190562
#226109092
C4×#145565182
#226083522
#226078222
#226077632
C73×#31072
#897173