Finite Group Databasebeta

← #33241316

Subgroups — #33241316

Order 18688.

602 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620470C211yesmaximal, characteristic, complemented
2#22612172C212yesmaximal, complemented
3#22612167C212yesmaximal, complemented
4#22612162C212yesmaximal, complemented
5#22612154C212yesmaximal, complemented
6#22611647C211yesmaximal, characteristic, complemented
7#22611643C211yesmaximal, characteristic, complemented
8#22611638C211yesmaximal, characteristic, complemented
9C2×#17988388C211yesmaximal, characteristic, complemented
10#22611528C211yesmaximal, characteristic, complemented
11C73×#3155C211yesmaximal, characteristic
12#17989215—21no—
13C2×#14556531C2211yescharacteristic
14C22×#11924003C2211yescharacteristic
15#17988996—22no—
16#17988992C2212yescomplemented
17#17988989C2212yescomplemented
18#17988629—21no—
19#17988625C2211yescharacteristic
20C2×#14556525C2211yescharacteristic
21#17988391C2211yescharacteristic, complemented
22#17988391C2211yescharacteristic, complemented
23#17988391C2212yescomplemented
24#17988388C2212yes—
25#17988388C2212yes—
26#17988388C2212yes—
27#17988388C2212yes—
28#17988388C2211yescharacteristic, complemented
29#17988388C2211yescharacteristic, complemented
30#17988388C2212yescomplemented
31#17988363C2212yescomplemented
32C4×#11924003C2212yescomplemented
33C2×AGL(1,ℤ/8)×C73—21no—
34C2×#135×C73C2211yescharacteristic
35C22×D8×C73C2211yescharacteristic
36#535×C73—21no—
37#496×C73—22no—
38#495×C73C2212yes—
39#493×C73C2212yes—
40#405×C73C2211yescharacteristic
41#14556591—22no—
42#14556590—41no—
43#14556583—22no—
44#14556582—41no—
45#14556568—42no—
46#14556550—21no—
47C2×C4⋊Dic73—21no—
48#14556531—22no—
49C2×#11924003—22no—
50C2×#11924003C2311yescharacteristic
51C2×#11924003C2312yes—
52C2×#11924003C2311yescharacteristic
53#14556525—21no—
54#14556525—22no—
55#14556525C2312yes—
56#14556525C2311yescharacteristic
57#14556525C2311yescharacteristic
58#14556523—21no—
59#14556523—22no—
60C2×#41×C73—21no—
61C22×D4×C73—21no—
62AGL(1,ℤ/8)×C73—44no—
63#135×C73—22no—
64#135×C73—22no—
65C2×Q16×C73C2311yescharacteristic
66C2×SD16×C73—22no—
67C2×SD16×C73—21no—
68C2×D8×C73—22no—
69C2×D8×C73—21no—
70C2×D8×C73—22no—
71C2×D8×C73—22no—
72C2×D8×C73C2312yes—
73C2×D8×C73C2311yescharacteristic
74C2×M16×C73D411yescharacteristic
75C22×C8×C73C2311yescharacteristic
76#127×C73—22no—
77#124×C73—22no—
78#121×C73—41no—
79#121×C73—41no—
80#120×C73—42no—
81#117×C73D411yescharacteristic
82#97×C73D412yes—
83C4×C8×C73C2312yes—
84C2×Dic146—42no—
85C22×Dic73—41no—
86C4⋊Dic73—21no—
87C4⋊Dic73—22no—
88C4⋊Dic73—21no—
89C4⋊Dic73—41no—
90C4⋊Dic73—42no—
91#11924007—42no—
92#11924007—42no—
93C22⋊Dic73—41no—
94C4×Dic73—41no—
95#11924003—21no—
96#11924003—21no—
97#11924003—22no—
98C24×C73—41no—
99#41×C73—42no—
100#41×C73—42no—
101C2×Q8×C73—21no—
102C2×D4×C73—22no—
103C2×D4×C73—21no—
104C2×D4×C73—41no—
105C2×D4×C73—42no—
106C2×D4×C73—41no—
107C2×D4×C73—42no—
108C2×D4×C73—42no—
109C2×D4×C73—42no—
110C22×C4×C73—41no—
111C22×C4×C73C2×D411yescharacteristic
112Q16×C73—21no—
113Q16×C73—21no—
114SD16×C73—42no—
115SD16×C73—44no—
116D8×C73—22no—
117D8×C73—22no—
118D8×C73—21no—
119D8×C73—21no—
120D8×C73—42no—
121D8×C73—44no—
122M16×C73SD1614yes—
123C2×C8×C73C2×D411yescharacteristic
124C2×C8×C73C2×D412yes—
125C2×C8×C73#4112yes—
126C2×C8×C73#4112yes—
127C2×C8×C73C2411yescharacteristic
128C4⋊C4×C73D811yescharacteristic
129C4⋊C4×C73D811yescharacteristic
130C22⋊C4×C73—42no—
131C22⋊C4×C73—42no—
132C22⋊C4×C73D812yes—
133C42×C73C2×D412yes—
134C23×C73—42no—
135C23×C73—41no—
136C23×C73—82no—
137C23×C73—161no—
138C23×C73—161no—
139C23×C73C2×D811yescharacteristic
140Q8×C73—41no—
141Q8×C73—41no—
142D4×C73—42no—
143D4×C73—41no—
144D4×C73—41no—
145D4×C73—42no—
146D4×C73—82no—
147D4×C73—82no—
148D4×C73—162no—
149D4×C73—162no—
150C2×C4×C73—41no—
151C2×C4×C73—82no—
152C2×C4×C73#12412yes—
153C2×C4×C73C2×D812yes—
154C2×C4×C73C2×D812yes—
155C2×C4×C73C2×D811yescharacteristic
156C2×C4×C73C2×SD1612yes—
157C2×C4×C73C22×D411yescharacteristic
158C2×Dic73—41no—
159C2×Dic73—41no—
160C2×Dic73—42no—
161C2×Dic73—161no—
162Dic146—162no—
163C8×C73#12412yes—
164C8×C73C2×SD1614yes—
165C8×C73C2×#4111yescharacteristic
166C8×C73C2×#4111yescharacteristic
167C22×C73—21no—
168C22×C73—21no—
169C22×C73—81no—
170C22×C73—82no—
171C22×C73—81no—
172C22×C73—82no—
173C22×C73—81no—
174C22×C73—82no—
175C22×C73—164no—
176C22×C73#48612yes—
177C22×C73C22×D811yescharacteristic
178C4×C73—22no—
179C4×C73—22no—
180C4×C73—81no—
181C4×C73#48612yes—
182C4×C73C2×#12411yescharacteristic
183C4×C73C22×SD1611yescharacteristic
184Dic73—81no—
185Dic73—82no—
186Dic73—81no—
187#33702—731nomaximal, Sylow
188C2×C73—22no—
189C2×C73—161no—
190C2×C73—162no—
191C2×C73C2×#48611yescharacteristic
192C2×C73#341511yescharacteristic
193C2×C73#341611yescharacteristic
194#3164—731no—
195#3155D7311yescharacteristic
196#2244—732no—
197#2243—732no—
198#2242—732no—
199#2240—732no—
200#2140—731no—
201#2139—731no—
202#2138—731no—
203C2×#357—731no—
204#2108—731no—
205C73#3370211yescharacteristic, complemented, Sylow
206C2×AGL(1,ℤ/8)—21no—
207C2×#135C2×D7311yescharacteristic
208C22×D8C2×D7311yescharacteristic
209#544—1461no—
210#535—21no—
211C2×M32—731no—
212C22×C16—731no—
213#501—1462no—
214#500—732no—
215#499—732no—
216#496—22no—
217#495C2×D7312yes—
218#493C2×D7312yes—
219#428—1461no—
220#427—731no—
221C2×#107—731no—
222#405C2×D7311yescharacteristic
223#358—731no—
224#358—731no—
225#358—732no—
226#357—732no—
227#357—732no—
228#357—732no—
229#357—732no—
230#357—731no—
231#357—731no—
232#357—732no—
233#346—732no—
234C4×C16—732no—
235C2×#41—21no—
236C22×D4—21no—
237AGL(1,ℤ/8)—44no—
238#135—22no—
239#135—22no—
240C2×Q16C22×D7311yescharacteristic
241C2×SD16—21no—
242C2×SD16—22no—
243C2×D8—21no—
244C2×D8—22no—
245C2×D8—22no—
246C2×D8—22no—
247C2×D8C22×D7312yes—
248C2×D8C22×D7311yescharacteristic
249C2×M16C22⋊D7311yescharacteristic
250C22×C8C22×D7311yescharacteristic
251C4⋊Q8—2921no—
252C4⋊Q8—1462no—
253#127—22no—
254#125—2921no—
255#125—1462no—
256#124—22no—
257C22⋊Q8—2922no—
258#121—41no—
259#121—41no—
260#120—42no—
261#117—1461no—
262#117C22⋊D7311yescharacteristic
263C2×C4⋊C4—1461no—
264M32—1462no—
265C2×C16—731no—
266C2×C16—732no—
267C2×C16—731no—
268C2×C16—1462no—
269#107—732no—
270#107—731no—
271#107—731no—
272#107—1461no—
273#107—1462no—
274#106—1461no—
275#106—1462no—
276#97C22⋊D7312yes—
277C4×C8C22×D7312yes—
278C24—41no—
279#41—42no—
280#41—42no—
281C2×Q8—21no—
282C2×Q8—2922no—
283C2×D4—22no—
284C2×D4—21no—
285C2×D4—41no—
286C2×D4—42no—
287C2×D4—41no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C22×C4—41no—
292C22×C4—2921no—
293C22×C4C2×C22⋊D7311yescharacteristic
294Q16—21no—
295Q16—21no—
296SD16—42no—
297SD16—44no—
298D8—22no—
299D8—22no—
300D8—21no—
301D8—21no—
302D8—42no—
303D8—44no—
304M16#1192401614yes—
305C2×C8C2×C22⋊D7311yescharacteristic
306C2×C8C2×C22⋊D7312yes—
307C2×C8#1192402712yes—
308C2×C8#1192402712yes—
309C2×C8C23×D7311yescharacteristic
310C4⋊C4—2922no—
311C4⋊C4—2922no—
312C4⋊C4—2921no—
313C4⋊C4—2922no—
314C4⋊C4—1461no—
315C4⋊C4—1462no—
316C4⋊C4—1461no—
317C4⋊C4#1192401311yescharacteristic
318C4⋊C4#1192401311yescharacteristic
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—2921no—
322C22⋊C4#1192401312yes—
323C42—2921no—
324C42C2×C22⋊D7312yes—
325C16—1461no—
326C16—1461no—
327C16—1462no—
328C23—42no—
329C23—41no—
330C23—82no—
331C23—161no—
332C23—161no—
333C23C2×#1192401311yescharacteristic
334Q8—41no—
335Q8—41no—
336Q8—11682no—
337D4—42no—
338D4—41no—
339D4—41no—
340D4—42no—
341D4—82no—
342D4—82no—
343D4—162no—
344D4—162no—
345C2×C4—41no—
346C2×C4—82no—
347C2×C4—2921no—
348C2×C4—2921no—
349C2×C4—2922no—
350C2×C4—11681no—
351C2×C4#1455657812yes—
352C2×C4C2×#1192401312yes—
353C2×C4C2×#1192401312yes—
354C2×C4C2×#1192401311yescharacteristic
355C2×C4C2×#1192401612yes—
356C2×C4C22×C22⋊D7311yescharacteristic
357C8#1455657812yes—
358C8C2×#1192401614yes—
359C8C2×#1192402711yescharacteristic
360C8C2×#1192402711yescharacteristic
361C22—21no—
362C22—21no—
363C22—81no—
364C22—82no—
365C22—81no—
366C22—82no—
367C22—81no—
368C22—82no—
369C22—164no—
370C22#1798893312yes—
371C22C22×#1192401311yescharacteristic
372C4—22no—
373C4—22no—
374C4—81no—
375C4—5842no—
376C4—5841no—
377C4—5841no—
378C4#1798893312yes—
379C4C2×#1455657811yescharacteristic
380C4C22×#1192401611yescharacteristic
381C2—22no—
382C2—162no—
383C2—161no—
384C2C2×#1798893311yescharacteristic
385C2#2262307011yescharacteristic
386C2#2262307811yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
Fitting, prime radicalC73×#3155C2
socleC22×C73C22×D8
hypercenter#3155D73
hypocenterC73#33702
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3155

Complements

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

NH ≅ G/Ntype
1. #22620470C2semidirect
2. #22612172C2semidirect
3. #22612167C2semidirect
4. #22612162C2semidirect
5. #22612154C2semidirect
6. #22611647C2semidirect
7. #22611643C2semidirect
8. #22611638C2semidirect
9. C2×#17988388C2semidirect
10. #22611528C2semidirect
16. #17988992C22semidirect
17. #17988989C22semidirect
21. #17988391C22semidirect
22. #17988391C22semidirect
23. #17988391C22semidirect
28. #17988388C22semidirect
29. #17988388C22semidirect
30. #17988388C22semidirect
31. #17988363C22semidirect
32. C4×#11924003C22semidirect
205. C73#33702semidirect

Maximal subgroups

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

subgroupindex
#226204702
#226121722
#226121672
#226121622
#226121542
#226116472
#226116432
#226116382
C2×#179883882
#226115282
C73×#31552
#3370273