Finite Group Databasebeta

← #33241332

Subgroups — #33241332

Order 18688.

570 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620425C211yesmaximal, characteristic
2#22612178C212yesmaximal, complemented
3#22612170C212yesmaximal, complemented
4#22612165C212yesmaximal, complemented
5#22612160C212yesmaximal, complemented
6#22611650C211yesmaximal, characteristic, complemented
7#22611641C211yesmaximal, characteristic
8#22611636C211yesmaximal, characteristic
9C2×#17988390C211yesmaximal, characteristic, complemented
10#22611528C211yesmaximal, characteristic, complemented
11C73×#3164C211yesmaximal, characteristic, complemented
12C2×#14556624—21no—
13C2×#14556613C2211yescharacteristic
14C22×Dic292C2211yescharacteristic
15#17989163—21no—
16C2×#14556531C2211yescharacteristic
17C22×#11924003C2211yescharacteristic
18#17988981—22no—
19#17988971C2212yes—
20#17988967C2212yes—
21#17988390C2212yes—
22#17988390C2212yes—
23#17988390C2211yescharacteristic
24#17988390C2211yescharacteristic
25#17988390C2212yes—
26#17988390C2212yes—
27#17988390C2212yes—
28#17988387C2211yescharacteristic
29#17988387C2211yescharacteristic
30#17988387C2212yes—
31#17988363C2212yes—
32C4×#11924003C2212yes—
33#544×C73—21no—
34#501×C73—22no—
35#500×C73C2212yes—
36#499×C73C2212yes—
37#428×C73—21no—
38#427×C73C2211yescharacteristic
39C2×#107×C73C2211yescharacteristic
40#405×C73C2211yescharacteristic
41C2×#11924028—21no—
42C22×Dic146—21no—
43#14556624—44no—
44#14556613—22no—
45#14556613—22no—
46C2×Dic292—22no—
47C2×Dic292—21no—
48C2×Dic292—22no—
49C2×Dic292—22no—
50C2×Dic292C2311yescharacteristic
51C2×Dic292C2312yes—
52C2×#11924017—22no—
53C2×#11924017—21no—
54C2×D584C2311yescharacteristic
55#14556591—22no—
56#14556579—22no—
57#14556575—42no—
58#14556571—41no—
59#14556571—41no—
60#14556531—22no—
61C2×#11924003—22no—
62C2×#11924003C2311yescharacteristic
63C2×#11924003C2312yes—
64C2×#11924003C2311yescharacteristic
65C2×M16×C73D411yescharacteristic
66C22×C8×C73C2311yescharacteristic
67C4⋊Q8×C73—22no—
68C4⋊Q8×C73—41no—
69#125×C73—22no—
70#125×C73—41no—
71C22⋊Q8×C73—42no—
72#117×C73—21no—
73#117×C73D411yescharacteristic
74C2×C4⋊C4×C73—21no—
75#107×C73—21no—
76#107×C73—22no—
77#107×C73C2312yes—
78#107×C73C2311yescharacteristic
79#107×C73C2311yescharacteristic
80#106×C73—21no—
81#106×C73—22no—
82#97×C73D412yes—
83C4×C8×C73C2312yes—
84#11924028—42no—
85#11924028—42no—
86C2×Dic146—22no—
87C2×Dic146—21no—
88C2×Dic146—42no—
89C2×Dic146—42no—
90C2×Dic146—41no—
91C2×D292—21no—
92C2×C22⋊D73—41no—
93C22×Dic73—41no—
94C2×C4×D73—41no—
95Dic292—22no—
96Dic292—22no—
97Dic292—21no—
98Dic292—21no—
99Dic292—42no—
100Dic292—44no—
101#11924017—42no—
102#11924017—44no—
103D584—21no—
104D584—21no—
105C4⋊Dic73—42no—
106C4⋊Dic73—42no—
107#11924005—42no—
108#11924005—42no—
109#11924003—21no—
110#11924003—21no—
111#11924003—22no—
112C2×Q8×C73—42no—
113C22×C4×C73—41no—
114C22×C4×C73C2×D411yescharacteristic
115M16×C73SD1614yes—
116C2×C8×C73C2×D411yescharacteristic
117C2×C8×C73C2×D412yes—
118C2×C8×C73#4112yes—
119C2×C8×C73#4112yes—
120C2×C8×C73C2411yescharacteristic
121C4⋊C4×C73—21no—
122C4⋊C4×C73—22no—
123C4⋊C4×C73—21no—
124C4⋊C4×C73—41no—
125C4⋊C4×C73—42no—
126C4⋊C4×C73—42no—
127C4⋊C4×C73—42no—
128C4⋊C4×C73D811yescharacteristic
129C4⋊C4×C73D811yescharacteristic
130C22⋊C4×C73—41no—
131C22⋊C4×C73D812yes—
132C42×C73—41no—
133C42×C73C2×D412yes—
134C23×C73C2×D811yescharacteristic
135C22×D73—41no—
136Q8×C73—162no—
137C2×C4×C73—41no—
138C2×C4×C73—41no—
139C2×C4×C73—42no—
140C2×C4×C73—161no—
141C2×C4×C73#12412yes—
142C2×C4×C73C2×D812yes—
143C2×C4×C73C2×D812yes—
144C2×C4×C73C2×D811yescharacteristic
145C2×C4×C73C2×SD1612yes—
146C2×C4×C73C22×D411yescharacteristic
147C22⋊D73—82no—
148C2×Dic73—42no—
149C2×Dic73—41no—
150C2×Dic73—82no—
151D292—41no—
152D292—41no—
153C4×D73—82no—
154Dic146—41no—
155Dic146—42no—
156Dic146—42no—
157Dic146—41no—
158Dic146—82no—
159C8×C73#12412yes—
160C8×C73C2×SD1614yes—
161C8×C73C2×#4111yescharacteristic
162C8×C73C2×#4111yescharacteristic
163C22×C73—21no—
164C22×C73—21no—
165C22×C73#48612yes—
166C22×C73C22×D811yescharacteristic
167C2×D73—81no—
168C2×D73—81no—
169C2×D73—81no—
170C4×C73—22no—
171C4×C73—22no—
172C4×C73—82no—
173C4×C73—81no—
174C4×C73—81no—
175C4×C73#48612yes—
176C4×C73C2×#12411yescharacteristic
177C4×C73C22×SD1611yescharacteristic
178Dic73—82no—
179Dic73—81no—
180#33703—731nomaximal, Sylow
181C2×C73—22no—
182C2×C73C2×#48611yescharacteristic
183C2×C73#341611yescharacteristic
184C2×C73#341711yescharacteristic
185D73—161no—
186#3164D7311yescharacteristic, complemented
187#3156—731no—
188#2245—732no—
189#2243—732no—
190#2242—732no—
191#2241—732no—
192#2141—731no—
193#2139—731no—
194#2138—731no—
195C2×#358—731no—
196#2108—731no—
197C73#3370311yescharacteristic, complemented, Sylow
198C2×#137—1461no—
199C2×#135—731no—
200C22×Q16—731no—
201#544—21no—
202#536—1461no—
203C2×M32—731no—
204C22×C16—731no—
205#501—22no—
206#500C2×D7312yes—
207#499C2×D7312yes—
208#497—1462no—
209#495—732no—
210#494—732no—
211#428—21no—
212#427C2×D7311yescharacteristic
213C2×#107C2×D7311yescharacteristic
214#405C2×D7311yescharacteristic
215#358—732no—
216#358—732no—
217#358—731no—
218#358—731no—
219#358—732no—
220#358—732no—
221#358—732no—
222#357—731no—
223#357—731no—
224#357—732no—
225#346—732no—
226C4×C16—732no—
227C2×#41—1461no—
228C22×Q8—1461no—
229#137—2924no—
230#135—1462no—
231#135—1462no—
232C2×Q16—732no—
233C2×Q16—731no—
234C2×Q16—1461no—
235C2×Q16—1462no—
236C2×Q16—1462no—
237C2×Q16—1462no—
238C2×SD16—1461no—
239C2×SD16—1462no—
240C2×D8—731no—
241C2×M16C22⋊D7311yescharacteristic
242C22×C8C22×D7311yescharacteristic
243C4⋊Q8—22no—
244C4⋊Q8—41no—
245C4⋊Q8—1462no—
246#125—22no—
247#125—41no—
248#124—1462no—
249#123—2922no—
250C22⋊Q8—42no—
251C22⋊Q8—2921no—
252C22⋊Q8—2921no—
253#117—21no—
254#117C22⋊D7311yescharacteristic
255C2×C4⋊C4—21no—
256M32—1462no—
257C2×C16—731no—
258C2×C16—732no—
259C2×C16—731no—
260C2×C16—1462no—
261#107—21no—
262#107—22no—
263#107C22×D7312yes—
264#107C22×D7311yescharacteristic
265#107C22×D7311yescharacteristic
266#106—21no—
267#106—22no—
268#97C22⋊D7312yes—
269C4×C8C22×D7312yes—
270#41—2922no—
271#41—2922no—
272C2×Q8—42no—
273C2×Q8—2921no—
274C2×Q8—2922no—
275C2×Q8—2922no—
276C2×Q8—1462no—
277C2×Q8—1461no—
278C2×D4—2921no—
279C2×D4—1461no—
280C22×C4—41no—
281C22×C4—2921no—
282C22×C4—2921no—
283C22×C4C2×C22⋊D7311yescharacteristic
284Q16—2922no—
285Q16—2924no—
286Q16—1462no—
287Q16—1462no—
288Q16—1461no—
289Q16—1461no—
290SD16—2922no—
291SD16—2924no—
292D8—1461no—
293D8—1461no—
294M16#1192401514yes—
295C2×C8C2×C22⋊D7311yescharacteristic
296C2×C8C2×C22⋊D7312yes—
297C2×C8#1192402912yes—
298C2×C8#1192402912yes—
299C2×C8C23×D7311yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—2922no—
308C4⋊C4—2922no—
309C4⋊C4#1192401311yescharacteristic
310C4⋊C4#1192401311yescharacteristic
311C22⋊C4—41no—
312C22⋊C4—2922no—
313C22⋊C4—2922no—
314C22⋊C4#1192401312yes—
315C42—41no—
316C42C2×C22⋊D7312yes—
317C16—1461no—
318C16—1461no—
319C16—1462no—
320C23—2921no—
321C23C2×#1192401311yescharacteristic
322Q8—162no—
323Q8—2922no—
324Q8—2921no—
325Q8—2921no—
326Q8—2922no—
327Q8—5842no—
328D4—2921no—
329D4—2921no—
330D4—5842no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—161no—
335C2×C4—2922no—
336C2×C4—2921no—
337C2×C4—5842no—
338C2×C4—5842no—
339C2×C4#1455658012yes—
340C2×C4C2×#1192401312yes—
341C2×C4C2×#1192401312yes—
342C2×C4C2×#1192401311yescharacteristic
343C2×C4C2×#1192401512yes—
344C2×C4C22×C22⋊D7311yescharacteristic
345C8#1455658012yes—
346C8C2×#1192401514yes—
347C8C2×#1192402911yescharacteristic
348C8C2×#1192402911yescharacteristic
349C22—21no—
350C22—21no—
351C22—5841no—
352C22—5841no—
353C22—5841no—
354C22#1798893512yes—
355C22C22×#1192401311yescharacteristic
356C4—22no—
357C4—22no—
358C4—82no—
359C4—81no—
360C4—81no—
361C4—5841no—
362C4—5842no—
363C4#1798893512yes—
364C4C2×#1455658011yescharacteristic
365C4C22×#1192401511yescharacteristic
366C2—22no—
367C2—11681no—
368C2C2×#1798893511yescharacteristic
369C2#2262307811yescharacteristic
370C2#2262308211yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#3164C2
socleC22×C73C22×D8
prime radicalC2×#14556613C22
hypercenter#3164D73
hypocenterC73#33703
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3164

Complements

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

NH ≅ G/Ntype
2. #22612178C2semidirect
3. #22612170C2semidirect
4. #22612165C2semidirect
5. #22612160C2semidirect
6. #22611650C2semidirect
9. C2×#17988390C2semidirect
10. #22611528C2semidirect
11. C73×#3164C2semidirect
186. #3164D73semidirect
197. C73#33703semidirect

Maximal subgroups

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

subgroupindex
#226204252
#226121782
#226121702
#226121652
#226121602
#226116502
#226116412
#226116362
C2×#179883902
#226115282
C73×#31642
#3370373