Finite Group Databasebeta

← #33003374

Subgroups — #33003374

Order 18688.

510 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556531C211yesmaximal, characteristic, complemented
2#22619512C211yesmaximal, characteristic
3C2×#17988396C212yesmaximal, complemented
4#22608038C212yesmaximal, complemented
5C73×#1807C211yesmaximal, characteristic, complemented
6C2×#14556613C2211yescharacteristic
7#17989098—21no—
8C2×#14556531—24no—
9C2×#14556531—21no—
10C2×#14556531C2212yes—
11C22×#11924003—21no—
12#17988828C411yescharacteristic, complemented
13#17988828C411yescharacteristic, complemented
14#17988823C411yescharacteristic, complemented
15#17988823C411yescharacteristic, complemented
16#17988816—21no—
17#17988816—21no—
18#17988625C2211yescharacteristic
19#17988584—21no—
20#17988396—24no—
21C23×C8×C73C2211yescharacteristic
22C2×#106×C73C2212yes—
23#380×C73—21no—
24#340×C73—21no—
25C2×#11924028—21no—
26#14556613—22no—
27#14556613—22no—
28C2×Dic292C2×C411yescharacteristic
29C2×#11924017—21no—
30C2×D584C2×C411yescharacteristic
31#14556574—41no—
32#14556570—21no—
33#14556570—21no—
34#14556565—21no—
35#14556565—21no—
36#14556560—41no—
37C4×C22⋊D73—41no—
38C4×C22⋊D73—41no—
39#14556550—21no—
40#14556531—24no—
41#14556531—22no—
42#14556531D414yescomplemented
43C2×#11924003—24no—
44C2×#11924003—22no—
45#14556525C2×C411yescharacteristic
46#14556525C2×C411yescharacteristic
47#14556523—21no—
48#14556516—21no—
49#14556516—21no—
50#14556513—21no—
51#14556513—21no—
52C23×C4×C73D411yescharacteristic
53C22×C8×C73—22no—
54C22×C8×C73—22no—
55C22×C8×C73C2×C411yescharacteristic
56C22×C8×C73C2×C411yescharacteristic
57C22×C8×C73D411yescharacteristic
58C22×C8×C73D411yescharacteristic
59C22×C8×C73D411yescharacteristic
60C22×C8×C73C2311yescharacteristic
61C2×C4⋊C4×C73—22no—
62C2×C22⋊C4×C73—41no—
63#106×C73—22no—
64#106×C73—22no—
65#95×C73—41no—
66#11924028—42no—
67#11924028—42no—
68C2×Dic146—21no—
69C2×D292—21no—
70C2×C22⋊D73—41no—
71C2×C22⋊D73—41no—
72C2×C22⋊D73—41no—
73C2×C4×D73—41no—
74Dic292—22no—
75#11924017—41no—
76#11924017—41no—
77D584—22no—
78C4⋊Dic73—21no—
79C4⋊Dic73—21no—
80#11924007—41no—
81#11924007—41no—
82C22⋊Dic73—41no—
83C22⋊Dic73—41no—
84C22⋊Dic73—41no—
85#11924005—41no—
86#11924005—41no—
87C4×Dic73—41no—
88#11924003—24no—
89C24×C73D811yescharacteristic
90C22×C4×C73—22no—
91C22×C4×C73—22no—
92C22×C4×C73—42no—
93C22×C4×C73C22⋊C411yescharacteristic
94C22×C4×C73C22⋊C411yescharacteristic
95C22×C4×C73D811yescharacteristic
96C22×C4×C73SD1611yescharacteristic
97C22×C4×C73SD1611yescharacteristic
98C22×C4×C73C2×D411yescharacteristic
99C2×C8×C73—22no—
100C2×C8×C73—22no—
101C2×C8×C73—22no—
102C2×C8×C73—21no—
103C2×C8×C73—21no—
104C2×C8×C73—21no—
105C2×C8×C73—21no—
106C2×C8×C73C22⋊C411yescharacteristic
107C2×C8×C73C22⋊C411yescharacteristic
108C2×C8×C73C22×C411yescharacteristic
109C2×C8×C73C2×D412yes—
110C2×C8×C73C2×D411yescharacteristic
111C2×C8×C73#4112yes—
112C4⋊C4×C73—42no—
113C4⋊C4×C73—42no—
114C22⋊C4×C73—82no—
115C23×C73—22no—
116C23×C73—22no—
117C23×C73—41no—
118C23×C73#10211yescharacteristic
119C23×C73#10211yescharacteristic
120C23×C73C2×D811yescharacteristic
121C22×D73—41no—
122C2×C4×C73—22no—
123C2×C4×C73—21no—
124C2×C4×C73—21no—
125C2×C4×C73—21no—
126C2×C4×C73—21no—
127C2×C4×C73—22no—
128C2×C4×C73—22no—
129C2×C4×C73—82no—
130C2×C4×C73—82no—
131C2×C4×C73—82no—
132C2×C4×C73#10211yescharacteristic
133C2×C4×C73#10211yescharacteristic
134C2×C4×C73C2×C22⋊C411yescharacteristic
135C2×C4×C73#12112yes—
136C2×C4×C73C2×SD1611yescharacteristic
137C2×C4×C73#13512yes—
138C22⋊D73—82no—
139C22⋊D73—82no—
140C22⋊D73—82no—
141C2×Dic73—41no—
142C2×Dic73—41no—
143C2×Dic73—41no—
144D292—41no—
145D292—41no—
146C4×D73—82no—
147Dic146—41no—
148Dic146—41no—
149C8×C73—21no—
150C8×C73—21no—
151C8×C73C4×D412yes—
152C8×C73#12011yescharacteristic
153C8×C73#12311yescharacteristic
154C22×C73—21no—
155C22×C73—21no—
156C22×C73—21no—
157C22×C73—21no—
158C22×C73—21no—
159C22×C73—21no—
160C22×C73—21no—
161C22×C73—44no—
162C22×C73#36012yescomplemented
163C22×C73C2×#10211yescharacteristic
164C22×C73#46612yes—
165C2×D73—81no—
166C2×D73—81no—
167C2×D73—81no—
168C4×C73—21no—
169C4×C73—21no—
170C4×C73—162no—
171C4×C73#38611yescharacteristic
172C4×C73#41611yescharacteristic
173C4×C73#46512yes—
174Dic73—81no—
175Dic73—81no—
176Dic73—81no—
177#13065—731nomaximal, Sylow
178C2×C73—22no—
179C2×C73—22no—
180C2×C73—41no—
181C2×C73#189311yescharacteristic
182C2×C73C2×#36011yescharacteristic
183C2×C73#214511yescharacteristic
184D73—161no—
185C22×M32—731no—
186#3050—731no—
187C2×#360—732no—
188#1807D7311yescharacteristic, complemented
189#1386—732no—
190C73#1306511yescharacteristic, complemented, Sylow
191C2×#135—731no—
192C23×C8C2×D7311yescharacteristic
193#525—1461no—
194C2×M32—732no—
195C2×M32—1464no—
196C2×M32—1461no—
197C22×C16—1461no—
198#467—731no—
199#467—731no—
200#466—731no—
201#466—731no—
202#465—1461no—
203#465—1461no—
204#427—731no—
205C2×#106C2×D7312yes—
206#416—1461no—
207#380—21no—
208#360—1464no—
209#340—21no—
210C2×#41—1461no—
211C23×C4C22⋊D7311yescharacteristic
212#135—1462no—
213#135—1462no—
214C2×Q16—731no—
215C2×SD16—1461no—
216C2×D8—731no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8C4×D7311yescharacteristic
220C22×C8C4×D7311yescharacteristic
221C22×C8D29211yescharacteristic
222C22×C8D29211yescharacteristic
223C22×C8C22⋊D7311yescharacteristic
224C22×C8C22×D7311yescharacteristic
225#123—2921no—
226C22⋊Q8—1461no—
227C22⋊Q8—1461no—
228#121—1461no—
229#121—1461no—
230#120—2921no—
231C4×D4—2921no—
232C4×D4—2921no—
233#117—1461no—
234C2×C4⋊C4—22no—
235C2×C22⋊C4—41no—
236M32—734no—
237M32—1464no—
238M32—1462no—
239C2×C16—1464no—
240C2×C16—1462no—
241#107—731no—
242#107—731no—
243#106—22no—
244#106—22no—
245#106—1461no—
246#103—1461no—
247#103—1461no—
248#102—1461no—
249#102—1461no—
250#95—41no—
251C24#1192401311yescharacteristic
252#41—2922no—
253#41—2922no—
254C2×Q8—1461no—
255C2×D4—2921no—
256C2×D4—2921no—
257C2×D4—2921no—
258C2×D4—1461no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—42no—
262C22×C4—2921no—
263C22×C4#1192400511yescharacteristic
264C22×C4#1192400511yescharacteristic
265C22×C4#1192401311yescharacteristic
266C22×C4#1192401511yescharacteristic
267C22×C4#1192401511yescharacteristic
268C22×C4C2×C22⋊D7311yescharacteristic
269Q16—1462no—
270SD16—2921no—
271SD16—2921no—
272D8—1462no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8#1192400511yescharacteristic
281C2×C8#1192400511yescharacteristic
282C2×C8C2×C4×D7311yescharacteristic
283C2×C8D4×D7312yes—
284C2×C8C2×D29211yescharacteristic
285C2×C8#1192402712yes—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—2921no—
289C4⋊C4—2921no—
290C4⋊C4—1461no—
291C4⋊C4—1461no—
292C22⋊C4—82no—
293C22⋊C4—2921no—
294C22⋊C4—2921no—
295C22⋊C4—2921no—
296C22⋊C4—2921no—
297C22⋊C4—2921no—
298C42—2921no—
299C16—1464no—
300C23—22no—
301C23—22no—
302C23—41no—
303C23—2921no—
304C23#1455651111yescharacteristic
305C23#1455651111yescharacteristic
306C23C2×#1192401311yescharacteristic
307Q8—2921no—
308Q8—2921no—
309D4—2921no—
310D4—2921no—
311D4—5842no—
312D4—5842no—
313D4—5842no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—2921no—
325C2×C4—2921no—
326C2×C4—2921no—
327C2×C4—5842no—
328C2×C4#1455651111yescharacteristic
329C2×C4#1455651111yescharacteristic
330C2×C4C2×#1192400511yescharacteristic
331C2×C4#1455656312yes—
332C2×C4C2×#1192401511yescharacteristic
333C2×C4#1455661112yes—
334C8—21no—
335C8—21no—
336C8#1455655112yes—
337C8#1455656111yescharacteristic
338C8#1455657511yescharacteristic
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—44no—
347C22—5841no—
348C22—5841no—
349C22—5841no—
350C22#1798839612yescomplemented
351C22C2×#1455651111yescharacteristic
352C22#1798882112yes—
353C4—21no—
354C4—21no—
355C4—162no—
356C4—5841no—
357C4—5841no—
358C4—5841no—
359C4#1798846411yescharacteristic
360C4#1798858111yescharacteristic
361C4#1798881812yes—
362C2—22no—
363C2—22no—
364C2—41no—
365C2—11681no—
366C2#2261035711yescharacteristic
367C2C2×#1798839611yescharacteristic
368C2#2261166611yescharacteristic
subgroupG/H
centerC22C2×#14556511
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC73×#1807C2
socleC22×C73C2×#102
prime radical#22619512C2
cosocleC22×C8×C73C23
hypercenter#1807D73
hypocenterC73#13065
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #1807

Complements

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

NH ≅ G/Ntype
1. C22×#14556531C2semidirect
3. C2×#17988396C2semidirect
4. #22608038C2semidirect
5. C73×#1807C2semidirect
12. #17988828C4semidirect
13. #17988828C4semidirect
14. #17988823C4semidirect
15. #17988823C4semidirect
42. #14556531D4semidirect
162. C22×C73#360semidirect
188. #1807D73semidirect
190. C73#13065semidirect
350. C22#17988396semidirect

Maximal subgroups

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

subgroupindex
C22×#145565312
#226195122
C2×#179883962
#226080382
C73×#18072
#1306573