Finite Group Databasebeta

← #33003376

Subgroups — #33003376

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
1#22619512C211yesmaximal, characteristic
2C2×#17988398C212yesmaximal, complemented
3#22609956C211yesmaximal, characteristic, complemented
4#22608037C212yesmaximal, complemented
5C22×M32×C73C211yesmaximal, characteristic, complemented
6C2×#14556613C2211yescharacteristic
7#17989098—21no—
8#17988828C411yescharacteristic, complemented
9#17988828C411yescharacteristic, complemented
10#17988823C411yescharacteristic, complemented
11#17988823C411yescharacteristic, complemented
12#17988816—21no—
13#17988816—21no—
14#17988625C2211yescharacteristic
15C2×#14556523C2212yes—
16#17988584—21no—
17#17988447—21no—
18#17988398—24no—
19#17988350—21no—
20C23×C8×C73C2211yescharacteristic
21C2×M32×C73—21no—
22C2×M32×C73—24no—
23C2×M32×C73C2212yes—
24C22×C16×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—
40C2×C4⋊Dic73—22no—
41C2×C22⋊Dic73—41no—
42#14556525C2×C411yescharacteristic
43#14556525C2×C411yescharacteristic
44#14556523—21no—
45#14556523—22no—
46#14556523—22no—
47#14556516—21no—
48#14556516—21no—
49#14556513—21no—
50#14556513—21no—
51#14556498—41no—
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
61M32×C73—24no—
62M32×C73—22no—
63M32×C73D414yescomplemented
64C2×C16×C73—24no—
65C2×C16×C73—22no—
66#11924028—42no—
67#11924028—42no—
68C2×Dic146—21no—
69C2×D292—21no—
70C2×C22⋊D73—41no—
71C2×C22⋊D73—41no—
72C2×C22⋊D73—41no—
73C22×Dic73—42no—
74C2×C4×D73—41no—
75Dic292—22no—
76#11924017—41no—
77#11924017—41no—
78D584—22no—
79C4⋊Dic73—21no—
80C4⋊Dic73—21no—
81C4⋊Dic73—42no—
82C4⋊Dic73—42no—
83#11924007—41no—
84#11924007—41no—
85C22⋊Dic73—41no—
86C22⋊Dic73—41no—
87C22⋊Dic73—41no—
88C22⋊Dic73—82no—
89#11924005—41no—
90#11924005—41no—
91C4×Dic73—41no—
92C24×C73D811yescharacteristic
93C22×C4×C73—22no—
94C22×C4×C73—22no—
95C22×C4×C73C22⋊C411yescharacteristic
96C22×C4×C73C22⋊C411yescharacteristic
97C22×C4×C73D811yescharacteristic
98C22×C4×C73SD1611yescharacteristic
99C22×C4×C73SD1611yescharacteristic
100C22×C4×C73C2×D411yescharacteristic
101C2×C8×C73—22no—
102C2×C8×C73—22no—
103C2×C8×C73—22no—
104C2×C8×C73—21no—
105C2×C8×C73—21no—
106C2×C8×C73—21no—
107C2×C8×C73—21no—
108C2×C8×C73C22⋊C411yescharacteristic
109C2×C8×C73C22⋊C411yescharacteristic
110C2×C8×C73C22×C411yescharacteristic
111C2×C8×C73C2×D412yes—
112C2×C8×C73C2×D411yescharacteristic
113C2×C8×C73#4112yes—
114C16×C73—24no—
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#10211yescharacteristic
130C2×C4×C73#10211yescharacteristic
131C2×C4×C73C2×C22⋊C411yescharacteristic
132C2×C4×C73#12112yes—
133C2×C4×C73C2×SD1611yescharacteristic
134C2×C4×C73#13512yes—
135C22⋊D73—82no—
136C22⋊D73—82no—
137C22⋊D73—82no—
138C2×Dic73—41no—
139C2×Dic73—41no—
140C2×Dic73—41no—
141C2×Dic73—82no—
142C2×Dic73—82no—
143C2×Dic73—82no—
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#38611yescharacteristic
171C4×C73#41611yescharacteristic
172C4×C73#46512yes—
173Dic73—81no—
174Dic73—81no—
175Dic73—81no—
176Dic73—162no—
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×M32D7311yescharacteristic, complemented
186#3050—731no—
187C2×#360—732no—
188#1807—731no—
189#1386—732no—
190C73#1306511yescharacteristic, complemented, Sylow
191C2×#135—731no—
192C23×C8C2×D7311yescharacteristic
193#525—1461no—
194C2×M32—21no—
195C2×M32—24no—
196C2×M32C2×D7312yes—
197C22×C16—21no—
198#467—731no—
199#467—731no—
200#466—731no—
201#466—731no—
202#465—1461no—
203#465—1461no—
204#427—731no—
205C2×#106—732no—
206#416—1461no—
207#380—1461no—
208#360—1464no—
209#340—1461no—
210C2×#41—1461no—
211C23×C4D29211yescharacteristic
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×C8C22⋊D7311yescharacteristic
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—1462no—
235C2×C22⋊C4—2921no—
236M32—24no—
237M32—22no—
238M32C22⋊D7314yescomplemented
239C2×C16—24no—
240C2×C16—22no—
241#107—731no—
242#107—731no—
243#106—1462no—
244#106—1462no—
245#106—1461no—
246#103—1461no—
247#103—1461no—
248#102—1461no—
249#102—1461no—
250#95—2921no—
251C24D58411yescharacteristic
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—2921no—
262C22×C4—2922no—
263C22×C4#1192400511yescharacteristic
264C22×C4#1192400511yescharacteristic
265C22×C4D58411yescharacteristic
266C22×C4#1192401711yescharacteristic
267C22×C4#1192401711yescharacteristic
268C22×C4C2×D29211yescharacteristic
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×C8C2×C22⋊D7312yes—
284C2×C8C2×C22⋊D7311yescharacteristic
285C2×C8#1192402812yes—
286C4⋊C4—2921no—
287C4⋊C4—2921no—
288C4⋊C4—2922no—
289C4⋊C4—2922no—
290C4⋊C4—1461no—
291C4⋊C4—1461no—
292C22⋊C4—2921no—
293C22⋊C4—2921no—
294C22⋊C4—2921no—
295C22⋊C4—2921no—
296C22⋊C4—2921no—
297C22⋊C4—5842no—
298C42—2921no—
299C16—24no—
300C23—22no—
301C23—22no—
302C23—41no—
303C23—2921no—
304C23#1455651311yescharacteristic
305C23#1455651311yescharacteristic
306C23C2×D58411yescharacteristic
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—2921no—
322C2×C4—2921no—
323C2×C4—2921no—
324C2×C4—5842no—
325C2×C4—5842no—
326C2×C4—5842no—
327C2×C4—5842no—
328C2×C4#1455651311yescharacteristic
329C2×C4#1455651311yescharacteristic
330C2×C4C2×#1192400511yescharacteristic
331C2×C4#1455656512yes—
332C2×C4C2×#1192401711yescharacteristic
333C2×C4#1455661312yes—
334C8—21no—
335C8—21no—
336C8C4×C22⋊D7312yes—
337C8#1455656011yescharacteristic
338C8#1455657411yescharacteristic
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—44no—
347C22—5841no—
348C22—5841no—
349C22—5841no—
350C22#1798839812yescomplemented
351C22C2×#1455651311yescharacteristic
352C22#1798882312yes—
353C4—21no—
354C4—21no—
355C4—5841no—
356C4—5841no—
357C4—5841no—
358C4—11682no—
359C4#1798846611yescharacteristic
360C4#1798858411yescharacteristic
361C4#1798881612yes—
362C2—22no—
363C2—22no—
364C2—41no—
365C2—11681no—
366C2#2261035911yescharacteristic
367C2C2×#1798839811yescharacteristic
368C2#2261166811yescharacteristic
subgroupG/H
centerC22C2×#14556513
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC22×M32×C73C2
socleC22×C73C2×#102
prime radical#22619512C2
cosocleC22×C8×C73C23
hypercenterC22×M32D73
hypocenterC73#13065
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C22×M32

Complements

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

NH ≅ G/Ntype
2. C2×#17988398C2semidirect
3. #22609956C2semidirect
4. #22608037C2semidirect
5. C22×M32×C73C2semidirect
8. #17988828C4semidirect
9. #17988828C4semidirect
10. #17988823C4semidirect
11. #17988823C4semidirect
63. M32×C73D4semidirect
162. C22×C73#360semidirect
185. C22×M32D73semidirect
190. C73#13065semidirect
238. M32C22⋊D73semidirect
350. C22#17988398semidirect

Maximal subgroups

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

subgroupindex
#226195122
C2×#179883982
#226099562
#226080372
C22×M32×C732
#1306573