Finite Group Databasebeta

← #32996321

Subgroups — #32996321

Order 18688.

470 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22613215C211yesmaximal, characteristic
2#22608249C211yesmaximal, characteristic
3#22607951C212yesmaximal, complemented
4#22607930C212yesmaximal, complemented
5C4×#99×C73C211yesmaximal, characteristic
6C2×C4×Dic146C411yescharacteristic, complemented
7C2×C22⋊#10596831—21no—
8#17988525—21no—
9#17988512C411yescharacteristic, complemented
10#17988512C2211yescharacteristic
11#17988501—21no—
12#17988499C411yescharacteristic, complemented
13#17988499C2211yescharacteristic
14C4×C22⋊Dic73—21no—
15C4×C22⋊Dic73—21no—
16#17988433C411yescharacteristic, complemented
17#17988313—21no—
18#17988311C2212yes—
19C2×C4×D4×C73C2211yescharacteristic
20C2×#99×C73—21no—
21C4×C22⋊C4×C73—21no—
22#342×C73C2212yes—
23C22×Dic146C2×C411yescharacteristic
24C22×#10596831—41no—
25C4×Dic146—41no—
26C4×Dic146—41no—
27C2×C4⋊Dic73C2×C411yescharacteristic
28C2×#11924007—21no—
29C2×C22⋊Dic73—21no—
30C2×C22⋊Dic73—21no—
31C2×C22⋊Dic73—21no—
32C2×C22⋊Dic73—21no—
33C2×C4×Dic73—21no—
34C2×C4×Dic73—21no—
35C22⋊#10596831—22no—
36C22⋊#10596831—41no—
37#14556498—21no—
38#14556498—21no—
39#14556498—21no—
40#14556498—21no—
41#14556498—21no—
42#14556498C2×C411yescharacteristic
43#14556498C2×C411yescharacteristic
44C22×D4×C73D411yescharacteristic
45C23×C4×C73D411yescharacteristic
46C23×C4×C73D411yescharacteristic
47C4×D4×C73—41no—
48C4×D4×C73—41no—
49C2×C4⋊C4×C73D411yescharacteristic
50C2×C22⋊C4×C73—22no—
51C2×C22⋊C4×C73C2×C411yescharacteristic
52C2×C22⋊C4×C73C2311yescharacteristic
53C2×C42×C73—41no—
54C2×C42×C73C2×C411yescharacteristic
55#99×C73—24no—
56#95×C73—41no—
57C2×Dic146—41no—
58C2×Dic146—42no—
59C22×Dic73—21no—
60C22×Dic73—21no—
61C22×Dic73—21no—
62C22×Dic73—21no—
63C2×#10596831—42no—
64C2×#10596831—82no—
65C4⋊Dic73—41no—
66#11924007—42no—
67C22⋊Dic73—42no—
68C22⋊Dic73—42no—
69C22⋊Dic73—42no—
70C22⋊Dic73—42no—
71C4×Dic73—42no—
72C4×Dic73—41no—
73C4×Dic73—41no—
74C24×C73C22⋊C411yescharacteristic
75C24×C73C2×D411yescharacteristic
76C2×D4×C73—22no—
77C2×D4×C73—22no—
78C2×D4×C73—41no—
79C22×C4×C73—21no—
80C22×C4×C73—21no—
81C22×C4×C73—21no—
82C22×C4×C73—21no—
83C22×C4×C73—42no—
84C22×C4×C73—42no—
85C22×C4×C73—42no—
86C22×C4×C73C22⋊C411yescharacteristic
87C22×C4×C73C22⋊C411yescharacteristic
88C22×C4×C73C22⋊C411yescharacteristic
89C22×C4×C73C22×C411yescharacteristic
90C22×C4×C73C2×D411yescharacteristic
91C4⋊C4×C73—41no—
92C22⋊C4×C73—24no—
93C22⋊C4×C73—41no—
94C22⋊C4×C73—41no—
95C22⋊C4×C73—42no—
96C42×C73—21no—
97C42×C73—21no—
98C42×C73—82no—
99C23×C73—21no—
100C23×C73—21no—
101C23×C73—21no—
102C23×C73—21no—
103C23×C73—41no—
104C23×C73—41no—
105C23×C73—41no—
106C23×C73—41no—
107C23×C73#10412yes—
108C23×C73C2×C22⋊C411yescharacteristic
109C23×C73#13512yes—
110D4×C73—44no—
111C2×C4×C73—22no—
112C2×C4×C73—21no—
113C2×C4×C73—21no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—21no—
117C2×C4×C73—21no—
118C2×C4×C73—42no—
119C2×C4×C73—42no—
120C2×C4×C73—41no—
121C2×C4×C73—41no—
122C2×C4×C73—41no—
123C2×C4×C73—44no—
124C2×C4×C73—81no—
125C2×C4×C73—81no—
126C2×C4×C73—82no—
127C2×C4×C73—82no—
128C2×C4×C73#9911yescharacteristic
129C2×C4×C73#9911yescharacteristic
130C2×Dic73—22no—
131C2×Dic73—41no—
132C2×Dic73—41no—
133C2×Dic73—41no—
134C2×Dic73—41no—
135C2×Dic73—41no—
136C2×Dic73—41no—
137C2×Dic73—41no—
138C2×Dic73—41no—
139C2×Dic73—41no—
140C2×Dic73—41no—
141C2×Dic73—41no—
142Dic146—42no—
143Dic146—42no—
144#10596831—82no—
145C22×C73—21no—
146C22×C73—21no—
147C22×C73—22no—
148C22×C73—22no—
149C22×C73—42no—
150C22×C73—42no—
151C22×C73—41no—
152C22×C73—41no—
153C22×C73—41no—
154C22×C73—41no—
155C22×C73—81no—
156C22×C73—81no—
157C22×C73C2×#9911yescharacteristic
158C22×C73#41611yescharacteristic
159C22×C73C2×#10411yescharacteristic
160C4×C73—21no—
161C4×C73—41no—
162C4×C73—41no—
163C4×C73—41no—
164C4×C73—41no—
165C4×C73—84no—
166C4×C73#35412yes—
167Dic73—42no—
168Dic73—81no—
169Dic73—81no—
170Dic73—81no—
171#11728—731nomaximal, Sylow
172C2×C73—21no—
173C2×C73—21no—
174C2×C73—42no—
175C2×C73—42no—
176C2×C73#150111yescharacteristic
177C2×C73#212011yescharacteristic
178C2×C73C2×#35411yescharacteristic
179#2398—731no—
180C4×#99D7311yescharacteristic
181#1459—731no—
182#1351—732no—
183#1344—732no—
184C73#1172811yescharacteristic, complemented, Sylow
185C2×C4×Q8—731no—
186C2×C4×D4C2×D7311yescharacteristic
187C2×#99—21no—
188C2×C22⋊C8—1461no—
189#400—1461no—
190#397—731no—
191#397—731no—
192#394—1461no—
193#393—731no—
194#393—731no—
195C4×C22⋊C4—21no—
196C4×C22⋊C4—1461no—
197C4×C22⋊C4—1461no—
198#376—731no—
199#342C2×D7312yes—
200#324—1461no—
201#323—732no—
202C22×Q8—731no—
203C22×D4C22⋊D7311yescharacteristic
204C23×C4D29211yescharacteristic
205C23×C4C22⋊D7311yescharacteristic
206C22×C8—2921no—
207C4×Q8—2921no—
208C4×Q8—2921no—
209C4×D4—41no—
210C4×D4—41no—
211C2×C4⋊C4—731no—
212C2×C4⋊C4—1461no—
213C2×C4⋊C4D29211yescharacteristic
214C2×C22⋊C4—22no—
215C2×C22⋊C4—1461no—
216C2×C22⋊C4—1461no—
217C2×C22⋊C4—1461no—
218C2×C22⋊C4—1461no—
219C2×C22⋊C4C4×D7311yescharacteristic
220C2×C22⋊C4C22×D7311yescharacteristic
221C2×C42—41no—
222C2×C42—1461no—
223C2×C42—1461no—
224C2×C42C4×D7311yescharacteristic
225#99—24no—
226C22⋊C8—2921no—
227C22⋊C8—1462no—
228#95—41no—
229#95—731no—
230#95—731no—
231#95—1461no—
232#95—1461no—
233#95—1461no—
234#95—1461no—
235#95—1461no—
236C24#1192400511yescharacteristic
237C24C2×C22⋊D7311yescharacteristic
238C2×Q8—2921no—
239C2×Q8—2922no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—1461no—
251C22×C4—1461no—
252C22×C4—1461no—
253C22×C4—1461no—
254C22×C4#1192400511yescharacteristic
255C22×C4#1192400511yescharacteristic
256C22×C4#1192400511yescharacteristic
257C22×C4C2×C4×D7311yescharacteristic
258C22×C4C2×D29211yescharacteristic
259C2×C8—2922no—
260C2×C8—5842no—
261C4⋊C4—41no—
262C4⋊C4—2921no—
263C4⋊C4—2922no—
264C22⋊C4—24no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—42no—
268C22⋊C4—2922no—
269C22⋊C4—2922no—
270C22⋊C4—2922no—
271C22⋊C4—2922no—
272C42—21no—
273C42—21no—
274C42—82no—
275C42—2922no—
276C42—2921no—
277C42—2921no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23#1455651912yes—
287C23C2×#1192400511yescharacteristic
288C23#1455661112yes—
289Q8—2922no—
290Q8—2922no—
291D4—44no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—44no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—2921no—
310C2×C4—2921no—
311C2×C4—2921no—
312C2×C4—2921no—
313C2×C4—2921no—
314C2×C4—2921no—
315C2×C4—2921no—
316C2×C4—2921no—
317C2×C4—2921no—
318C2×C4—2921no—
319C2×C4—2921no—
320C2×C4—1462no—
321C2×C4#1455650511yescharacteristic
322C2×C4#1455650511yescharacteristic
323C8—5842no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—22no—
328C22—42no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—81no—
335C22—81no—
336C22C2×#1455650511yescharacteristic
337C22#1798858111yescharacteristic
338C22C2×#1455651911yescharacteristic
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—84no—
345C4—2922no—
346C4—5841no—
347C4—5841no—
348C4—5841no—
349C4#1798838112yes—
350C2—21no—
351C2—21no—
352C2—42no—
353C2—42no—
354C2#2260842011yescharacteristic
355C2#2261157011yescharacteristic
356C2C2×#1798838111yescharacteristic
subgroupG/H
centerC2×C4#14556505
commutatorC22×C4×C73C22×C4
FrattiniC2×C22⋊C4C22×D73
FittingC4×#99×C73C2
socleC22×C73C2×#99
prime radicalC22×D4×C73D4
cosocleC2×C22⋊C4×C73C23
hypercenterC4×#99D73
hypocenterC73#11728
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #22607951C2semidirect
4. #22607930C2semidirect
6. C2×C4×Dic146C4semidirect
9. #17988512C4semidirect
12. #17988499C4semidirect
16. #17988433C4semidirect
184. C73#11728semidirect

Maximal subgroups

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

subgroupindex
#226132152
#226082492
#226079512
#226079302
C4×#99×C732
#1172873