Finite Group Databasebeta

← #32996608

Subgroups — #32996608

Order 18688.

462 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 83 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#22615107C211yesmaximal, characteristic
2#22608272C211yesmaximal, characteristic
3#22607954C212yesmaximal, complemented
4#22607951C212yesmaximal, complemented
5C4×#99×C73C211yesmaximal, characteristic
6C2×#14556570C411yescharacteristic, complemented
7#17988644—21no—
8#17988526C411yescharacteristic, complemented
9#17988523C2211yescharacteristic
10#17988507C411yescharacteristic, complemented
11#17988499C2211yescharacteristic
12#17988471—21no—
13#17988467—21no—
14#17988460—21no—
15#17988445—21no—
16C2×#14556498C411yescharacteristic, complemented
17#17988311—21no—
18#17988311C2212yes—
19C2×C4×D4×C73C2211yescharacteristic
20C2×#99×C73—21no—
21C4×C22⋊C4×C73—21no—
22#342×C73C2212yes—
23C22×Dic146C2×C411yescharacteristic
24C23×Dic73—21no—
25#14556570—41no—
26#14556570—41no—
27C2×C4⋊Dic73C2×C411yescharacteristic
28C2×#11924007—21no—
29C2×#11924007—21no—
30C2×#11924007—21no—
31C2×C22⋊Dic73—21no—
32C2×C22⋊Dic73—21no—
33C2×C22⋊Dic73—21no—
34C4⋊#10596831—41no—
35C22⋊#10596831—22no—
36#14556502—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×C73C2×C411yescharacteristic
46C23×C4×C73D411yescharacteristic
47C4×D4×C73—41no—
48C4×D4×C73—41no—
49C2×C4⋊C4×C73C2×C411yescharacteristic
50C2×C22⋊C4×C73—22no—
51C2×C22⋊C4×C73D411yescharacteristic
52C2×C22⋊C4×C73C2311yescharacteristic
53C2×C42×C73—41no—
54C2×C42×C73D411yescharacteristic
55#99×C73—24no—
56#95×C73—41no—
57C2×Dic146—41no—
58C2×Dic146—81no—
59C22×Dic73—21no—
60C22×Dic73—21no—
61C22×Dic73—21no—
62C22×Dic73—21no—
63C22×Dic73—42no—
64C22×Dic73—42no—
65C22×Dic73—81no—
66C2×#10596831—42no—
67C4⋊Dic73—41no—
68#11924007—42no—
69#11924007—41no—
70#11924007—41no—
71#11924007—81no—
72C22⋊Dic73—42no—
73C22⋊Dic73—42no—
74C22⋊Dic73—81no—
75C24×C73C22⋊C411yescharacteristic
76C24×C73C2×D411yescharacteristic
77C2×D4×C73—22no—
78C2×D4×C73—22no—
79C2×D4×C73—41no—
80C22×C4×C73—21no—
81C22×C4×C73—21no—
82C22×C4×C73—21no—
83C22×C4×C73—21no—
84C22×C4×C73—42no—
85C22×C4×C73—81no—
86C22×C4×C73—81no—
87C22×C4×C73C22⋊C411yescharacteristic
88C22×C4×C73C22⋊C411yescharacteristic
89C22×C4×C73C22⋊C411yescharacteristic
90C22×C4×C73C22×C411yescharacteristic
91C22×C4×C73C2×D411yescharacteristic
92C4⋊C4×C73—41no—
93C22⋊C4×C73—24no—
94C22⋊C4×C73—41no—
95C22⋊C4×C73—41no—
96C22⋊C4×C73—42no—
97C42×C73—21no—
98C42×C73—21no—
99C42×C73—82no—
100C23×C73—21no—
101C23×C73—21no—
102C23×C73—21no—
103C23×C73—21no—
104C23×C73—41no—
105C23×C73—41no—
106C23×C73—41no—
107C23×C73—41no—
108C23×C73#10412yes—
109C23×C73C2×C22⋊C411yescharacteristic
110C23×C73#13712yes—
111D4×C73—44no—
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—41no—
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—81no—
127C2×C4×C73—82no—
128C2×C4×C73—82no—
129C2×C4×C73#9911yescharacteristic
130C2×C4×C73#9911yescharacteristic
131C2×Dic73—22no—
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—
142C2×Dic73—41no—
143C2×Dic73—84no—
144Dic146—82no—
145#10596831—82no—
146C22×C73—21no—
147C22×C73—21no—
148C22×C73—22no—
149C22×C73—22no—
150C22×C73—42no—
151C22×C73—42no—
152C22×C73—42no—
153C22×C73—41no—
154C22×C73—41no—
155C22×C73—41no—
156C22×C73—41no—
157C22×C73—81no—
158C22×C73C2×#9911yescharacteristic
159C22×C73#41911yescharacteristic
160C22×C73C2×#10411yescharacteristic
161C4×C73—21no—
162C4×C73—21no—
163C4×C73—41no—
164C4×C73—41no—
165C4×C73—41no—
166C4×C73—41no—
167C4×C73—84no—
168Dic73—42no—
169Dic73—81no—
170Dic73—81no—
171Dic73—81no—
172#11789—731nomaximal, Sylow
173C2×C73—21no—
174C2×C73—21no—
175C2×C73—42no—
176C2×C73—42no—
177C2×C73#150811yescharacteristic
178C2×C73#212111yescharacteristic
179C2×C73#212611yescharacteristic
180#2564—731no—
181C4×#99D7311yescharacteristic
182#1466—731no—
183#1352—732no—
184#1351—732no—
185C73#1178911yescharacteristic, complemented, Sylow
186C2×C22⋊Q8—731no—
187C2×C4×D4C2×D7311yescharacteristic
188#432—1461no—
189C2×#99—21no—
190#400—731no—
191#399—731no—
192#395—731no—
193#393—731no—
194#387—1461no—
195#386—1461no—
196#385—1461no—
197#380—1461no—
198C4×C22⋊C4—21no—
199C2×#95—731no—
200#342C2×D7312yes—
201#323—732no—
202#323—1461no—
203C22×Q8—731no—
204C22×D4C22⋊D7311yescharacteristic
205C23×C4—1461no—
206C23×C4C4×D7311yescharacteristic
207C23×C4C22⋊D7311yescharacteristic
208C22⋊Q8—2921no—
209C22⋊Q8—2921no—
210C4×D4—41no—
211C4×D4—41no—
212C2×C4⋊C4—731no—
213C2×C4⋊C4—1461no—
214C2×C4⋊C4—1461no—
215C2×C4⋊C4—1461no—
216C2×C4⋊C4C4×D7311yescharacteristic
217C2×C22⋊C4—22no—
218C2×C22⋊C4—1461no—
219C2×C22⋊C4—1461no—
220C2×C22⋊C4—1461no—
221C2×C22⋊C4D29211yescharacteristic
222C2×C22⋊C4C22×D7311yescharacteristic
223C2×C42—41no—
224C2×C42D29211yescharacteristic
225C4⋊C8—2921no—
226#99—24no—
227C22⋊C8—1462no—
228#97—2921no—
229#95—41no—
230#95—731no—
231#95—731no—
232#95—1461no—
233#95—1461no—
234#95—1461no—
235#95—1461no—
236#95—1461no—
237C24#1192400511yescharacteristic
238C24C2×C22⋊D7311yescharacteristic
239C2×Q8—2921no—
240C2×Q8—5841no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—41no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—42no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4—2922no—
252C22×C4—2922no—
253C22×C4—5841no—
254C22×C4—1461no—
255C22×C4—1461no—
256C22×C4—1461no—
257C22×C4—1461no—
258C22×C4#1192400511yescharacteristic
259C22×C4#1192400511yescharacteristic
260C22×C4#1192400511yescharacteristic
261C22×C4C2×C4×D7311yescharacteristic
262C22×C4C2×D29211yescharacteristic
263C2×C8—2922no—
264C4⋊C4—41no—
265C4⋊C4—2921no—
266C4⋊C4—2922no—
267C4⋊C4—2921no—
268C4⋊C4—2921no—
269C4⋊C4—5841no—
270C22⋊C4—24no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—42no—
274C22⋊C4—2922no—
275C22⋊C4—2922no—
276C22⋊C4—5841no—
277C42—21no—
278C42—21no—
279C42—82no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23#1455651912yes—
289C23C2×#1192400511yescharacteristic
290C23#1455662112yes—
291Q8—5842no—
292D4—44no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—44no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—82no—
309C2×C4—82no—
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—2921no—
321C2×C4—5844no—
322C2×C4—1462no—
323C2×C4#1455650511yescharacteristic
324C2×C4#1455650511yescharacteristic
325C8—5842no—
326C22—21no—
327C22—21no—
328C22—22no—
329C22—22no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22C2×#1455650511yescharacteristic
339C22#1798859511yescharacteristic
340C22C2×#1455651911yescharacteristic
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—84no—
348C4—2922no—
349C4—5841no—
350C4—5841no—
351C4—5841no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—42no—
356C2#2260845011yescharacteristic
357C2#2261157511yescharacteristic
358C2#2261159411yescharacteristic
subgroupG/H
centerC22C2×#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#11789
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #22607954C2semidirect
4. #22607951C2semidirect
6. C2×#14556570C4semidirect
8. #17988526C4semidirect
10. #17988507C4semidirect
16. C2×#14556498C4semidirect
185. C73#11789semidirect

Maximal subgroups

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

subgroupindex
#226151072
#226082722
#226079542
#226079512
C4×#99×C732
#1178973