Finite Group Databasebeta

← #32996339

Subgroups — #32996339

Order 18688.

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

normal subgroups only

This group has 436 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#22615848C211yesmaximal, characteristic
2C4×#14556506C211yesmaximal, characteristic
3#22607953C212yesmaximal, complemented
4#22607932C212yesmaximal, complemented
5C73×#1460C211yesmaximal, characteristic
6C2×#14556570C411yescharacteristic, complemented
7C2×#14556506—21no—
8#17988517C411yescharacteristic, complemented
9#17988515—21no—
10#17988512C2211yescharacteristic
11#17988507C411yescharacteristic, complemented
12#17988501—21no—
13#17988499C2211yescharacteristic
14#17988471—21no—
15#17988460—21no—
16#17988450C411yescharacteristic, complemented
17C4×C22⋊Dic73—21no—
18#17988354C2212yes—
19C2×C4×D4×C73C2211yescharacteristic
20#431×C73—21no—
21#323×C73—21no—
22#323×C73C2212yes—
23C22×Dic146C2×C411yescharacteristic
24#14556570—41no—
25#14556570—41no—
26C2×C4⋊Dic73C2×C411yescharacteristic
27C2×#11924007—21no—
28C2×#11924007—21no—
29C2×#11924007—21no—
30C2×C22⋊Dic73—21no—
31C2×C22⋊Dic73—21no—
32C2×C22⋊Dic73—22no—
33C2×C22⋊Dic73—21no—
34C2×C22⋊Dic73—21no—
35C2×C4×Dic73—41no—
36#14556506—24no—
37#14556498—21no—
38#14556498—21no—
39#14556498—21no—
40#14556498—21no—
41#14556498—21no—
42#14556498—41no—
43#14556498C2×C411yescharacteristic
44#14556498C2×C411yescharacteristic
45C22×D4×C73D411yescharacteristic
46C23×C4×C73C2×C411yescharacteristic
47C23×C4×C73D411yescharacteristic
48C4×D4×C73—41no—
49C4×D4×C73—41no—
50C2×C4⋊C4×C73C2×C411yescharacteristic
51C2×C22⋊C4×C73D411yescharacteristic
52C2×C22⋊C4×C73C2311yescharacteristic
53C2×C42×C73D411yescharacteristic
54C4⋊C8×C73—41no—
55C22⋊C8×C73—22no—
56C4×C8×C73—41no—
57C2×Dic146—41no—
58C2×Dic146—81no—
59C22×Dic73—21no—
60C22×Dic73—21no—
61C22×Dic73—21no—
62C22×Dic73—21no—
63C22×Dic73—42no—
64C4⋊Dic73—41no—
65#11924007—41no—
66#11924007—41no—
67#11924007—42no—
68#11924007—81no—
69C22⋊Dic73—24no—
70C22⋊Dic73—42no—
71C22⋊Dic73—42no—
72C22⋊Dic73—42no—
73C22⋊Dic73—81no—
74C22⋊Dic73—81no—
75C4×Dic73—82no—
76C24×C73C22⋊C411yescharacteristic
77C24×C73C2×D411yescharacteristic
78C2×D4×C73—22no—
79C2×D4×C73—22no—
80C2×D4×C73—41no—
81C22×C4×C73—21no—
82C22×C4×C73—21no—
83C22×C4×C73—21no—
84C22×C4×C73—21no—
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
92C2×C8×C73—42no—
93C4⋊C4×C73—41no—
94C22⋊C4×C73—41no—
95C22⋊C4×C73—41no—
96C42×C73—21no—
97C42×C73—21no—
98C23×C73—21no—
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×C73C2×C22⋊C411yescharacteristic
108C23×C73#13512yes—
109D4×C73—44no—
110C2×C4×C73—21no—
111C2×C4×C73—21no—
112C2×C4×C73—21no—
113C2×C4×C73—21no—
114C2×C4×C73—21no—
115C2×C4×C73—41no—
116C2×C4×C73—42no—
117C2×C4×C73—41no—
118C2×C4×C73—41no—
119C2×C4×C73—41no—
120C2×C4×C73—81no—
121C2×C4×C73—81no—
122C2×C4×C73—81no—
123C2×C4×C73#9911yescharacteristic
124C2×C4×C73#9911yescharacteristic
125C2×C4×C73#10412yes—
126C2×Dic73—44no—
127C2×Dic73—41no—
128C2×Dic73—41no—
129C2×Dic73—41no—
130C2×Dic73—41no—
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—82no—
140C2×Dic73—82no—
141Dic146—82no—
142C8×C73—82no—
143C22×C73—21no—
144C22×C73—21no—
145C22×C73—22no—
146C22×C73—42no—
147C22×C73—41no—
148C22×C73—41no—
149C22×C73—41no—
150C22×C73—41no—
151C22×C73—41no—
152C22×C73—81no—
153C22×C73—81no—
154C22×C73—81no—
155C22×C73C2×#9911yescharacteristic
156C22×C73#41611yescharacteristic
157C22×C73C2×#10411yescharacteristic
158C4×C73—21no—
159C4×C73—21no—
160C4×C73—22no—
161C4×C73—41no—
162C4×C73—41no—
163C4×C73—41no—
164Dic73—81no—
165Dic73—81no—
166Dic73—84no—
167Dic73—81no—
168Dic73—81no—
169#11731—731nomaximal, Sylow
170C2×C73—21no—
171C2×C73—21no—
172C2×C73—42no—
173C2×C73—81no—
174C2×C73#150111yescharacteristic
175C2×C73#212111yescharacteristic
176C2×C73#212911yescharacteristic
177#2638—731no—
178C4×#99—731no—
179#1460D7311yescharacteristic
180#1351—732no—
181#1344—732no—
182C73#1173111yescharacteristic, complemented, Sylow
183C2×C22⋊Q8—731no—
184C2×C4×D4C2×D7311yescharacteristic
185#431—21no—
186C2×#99—1461no—
187#398—731no—
188#397—731no—
189#397—1461no—
190#395—731no—
191#394—1461no—
192#393—731no—
193#387—1461no—
194#385—1461no—
195#381—731no—
196C4×C22⋊C4—1461no—
197#342—732no—
198#323—21no—
199#323C2×D7312yes—
200C22×Q8—731no—
201C22×D4D29211yescharacteristic
202C23×C4C4×D7311yescharacteristic
203C23×C4D29211yescharacteristic
204C22⋊Q8—2921no—
205C22⋊Q8—2921no—
206C4×D4—41no—
207C4×D4—41no—
208C2×C4⋊C4—731no—
209C2×C4⋊C4—1461no—
210C2×C4⋊C4—1461no—
211C2×C4⋊C4—1461no—
212C2×C4⋊C4C4×D7311yescharacteristic
213C2×C22⋊C4—1461no—
214C2×C22⋊C4—1461no—
215C2×C22⋊C4—1462no—
216C2×C22⋊C4—1461no—
217C2×C22⋊C4—1461no—
218C2×C22⋊C4C22⋊D7311yescharacteristic
219C2×C22⋊C4C22×D7311yescharacteristic
220C2×C42—2921no—
221C2×C42C22⋊D7311yescharacteristic
222C4⋊C8—41no—
223#99—1464no—
224C22⋊C8—22no—
225C4×C8—41no—
226#95—2921no—
227#95—731no—
228#95—731no—
229#95—1461no—
230#95—1461no—
231#95—1461no—
232#95—1461no—
233#95—1461no—
234C24#1192400511yescharacteristic
235C24C2×D29211yescharacteristic
236C2×Q8—2921no—
237C2×Q8—5841no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—41no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—2922no—
248C22×C4—1461no—
249C22×C4—1461no—
250C22×C4—1461no—
251C22×C4—1461no—
252C22×C4#1192400511yescharacteristic
253C22×C4#1192400511yescharacteristic
254C22×C4#1192400511yescharacteristic
255C22×C4C2×C4×D7311yescharacteristic
256C22×C4C2×C22⋊D7311yescharacteristic
257C2×C8—42no—
258C4⋊C4—41no—
259C4⋊C4—2921no—
260C4⋊C4—2921no—
261C4⋊C4—2921no—
262C4⋊C4—2922no—
263C4⋊C4—5841no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—2922no—
267C22⋊C4—2922no—
268C22⋊C4—2922no—
269C22⋊C4—5841no—
270C22⋊C4—5841no—
271C22⋊C4—1464no—
272C42—21no—
273C42—21no—
274C42—5842no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23C2×#1192400511yescharacteristic
285C23#1455661312yes—
286Q8—5842no—
287D4—44no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—2924no—
302C2×C4—2921no—
303C2×C4—2921no—
304C2×C4—2921no—
305C2×C4—2921no—
306C2×C4—2921no—
307C2×C4—2921no—
308C2×C4—2921no—
309C2×C4—2921no—
310C2×C4—2921no—
311C2×C4—2921no—
312C2×C4—2921no—
313C2×C4—2921no—
314C2×C4—5842no—
315C2×C4—5842no—
316C2×C4#1455650511yescharacteristic
317C2×C4#1455650511yescharacteristic
318C2×C4#1455651712yes—
319C8—82no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22C2×#1455650511yescharacteristic
333C22#1798858411yescharacteristic
334C22C2×#1455651711yescharacteristic
335C4—22no—
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—5841no—
342C4—5841no—
343C4—5844no—
344C4—5841no—
345C4—5841no—
346C2—21no—
347C2—21no—
348C2—42no—
349C2—81no—
350C2#2260842311yescharacteristic
351C2#2261157811yescharacteristic
352C2#2261160511yescharacteristic
subgroupG/H
centerC22C2×#14556505
commutatorC22×C4×C73C22×C4
FrattiniC2×C22⋊C4C22×D73
FittingC73×#1460C2
socleC22×C73C2×#99
prime radicalC22×D4×C73D4
cosocleC2×C22⋊C4×C73C23
hypercenter#1460D73
hypocenterC73#11731
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1460

Complements

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

NH ≅ G/Ntype
3. #22607953C2semidirect
4. #22607932C2semidirect
6. C2×#14556570C4semidirect
8. #17988517C4semidirect
11. #17988507C4semidirect
16. #17988450C4semidirect
182. C73#11731semidirect

Maximal subgroups

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

subgroupindex
#226158482
C4×#145565062
#226079532
#226079322
C73×#14602
#1173173