Finite Group Databasebeta

← #33240810

Subgroups — #33240810

Order 18688.

632 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988823C211yesmaximal, characteristic
2#22612122C214yesmaximal, complemented
3#22612116C212yesmaximal, complemented
4#22612111C212yesmaximal, complemented
5#22611720C212yesmaximal, complemented
6#22611646C212yesmaximal, complemented
7#22611542C211yesmaximal, characteristic, complemented
8C2×#427×C73C211yesmaximal, characteristic, complemented
9C22×D584C2211yescharacteristic
10C2×#14556565—21no—
11C2×#14556531C2212yes—
12#17988823—21no—
13#17988823—21no—
14#17988823C2212yes—
15#17988823C2212yes—
16C2×#14556525C2211yescharacteristic
17C2×#14556513—21no—
18#17988414C2214yescomplemented
19#17988412C2214yescomplemented
20#17988387C2214yescomplemented
21#17988387C2214yescomplemented
22C22⋊#11924003C2212yescomplemented
23C22⋊#11924003C2212yescomplemented
24C23×C8×C73C2211yescharacteristic
25C2×#117×C73—21no—
26#427×C73—21no—
27#427×C73—21no—
28#427×C73C2212yes—
29#427×C73C2212yescomplemented
30C2×#107×C73C2211yescharacteristic, complemented
31C2×#107×C73C2211yescharacteristic, complemented
32C2×#106×C73—21no—
33C22×D292—21no—
34C22×C22⋊D73—41no—
35C2×D584—21no—
36C2×D584—24no—
37C2×D584C2312yes—
38#14556565—22no—
39#14556565—22no—
40#14556565—41no—
41#14556565—41no—
42C2×C4⋊Dic73—21no—
43C2×#11924005—41no—
44#14556531—24no—
45C2×#11924003C2314yes—
46#14556525—21no—
47#14556525C2312yes—
48#14556513—22no—
49#14556513—22no—
50C23×C4×C73D411yescharacteristic
51C22×C8×C73—24no—
52C22×C8×C73D411yescharacteristic
53C22×C8×C73D411yescharacteristic
54C22×C8×C73D411yescharacteristic
55C22×C8×C73C2311yescharacteristic
56C22×C8×C73C2311yescharacteristic
57C22×C8×C73C2311yescharacteristic
58#117×C73—22no—
59#117×C73—22no—
60#117×C73—41no—
61#117×C73—41no—
62C2×C4⋊C4×C73—21no—
63C2×C4⋊C4×C73—21no—
64C2×C22⋊C4×C73—41no—
65C2×C42×C73—41no—
66#107×C73—21no—
67#107×C73—21no—
68#107×C73C2312yes—
69#107×C73C2312yes—
70#106×C73—22no—
71#106×C73—22no—
72C23×D73—41no—
73C2×D292—22no—
74C2×D292—42no—
75C2×D292—42no—
76C2×D292—41no—
77C2×C22⋊D73—42no—
78C2×C22⋊D73—42no—
79C2×C22⋊D73—84no—
80C22×Dic73—41no—
81D584—24no—
82D584—42no—
83C4⋊Dic73—22no—
84C4⋊Dic73—41no—
85#11924005—42no—
86#11924005—42no—
87#11924003—24no—
88C24×C73D811yescharacteristic
89C22×C4×C73—22no—
90C22×C4×C73—22no—
91C22×C4×C73—41no—
92C22×C4×C73—41no—
93C22×C4×C73D811yescharacteristic
94C22×C4×C73D811yescharacteristic
95C22×C4×C73D811yescharacteristic
96C22×C4×C73C2×D411yescharacteristic
97C22×C4×C73C2×D411yescharacteristic
98C22×C4×C73C2×D411yescharacteristic
99C2×C8×C73—22no—
100C2×C8×C73—22no—
101C2×C8×C73—22no—
102C2×C8×C73—22no—
103C2×C8×C73—22no—
104C2×C8×C73C2×D411yescharacteristic
105C2×C8×C73C2×D411yescharacteristic
106C2×C8×C73C2×D411yescharacteristic
107C2×C8×C73#4114yes—
108C2×C8×C73C2411yescharacteristic
109C4⋊C4×C73—22no—
110C4⋊C4×C73—22no—
111C4⋊C4×C73—41no—
112C4⋊C4×C73—41no—
113C22⋊C4×C73—42no—
114C22⋊C4×C73—42no—
115C42×C73—42no—
116C42×C73—42no—
117C23×C73—22no—
118C23×C73—22no—
119C23×C73—41no—
120C23×C73C2×D811yescharacteristic
121C23×C73C2×D811yescharacteristic
122C23×C73C2×D811yescharacteristic
123C22×D73—42no—
124C22×D73—82no—
125C22×D73—82no—
126C22×D73—82no—
127C2×C4×C73—22no—
128C2×C4×C73—21no—
129C2×C4×C73—21no—
130C2×C4×C73—21no—
131C2×C4×C73—21no—
132C2×C4×C73—22no—
133C2×C4×C73—22no—
134C2×C4×C73—42no—
135C2×C4×C73—42no—
136C2×C4×C73—82no—
137C2×C4×C73—82no—
138C2×C4×C73#12312yes—
139C2×C4×C73C2×D811yescharacteristic
140C2×C4×C73C2×D811yescharacteristic
141C2×C4×C73C2×D811yescharacteristic
142C2×C4×C73#13712yes—
143C2×C4×C73C22×D411yescharacteristic
144C22⋊D73—84no—
145C22⋊D73—84no—
146C2×Dic73—42no—
147C2×Dic73—82no—
148D292—42no—
149D292—42no—
150D292—82no—
151C8×C73—21no—
152C8×C73—21no—
153C8×C73#12312yes—
154C8×C73C2×#4112yes—
155C22×C73—21no—
156C22×C73—21no—
157C22×C73—21no—
158C22×C73—21no—
159C22×C73—21no—
160C22×C73—21no—
161C22×C73—21no—
162C22×C73—44no—
163C22×C73#48012yes—
164C22×C73#50912yes—
165C22×C73C22×D811yescharacteristic
166C2×D73—82no—
167C2×D73—82no—
168C2×D73—82no—
169C2×D73—164no—
170C4×C73—21no—
171C4×C73—21no—
172C4×C73—82no—
173C4×C73—82no—
174C4×C73#48012yes—
175C4×C73C2×#12311yescharacteristic
176C4×C73C2×#13711yescharacteristic
177Dic73—82no—
178#33648—731nomaximal, Sylow
179C2×C73—22no—
180C2×C73—22no—
181C2×C73—41no—
182C2×C73C2×#48011yescharacteristic
183C2×C73C2×#50911yescharacteristic
184C2×C73#341411yescharacteristic
185D73—162no—
186C2×#466—731no—
187C2×#427D7311yescharacteristic, complemented
188#2234—734no—
189#2233—732no—
190#2232—732no—
191#2159—732no—
192#2140—732no—
193#2112—731no—
194C73#3364811yescharacteristic, complemented, Sylow
195C22×D8—731no—
196C23×C8C2×D7311yescharacteristic
197C2×#121—1461no—
198C2×#117—21no—
199C2×M32—732no—
200#466—732no—
201#466—732no—
202#466—1461no—
203#466—1461no—
204#427—21no—
205#427—21no—
206#427C2×D7312yes—
207#427C2×D7312yescomplemented
208C2×#107—731no—
209C2×#107C2×D7311yescharacteristic, complemented
210C2×#107C2×D7311yescharacteristic, complemented
211C2×#106—21no—
212C2×#102—1461no—
213#367—734no—
214#366—734no—
215#357—734no—
216#357—734no—
217C22⋊C16—732no—
218C22⋊C16—732no—
219C22×D4—2921no—
220C22×D4—1461no—
221C23×C4C22⋊D7311yescharacteristic
222C2×D8—732no—
223C2×D8—1461no—
224C2×D8—1464no—
225C22×C8—24no—
226C22×C8C22⋊D7311yescharacteristic
227C22×C8C22⋊D7311yescharacteristic
228C22×C8C22⋊D7311yescharacteristic
229C22×C8C22×D7311yescharacteristic
230C22×C8C22×D7311yescharacteristic
231C22×C8C22×D7311yescharacteristic
232#121—2921no—
233#121—2921no—
234#121—1462no—
235#121—1462no—
236#117—22no—
237#117—22no—
238#117—41no—
239#117—41no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—1461no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—2921no—
245C2×C42—41no—
246M32—1464no—
247C2×C16—734no—
248#107—21no—
249#107—21no—
250#107—732no—
251#107—1461no—
252#107C22×D7312yes—
253#107C22×D7312yes—
254#106—22no—
255#106—22no—
256#102—1462no—
257#102—1462no—
258C24—2921no—
259C24#1192401311yescharacteristic
260C2×D4—2922no—
261C2×D4—2922no—
262C2×D4—2922no—
263C2×D4—2921no—
264C2×D4—2922no—
265C2×D4—5844no—
266C2×D4—1462no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—2921no—
272C22×C4#1192401311yescharacteristic
273C22×C4#1192401311yescharacteristic
274C22×C4#1192401311yescharacteristic
275C22×C4C2×C22⋊D7311yescharacteristic
276C22×C4C2×C22⋊D7311yescharacteristic
277C22×C4C2×C22⋊D7311yescharacteristic
278D8—2922no—
279D8—1464no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8C2×C22⋊D7311yescharacteristic
286C2×C8C2×C22⋊D7311yescharacteristic
287C2×C8C2×C22⋊D7311yescharacteristic
288C2×C8#1192402814yes—
289C2×C8C23×D7311yescharacteristic
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—2921no—
295C4⋊C4—1462no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—2922no—
299C22⋊C4—2922no—
300C42—42no—
301C42—42no—
302C16—1464no—
303C23—22no—
304C23—22no—
305C23—41no—
306C23—2922no—
307C23—5842no—
308C23—5842no—
309C23—5842no—
310C23C2×#1192401311yescharacteristic
311C23C2×#1192401311yescharacteristic
312C23C2×#1192401311yescharacteristic
313D4—2922no—
314D4—2922no—
315D4—5844no—
316D4—5844no—
317D4—5842no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—2922no—
330C2×C4—5842no—
331C2×C4#1455657412yes—
332C2×C4C2×#1192401311yescharacteristic
333C2×C4C2×#1192401311yescharacteristic
334C2×C4C2×#1192401311yescharacteristic
335C2×C4#1455662212yes—
336C2×C4C22×C22⋊D7311yescharacteristic
337C8—21no—
338C8—21no—
339C8#1455657412yes—
340C8C2×#1192402812yes—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—44no—
349C22—5842no—
350C22—5842no—
351C22—5842no—
352C22—11684no—
353C22#1798890012yes—
354C22#1798902212yes—
355C22C22×#1192401311yescharacteristic
356C4—21no—
357C4—21no—
358C4—82no—
359C4—82no—
360C4—5842no—
361C4#1798890012yes—
362C4C2×#1455657411yescharacteristic
363C4C2×#1455662211yescharacteristic
364C2—22no—
365C2—22no—
366C2—41no—
367C2—11682no—
368C2C2×#1798890011yescharacteristic
369C2C2×#1798902211yescharacteristic
370C2#2262306511yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC2×#427×C73C2
socleC22×C73C22×D8
prime radicalC2×#17988823C2
hypercenterC2×#427D73
hypocenterC73#33648
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
2. #22612122C2semidirect
3. #22612116C2semidirect
4. #22612111C2semidirect
5. #22611720C2semidirect
6. #22611646C2semidirect
7. #22611542C2semidirect
8. C2×#427×C73C2semidirect
18. #17988414C22semidirect
19. #17988412C22semidirect
20. #17988387C22semidirect
21. #17988387C22semidirect
22. C22⋊#11924003C22semidirect
23. C22⋊#11924003C22semidirect
29. #427×C73C22semidirect
30. C2×#107×C73C22semidirect
31. C2×#107×C73C22semidirect
187. C2×#427D73semidirect
194. C73#33648semidirect
207. #427C2×D73semidirect
209. C2×#107C2×D73semidirect
210. C2×#107C2×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179888232
#226121222
#226121162
#226121112
#226117202
#226116462
#226115422
C2×#427×C732
#3364873