Finite Group Databasebeta

← #32986163

Subgroups — #32986163

Order 18688.

640 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22609911C211yesmaximal, characteristic
2C4×#14556498C211yesmaximal, characteristic
3#22607809C212yesmaximal
4#22607809C212yesmaximal
5C73×#2851C211yesmaximal, characteristic
6C22×C4×Dic73—21no—
7C2×C4⋊#10596831—21no—
8C2×C22⋊#10596831C2212yes—
9C2×#14556498—21no—
10C2×#14556498C2212yes—
11#17988342—21no—
12C2×#125×C73C411yescharacteristic, complemented
13C22×C42×C73C2211yescharacteristic
14#389×C73C411yescharacteristic, complemented
15#389×C73C411yescharacteristic, complemented
16#387×C73—21no—
17#387×C73—21no—
18#385×C73—21no—
19#385×C73—21no—
20#382×C73C411yescharacteristic, complemented
21#380×C73C2211yescharacteristic
22#380×C73C2211yescharacteristic
23C23×Dic73—22no—
24C22×#10596831—22no—
25C2×C4×Dic73—24no—
26C2×C4×Dic73—42no—
27C2×C4×Dic73—42no—
28C4⋊#10596831—42no—
29C22⋊#10596831—24no—
30#14556498—24no—
31#14556498—22no—
32#14556498Q814yes—
33C23×C4×C73D411yescharacteristic
34C23×C4×C73D411yescharacteristic
35C23×C4×C73C2311yescharacteristic
36#125×C73—42no—
37C2×C4⋊C4×C73—21no—
38C2×C4⋊C4×C73—21no—
39C2×C4⋊C4×C73—21no—
40C2×C4⋊C4×C73—21no—
41C2×C4⋊C4×C73C2×C411yescharacteristic
42C2×C4⋊C4×C73C2×C411yescharacteristic
43C2×C4⋊C4×C73C2×C411yescharacteristic
44C2×C4⋊C4×C73C2×C411yescharacteristic
45C2×C22⋊C4×C73—21no—
46C2×C22⋊C4×C73—21no—
47C2×C42×C73—42no—
48C2×C42×C73C2×C411yescharacteristic
49C2×C42×C73C2×C411yescharacteristic
50C2×C42×C73D411yescharacteristic
51C2×C42×C73D411yescharacteristic
52#95×C73—21no—
53#95×C73—21no—
54#95×C73—21no—
55#95×C73—21no—
56#95×C73—21no—
57#95×C73—21no—
58C22×Dic73—24no—
59C22×Dic73—24no—
60C22×Dic73—24no—
61C22×Dic73—42no—
62C22×Dic73—42no—
63C22×Dic73—42no—
64C22×Dic73—42no—
65C2×#10596831—42no—
66C2×#10596831—42no—
67C2×#10596831—42no—
68C4×Dic73—48no—
69C24×C73C2×D411yescharacteristic
70C22×C4×C73—21no—
71C22×C4×C73—21no—
72C22×C4×C73—22no—
73C22×C4×C73—22no—
74C22×C4×C73—22no—
75C22×C4×C73—22no—
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73—21no—
79C22×C4×C73—21no—
80C22×C4×C73—41no—
81C22×C4×C73—41no—
82C22×C4×C73—41no—
83C22×C4×C73C22⋊C411yescharacteristic
84C22×C4×C73C22⋊C411yescharacteristic
85C22×C4×C73C22⋊C411yescharacteristic
86C22×C4×C73C22⋊C411yescharacteristic
87C22×C4×C73C22×C411yescharacteristic
88C22×C4×C73C2×D411yescharacteristic
89C22×C4×C73C2×Q812yes—
90C22×C4×C73#4112yes—
91C4⋊C4×C73—41no—
92C4⋊C4×C73—41no—
93C4⋊C4×C73—41no—
94C4⋊C4×C73—41no—
95C4⋊C4×C73—42no—
96C4⋊C4×C73—42no—
97C4⋊C4×C73—42no—
98C4⋊C4×C73—42no—
99C22⋊C4×C73—42no—
100C22⋊C4×C73—41no—
101C22⋊C4×C73—41no—
102C42×C73—22no—
103C42×C73—22no—
104C42×C73—41no—
105C42×C73—41no—
106C23×C73—21no—
107C23×C73—21no—
108C23×C73—21no—
109C23×C73—21no—
110C23×C73C2×C22⋊C411yescharacteristic
111C23×C73C22⋊Q812yes—
112C23×C73#13512yes—
113C23×C73AGL(1,ℤ/8)11yescharacteristic
114C23×C73#13711yescharacteristic
115C2×C4×C73—24no—
116C2×C4×C73—24no—
117C2×C4×C73—21no—
118C2×C4×C73—21no—
119C2×C4×C73—21no—
120C2×C4×C73—21no—
121C2×C4×C73—21no—
122C2×C4×C73—21no—
123C2×C4×C73—21no—
124C2×C4×C73—21no—
125C2×C4×C73—21no—
126C2×C4×C73—21no—
127C2×C4×C73—21no—
128C2×C4×C73—21no—
129C2×C4×C73—21no—
130C2×C4×C73—42no—
131C2×C4×C73—42no—
132C2×C4×C73—44no—
133C2×C4×C73—41no—
134C2×C4×C73—41no—
135C2×C4×C73—41no—
136C2×C4×C73—41no—
137C2×C4×C73—41no—
138C2×C4×C73—41no—
139C2×C4×C73—41no—
140C2×C4×C73—41no—
141C2×C4×C73—41no—
142C2×C4×C73—41no—
143C2×C4×C73—41no—
144C2×C4×C73—41no—
145C2×C4×C73#9912yes—
146C2×C4×C73#10412yes—
147C2×C4×C73#10412yes—
148C2×C4×C73C4×Q812yes—
149C2×C4×C73#12411yescharacteristic
150C2×C4×C73C4⋊Q811yescharacteristic
151C2×Dic73—28no—
152C2×Dic73—44no—
153C2×Dic73—44no—
154C2×Dic73—44no—
155C2×Dic73—44no—
156C2×Dic73—44no—
157C2×Dic73—44no—
158#10596831—82no—
159C22×C73—22no—
160C22×C73—22no—
161C22×C73—22no—
162C22×C73—22no—
163C22×C73—21no—
164C22×C73—21no—
165C22×C73—21no—
166C22×C73—21no—
167C22×C73#39111yescharacteristic
168C22×C73C2×#9911yescharacteristic
169C22×C73#41011yescharacteristic
170C22×C73#41611yescharacteristic
171C22×C73#41711yescharacteristic
172C22×C73C2×#10411yescharacteristic
173C22×C73C2×#10411yescharacteristic
174C22×C73#47812yes—
175C22×C73#47912yes—
176C4×C73—21no—
177C4×C73—22no—
178C4×C73—21no—
179C4×C73—22no—
180C4×C73—22no—
181C4×C73—22no—
182C4×C73—41no—
183C4×C73—81no—
184C4×C73—81no—
185C4×C73—81no—
186C4×C73—81no—
187Dic73—48no—
188#9910—731nomaximal, Sylow
189C2×C73—22no—
190C2×C73—22no—
191C2×C73#150011yescharacteristic
192C2×C73#150111yescharacteristic
193C2×C73#150411yescharacteristic
194C2×C73#150911yescharacteristic
195C2×C73#198911yescharacteristic
196C2×C73#199411yescharacteristic
197C2×C73#199611yescharacteristic
198#2851D7311yescharacteristic
199#1797—731no—
200C4×#95—731no—
201#1294—732no—
202#1294—732no—
203C73#991011yescharacteristic, complemented, Sylow
204C2×#125Dic7311yescharacteristic, complemented
205C22×C42—1461no—
206C22×C42C2×D7311yescharacteristic
207C2×C4⋊C8—1461no—
208C2×C22⋊C8—732no—
209#389Dic7311yescharacteristic, complemented
210#389Dic7311yescharacteristic, complemented
211#387—21no—
212#387—21no—
213#385—21no—
214#385—21no—
215#382Dic7311yescharacteristic, complemented
216#380C2×D7311yescharacteristic
217#380C2×D7311yescharacteristic
218C2×#95—732no—
219C2×#95—1461no—
220#336—1461no—
221C23×C4—1462no—
222C23×C4C22⋊D7311yescharacteristic
223C23×C4C22⋊D7311yescharacteristic
224C23×C4C22×D7311yescharacteristic
225C22×C8—1462no—
226#125—42no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4C2×Dic7311yescharacteristic
232C2×C4⋊C4C2×Dic7311yescharacteristic
233C2×C4⋊C4C2×Dic7311yescharacteristic
234C2×C4⋊C4C2×Dic7311yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42—42no—
238C2×C42—2922no—
239C2×C42—2922no—
240C2×C42—1464no—
241C2×C42C2×Dic7311yescharacteristic
242C2×C42C2×Dic7311yescharacteristic
243C2×C42C22⋊D7311yescharacteristic
244C2×C42C22⋊D7311yescharacteristic
245C4⋊C8—2922no—
246C22⋊C8—1464no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—734no—
254#95—1464no—
255#95—1462no—
256C24C2×C22⋊D7311yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—2922no—
271C22×C4—2922no—
272C22×C4—2922no—
273C22×C4—2922no—
274C22×C4—1464no—
275C22×C4—1464no—
276C22×C4—1464no—
277C22×C4C22⋊Dic7311yescharacteristic
278C22×C4C22⋊Dic7311yescharacteristic
279C22×C4C22⋊Dic7311yescharacteristic
280C22×C4C22⋊Dic7311yescharacteristic
281C22×C4C22×Dic7311yescharacteristic
282C22×C4C2×C22⋊D7311yescharacteristic
283C22×C4Q8×D7312yes—
284C22×C4#1192402912yes—
285C2×C8—2922no—
286C2×C8—2922no—
287C2×C8—2922no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C42—22no—
300C42—22no—
301C42—41no—
302C42—41no—
303C42—2928no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23C2×C22⋊Dic7311yescharacteristic
309C23#1455656712yes—
310C23#1455661112yes—
311C23#1455661611yescharacteristic
312C23#1455662111yescharacteristic
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—2924no—
344C2×C4—2924no—
345C2×C4—2924no—
346C2×C4—2924no—
347C2×C4—2924no—
348C2×C4—2924no—
349C2×C4—1468no—
350C2×C4#1455650612yes—
351C2×C4#1455651812yes—
352C2×C4#1455651812yes—
353C2×C4Q8×Dic7312yes—
354C2×C4#1455658011yescharacteristic
355C2×C4#1455658911yescharacteristic
356C8—5842no—
357C22—22no—
358C22—22no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—22no—
364C22—22no—
365C22#1798849411yescharacteristic
366C22C2×#1455650611yescharacteristic
367C22#1798856111yescharacteristic
368C22#1798858311yescharacteristic
369C22#1798858811yescharacteristic
370C22C2×#1455651811yescharacteristic
371C22C2×#1455651811yescharacteristic
372C22#1798888412yes—
373C22#1798889112yes—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—21no—
378C4—22no—
379C4—21no—
380C4—41no—
381C4—81no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4—2928no—
386C2—22no—
387C2—22no—
388C2#2260841711yescharacteristic
389C2#2260842211yescharacteristic
390C2#2260843511yescharacteristic
391C2#2260845611yescharacteristic
392C2#2261088911yescharacteristic
393C2#2261090911yescharacteristic
394C2#2261091711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic73
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#2851C2
socleC23×C73C2×C22⋊C4
prime radicalC24×C73C2×D4
cosocleC23×C4×C73C23
hypercenter#2851D73
hypocenterC73#9910
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
12. C2×#125×C73C4semidirect
14. #389×C73C4semidirect
15. #389×C73C4semidirect
20. #382×C73C4semidirect
203. C73#9910semidirect
204. C2×#125Dic73semidirect
209. #389Dic73semidirect
210. #389Dic73semidirect
215. #382Dic73semidirect

Maximal subgroups

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

subgroupindex
#226099112
C4×#145564982
#226078092
C73×#28512
#991073