Finite Group Databasebeta

← #33001126

Subgroups — #33001126

Order 18688.

548 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988377C212yesmaximal
2C4×#14556508C211yesmaximal, characteristic
3C4×#14556506C211yesmaximal, characteristic
4#22608049C212yesmaximal
5C73×#2322C211yesmaximal, characteristic
6C2×#14556508C2212yes—
7C2×#14556506C2212yes—
8C4×#11924012—21no—
9C4×C22⋊Dic73—21no—
10#17988377C418yescomplemented
11#17988356—21no—
12#17988354—21no—
13C2×#123×C73C2211yescharacteristic
14C2×C4×D4×C73C2211yescharacteristic
15C22×C4⋊C4×C73C411yescharacteristic, complemented
16#389×C73—21no—
17#388×C73—21no—
18#387×C73—21no—
19#386×C73C2211yescharacteristic
20#385×C73—21no—
21#385×C73C411yescharacteristic, complemented
22#381×C73C411yescharacteristic, complemented
23C4×C22⋊C4×C73C411yescharacteristic, complemented
24C2×#11924012—22no—
25C2×C22⋊Dic73—22no—
26C2×C4×Dic73—41no—
27#14556508—22no—
28#14556508C2×C414yes—
29#14556506—22no—
30#14556506C2×C414yes—
31#14556502—41no—
32C4×#10596831—41no—
33#14556498—41no—
34C22×D4×C73C2311yescharacteristic
35C23×C4×C73C2×C411yescharacteristic
36C23×C4×C73C2×C411yescharacteristic
37C23×C4×C73C2×C411yescharacteristic
38C23×C4×C73D411yescharacteristic
39#123×C73—22no—
40#123×C73C2×C414yes—
41C4×D4×C73—41no—
42C4×D4×C73—41no—
43C2×C4⋊C4×C73—21no—
44C2×C4⋊C4×C73—21no—
45C2×C4⋊C4×C73—21no—
46C2×C4⋊C4×C73—21no—
47C2×C4⋊C4×C73—41no—
48C2×C4⋊C4×C73—41no—
49C2×C4⋊C4×C73D411yescharacteristic
50C2×C22⋊C4×C73—21no—
51C2×C22⋊C4×C73—21no—
52C2×C22⋊C4×C73C2×C411yescharacteristic
53C2×C22⋊C4×C73C2×C411yescharacteristic
54C2×C22⋊C4×C73C2×C411yescharacteristic
55C2×C22⋊C4×C73D411yescharacteristic
56C2×C42×C73—21no—
57C2×C42×C73D411yescharacteristic
58#95×C73—21no—
59#95×C73—21no—
60#95×C73—21no—
61#95×C73—21no—
62#95×C73—21no—
63C22×Dic73—42no—
64#11924012—24no—
65#11924012—42no—
66C2×#10596831—42no—
67C22⋊Dic73—24no—
68C22⋊Dic73—42no—
69C4×Dic73—82no—
70C24×C73C22×C411yescharacteristic
71C24×C73C2×D411yescharacteristic
72C2×D4×C73—41no—
73C2×D4×C73—41no—
74C2×D4×C73C22×C412yes—
75C2×D4×C73#4112yes—
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73—21no—
79C22×C4×C73—21no—
80C22×C4×C73—21no—
81C22×C4×C73—21no—
82C22×C4×C73—21no—
83C22×C4×C73—21no—
84C22×C4×C73—21no—
85C22×C4×C73—41no—
86C22×C4×C73—41no—
87C22×C4×C73—41no—
88C22×C4×C73—81no—
89C22×C4×C73—81no—
90C22×C4×C73—81no—
91C22×C4×C73C4212yescomplemented
92C22×C4×C73C22⋊C411yescharacteristic
93C22×C4×C73C22⋊C411yescharacteristic
94C22×C4×C73C22⋊C411yescharacteristic
95C22×C4×C73C22⋊C411yescharacteristic
96C22×C4×C73C2×D411yescharacteristic
97C4⋊C4×C73—24no—
98C4⋊C4×C73—41no—
99C4⋊C4×C73—42no—
100C4⋊C4×C73—41no—
101C4⋊C4×C73—41no—
102C4⋊C4×C73—81no—
103C22⋊C4×C73—21no—
104C22⋊C4×C73—24no—
105C22⋊C4×C73—41no—
106C22⋊C4×C73—41no—
107C22⋊C4×C73—41no—
108C22⋊C4×C73—81no—
109C22⋊C4×C73C4212yescomplemented
110C42×C73—21no—
111C42×C73—21no—
112C42×C73—81no—
113C23×C73—21no—
114C23×C73—21no—
115C23×C73—21no—
116C23×C73—21no—
117C23×C73—41no—
118C23×C73—41no—
119C23×C73—41no—
120C23×C73—41no—
121C23×C73C2×C4211yescharacteristic
122C23×C73C2×C22⋊C411yescharacteristic
123C23×C73#11711yescharacteristic
124C23×C73C4×D412yes—
125D4×C73—42no—
126D4×C73—42no—
127C2×C4×C73—21no—
128C2×C4×C73—21no—
129C2×C4×C73—22no—
130C2×C4×C73—22no—
131C2×C4×C73—21no—
132C2×C4×C73—21no—
133C2×C4×C73—21no—
134C2×C4×C73—41no—
135C2×C4×C73—42no—
136C2×C4×C73—41no—
137C2×C4×C73—41no—
138C2×C4×C73—42no—
139C2×C4×C73—41no—
140C2×C4×C73—41no—
141C2×C4×C73—41no—
142C2×C4×C73—41no—
143C2×C4×C73—42no—
144C2×C4×C73—41no—
145C2×C4×C73—41no—
146C2×C4×C73—41no—
147C2×C4×C73—41no—
148C2×C4×C73—41no—
149C2×C4×C73—41no—
150C2×C4×C73—41no—
151C2×C4×C73—81no—
152C2×C4×C73—81no—
153C2×C4×C73—81no—
154C2×C4×C73—82no—
155C2×C4×C73#9911yescharacteristic
156C2×C4×C73#9911yescharacteristic
157C2×C4×C73C4×D412yes—
158C2×Dic73—44no—
159C2×Dic73—82no—
160C2×Dic73—82no—
161#10596831—44no—
162C22×C73—21no—
163C22×C73—21no—
164C22×C73—22no—
165C22×C73—21no—
166C22×C73—21no—
167C22×C73—41no—
168C22×C73—41no—
169C22×C73—42no—
170C22×C73—42no—
171C22×C73—41no—
172C22×C73—41no—
173C22×C73—41no—
174C22×C73—41no—
175C22×C73—81no—
176C22×C73C4×C22⋊C411yescharacteristic
177C22×C73C2×#9911yescharacteristic
178C22×C73#41011yescharacteristic
179C4×C73—21no—
180C4×C73—21no—
181C4×C73—22no—
182C4×C73—42no—
183C4×C73—41no—
184C4×C73—41no—
185C4×C73—41no—
186C4×C73—42no—
187C4×C73—81no—
188C4×C73—81no—
189Dic73—84no—
190#12621—731nomaximal, Sylow
191C2×C73—21no—
192C2×C73—21no—
193C2×C73—42no—
194C2×C73—41no—
195C2×C73—41no—
196C2×C73C4×#9911yescharacteristic
197C2×C73#212111yescharacteristic
198C2×C73#212311yescharacteristic
199#2322D7311yescharacteristic
200C2×#352—732no—
201C4×#100—731no—
202C4×#99—731no—
203#1390—732no—
204C73#1262111yescharacteristic, complemented, Sylow
205C2×#123C2×D7311yescharacteristic
206C2×C4×D4C2×D7311yescharacteristic
207C22×C4⋊C4Dic7311yescharacteristic, complemented
208C2×#100—732no—
209C2×#99—732no—
210C4×M16—1461no—
211#389—21no—
212#388—21no—
213#387—21no—
214#386C2×D7311yescharacteristic
215#385—21no—
216#385Dic7311yescharacteristic, complemented
217#381Dic7311yescharacteristic, complemented
218C4×C22⋊C4—1461no—
219C4×C22⋊C4Dic7311yescharacteristic, complemented
220#352—738no—
221#343—1461no—
222#342—1461no—
223C22×D4C22×D7311yescharacteristic
224C23×C4C2×Dic7311yescharacteristic
225C23×C4C2×Dic7311yescharacteristic
226C23×C4C2×Dic7311yescharacteristic
227C23×C4C22⋊D7311yescharacteristic
228C2×M16—1462no—
229#123—22no—
230#123C4×D7314yes—
231C4×D4—41no—
232C4×D4—41no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4C22⋊D7311yescharacteristic
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—1462no—
243C2×C22⋊C4C2×Dic7311yescharacteristic
244C2×C22⋊C4C2×Dic7311yescharacteristic
245C2×C22⋊C4C2×Dic7311yescharacteristic
246C2×C22⋊C4C22⋊D7311yescharacteristic
247C2×C42—21no—
248C2×C42—2921no—
249C2×C42C22⋊D7311yescharacteristic
250#100—734no—
251#100—1462no—
252#99—734no—
253#99—1462no—
254#97—2921no—
255C4×C8—2921no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—2921no—
262C24C22×Dic7311yescharacteristic
263C24C2×C22⋊D7311yescharacteristic
264C2×D4—41no—
265C2×D4—41no—
266C2×D4C2×C4×D7312yes—
267C2×D4#1192402812yes—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—81no—
281C22×C4—81no—
282C22×C4—81no—
283C22×C4—2922no—
284C22×C4C4×Dic7312yescomplemented
285C22×C4C22⋊Dic7311yescharacteristic
286C22×C4C22⋊Dic7311yescharacteristic
287C22×C4C22⋊Dic7311yescharacteristic
288C22×C4C22⋊Dic7311yescharacteristic
289C22×C4C2×C22⋊D7311yescharacteristic
290M16—2922no—
291M16—1464no—
292C2×C8—2922no—
293C4⋊C4—24no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—81no—
299C22⋊C4—21no—
300C22⋊C4—24no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—81no—
305C22⋊C4—2922no—
306C22⋊C4—1464no—
307C22⋊C4C4×Dic7312yescomplemented
308C42—21no—
309C42—21no—
310C42—81no—
311C42—5842no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23C2×C4×Dic7311yescharacteristic
321C23C2×C22⋊Dic7311yescharacteristic
322C23#1455655011yescharacteristic
323C23C4×C22⋊D7312yes—
324D4—42no—
325D4—42no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—82no—
354C2×C4—2924no—
355C2×C4—5842no—
356C2×C4—5842no—
357C2×C4#1455650611yescharacteristic
358C2×C4#1455650611yescharacteristic
359C2×C4C4×C22⋊D7312yes—
360C8—2924no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—42no—
369C22—42no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—81no—
375C22C4×C22⋊Dic7311yescharacteristic
376C22C2×#1455650611yescharacteristic
377C22#1798856111yescharacteristic
378C4—22no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—42no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—81no—
387C4—81no—
388C4—5844no—
389C2—21no—
390C2—21no—
391C2—42no—
392C2—41no—
393C2—41no—
394C2C4×#1455650611yescharacteristic
395C2#2261157711yescharacteristic
396C2#2261158511yescharacteristic
subgroupG/H
centerC22C2×#14556506
commutatorC23×C73C2×C42
FrattiniC22×D4C22×D73
FittingC73×#2322C2
socleC22×C73C2×#99
prime radical, cosocleC22×D4×C73C23
hypercenter#2322D73
hypocenterC73#12621
LCSG ▷ C23×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2322

Complements

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

NH ≅ G/Ntype
10. #17988377C4semidirect
15. C22×C4⋊C4×C73C4semidirect
21. #385×C73C4semidirect
22. #381×C73C4semidirect
23. C4×C22⋊C4×C73C4semidirect
91. C22×C4×C73C42semidirect
109. C22⋊C4×C73C42semidirect
204. C73#12621semidirect
207. C22×C4⋊C4Dic73semidirect
216. #385Dic73semidirect
217. #381Dic73semidirect
219. C4×C22⋊C4Dic73semidirect
284. C22×C4C4×Dic73semidirect
307. C22⋊C4C4×Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179883772
C4×#145565082
C4×#145565062
#226080492
C73×#23222
#1262173