Finite Group Databasebeta

← #33003360

Subgroups — #33003360

Order 18688.

506 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
1C22×#14556530C211yesmaximal, characteristic
2C2×#17988829C211yesmaximal, characteristic
3C2×#17988392C211yesmaximal, characteristic, complemented
4C2×#17988390C211yesmaximal, characteristic, complemented
5#22609957C211yesmaximal, characteristic
6#22608022C211yesmaximal, characteristic, complemented
7C73×#1380C211yesmaximal, characteristic, complemented
8C22×Dic292C2211yescharacteristic
9C22×C8×D73C2211yescharacteristic
10C2×#14556571—21no—
11C2×#14556530—22no—
12C2×#14556530—22no—
13C2×#14556530—22no—
14C22×#11924003C2211yescharacteristic
15#17988829—22no—
16#17988829C414yescomplemented
17C2×#14556525C2211yescharacteristic
18C2×#14556514—21no—
19#17988444—21no—
20#17988392—22no—
21#17988390—22no—
22#17988349—21no—
23C22×C16×C73C2211yescharacteristic
24C2×#107×C73C2211yescharacteristic
25C2×#107×C73C2211yescharacteristic
26C22×Dic146—21no—
27C22×C4×D73D411yescharacteristic
28C2×Dic292—21no—
29C2×Dic292—21no—
30C2×Dic292—21no—
31C2×Dic292—21no—
32C2×Dic292—21no—
33C2×Dic292C2×C412yes—
34C22×#10596831D411yescharacteristic
35C2×C8×D73—21no—
36C2×C8×D73—21no—
37C2×C8×D73—21no—
38C2×C8×D73—21no—
39C2×C8×D73C2×C412yes—
40C2×C8×D73D412yes—
41#14556571—24no—
42#14556571—42no—
43C2×C4⋊Dic73—21no—
44C2×C4⋊Dic73—41no—
45C2×#11924005—41no—
46C2×#11924005—41no—
47#14556530—28no—
48C2×#11924003—21no—
49C2×#11924003—21no—
50C2×#11924003D412yes—
51#14556525—21no—
52#14556525—21no—
53#14556514—24no—
54#14556498—41no—
55C22×C8×C73C2311yescharacteristic
56C2×C4⋊C4×C73—21no—
57C2×C4⋊C4×C73—21no—
58C2×C16×C73—21no—
59C2×C16×C73—21no—
60C2×C16×C73D412yes—
61#107×C73—21no—
62#107×C73—21no—
63#107×C73—21no—
64#107×C73C2×C412yes—
65C23×D73D811yescharacteristic
66C2×Dic146—22no—
67C2×Dic146—41no—
68C2×Dic146—41no—
69C2×Dic146—41no—
70C2×Dic146—41no—
71C2×Dic146—41no—
72C22×Dic73—41no—
73C22×Dic73—41no—
74C22×Dic73D811yescharacteristic
75C2×C4×D73—21no—
76C2×C4×D73—21no—
77C2×C4×D73—21no—
78C2×C4×D73—21no—
79C2×C4×D73C22⋊C412yes—
80C2×C4×D73SD1612yes—
81Dic292—22no—
82Dic292—22no—
83Dic292—41no—
84Dic292—41no—
85C2×#10596831—21no—
86C2×#10596831—21no—
87C2×#10596831C22⋊C412yes—
88C8×D73—22no—
89C8×D73—22no—
90C8×D73—22no—
91C8×D73—22no—
92C4⋊Dic73—44no—
93C4⋊Dic73—41no—
94C4⋊Dic73—41no—
95#11924005—44no—
96#11924005—82no—
97#11924003—22no—
98C22×C4×C73—41no—
99C22×C4×C73—41no—
100C22×C4×C73C2×D411yescharacteristic
101C2×C8×C73C22×C411yescharacteristic
102C2×C8×C73C2×D411yescharacteristic
103C2×C8×C73C2×D411yescharacteristic
104C2×C8×C73C2×D411yescharacteristic
105C2×C8×C73#4111yescharacteristic
106C2×C8×C73#4111yescharacteristic
107C4⋊C4×C73—22no—
108C4⋊C4×C73—41no—
109C4⋊C4×C73—41no—
110C4⋊C4×C73—41no—
111C16×C73—22no—
112C23×C73C2×D811yescharacteristic
113C22×D73—21no—
114C22×D73—21no—
115C22×D73—21no—
116C22×D73—21no—
117C22×D73—21no—
118C22×D73—21no—
119C22×D73#10212yes—
120C2×C4×C73—42no—
121C2×C4×C73—81no—
122C2×C4×C73—81no—
123C2×C4×C73—81no—
124C2×C4×C73—81no—
125C2×C4×C73—81no—
126C2×C4×C73C2×C22⋊C411yescharacteristic
127C2×C4×C73#12111yescharacteristic
128C2×C4×C73#12111yescharacteristic
129C2×C4×C73C2×SD1611yescharacteristic
130C2×C4×C73#13511yescharacteristic
131C2×C4×C73#13511yescharacteristic
132C2×Dic73—21no—
133C2×Dic73—21no—
134C2×Dic73—42no—
135C2×Dic73—81no—
136C2×Dic73—81no—
137C2×Dic73—81no—
138C2×Dic73—81no—
139C2×Dic73—81no—
140C2×Dic73#10212yes—
141C4×D73—22no—
142C4×D73—22no—
143C4×D73—22no—
144C4×D73—22no—
145Dic146—42no—
146Dic146—42no—
147Dic146—81no—
148Dic146—81no—
149C8×C73C4×D411yescharacteristic
150C8×C73C4×D411yescharacteristic
151C8×C73#12011yescharacteristic
152C8×C73#12311yescharacteristic
153#10596831—22no—
154C22×C73C2×#10211yescharacteristic
155C22×C73#46611yescharacteristic
156C22×C73#46611yescharacteristic
157C22×C73#50911yescharacteristic
158C22×C73#51011yescharacteristic
159C22×C73#51011yescharacteristic
160C22×C73#51011yescharacteristic
161C2×D73—22no—
162C2×D73—22no—
163C2×D73—22no—
164C2×D73—41no—
165C2×D73—41no—
166C2×D73—41no—
167C2×D73—41no—
168C4×C73—82no—
169C4×C73—161no—
170C4×C73#38611yescharacteristic
171C4×C73#41611yescharacteristic
172C4×C73#46511yescharacteristic
173C4×C73#46511yescharacteristic
174Dic73—22no—
175Dic73—82no—
176Dic73—161no—
177#13062—731nomaximal, Sylow
178C2×C73#189311yescharacteristic
179C2×C73#213911yescharacteristic
180C2×C73#214111yescharacteristic
181C2×C73#221911yescharacteristic
182C2×C73#221911yescharacteristic
183C2×C73#222011yescharacteristic
184C2×C73#222011yescharacteristic
185D73—42no—
186C22×M32—731no—
187C2×#467—731no—
188C2×#358—731no—
189C2×#358—731no—
190#1808—731no—
191#1380—731no—
192#1380D7311yescharacteristic, complemented
193C73#1306211yescharacteristic, complemented, Sylow
194C22×Q16—731no—
195C23×C8—731no—
196C2×C22⋊Q8—1461no—
197C2×M32—1462no—
198C2×M32—1462no—
199C2×M32—1462no—
200C22×C16—731no—
201C22×C16C2×D7311yescharacteristic
202#467—734no—
203#467—1462no—
204C2×#107—731no—
205C2×#107C2×D7311yescharacteristic
206C2×#107C2×D7311yescharacteristic
207C2×#103—1461no—
208#380—1461no—
209#358—1462no—
210#358—1462no—
211#340—1461no—
212C22×Q8—1461no—
213C23×C4—731no—
214C2×Q16—732no—
215C2×Q16—1461no—
216C2×Q16—1461no—
217C2×Q16—1461no—
218C2×Q16—1461no—
219C2×Q16—1461no—
220C22×C8—731no—
221C22×C8—732no—
222C22×C8—732no—
223C22×C8—1461no—
224C22×C8—1461no—
225C22×C8—1461no—
226C22×C8—1461no—
227C22×C8C22×D7311yescharacteristic
228C22⋊Q8—2922no—
229C22⋊Q8—1464no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—2921no—
233C2×C4⋊C4—1461no—
234C2×C22⋊C4—2921no—
235C2×C22⋊C4—2921no—
236M32—1468no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—732no—
240C2×C16—1461no—
241C2×C16—1461no—
242C2×C16D29212yes—
243#107—21no—
244#107—21no—
245#107—21no—
246#107—1461no—
247#107—1461no—
248#107C4×D7312yes—
249#103—1464no—
250#95—2921no—
251C24—731no—
252C2×Q8—2921no—
253C2×Q8—2921no—
254C2×Q8—2921no—
255C2×Q8—2921no—
256C2×Q8—2921no—
257C2×Q8—1462no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—2921no—
261C22×C4—2921no—
262C22×C4—731no—
263C22×C4—732no—
264C22×C4—732no—
265C22×C4—1461no—
266C22×C4—1461no—
267C22×C4—1461no—
268C22×C4—1461no—
269C22×C4D4×D7311yescharacteristic
270Q16—2921no—
271Q16—2921no—
272Q16—1462no—
273Q16—1462no—
274C2×C8—732no—
275C2×C8—1461no—
276C2×C8—1461no—
277C2×C8—1462no—
278C2×C8—1462no—
279C2×C8—1462no—
280C2×C8—1462no—
281C2×C8C2×C4×D7311yescharacteristic
282C2×C8D4×D7311yescharacteristic
283C2×C8D4×D7311yescharacteristic
284C2×C8C2×D29211yescharacteristic
285C2×C8#1192402811yescharacteristic
286C2×C8#1192402911yescharacteristic
287C4⋊C4—22no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—2924no—
292C4⋊C4—2921no—
293C4⋊C4—2921no—
294C22⋊C4—2924no—
295C22⋊C4—5842no—
296C16—22no—
297C16—1462no—
298C23—732no—
299C23—1461no—
300C23—1461no—
301C23—1461no—
302C23—1461no—
303C23—1461no—
304C23—1461no—
305C23D8×D7311yescharacteristic
306Q8—2922no—
307Q8—2922no—
308Q8—5841no—
309Q8—5841no—
310C2×C4—42no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—2922no—
317C2×C4—5841no—
318C2×C4—5841no—
319C2×C4—5841no—
320C2×C4—5841no—
321C2×C4—5841no—
322C2×C4—732no—
323C2×C4—1461no—
324C2×C4—1461no—
325C2×C4—1462no—
326C2×C4—1462no—
327C2×C4—1462no—
328C2×C4—1462no—
329C2×C4C22⋊C4×D7311yescharacteristic
330C2×C4#1455656411yescharacteristic
331C2×C4#1455656611yescharacteristic
332C2×C4SD16×D7311yescharacteristic
333C2×C4#1455661211yescharacteristic
334C2×C4#1455661411yescharacteristic
335C8—1462no—
336C8#1455655311yescharacteristic
337C8C4×D29211yescharacteristic
338C8#1455656111yescharacteristic
339C8#1455657611yescharacteristic
340C22—2921no—
341C22—2921no—
342C22—2921no—
343C22—2921no—
344C22—1462no—
345C22—1462no—
346C22—1462no—
347C22#102×D7311yescharacteristic
348C22#1798882211yescharacteristic
349C22#1798882411yescharacteristic
350C22#1798902111yescharacteristic
351C22#1798902611yescharacteristic
352C22#1798902611yescharacteristic
353C22#1798902911yescharacteristic
354C4—82no—
355C4—161no—
356C4—5842no—
357C4—11681no—
358C4—1462no—
359C4#1798846511yescharacteristic
360C4#1798858211yescharacteristic
361C4#1798881711yescharacteristic
362C4#1798881911yescharacteristic
363C2—2922no—
364C2#2261035811yescharacteristic
365C2#2261164011yescharacteristic
366C2#2261164911yescharacteristic
367C2#2261203811yescharacteristic
368C2#2261204011yescharacteristic
369C2#2261204311yescharacteristic
370C2#2261204511yescharacteristic
subgroupG/H
centerC23D8×D73
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC73×#1380C2
socleC23×C73C2×D8
prime radicalC23×D73D8
cosocleC22×C8×C73C23
hypercenter#1380D73
hypocenterC73#13062
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
3. C2×#17988392C2semidirect
4. C2×#17988390C2semidirect
6. #22608022C2semidirect
7. C73×#1380C2semidirect
16. #17988829C4semidirect
192. #1380D73semidirect
193. C73#13062semidirect

Maximal subgroups

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

subgroupindex
C22×#145565302
C2×#179888292
C2×#179883922
C2×#179883902
#226099572
#226080222
C73×#13802
#1306273