Finite Group Databasebeta

← #32979889

Subgroups — #32979889

Order 18688.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619044C211yesmaximal, characteristic
2#22610048C211yesmaximal, characteristic, complemented
3C4×#14556519C211yesmaximal, characteristic
4#22608684C211yesmaximal, characteristic, complemented
5#22608319C211yesmaximal, characteristic
6#22607780C211yesmaximal, characteristic, complemented
7C73×#1276C211yesmaximal, characteristic, complemented
8C4×#11924028C2211yescharacteristic
9#17988717—21no—
10#17988666—21no—
11#17988655—22no—
12#17988643—21no—
13#17988609—21no—
14C2×C4⋊#10596831C2211yescharacteristic
15C2×C4⋊#10596831C2211yescharacteristic
16C2×#14556519C411yescharacteristic
17C2×#14556519C411yescharacteristic
18C4×#11924012C2211yescharacteristic
19#17988347C411yescharacteristic
20#17988347C411yescharacteristic
21#17988342—22no—
22#17988339—21no—
23#17988337—21no—
24#17988319—21no—
25#17988316—21no—
26#431×C73C2211yescharacteristic
27#431×C73C2211yescharacteristic
28C43×C73C2211yescharacteristic
29C2×#11924028C2×C411yescharacteristic
30C2×#11924012C2×C411yescharacteristic
31C2×#11924012C2×C411yescharacteristic
32C2×#11924011—22no—
33C22×#10596831—21no—
34C22×#10596831D412yes—
35C4×Dic146D411yescharacteristic
36C4×Dic146Q811yescharacteristic
37C4×D292D411yescharacteristic
38C4×D292Q811yescharacteristic
39C4×C22⋊D73—21no—
40C4×C22⋊D73—21no—
41#14556550C2×C411yescharacteristic
42#14556548—21no—
43C42×D73—21no—
44C4⋊#10596831—21no—
45C4⋊#10596831—21no—
46C4⋊#10596831—21no—
47C4⋊#10596831—21no—
48#14556520—22no—
49#14556519—22no—
50#14556519—22no—
51#14556503—22no—
52#14556502—21no—
53#14556501—22no—
54C4×#10596831—21no—
55C2×C42×C73—21no—
56C2×C42×C73—21no—
57C2×C42×C73C2×C411yescharacteristic
58C2×C42×C73C2×C411yescharacteristic
59C2×C42×C73C2311yescharacteristic
60C4⋊C8×C73—21no—
61C4⋊C8×C73D411yescharacteristic
62C4⋊C8×C73D411yescharacteristic
63C22⋊C8×C73—21no—
64C22⋊C8×C73—21no—
65C4×C8×C73—21no—
66C4×C8×C73—21no—
67#11924028—21no—
68#11924028—21no—
69#11924028—21no—
70#11924028—21no—
71C2×Dic146C4⋊C411yescharacteristic
72C2×D292C4⋊C411yescharacteristic
73C2×C22⋊D73—21no—
74C2×C4×D73—21no—
75C2×C4×D73—21no—
76C2×C4×D73—21no—
77#11924012—21no—
78#11924012—21no—
79#11924012—21no—
80#11924012M1612yescomplemented
81#11924011—82no—
82C2×#10596831—21no—
83C2×#10596831—21no—
84C2×#10596831—22no—
85C2×#10596831—42no—
86C2×#10596831—42no—
87C2×#10596831—41no—
88C4⋊Dic73C4⋊C411yescharacteristic
89C4⋊Dic73C4⋊C411yescharacteristic
90#11924007—21no—
91#11924007—21no—
92C22⋊Dic73—21no—
93#11924005—21no—
94#11924005—21no—
95C4×Dic73—21no—
96C4×Dic73—21no—
97C4×Dic73—21no—
98C22×C4×C73—21no—
99C22×C4×C73—21no—
100C22×C4×C73C22×C411yescharacteristic
101C22×C4×C73C2×D411yescharacteristic
102C22×C4×C73#4111yescharacteristic
103C2×C8×C73—21no—
104C2×C8×C73—21no—
105C2×C8×C73—41no—
106C42×C73—21no—
107C42×C73—21no—
108C42×C73—21no—
109C42×C73—41no—
110C42×C73—41no—
111C42×C73—42no—
112C42×C73M1612yescomplemented
113C42×C73C2×D411yescharacteristic
114C42×C73C2×D411yescharacteristic
115C42×C73C2×Q811yescharacteristic
116C42×C73#4111yescharacteristic
117C23×C73C4×D411yescharacteristic
118C22×D73—21no—
119C2×C4×C73—21no—
120C2×C4×C73—21no—
121C2×C4×C73—22no—
122C2×C4×C73—22no—
123C2×C4×C73—41no—
124C2×C4×C73—41no—
125C2×C4×C73—41no—
126C2×C4×C73—41no—
127C2×C4×C73—41no—
128C2×C4×C73C2×C4⋊C411yescharacteristic
129C2×C4×C73C4×D411yescharacteristic
130C2×C4×C73#12011yescharacteristic
131C2×C4×C73C22⋊Q811yescharacteristic
132C2×C4×C73C22⋊Q811yescharacteristic
133C2×C4×C73#12311yescharacteristic
134C2×C4×C73C2×M1611yescharacteristic
135C2×C4×C73C2×M1611yescharacteristic
136C2×C4×C73#13111yescharacteristic
137C2×C4×C73#13111yescharacteristic
138C22⋊D73—41no—
139C22⋊D73—41no—
140C2×Dic73—21no—
141C2×Dic73—21no—
142C2×Dic73—21no—
143D292—21no—
144D292—21no—
145C4×D73—41no—
146C4×D73—41no—
147C4×D73—41no—
148C4×D73—41no—
149C4×D73—81no—
150Dic146—21no—
151Dic146—21no—
152C8×C73—41no—
153C8×C73—41no—
154C8×C73—81no—
155#10596831—22no—
156#10596831—41no—
157#10596831—42no—
158#10596831—81no—
159C22×C73—21no—
160C22×C73—21no—
161C22×C73#38511yescharacteristic
162C22×C73#43511yescharacteristic
163C22×C73#43611yescharacteristic
164C2×D73—41no—
165C2×D73—41no—
166C2×D73—41no—
167C4×C73—21no—
168C4×C73—21no—
169C4×C73—21no—
170C4×C73—42no—
171C4×C73—42no—
172C4×C73#42311yescharacteristic
173C4×C73#42411yescharacteristic
174C4×C73#43511yescharacteristic
175C4×C73#43611yescharacteristic
176C4×C73#45411yescharacteristic
177C4×C73#45511yescharacteristic
178Dic73—41no—
179Dic73—41no—
180Dic73—41no—
181#8842—731nomaximal, Sylow
182C2×C73—21no—
183C2×C73—21no—
184C2×C73#185111yescharacteristic
185C2×C73#186011yescharacteristic
186C2×C73#186111yescharacteristic
187D73—81no—
188#2989—731no—
189#1831—731no—
190C4×#104—731no—
191#1565—731no—
192#1477—731no—
193#1283—731no—
194#1276D7311yescharacteristic, complemented
195C73#884211yescharacteristic, complemented, Sylow
196C4×#41—731no—
197#446—1461no—
198#436—1461no—
199#435—1462no—
200#432—1461no—
201#431C2×D7311yescharacteristic
202#431C2×D7311yescharacteristic
203#423—1461no—
204C2×C4⋊C8—731no—
205C2×C4⋊C8—731no—
206C2×#104—731no—
207C2×#104—731no—
208C4×M16—731no—
209C43C2×D7311yescharacteristic
210#339—731no—
211#339—731no—
212#336—1462no—
213#335—1461no—
214#334—1461no—
215Q8⋊C8—1461no—
216D4⋊C8—1461no—
217C2×#41—731no—
218C2×M16—731no—
219C2×M16—731no—
220C2×M16—1462no—
221C22×C8—732no—
222C22×C8—1461no—
223C4×Q8—731no—
224C4×Q8—731no—
225C4×D4—731no—
226C4×D4—731no—
227C4×D4—1461no—
228C4×D4—1461no—
229#117—731no—
230#117—1461no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—1461no—
234C2×C42C4×D7311yescharacteristic
235C2×C42C4×D7311yescharacteristic
236C2×C42C22×D7311yescharacteristic
237C4⋊C8—21no—
238C4⋊C8—1462no—
239C4⋊C8—1461no—
240C4⋊C8—1461no—
241C4⋊C8—1461no—
242C4⋊C8—1461no—
243C4⋊C8D29211yescharacteristic
244C4⋊C8D29211yescharacteristic
245#104—1462no—
246#104—1462no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C22⋊C8—1462no—
250#97—1462no—
251#97—1461no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—1461no—
255#41—1461no—
256#41—1461no—
257#41—1461no—
258#41—1461no—
259C2×Q8—731no—
260C2×D4—731no—
261C2×D4—1461no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—1461no—
265C22×C4—1461no—
266C22×C4—1461no—
267C22×C4C2×C4×D7311yescharacteristic
268C22×C4D4×D7311yescharacteristic
269C22×C4#1192402711yescharacteristic
270M16—5842no—
271M16—732no—
272M16—1461no—
273M16—1461no—
274M16—1461no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—41no—
278C2×C8—2922no—
279C2×C8—2922no—
280C2×C8—2921no—
281C2×C8—1461no—
282C2×C8—1462no—
283C2×C8—1461no—
284C4⋊C4—731no—
285C4⋊C4—731no—
286C4⋊C4—1461no—
287C4⋊C4—1461no—
288C22⋊C4—1461no—
289C22⋊C4—1461no—
290C22⋊C4—1461no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—41no—
295C42—41no—
296C42—42no—
297C42—1461no—
298C42—1461no—
299C42—1461no—
300C42#1192401112yescomplemented
301C42D4×D7311yescharacteristic
302C42C2×D29211yescharacteristic
303C42Q8×D7311yescharacteristic
304C42#1192402811yescharacteristic
305C23—1461no—
306C23#1455655111yescharacteristic
307Q8—1461no—
308Q8—1461no—
309D4—2921no—
310D4—2921no—
311D4—1461no—
312D4—1461no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—2921no—
323C2×C4—2921no—
324C2×C4—2921no—
325C2×C4—2921no—
326C2×C4—5841no—
327C2×C4—1461no—
328C2×C4—1461no—
329C2×C4—1461no—
330C2×C4C4⋊C4×D7311yescharacteristic
331C2×C4C4×D29211yescharacteristic
332C2×C4#1455656111yescharacteristic
333C2×C4#1455656911yescharacteristic
334C2×C4#1455657111yescharacteristic
335C2×C4#1455657311yescharacteristic
336C2×C4M16×D7311yescharacteristic
337C2×C4C2×#1192401111yescharacteristic
338C2×C4#1455659711yescharacteristic
339C2×C4#1455659911yescharacteristic
340C8—41no—
341C8—41no—
342C8—81no—
343C8—2921no—
344C8—2922no—
345C8—5841no—
346C8—1462no—
347C22—21no—
348C22—21no—
349C22—2921no—
350C22—2921no—
351C22—2921no—
352C22#1798846111yescharacteristic
353C22#1798865511yescharacteristic
354C22#1798866211yescharacteristic
355C4—21no—
356C4—21no—
357C4—21no—
358C4—42no—
359C4—42no—
360C4—2921no—
361C4—2921no—
362C4—2921no—
363C4#1798860911yescharacteristic
364C4#1798861211yescharacteristic
365C4#1798866011yescharacteristic
366C4#1798866611yescharacteristic
367C4#1798876111yescharacteristic
368C4#1798876611yescharacteristic
369C2—21no—
370C2—21no—
371C2—5841no—
372C2#2261011511yescharacteristic
373C2#2261016511yescharacteristic
374C2#2261017211yescharacteristic
subgroupG/H
centerC2×C4#14556561
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1276C2
socleC22×C73#385
prime radicalC2×#11924028C2×C4
cosocleC2×C42×C73C23
hypercenter#1276D73
hypocenterC73#8842
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
2. #22610048C2semidirect
4. #22608684C2semidirect
6. #22607780C2semidirect
7. C73×#1276C2semidirect
80. #11924012M16semidirect
112. C42×C73M16semidirect
194. #1276D73semidirect
195. C73#8842semidirect
300. C42#11924011semidirect

Maximal subgroups

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

subgroupindex
#226190442
#226100482
C4×#145565192
#226086842
#226083192
#226077802
C73×#12762
#884273