Finite Group Databasebeta

← #32981416

Subgroups — #32981416

Order 18688.

512 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988685C211yesmaximal, characteristic
2#22608326C211yesmaximal, characteristic, complemented
3#22608323C211yesmaximal, characteristic
4#22608251C211yesmaximal, characteristic
5#22607775C211yesmaximal, characteristic, complemented
6#22607761C211yesmaximal, characteristic, complemented
7C73×#2982C211yesmaximal, characteristic, complemented
8C22×#11924015C411yescharacteristic
9C2×C4×D292C2211yescharacteristic
10#17988685—22no—
11#17988685—22no—
12#17988640C2211yescharacteristic
13#17988640C2211yescharacteristic
14C2×#14556524C411yescharacteristic
15C2×#14556515C411yescharacteristic
16C2×#14556511C411yescharacteristic
17C2×#14556502C2211yescharacteristic
18Q8⋊#10596831—22no—
19#17988316—22no—
20#17988310—22no—
21C2×C4×Q8×C73C2211yescharacteristic
22C8×Q8×C73—22no—
23C8×Q8×C73—22no—
24#431×C73—22no—
25#431×C73C2211yescharacteristic
26#431×C73C2211yescharacteristic
27C22×D292C2×C411yescharacteristic
28C22×C4×D73—21no—
29C2×#11924015—22no—
30C2×#11924015—22no—
31C2×#11924015C814yescomplemented
32C22×#10596831C2×C411yescharacteristic
33C22×#10596831C2×C411yescharacteristic
34C4×D292—21no—
35C4×D292—21no—
36C4×D292—21no—
37C4×D292—21no—
38C2×C4⋊Dic73C2×C411yescharacteristic
39C2×#11924005—21no—
40#14556524—22no—
41C4⋊#10596831—21no—
42C4⋊#10596831—21no—
43#14556515—22no—
44#14556511—22no—
45C22⋊#10596831—21no—
46C22⋊#10596831—21no—
47#14556502—21no—
48#14556502—21no—
49C4×#10596831—21no—
50C4×#10596831—21no—
51C22×Q8×C73C2×C411yescharacteristic
52C4×Q8×C73—21no—
53C4×Q8×C73—21no—
54C4×Q8×C73D412yes—
55C4×Q8×C73D412yes—
56C2×C4⋊C4×C73—21no—
57C2×C4⋊C4×C73C2×C411yescharacteristic
58C2×C42×C73—21no—
59C2×C42×C73C2311yescharacteristic
60C4⋊C8×C73—24no—
61C4⋊C8×C73—21no—
62C4⋊C8×C73—21no—
63C22⋊C8×C73—21no—
64C22⋊C8×C73D412yes—
65C4×C8×C73—24no—
66C4×C8×C73—21no—
67C4×C8×C73D412yes—
68C23×D73—21no—
69C2×D292—21no—
70C2×D292—21no—
71C2×D292—21no—
72C2×D292—21no—
73C2×D292—21no—
74C2×D292C2×C812yes—
75C22×Dic73—21no—
76C2×C4×D73—41no—
77C2×C4×D73—41no—
78C2×C4×D73—82no—
79#11924015—28no—
80C2×#10596831—21no—
81C2×#10596831—21no—
82C2×#10596831—21no—
83C2×#10596831—21no—
84C2×#10596831—21no—
85C2×#10596831—41no—
86C2×#10596831—41no—
87C2×#10596831C2×C812yes—
88C4⋊Dic73—21no—
89C4⋊Dic73—21no—
90#11924005—41no—
91#11924005—41no—
92C2×Q8×C73—21no—
93C2×Q8×C73—21no—
94C2×Q8×C73—21no—
95C2×Q8×C73C22⋊C412yes—
96C2×Q8×C73C2×C812yes—
97C2×Q8×C73M1612yes—
98C22×C4×C73—21no—
99C22×C4×C73—21no—
100C22×C4×C73C22×C411yescharacteristic
101C22×C4×C73C2×D411yescharacteristic
102C22×C4×C73#4111yescharacteristic
103C2×C8×C73—22no—
104C2×C8×C73—22no—
105C4⋊C4×C73—21no—
106C4⋊C4×C73—22no—
107C4⋊C4×C73—22no—
108C4⋊C4×C73C22⋊C412yes—
109C42×C73—22no—
110C42×C73—22no—
111C42×C73C2×D411yescharacteristic
112C42×C73C2×D411yescharacteristic
113C42×C73C2×D411yescharacteristic
114C42×C73#4111yescharacteristic
115C23×C73C4×D411yescharacteristic
116C22×D73—22no—
117C22×D73—41no—
118C22×D73—41no—
119C22×D73—81no—
120C22×D73—81no—
121Q8×C73—22no—
122Q8×C73—22no—
123Q8×C73—22no—
124Q8×C73C22⋊C814yes—
125C2×C4×C73—22no—
126C2×C4×C73—22no—
127C2×C4×C73—22no—
128C2×C4×C73—22no—
129C2×C4×C73—41no—
130C2×C4×C73—41no—
131C2×C4×C73—81no—
132C2×C4×C73C2×C22⋊C411yescharacteristic
133C2×C4×C73C4×D411yescharacteristic
134C2×C4×C73#12011yescharacteristic
135C2×C4×C73#12111yescharacteristic
136C2×C4×C73#12111yescharacteristic
137C2×C4×C73#12311yescharacteristic
138C2×C4×C73C22×C811yescharacteristic
139C2×C4×C73C2×M1611yescharacteristic
140C2×C4×C73#13111yescharacteristic
141C2×C4×C73#13111yescharacteristic
142C2×Dic73—41no—
143C2×Dic73—81no—
144D292—22no—
145D292—22no—
146D292—41no—
147D292—41no—
148C4×D73—82no—
149C4×D73—82no—
150C8×C73—42no—
151C8×C73—42no—
152#10596831—22no—
153#10596831—41no—
154#10596831—81no—
155#10596831—81no—
156C22×C73—21no—
157C22×C73—21no—
158C22×C73#38611yescharacteristic
159C22×C73C8×D411yescharacteristic
160C22×C73#43511yescharacteristic
161C2×D73—42no—
162C2×D73—82no—
163C2×D73—81no—
164C2×D73—81no—
165C2×D73—82no—
166C4×C73—24no—
167C4×C73—21no—
168C4×C73—21no—
169C4×C73—21no—
170C4×C73—21no—
171C4×C73—42no—
172C4×C73C2×C22⋊C811yescharacteristic
173C4×C73#40811yescharacteristic
174C4×C73C8×D411yescharacteristic
175C4×C73#43511yescharacteristic
176Dic73—81no—
177#9090—731nomaximal, Sylow
178C2×C73—21no—
179C2×C73—21no—
180C2×C73#186911yescharacteristic
181C2×C73#190311yescharacteristic
182C2×C73#190511yescharacteristic
183D73—82no—
184#2982D7311yescharacteristic, complemented
185C2×#440—731no—
186#1479—731no—
187#1478—731no—
188#1460—731no—
189#1281—731no—
190#1275—731no—
191C73#909011yescharacteristic, complemented, Sylow
192C22×SD16—731no—
193C2×C4×Q8C2×D7311yescharacteristic
194C2×C4×D4—731no—
195C8×Q8—22no—
196C8×Q8—22no—
197#440—1462no—
198#440—1462no—
199#431—22no—
200#431—731no—
201#431—731no—
202#431C2×D7311yescharacteristic
203#431C2×D7311yescharacteristic
204C2×#107—731no—
205C2×#103—731no—
206C2×#102—731no—
207C2×#97—731no—
208Q8⋊C8—1462no—
209D4⋊C8—1462no—
210#323—1462no—
211C22×Q8C4×D7311yescharacteristic
212C22×D4—731no—
213C23×C4—1461no—
214C2×SD16—734no—
215C2×SD16—1462no—
216C2×SD16—1462no—
217C22×C8—731no—
218C22×C8—731no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8D29212yes—
222C4×Q8C22⋊D7312yes—
223C4×D4—1461no—
224C4×D4—1461no—
225C4×D4—1461no—
226C4×D4—1461no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—731no—
229C2×C4⋊C4C4×D7311yescharacteristic
230C2×C22⋊C4—1461no—
231C2×C42—21no—
232C2×C42C22×D7311yescharacteristic
233#107—1462no—
234C4⋊C8—24no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—1461no—
238C4⋊C8—1461no—
239#103—1462no—
240#102—1462no—
241C22⋊C8—21no—
242C22⋊C8—1461no—
243C22⋊C8—1461no—
244C22⋊C8C22⋊D7312yes—
245#97—1461no—
246#97—1461no—
247C4×C8—24no—
248C4×C8—21no—
249C4×C8—1461no—
250C4×C8—1461no—
251C4×C8C22⋊D7312yes—
252C24—1461no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8#1192400512yes—
257C2×Q8C8×D7312yes—
258C2×Q8#1192401112yes—
259C2×D4—732no—
260C2×D4—1461no—
261C2×D4—1461no—
262C2×D4—1461no—
263C2×D4—1461no—
264C2×D4—1461no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—2921no—
268C22×C4—2921no—
269C22×C4—5842no—
270C22×C4—1461no—
271C22×C4C2×C4×D7311yescharacteristic
272C22×C4C2×C22⋊D7311yescharacteristic
273C22×C4#1192402811yescharacteristic
274SD16—1468no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—2921no—
278C2×C8—2921no—
279C2×C8—732no—
280C2×C8—1461no—
281C2×C8—1461no—
282C2×C8—1461no—
283C2×C8—1461no—
284C2×C8—1461no—
285C4⋊C4—21no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—1461no—
289C4⋊C4—1461no—
290C4⋊C4#1192400512yes—
291C22⋊C4—2921no—
292C22⋊C4—2921no—
293C42—22no—
294C42—22no—
295C42C2×C22⋊D7311yescharacteristic
296C42C2×C22⋊D7311yescharacteristic
297C42C2×D29211yescharacteristic
298C42#1192402811yescharacteristic
299C23—2921no—
300C23—2921no—
301C23—5841no—
302C23—5841no—
303C23—1462no—
304C23C4×C22⋊D7311yescharacteristic
305Q8—22no—
306Q8—22no—
307Q8—22no—
308Q8#1455650314yes—
309D4—2921no—
310D4—2921no—
311D4—1462no—
312D4—1462no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—2921no—
321C2×C4—5842no—
322C2×C4—5842no—
323C2×C4—5841no—
324C2×C4C2×#1192400511yescharacteristic
325C2×C4C4×C22⋊D7311yescharacteristic
326C2×C4#1455656011yescharacteristic
327C2×C4#1455656511yescharacteristic
328C2×C4#1455656511yescharacteristic
329C2×C4#1455657411yescharacteristic
330C2×C4C2×C8×D7311yescharacteristic
331C2×C4C2×#1192401111yescharacteristic
332C2×C4#1455659711yescharacteristic
333C2×C4#1455659711yescharacteristic
334C8—42no—
335C8—42no—
336C8—2921no—
337C8—5841no—
338C8—5841no—
339C8—1462no—
340C22—21no—
341C22—21no—
342C22—2922no—
343C22—5842no—
344C22—5842no—
345C22—5841no—
346C22—5841no—
347C22#1798846611yescharacteristic
348C22C8×C22⋊D7311yescharacteristic
349C22#1798865811yescharacteristic
350C4—24no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—42no—
356C4—5841no—
357C4C2×#1455650311yescharacteristic
358C4#1798855311yescharacteristic
359C4C8×C22⋊D7311yescharacteristic
360C4#1798865811yescharacteristic
361C2—21no—
362C2—21no—
363C2—5842no—
364C2#2261022311yescharacteristic
365C2#2261041111yescharacteristic
366C2#2261042511yescharacteristic
subgroupG/H
centerC22#17988466
commutatorC2×C4×C73C22×C8
FrattiniC2×C42C22×D73
FittingC73×#2982C2
socleC22×C73#386
prime radicalC22×D292C2×C4
cosocleC2×C42×C73C23
hypercenter#2982D73
hypocenterC73#9090
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. #22608326C2semidirect
5. #22607775C2semidirect
6. #22607761C2semidirect
7. C73×#2982C2semidirect
31. C2×#11924015C8semidirect
184. #2982D73semidirect
191. C73#9090semidirect

Maximal subgroups

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

subgroupindex
C2×#179886852
#226083262
#226083232
#226082512
#226077752
#226077612
C73×#29822
#909073