Finite Group Databasebeta

← #26184730

Subgroups — #26184730

Order 12032.

1172 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252499C211yesmaximal, characteristic, complemented
2#20251956C211yesmaximal, characteristic, complemented
3#20242065C213yesmaximal, complemented
4#20242060C213yesmaximal, complemented
5#20242037C213yesmaximal, complemented
6C2×#15688869C213yesmaximal, complemented
7C47×#3499C211yesmaximal, characteristic
8C2×#13094031—21no—
9Q8×C47⋊C8—21no—
10D4×C47⋊C8—23no—
11#15689176C2213yes—
12#15689150—23no—
13#15689147C2213yes—
14C2×C4⋊(C47⋊C8)C2213yes—
15C2×C4⋊(C47⋊C8)C2213yes—
16#15688869C2213yes—
17#15688869C2216yes—
18#15688869C2213yes—
19#15688867C2211yescharacteristic
20#15688867C2213yes—
21C2×2−1+4×C47—21no—
22D4×Q8×C47—21no—
23D4×Q8×C47C412yes—
24C47×#547—23no—
25C47×#547C416yes—
26C47×#536—23no—
27C47×#532C2213yes—
28C2×C4⋊Q8×C47C2213yes—
29C4×#41×C47C2211yescharacteristic
30#13094031—44no—
31C2×#10625329—23no—
32C22×C47⋊C8—23no—
33C22×C47⋊C8—23no—
34C4⋊(C47⋊C8)—23no—
35C4⋊(C47⋊C8)—23no—
36C4⋊(C47⋊C8)—23no—
37C4⋊(C47⋊C8)C2311yescharacteristic
38C4⋊(C47⋊C8)C2313yes—
39C4⋊(C47⋊C8)C2313yes—
40C4⋊(C47⋊C8)C2311yescharacteristic
41C22⋊(C47⋊C8)—23no—
42C4×C47⋊C8—23no—
432−1+4×C47—44no—
442−1+4×C47—44no—
45C2×#41×C47—26no—
46C2×#41×C47—23no—
47C2×#41×C47D411yescharacteristic
48C22×Q8×C47—22no—
49C22×Q8×C47—23no—
50C4⋊Q8×C47—23no—
51C4⋊Q8×C47—23no—
52C4⋊Q8×C47C2×C416yes—
53C4⋊Q8×C47C2313yes—
54#127×C47C2311yescharacteristic
55#124×C47—23no—
56#123×C47—212no—
57C22⋊Q8×C47—26no—
58C22⋊Q8×C47—26no—
59#121×C47—23no—
60#121×C47C2×C416yes—
61C4×Q8×C47C2×C411yescharacteristic
62C4×Q8×C47D411yescharacteristic
63C4×D4×C47—23no—
64C4×D4×C47C2×C416yes—
65C4×D4×C47C2×C413yes—
66C4×D4×C47D413yes—
67#117×C47D413yes—
68C2×C4⋊C4×C47—23no—
69C2×C4⋊C4×C47C2×C416yes—
70C2×C42×C47C2313yes—
71#41×C47—424no—
72#41×C47—46no—
73#41×C47—46no—
74#41×C47D814yes—
75#41×C47SD1614yes—
76C2×Q8×C47—26no—
77C2×Q8×C47—23no—
78C2×Q8×C47—48no—
79C2×Q8×C47—46no—
80C2×Q8×C47—46no—
81C2×Q8×C47C22⋊C411yescharacteristic
82C2×D4×C47—26no—
83C2×D4×C47—21no—
84C2×D4×C47—21no—
85C2×D4×C47C22⋊C413yes—
86C2×D4×C47C22×C412yes—
87C22×C4×C47—26no—
88C22×C4×C47—26no—
89C22×C4×C47C22×C413yes—
90C22×C4×C47C2×D413yes—
91C22×C4×C47C2×D413yes—
92C4⋊C4×C47—26no—
93C4⋊C4×C47—26no—
94C4⋊C4×C47—23no—
95C4⋊C4×C47C22⋊C413yes—
96C4⋊C4×C47C22⋊C413yes—
97C4⋊C4×C47C22×C416yes—
98C22⋊C4×C47—26no—
99C22⋊C4×C47—26no—
100C22⋊C4×C47C22⋊C416yes—
101C42×C47C22×C413yes—
102C42×C47C2×D413yes—
103C42×C47C2×D413yes—
104C42×C47C2411yescharacteristic
105#10625329—46no—
106C2×C47⋊C8—21no—
107C2×C47⋊C8—23no—
108C2×C47⋊C8—21no—
109C2×C47⋊C8—23no—
110C2×C47⋊C8—46no—
111C2×C47⋊C8—83no—
112C23×C47—22no—
113C23×C47C2×C22⋊C413yes—
114Q8×C47—43no—
115Q8×C47—43no—
116Q8×C47—412no—
117Q8×C47#10214yes—
118D4×C47—42no—
119D4×C47—412no—
120D4×C47—41no—
121D4×C47—41no—
122D4×C47—41no—
123D4×C47—41no—
124D4×C47#102112yes—
125C2×C4×C47—23no—
126C2×C4×C47—23no—
127C2×C4×C47—26no—
128C2×C4×C47—43no—
129C2×C4×C47—412no—
130C2×C4×C47—412no—
131C2×C4×C47C2×C22⋊C413yes—
132C2×C4×C47C2×C22⋊C413yes—
133C2×C4×C47C2×C22⋊C413yes—
134C2×C4×C47C2×D816yes—
135C2×C4×C47C2×SD1616yes—
136C2×C4×C47C23×C411yescharacteristic
137C2×C4×C47C22×D411yescharacteristic
138C2×C4×C47C22×D411yescharacteristic
139C47⋊C8—41no—
140C47⋊C8—43no—
141C47⋊C8—43no—
142C47⋊C8—41no—
143#17721—471nomaximal, Sylow
144C22×C47—23no—
145C22×C47—23no—
146C22×C47—42no—
147C22×C47—42no—
148C22×C47—42no—
149C22×C47C2×#10216yes—
150C22×C47C22×C22⋊C411yescharacteristic
151C4×C47—21no—
152C4×C47—23no—
153C4×C47—23no—
154C4×C47—46no—
155C4×C47C2×#10216yes—
156C4×C47C22×D811yescharacteristic
157C4×C47C22×SD1611yescharacteristic
158C4×C47#57711yescharacteristic
159C4×C47#57811yescharacteristic
160#3499D4711yescharacteristic
161#2988—471no—
162#2906—471no—
163#1547—473no—
164#1546—473no—
165#1540—473no—
166C2×#332—473no—
167C2×C47—26no—
168C2×C47—82no—
169C2×C47#288911yescharacteristic
170C2×C47C22×#10211yescharacteristic
171C2×C47#289811yescharacteristic
172C2×2−1+4—21no—
173C2×#131—941no—
174D4×Q8—21no—
175D4×Q8Dic4712yes—
176#547—23no—
177#547Dic4716yes—
178#536—23no—
179#532C2×D4713yes—
180C2×C4⋊Q8C2×D4713yes—
181C4×#41C2×D4711yescharacteristic
182C8×Q8—941no—
183C8×D4—943no—
184#431—473no—
185#424—943no—
186#423—473no—
187C2×C4⋊C8—473no—
188C2×C4⋊C8—473no—
189#332—473no—
190#332—476no—
191#332—473no—
192#331—471no—
193#331—473no—
194C47#1772111yescharacteristic, complemented, Sylow
1952−1+4—44no—
1962−1+4—44no—
197C2×#41—26no—
198C2×#41—23no—
199C2×#41C22⋊D4711yescharacteristic
200C22×Q8—22no—
201C22×Q8—23no—
202#131—1884no—
203C2×M16—943no—
204C22×C8—943no—
205C22×C8—943no—
206C4⋊Q8—23no—
207C4⋊Q8—23no—
208C4⋊Q8C2×Dic4716yes—
209C4⋊Q8C22×D4713yes—
210#127C22×D4711yescharacteristic
211#124—23no—
212#123—212no—
213C22⋊Q8—26no—
214C22⋊Q8—26no—
215#121—23no—
216#121C2×Dic4716yes—
217C4×Q8C2×Dic4711yescharacteristic
218C4×Q8C22⋊D4711yescharacteristic
219C4×D4—23no—
220C4×D4C2×Dic4716yes—
221C4×D4C2×Dic4713yes—
222C4×D4C22⋊D4713yes—
223#117C22⋊D4713yes—
224C2×C4⋊C4—23no—
225C2×C4⋊C4C2×Dic4716yes—
226C2×C42C22×D4713yes—
227C4⋊C8—471no—
228C4⋊C8—473no—
229C4⋊C8—473no—
230C4⋊C8—471no—
231C4⋊C8—943no—
232C4⋊C8—943no—
233C4⋊C8—943no—
234C22⋊C8—943no—
235C4×C8—943no—
236#41—424no—
237#41—46no—
238#41—46no—
239#41#1062533414yes—
240#41#1062533514yes—
241C2×Q8—26no—
242C2×Q8—23no—
243C2×Q8—48no—
244C2×Q8—46no—
245C2×Q8—46no—
246C2×Q8C22⋊Dic4711yescharacteristic
247C2×D4—26no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4C22⋊Dic4713yes—
251C2×D4C22×Dic4712yes—
252C22×C4—26no—
253C22×C4—26no—
254C22×C4C22×Dic4713yes—
255C22×C4C2×C22⋊D4713yes—
256C22×C4C2×C22⋊D4713yes—
257M16—1886no—
258C2×C8—941no—
259C2×C8—943no—
260C2×C8—941no—
261C2×C8—943no—
262C2×C8—3763no—
263C2×C8—1886no—
264C4⋊C4—26no—
265C4⋊C4—26no—
266C4⋊C4—23no—
267C4⋊C4C22⋊Dic4713yes—
268C4⋊C4C22⋊Dic4713yes—
269C4⋊C4C22×Dic4716yes—
270C22⋊C4—26no—
271C22⋊C4—26no—
272C22⋊C4C22⋊Dic4716yes—
273C42C22×Dic4713yes—
274C42C2×C22⋊D4713yes—
275C42C2×C22⋊D4713yes—
276C42C23×D4711yescharacteristic
277C23—22no—
278C23C2×C22⋊Dic4713yes—
279Q8—412no—
280Q8—43no—
281Q8—43no—
282Q8#1309394514yes—
283D4—42no—
284D4—412no—
285D4—41no—
286D4—41no—
287D4—41no—
288D4—41no—
289D4#13093945112yes—
290C2×C4—23no—
291C2×C4—23no—
292C2×C4—26no—
293C2×C4—43no—
294C2×C4—412no—
295C2×C4—412no—
296C2×C4C2×C22⋊Dic4713yes—
297C2×C4C2×C22⋊Dic4713yes—
298C2×C4C2×C22⋊Dic4713yes—
299C2×C4C2×#1062533416yes—
300C2×C4C2×#1062533516yes—
301C2×C4C23×Dic4711yescharacteristic
302C2×C4C22×C22⋊D4711yescharacteristic
303C2×C4C22×C22⋊D4711yescharacteristic
304C8—1881no—
305C8—1883no—
306C8—1883no—
307C8—1881no—
308C22—23no—
309C22—23no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22C2×#1309394516yes—
314C22C22×C22⋊Dic4711yescharacteristic
315C4—23no—
316C4—21no—
317C4—23no—
318C4—46no—
319C4C2×#1309394516yes—
320C4C22×#1062533411yescharacteristic
321C4C22×#1062533511yescharacteristic
322C4#1568993111yescharacteristic
323C4#1568993811yescharacteristic
324C2—26no—
325C2—82no—
326C2#2025187111yescharacteristic
327C2C22×#1309394511yescharacteristic
328C2#2025192111yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
Fitting, prime radicalC47×#3499C2
socleC22×C47C22×C22⋊C4
cosocleC42×C47C24
hypercenter#3499D47
hypocenterC47#17721
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3499

Complements

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

NH ≅ G/Ntype
1. #20252499C2semidirect
2. #20251956C2semidirect
3. #20242065C2semidirect
4. #20242060C2semidirect
5. #20242037C2semidirect
6. C2×#15688869C2semidirect
194. C47#17721semidirect

Maximal subgroups

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

subgroupindex
#202524992
#202519562
#202420652
#202420602
#202420372
C2×#156888692
C47×#34992
#1772147