Finite Group Databasebeta

← #28557141

Subgroups — #28557141

Order 15104.

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#21433207C211yesmaximal, characteristic, complemented
2#21432664C211yesmaximal, characteristic, complemented
3#21422773C213yesmaximal, complemented
4#21422768C213yesmaximal, complemented
5#21422745C213yesmaximal, complemented
6C2×#16837148C213yesmaximal, complemented
7C59×#3499C211yesmaximal, characteristic
8C2×#14273734—21no—
9Q8×C59⋊C8—21no—
10D4×C59⋊C8—23no—
11#16837455C2213yes—
12#16837429—23no—
13#16837426C2213yes—
14C2×C4⋊(C59⋊C8)C2213yes—
15C2×C4⋊(C59⋊C8)C2213yes—
16#16837148C2213yes—
17#16837148C2216yes—
18#16837148C2213yes—
19#16837146C2211yescharacteristic
20#16837146C2213yes—
21C2×2−1+4×C59—21no—
22D4×Q8×C59—21no—
23D4×Q8×C59C412yes—
24C59×#547—23no—
25C59×#547C416yes—
26C59×#536—23no—
27C59×#532C2213yes—
28C2×C4⋊Q8×C59C2213yes—
29C4×#41×C59C2211yescharacteristic
30#14273734—44no—
31C2×#11745664—23no—
32C22×C59⋊C8—23no—
33C22×C59⋊C8—23no—
34C4⋊(C59⋊C8)—23no—
35C4⋊(C59⋊C8)—23no—
36C4⋊(C59⋊C8)—23no—
37C4⋊(C59⋊C8)C2311yescharacteristic
38C4⋊(C59⋊C8)C2313yes—
39C4⋊(C59⋊C8)C2313yes—
40C4⋊(C59⋊C8)C2311yescharacteristic
41C22⋊(C59⋊C8)—23no—
42C4×C59⋊C8—23no—
432−1+4×C59—44no—
442−1+4×C59—44no—
45C2×#41×C59—26no—
46C2×#41×C59—23no—
47C2×#41×C59D411yescharacteristic
48C22×Q8×C59—22no—
49C22×Q8×C59—23no—
50C4⋊Q8×C59—23no—
51C4⋊Q8×C59—23no—
52C4⋊Q8×C59C2×C416yes—
53C4⋊Q8×C59C2313yes—
54#127×C59C2311yescharacteristic
55#124×C59—23no—
56#123×C59—212no—
57C22⋊Q8×C59—26no—
58C22⋊Q8×C59—26no—
59#121×C59—23no—
60#121×C59C2×C416yes—
61C4×Q8×C59C2×C411yescharacteristic
62C4×Q8×C59D411yescharacteristic
63C4×D4×C59—23no—
64C4×D4×C59C2×C416yes—
65C4×D4×C59C2×C413yes—
66C4×D4×C59D413yes—
67#117×C59D413yes—
68C2×C4⋊C4×C59—23no—
69C2×C4⋊C4×C59C2×C416yes—
70C2×C42×C59C2313yes—
71#41×C59—424no—
72#41×C59—46no—
73#41×C59—46no—
74#41×C59D814yes—
75#41×C59SD1614yes—
76C2×Q8×C59—26no—
77C2×Q8×C59—23no—
78C2×Q8×C59—48no—
79C2×Q8×C59—46no—
80C2×Q8×C59—46no—
81C2×Q8×C59C22⋊C411yescharacteristic
82C2×D4×C59—26no—
83C2×D4×C59—21no—
84C2×D4×C59—21no—
85C2×D4×C59C22⋊C413yes—
86C2×D4×C59C22×C412yes—
87C22×C4×C59—26no—
88C22×C4×C59—26no—
89C22×C4×C59C22×C413yes—
90C22×C4×C59C2×D413yes—
91C22×C4×C59C2×D413yes—
92C4⋊C4×C59—26no—
93C4⋊C4×C59—26no—
94C4⋊C4×C59—23no—
95C4⋊C4×C59C22⋊C413yes—
96C4⋊C4×C59C22⋊C413yes—
97C4⋊C4×C59C22×C416yes—
98C22⋊C4×C59—26no—
99C22⋊C4×C59—26no—
100C22⋊C4×C59C22⋊C416yes—
101C42×C59C22×C413yes—
102C42×C59C2×D413yes—
103C42×C59C2×D413yes—
104C42×C59C2411yescharacteristic
105#11745664—46no—
106C2×C59⋊C8—21no—
107C2×C59⋊C8—23no—
108C2×C59⋊C8—21no—
109C2×C59⋊C8—23no—
110C2×C59⋊C8—46no—
111C2×C59⋊C8—83no—
112C23×C59—22no—
113C23×C59C2×C22⋊C413yes—
114Q8×C59—43no—
115Q8×C59—43no—
116Q8×C59—412no—
117Q8×C59#10214yes—
118D4×C59—42no—
119D4×C59—412no—
120D4×C59—41no—
121D4×C59—41no—
122D4×C59—41no—
123D4×C59—41no—
124D4×C59#102112yes—
125C2×C4×C59—23no—
126C2×C4×C59—23no—
127C2×C4×C59—26no—
128C2×C4×C59—43no—
129C2×C4×C59—412no—
130C2×C4×C59—412no—
131C2×C4×C59C2×C22⋊C413yes—
132C2×C4×C59C2×C22⋊C413yes—
133C2×C4×C59C2×C22⋊C413yes—
134C2×C4×C59C2×D816yes—
135C2×C4×C59C2×SD1616yes—
136C2×C4×C59C23×C411yescharacteristic
137C2×C4×C59C22×D411yescharacteristic
138C2×C4×C59C22×D411yescharacteristic
139C59⋊C8—41no—
140C59⋊C8—43no—
141C59⋊C8—43no—
142C59⋊C8—41no—
143#17721—591nomaximal, Sylow
144C22×C59—23no—
145C22×C59—23no—
146C22×C59—42no—
147C22×C59—42no—
148C22×C59—42no—
149C22×C59C2×#10216yes—
150C22×C59C22×C22⋊C411yescharacteristic
151C4×C59—21no—
152C4×C59—23no—
153C4×C59—23no—
154C4×C59—46no—
155C4×C59C2×#10216yes—
156C4×C59C22×D811yescharacteristic
157C4×C59C22×SD1611yescharacteristic
158C4×C59#57711yescharacteristic
159C4×C59#57811yescharacteristic
160#3499D5911yescharacteristic
161#2988—591no—
162#2906—591no—
163#1547—593no—
164#1546—593no—
165#1540—593no—
166C2×#332—593no—
167C2×C59—26no—
168C2×C59—82no—
169C2×C59#288911yescharacteristic
170C2×C59C22×#10211yescharacteristic
171C2×C59#289811yescharacteristic
172C2×2−1+4—21no—
173C2×#131—1181no—
174D4×Q8—21no—
175D4×Q8Dic5912yes—
176#547—23no—
177#547Dic5916yes—
178#536—23no—
179#532C2×D5913yes—
180C2×C4⋊Q8C2×D5913yes—
181C4×#41C2×D5911yescharacteristic
182C8×Q8—1181no—
183C8×D4—1183no—
184#431—593no—
185#424—1183no—
186#423—593no—
187C2×C4⋊C8—593no—
188C2×C4⋊C8—593no—
189#332—593no—
190#332—596no—
191#332—593no—
192#331—591no—
193#331—593no—
194C59#1772111yescharacteristic, complemented, Sylow
1952−1+4—44no—
1962−1+4—44no—
197C2×#41—26no—
198C2×#41—23no—
199C2×#41C22⋊D5911yescharacteristic
200C22×Q8—22no—
201C22×Q8—23no—
202#131—2364no—
203C2×M16—1183no—
204C22×C8—1183no—
205C22×C8—1183no—
206C4⋊Q8—23no—
207C4⋊Q8—23no—
208C4⋊Q8C2×Dic5916yes—
209C4⋊Q8C22×D5913yes—
210#127C22×D5911yescharacteristic
211#124—23no—
212#123—212no—
213C22⋊Q8—26no—
214C22⋊Q8—26no—
215#121—23no—
216#121C2×Dic5916yes—
217C4×Q8C2×Dic5911yescharacteristic
218C4×Q8C22⋊D5911yescharacteristic
219C4×D4—23no—
220C4×D4C2×Dic5916yes—
221C4×D4C2×Dic5913yes—
222C4×D4C22⋊D5913yes—
223#117C22⋊D5913yes—
224C2×C4⋊C4—23no—
225C2×C4⋊C4C2×Dic5916yes—
226C2×C42C22×D5913yes—
227C4⋊C8—591no—
228C4⋊C8—593no—
229C4⋊C8—593no—
230C4⋊C8—591no—
231C4⋊C8—1183no—
232C4⋊C8—1183no—
233C4⋊C8—1183no—
234C22⋊C8—1183no—
235C4×C8—1183no—
236#41—424no—
237#41—46no—
238#41—46no—
239#41#1174566914yes—
240#41#1174567014yes—
241C2×Q8—26no—
242C2×Q8—23no—
243C2×Q8—48no—
244C2×Q8—46no—
245C2×Q8—46no—
246C2×Q8C22⋊Dic5911yescharacteristic
247C2×D4—26no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4C22⋊Dic5913yes—
251C2×D4C22×Dic5912yes—
252C22×C4—26no—
253C22×C4—26no—
254C22×C4C22×Dic5913yes—
255C22×C4C2×C22⋊D5913yes—
256C22×C4C2×C22⋊D5913yes—
257M16—2366no—
258C2×C8—1181no—
259C2×C8—1183no—
260C2×C8—1181no—
261C2×C8—1183no—
262C2×C8—4723no—
263C2×C8—2366no—
264C4⋊C4—26no—
265C4⋊C4—26no—
266C4⋊C4—23no—
267C4⋊C4C22⋊Dic5913yes—
268C4⋊C4C22⋊Dic5913yes—
269C4⋊C4C22×Dic5916yes—
270C22⋊C4—26no—
271C22⋊C4—26no—
272C22⋊C4C22⋊Dic5916yes—
273C42C22×Dic5913yes—
274C42C2×C22⋊D5913yes—
275C42C2×C22⋊D5913yes—
276C42C23×D5911yescharacteristic
277C23—22no—
278C23C2×C22⋊Dic5913yes—
279Q8—412no—
280Q8—43no—
281Q8—43no—
282Q8#1427364814yes—
283D4—42no—
284D4—412no—
285D4—41no—
286D4—41no—
287D4—41no—
288D4—41no—
289D4#14273648112yes—
290C2×C4—23no—
291C2×C4—23no—
292C2×C4—26no—
293C2×C4—43no—
294C2×C4—412no—
295C2×C4—412no—
296C2×C4C2×C22⋊Dic5913yes—
297C2×C4C2×C22⋊Dic5913yes—
298C2×C4C2×C22⋊Dic5913yes—
299C2×C4C2×#1174566916yes—
300C2×C4C2×#1174567016yes—
301C2×C4C23×Dic5911yescharacteristic
302C2×C4C22×C22⋊D5911yescharacteristic
303C2×C4C22×C22⋊D5911yescharacteristic
304C8—2361no—
305C8—2363no—
306C8—2363no—
307C8—2361no—
308C22—23no—
309C22—23no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22C2×#1427364816yes—
314C22C22×C22⋊Dic5911yescharacteristic
315C4—21no—
316C4—23no—
317C4—23no—
318C4—46no—
319C4C2×#1427364816yes—
320C4C22×#1174566911yescharacteristic
321C4C22×#1174567011yescharacteristic
322C4#1683821011yescharacteristic
323C4#1683821711yescharacteristic
324C2—26no—
325C2—82no—
326C2#2143257911yescharacteristic
327C2C22×#1427364811yescharacteristic
328C2#2143262911yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
Fitting, prime radicalC59×#3499C2
socleC22×C59C22×C22⋊C4
cosocleC42×C59C24
hypercenter#3499D59
hypocenterC59#17721
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3499

Complements

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

NH ≅ G/Ntype
1. #21433207C2semidirect
2. #21432664C2semidirect
3. #21422773C2semidirect
4. #21422768C2semidirect
5. #21422745C2semidirect
6. C2×#16837148C2semidirect
194. C59#17721semidirect

Maximal subgroups

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

subgroupindex
#214332072
#214326642
#214227732
#214227682
#214227452
C2×#168371482
C59×#34992
#1772159