Finite Group Databasebeta

← #28486117

Subgroups — #28486117

Order 15104.

554 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433181C211yesmaximal, characteristic, complemented
2#21429095C211yesmaximal, characteristic
3#21422765C212yesmaximal
4#21421997C212yesmaximal, complemented
5C59×#1801C211yesmaximal, characteristic, complemented
6C2×#14273724C2211yescharacteristic
7C2×#14273716—21no—
8#16837533—24no—
9#16837459—22no—
10#16837455C2212yes—
11#16837319C414yes—
12#16837304—22no—
13#16837303—21no—
14C4×C4⋊Dic59C2211yescharacteristic
15#16837145—24no—
16C2×C4×Q8×C59C2211yescharacteristic
17C59×#423C2212yes—
18C2×#101×C59—21no—
19C59×#343—21no—
20C22×D236—21no—
21C22×D236—21no—
22C22×D236C2×C412yes—
23#14273724—22no—
24#14273724—22no—
25#14273716—44no—
26C2×C4⋊Dic59—21no—
27C2×(C2×D59)⋊C4—24no—
28C2×(C2×D59)⋊C4—24no—
29C2×C4×Dic59—21no—
30C4⋊(C59⋊C8)—24no—
31C4⋊(C59⋊C8)D414yes—
32C22⋊(C59⋊C8)—22no—
33#14273638—24no—
34C4×C59⋊C8—22no—
35C59⋊#95C2×C412yes—
36C22×Q8×C59D411yescharacteristic
37C2×M16×C59—22no—
38C4×Q8×C59—22no—
39C4×Q8×C59—22no—
40C2×C4⋊C4×C59C2×C412yes—
41C2×C4⋊C4×C59D411yescharacteristic
42C2×C42×C59D412yes—
43C2×C42×C59C2311yescharacteristic
44C4⋊C8×C59—22no—
45C4⋊C8×C59—22no—
46#101×C59—42no—
47C23×D59—22no—
48C2×Q8×C59—22no—
49C2×Q8×C59—21no—
50C2×Q8×C59—22no—
51C2×Q8×C59—21no—
52C22×C4×C59C22⋊C412yes—
53C22×C4×C59C22⋊C412yes—
54C22×C4×C59C22×C411yescharacteristic
55C22×C4×C59C2×D411yescharacteristic
56C22×C4×C59C2×D411yescharacteristic
57C22×Dic59—22no—
58C2×D236—24no—
59C2×D236—44no—
60C2×D236—41no—
61C2×D236—42no—
62C2×D236—41no—
63C2×D236—42no—
64C2×D236—41no—
65C2×D236—41no—
66C2×D236—82no—
67M16×C59—42no—
68M16×C59—42no—
69C2×C8×C59—42no—
70C4⋊C4×C59—24no—
71C4⋊C4×C59—22no—
72C42×C59—22no—
73C42×C59—22no—
74C42×C59C2×D412yes—
75C42×C59#4112yes—
76(C2×D59)⋊C4—48no—
77(C2×D59)⋊C4—84no—
78C4⋊Dic59—42no—
79C4×Dic59—42no—
80C2×C59⋊C8—24no—
81C23×C59C2×C22⋊C411yescharacteristic
82C22×D59—44no—
83C22×D59—44no—
84C22×D59—42no—
85C22×D59—42no—
86C22×D59—42no—
87Q8×C59—42no—
88Q8×C59—42no—
89C2×C4×C59—22no—
90C2×C4×C59—22no—
91C2×C4×C59—22no—
92C2×C4×C59—22no—
93C2×C4×C59—22no—
94C2×C4×C59—41no—
95C2×C4×C59—42no—
96C2×C4×C59C4×D412yes—
97C2×C4×C59#12011yescharacteristic
98C2×C4×C59#12112yes—
99C2×C4×C59#12311yescharacteristic
100C2×C4×C59AGL(1,ℤ/8)12yes—
101C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
102C2×C4×C59#13711yescharacteristic
103C2×Dic59—42no—
104C2×Dic59—42no—
105C2×Dic59—42no—
106D236—84no—
107D236—82no—
108D236—82no—
109D236—82no—
110D236—82no—
111C8×C59—82no—
112C59⋊C8—44no—
113#9406—591nomaximal, Sylow
114C22×C59—21no—
115C22×C59—21no—
116C22×C59#38611yescharacteristic
117C22×C59#41711yescharacteristic
118C22×C59#41811yescharacteristic
119C2×D59—84no—
120C2×D59—84no—
121C2×D59—82no—
122C2×D59—82no—
123C2×D59—82no—
124C4×C59—21no—
125C4×C59—22no—
126C4×C59—42no—
127C4×C59—42no—
128C4×C59#45412yes—
129C4×C59#46812yes—
130C4×C59#46912yes—
131Dic59—82no—
132#2983—591no—
133#2548—591no—
134#1801D5911yescharacteristic, complemented
135#1545—592no—
136#1303—592no—
137C2×C59—21no—
138C2×C59—21no—
139C2×C59#188811yescharacteristic
140C2×C59#189511yescharacteristic
141C2×C59#191411yescharacteristic
142D59—162no—
143C2×#127—591no—
144C2×#124—1181no—
145C2×C4×Q8C2×D5911yescharacteristic
146#446—1184no—
147#432—1182no—
148#431—592no—
149#423C2×D5912yes—
150C2×#101—21no—
151#394—594no—
152#390—1181no—
153#390—1182no—
154C4×C4⋊C4—591no—
155#343—21no—
156#331—1184no—
157C59#940611yescharacteristic, complemented, Sylow
158C22×Q8C22⋊D5911yescharacteristic
159C22×D4—592no—
160C22×D4—1181no—
161C22×D4—1181no—
162C2×M16—22no—
163#127—1182no—
164#127—1182no—
165#124—2364no—
166C4×Q8—22no—
167C4×Q8—22no—
168C2×C4⋊C4—1181no—
169C2×C4⋊C4C4×D5912yes—
170C2×C4⋊C4C22⋊D5911yescharacteristic
171C2×C22⋊C4—1184no—
172C2×C22⋊C4—1184no—
173C2×C42—1181no—
174C2×C42D23612yes—
175C2×C42C22×D5911yescharacteristic
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—594no—
179C4⋊C8—1184no—
180#101—42no—
181C22⋊C8—1182no—
182#97—1184no—
183C4×C8—1182no—
184#95—592no—
185C24—1182no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×D4—1184no—
191C2×D4—4722no—
192C2×D4—2364no—
193C2×D4—2361no—
194C2×D4—2362no—
195C2×D4—2361no—
196C2×D4—2362no—
197C2×D4—2361no—
198C2×D4—2361no—
199C22×C4—1182no—
200C22×C4(C2×D59)⋊C412yes—
201C22×C4(C2×D59)⋊C412yes—
202C22×C4C2×C4×D5911yescharacteristic
203C22×C4C2×D23611yescharacteristic
204C22×C4C2×C22⋊D5911yescharacteristic
205M16—42no—
206M16—42no—
207C2×C8—42no—
208C2×C8—1184no—
209C4⋊C4—24no—
210C4⋊C4—22no—
211C4⋊C4—2362no—
212C22⋊C4—4724no—
213C22⋊C4—2368no—
214C42—22no—
215C42—22no—
216C42—2362no—
217C42D4×D5912yes—
218C42#1174568712yes—
219C23—2364no—
220C23—2364no—
221C23—2362no—
222C23—2362no—
223C23—2362no—
224C23C2×(C2×D59)⋊C411yescharacteristic
225Q8—42no—
226Q8—42no—
227D4—4724no—
228D4—4722no—
229D4—4722no—
230D4—4722no—
231D4—4722no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—2362no—
240C2×C4—2362no—
241C2×C4—2362no—
242C2×C4(C22⋊D59)⋊C412yes—
243C2×C4#1427369711yescharacteristic
244C2×C4#1427369912yes—
245C2×C4#1427371111yescharacteristic
246C2×C4#1427375212yes—
247C2×C4#1427375311yescharacteristic
248C2×C4#1427375811yescharacteristic
249C8—82no—
250C8—2364no—
251C22—21no—
252C22—21no—
253C22—4724no—
254C22—4724no—
255C22—4722no—
256C22—4722no—
257C22—4722no—
258C22#1683727911yescharacteristic
259C22#1683740111yescharacteristic
260C22#1683740611yescharacteristic
261C4—22no—
262C4—21no—
263C4—42no—
264C4—42no—
265C4—4722no—
266C4#1683757712yes—
267C4#1683765012yes—
268C4#1683765312yes—
269C2—21no—
270C2—21no—
271C2—9442no—
272C2#2142449711yescharacteristic
273C2#2142453211yescharacteristic
274C2#2142464111yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1801C2
socleC22×C59#386
prime radicalC2×#14273724C22
cosocleC2×C42×C59C23
hypercenter#1801D59
hypocenterC59#9406
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
1. #21433181C2semidirect
4. #21421997C2semidirect
5. C59×#1801C2semidirect
134. #1801D59semidirect
157. C59#9406semidirect

Maximal subgroups

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

subgroupindex
#214331812
#214290952
#214227652
#214219972
C59×#18012
#940659