Finite Group Databasebeta

← #28476201

Subgroups — #28476201

Order 15104.

304 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 41 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#21425183C211yesmaximal, characteristic
2#21422030C211yesmaximal, characteristic, complemented
3C59×#1394C211yesmaximal, characteristic, complemented
4C22×#11745671—21no—
5C2×#14273724C411yescharacteristic
6C2×#14273647—21no—
7#16837157C411yescharacteristic
8#16837128—21no—
9C2×#101×C59—21no—
10C59×#391C2211yescharacteristic
11C22×D236—21no—
12C22×D236—41no—
13C2×#11745671—42no—
14C2×#11745671—42no—
15C2×#11745671—41no—
16C2×#11745671—41no—
17C22×C59⋊C8—21no—
18#14273724—42no—
19#14273647—42no—
20C22⋊(C59⋊C8)—41no—
21C22×Q8×C59D411yescharacteristic
22C2×M16×C59—41no—
23C2×C4⋊C4×C59D411yescharacteristic
24C2×C42×C59C2×C411yescharacteristic
25#101×C59—22no—
26#101×C59—41no—
27#95×C59—41no—
28C23×D59—41no—
29C2×Q8×C59—21no—
30C2×Q8×C59—41no—
31C2×Q8×C59—41no—
32C2×Q8×C59D811yescharacteristic
33C2×Q8×C59SD1611yescharacteristic
34C22×C4×C59—21no—
35C22×C4×C59—21no—
36C22×C4×C59—21no—
37C22×C4×C59C22⋊C411yescharacteristic
38C2×D236—22no—
39C2×D236—41no—
40C2×D236—41no—
41C2×D236—41no—
42C2×D236—41no—
43C2×D236—41no—
44C2×D236—42no—
45C2×D236—42no—
46C2×D236—82no—
47C2×D236—82no—
48M16×C59—42no—
49M16×C59—81no—
50C2×C8×C59—81no—
51C4⋊C4×C59—41no—
52C42×C59—41no—
53#11745671—44no—
54#11745671—44no—
55C2×C59⋊C8—22no—
56C2×C59⋊C8—41no—
57C2×C59⋊C8—41no—
58C2×C59⋊C8—81no—
59C23×C59#9911yescharacteristic
60C22×D59—42no—
61C22×D59—82no—
62C22×D59—82no—
63C22×D59—81no—
64C22×D59—81no—
65Q8×C59—41no—
66Q8×C59—41no—
67Q8×C59—81no—
68C2×C4×C59—21no—
69C2×C4×C59—21no—
70C2×C4×C59—22no—
71C2×C4×C59—21no—
72C2×C4×C59—21no—
73C2×C4×C59—41no—
74C2×C4×C59—41no—
75C2×C4×C59—41no—
76C2×C4×C59—41no—
77C2×C4×C59—41no—
78C2×C4×C59—81no—
79C2×C4×C59#10211yescharacteristic
80C2×C4×C59#10411yescharacteristic
81D236—42no—
82D236—42no—
83D236—81no—
84D236—81no—
85D236—84no—
86D236—84no—
87C8×C59—82no—
88C59⋊C8—42no—
89C59⋊C8—161no—
90#7404—591nomaximal, Sylow
91C22×C59—21no—
92C22×C59—21no—
93C22×C59#32711yescharacteristic
94C22×C59AΣL(1,𝔽16)11yescharacteristic
95C22×C59#35311yescharacteristic
96C2×D59—84no—
97C2×D59—82no—
98C2×D59—162no—
99C2×D59—161no—
100C2×D59—161no—
101C4×C59—21no—
102C4×C59—21no—
103C4×C59—42no—
104C4×C59—41no—
105C4×C59—41no—
106C4×C59—81no—
107#2014—591no—
108#1394D5911yescharacteristic, complemented
109#1319—591no—
110C2×C59—21no—
111C2×C59—21no—
112C2×C59#134311yescharacteristic
113C2×C59#140611yescharacteristic
114C2×C59#141011yescharacteristic
115D59—162no—
116C22×SD16—1181no—
117C2×#127—591no—
118C2×#102—1181no—
119C2×#101—21no—
120#391C2×D5911yescharacteristic
121#336—591no—
122#324—1181no—
123C59#740411yescharacteristic, complemented, Sylow
124C22×Q8C22⋊D5911yescharacteristic
125C22×D4—1181no—
126C22×D4—2361no—
127C2×SD16—2362no—
128C2×SD16—2362no—
129C2×SD16—2361no—
130C2×SD16—2361no—
131C2×M16—41no—
132C22×C8—1181no—
133#127—2362no—
134C2×C4⋊C4D23611yescharacteristic
135C2×C42C4×D5911yescharacteristic
136#102—2362no—
137#101—22no—
138#101—41no—
139C22⋊C8—2361no—
140#95—41no—
141C24—2361no—
142C2×Q8—21no—
143C2×Q8—41no—
144C2×Q8—41no—
145C2×Q8#1174566911yescharacteristic
146C2×Q8#1174567111yescharacteristic
147C2×D4—1182no—
148C2×D4—4722no—
149C2×D4—4722no—
150C2×D4—2361no—
151C2×D4—2361no—
152C2×D4—2361no—
153C2×D4—2362no—
154C2×D4—2362no—
155C2×D4—2361no—
156C2×D4—2361no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4(C2×D59)⋊C411yescharacteristic
161SD16—2364no—
162SD16—2364no—
163M16—42no—
164M16—81no—
165C2×C8—81no—
166C2×C8—1182no—
167C2×C8—4721no—
168C2×C8—2361no—
169C2×C8—2361no—
170C4⋊C4—41no—
171C42—41no—
172C23—4722no—
173C23—4722no—
174C23—4721no—
175C23—4721no—
176C23—2362no—
177C23#1427364111yescharacteristic
178Q8—41no—
179Q8—41no—
180Q8—81no—
181D4—4721no—
182D4—4721no—
183D4—4724no—
184D4—4724no—
185D4—2362no—
186D4—2362no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—81no—
198C2×C4#1427364711yescharacteristic
199C2×C4#1427365511yescharacteristic
200C8—82no—
201C8—9441no—
202C8—2362no—
203C22—21no—
204C22—21no—
205C22—9442no—
206C22—9441no—
207C22—9441no—
208C22—4724no—
209C22—4722no—
210C22#1683713511yescharacteristic
211C22#1683718811yescharacteristic
212C22#1683719411yescharacteristic
213C4—21no—
214C4—21no—
215C4—42no—
216C4—41no—
217C4—41no—
218C4—81no—
219C2—21no—
220C2—21no—
221C2—9442no—
222C2#2142209211yescharacteristic
223C2#2142225411yescharacteristic
224C2#2142226211yescharacteristic
subgroupG/H
centerC22#16837194
commutatorC2×C42×C59C2×C4
Frattini#391C2×D59
FittingC59×#1394C2
socleC22×C59#353
prime radicalC2×#14273724C4
cosocleC59×#391C22
hypercenter#1394D59
hypocenterC59#7404
LCSG ▷ C2×C42×C59 ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #391 ◁ #1394

Complements

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

NH ≅ G/Ntype
2. #21422030C2semidirect
3. C59×#1394C2semidirect
108. #1394D59semidirect
123. C59#7404semidirect

Maximal subgroups

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

subgroupindex
#214251832
#214220302
C59×#13942
#740459