Finite Group Databasebeta

← #28498014

Subgroups — #28498014

Order 15104.

364 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 91 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#21432804C211yesmaximal, characteristic
2#21425892C212yesmaximal, complemented
3#21422022C211yesmaximal, characteristic, complemented
4#21422021C212yesmaximal, complemented
5C59×#1315C211yesmaximal, characteristic, complemented
6C2×#14273735C411yescharacteristic
7C2×#14273734C411yescharacteristic
8#16837948C411yescharacteristic
9#16837477—22no—
10(C22⋊D59)⋊C8—22no—
11#16837425C411yescharacteristic
12#16837368C2211yescharacteristic
13C2×C8×Dic59C2211yescharacteristic
14Dic59⋊C16—22no—
15C22⋊(C59⋊C16)C2212yes—
16C16×Dic59—22no—
17C59×#408C2211yescharacteristic
18C59×C22⋊C16C2212yes—
19C2×#11745685C2×C411yescharacteristic
20#14273735—42no—
21#14273734—42no—
22C2×#11745664—21no—
23C2×#11745664C2×C411yescharacteristic
24C2×#11745659—21no—
25C22×C59⋊C8—21no—
26C22×C59⋊C8C2×C411yescharacteristic
27C2×C8×D59—21no—
28#14273725C811yescharacteristic
29#14273723C811yescharacteristic
30#14273716C811yescharacteristic
31#14273714C811yescharacteristic
32#14273699—22no—
33C4×C22⋊D59—22no—
34C2×C4×Dic59C2×C411yescharacteristic
35C2×C59⋊C16—22no—
36Dic59⋊C8—22no—
37(C2×D59)⋊C8—22no—
38C8×Dic59D412yes—
39C8×Dic59D412yes—
40C2×#41×C59C2×C411yescharacteristic
41C2×M16×C59C2×C411yescharacteristic
42C22×C8×C59C2311yescharacteristic
43C2×C16×C59—22no—
44C22⋊C8×C59—22no—
45#41×C59—41no—
46#41×C59—41no—
47C2×Q8×C59C2×C811yescharacteristic
48C2×D4×C59—21no—
49C2×D4×C59C2×C811yescharacteristic
50C22×C4×C59—21no—
51C22×C4×C59C22×C411yescharacteristic
52C2×C22⋊D59—21no—
53C2×C22⋊D59—22no—
54C22×Dic59M1612yes—
55#11745685—41no—
56#11745685—41no—
57C2×D236C2×C811yescharacteristic
58C2×C4×D59—21no—
59C2×Dic118C2×C811yescharacteristic
60M16×C59—41no—
61C2×C8×C59—21no—
62C2×C8×C59—41no—
63C2×C8×C59C2×D412yes—
64C22⋊Dic59—22no—
65(C2×D59)⋊C4—22no—
66Dic59⋊C4—22no—
67C4×Dic59C22⋊C412yes—
68C4×Dic59C2×C811yescharacteristic
69C4×Dic59C2×C811yescharacteristic
70#11745664—42no—
71#11745664—41no—
72C2×C59⋊C8—22no—
73C2×C59⋊C8—21no—
74C2×C59⋊C8—42no—
75C2×C59⋊C8C22⋊C411yescharacteristic
76C2×C59⋊C8C22⋊C411yescharacteristic
77#11745659—42no—
78C8×D59—42no—
79C16×C59—42no—
80C59⋊C16—42no—
81C23×C59—21no—
82C23×C59C2×M1611yescharacteristic
83C22×D59—21no—
84Q8×C59—41no—
85D4×C59—41no—
86D4×C59—42no—
87C2×C4×C59—21no—
88C2×C4×C59—21no—
89C2×C4×C59—21no—
90C2×C4×C59—41no—
91C2×C4×C59—41no—
92C2×C4×C59C2×C22⋊C411yescharacteristic
93C2×C4×C59C22×C811yescharacteristic
94C22⋊D59—42no—
95C22⋊D59—44no—
96C2×Dic59—24no—
97C2×Dic59—21no—
98C2×Dic59C22⋊C812yes—
99C2×Dic59C22⋊C812yes—
100D236—41no—
101C4×D59—41no—
102C4×D59—41no—
103Dic118—41no—
104C8×C59—22no—
105C8×C59—41no—
106C59⋊C8—21no—
107C59⋊C8—41no—
108C59⋊C8#10012yes—
109#11490—591nomaximal, Sylow
110C22×C59—22no—
111C22×C59—21no—
112C22×C59—42no—
113C22×C59—41no—
114C22×C59C2×C22⋊C811yescharacteristic
115C2×D59—42no—
116C2×D59—41no—
117C4×C59—21no—
118C4×C59—41no—
119C4×C59#41011yescharacteristic
120C4×C59C2×#10011yescharacteristic
121Dic59—22no—
122Dic59—41no—
123Dic59#35014yes—
124#2931—591no—
125#2162—592no—
126#1315—591no—
127#1315—592no—
128#1315D5911yescharacteristic, complemented
129C2×C59—41no—
130C2×C59—81no—
131C2×C59#146011yescharacteristic
132C2×C59C2×#35012yes—
133D59—81no—
134C2×#131—591no—
135C2×#131—591no—
136#532—591no—
137#436—1182no—
138C8×D4—1182no—
139#423—591no—
140#408—591no—
141#408C2×D5911yescharacteristic
142C2×C4×C8—591no—
143C4⋊C16—1182no—
144C22⋊C16—592no—
145C22⋊C16C2×D5912yes—
146C4×C16—1182no—
147C59#1149011yescharacteristic, complemented, Sylow
148C2×#41—591no—
149C2×#41C4×D5911yescharacteristic
150#131—2362no—
151#131—2362no—
152C2×M16—591no—
153C2×M16—1181no—
154C2×M16—1181no—
155C2×M16C4×D5911yescharacteristic
156C22×C8—591no—
157C22×C8—1181no—
158C22×C8—1181no—
159C22×C8C22×D5911yescharacteristic
160C4⋊Q8—591no—
161#127—591no—
162#124—591no—
163#124—591no—
164#121—1182no—
165C4×D4—1182no—
166C2×C42—591no—
167C2×C16—22no—
168C2×C16—1182no—
169C4⋊C8—1182no—
170C22⋊C8—22no—
171C22⋊C8—1182no—
172C4×C8—592no—
173C4×C8—592no—
174#41—41no—
175#41—41no—
176#41—2361no—
177#41—2361no—
178C2×Q8—591no—
179C2×Q8C8×D5911yescharacteristic
180C2×D4—21no—
181C2×D4—591no—
182C2×D4—1182no—
183C2×D4—1181no—
184C2×D4C8×D5911yescharacteristic
185C22×C4—21no—
186C22×C4—592no—
187C22×C4—1181no—
188C22×C4C2×C4×D5911yescharacteristic
189M16—41no—
190M16—2362no—
191M16—2362no—
192M16—2361no—
193C2×C8—21no—
194C2×C8—41no—
195C2×C8—591no—
196C2×C8—591no—
197C2×C8—1182no—
198C2×C8—1181no—
199C2×C8—2362no—
200C2×C8—2362no—
201C2×C8D4×D5912yes—
202C4⋊C4—1182no—
203C22⋊C4—1182no—
204C22⋊C4—1182no—
205C42—592no—
206C42—591no—
207C42—591no—
208C16—42no—
209C16—2362no—
210C23—21no—
211C23—1181no—
212C23M16×D5911yescharacteristic
213Q8—41no—
214Q8—2361no—
215D4—41no—
216D4—42no—
217D4—2361no—
218D4—2364no—
219D4—2362no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—592no—
226C2×C4—592no—
227C2×C4—1184no—
228C2×C4—1181no—
229C2×C4—2361no—
230C2×C4—2361no—
231C2×C4C22⋊C4×D5911yescharacteristic
232C2×C4C2×C8×D5911yescharacteristic
233C8—22no—
234C8—41no—
235C8—592no—
236C8—1181no—
237C8—2361no—
238C22—22no—
239C22—21no—
240C22—42no—
241C22—41no—
242C22—2362no—
243C22—2361no—
244C22C22⋊C8×D5911yescharacteristic
245C4—21no—
246C4—41no—
247C4—594no—
248C4—1182no—
249C4—2361no—
250C4#1683737411yescharacteristic
251C4#100×D5911yescharacteristic
252C2—41no—
253C2—81no—
254C2—4721no—
255C2#2142241511yescharacteristic
256C2#350×D5912yes—
subgroupG/H
centerC2×C4C22⋊C4×D59
commutatorC2×C4×C59C22×C8
FrattiniC22×C8C22×D59
FittingC59×#1315C2
socleC22×C59C2×C22⋊C8
prime radical#16837948C4
cosocleC22×C8×C59C23
hypercenter#1315D59
hypocenterC59#11490
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1315

Complements

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

NH ≅ G/Ntype
2. #21425892C2semidirect
3. #21422022C2semidirect
4. #21422021C2semidirect
5. C59×#1315C2semidirect
128. #1315D59semidirect
147. C59#11490semidirect

Maximal subgroups

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

subgroupindex
#214328042
#214258922
#214220222
#214220212
C59×#13152
#1149059