Finite Group Databasebeta

← #28501951

Subgroups — #28501951

Order 15104.

354 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 67 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#21435406C211yesmaximal, characteristic
2#21426276C212yesmaximal, complemented
3#21422104C212yesmaximal, complemented
4#21422036C211yesmaximal, characteristic, complemented
5C59×#2086C211yesmaximal, characteristic, complemented
6#16837952—21no—
7#16837683—22no—
8#16837638—21no—
9#16837638C2212yes—
10#16837598—22no—
11#16837227—22no—
12#16837202—22no—
13C22⋊(C59⋊C16)C2212yes—
14C2×#107×C59C2211yescharacteristic
15C2×C4⋊C8×C59C2211yescharacteristic
16C59×#389—21no—
17C59×#340—21no—
18C59×#340C2211yescharacteristic
19C2×#11745685—41no—
20C2×D472—21no—
21C2×D472D412yes—
22#14273727—21no—
23#14273724—21no—
24#14273715—41no—
25#14273701—22no—
26#14273701—41no—
27C4×D236—22no—
28C4×D236—41no—
29C2×C59⋊C16—22no—
30#14273661—21no—
31#14273661D412yes—
32#14273649—22no—
33#14273649—22no—
34C22×C8×C59D411yescharacteristic
35C22×C8×C59C2311yescharacteristic
36C2×C4⋊C4×C59—21no—
37C2×C4⋊C4×C59—21no—
38C2×C4⋊C4×C59—41no—
39C2×C42×C59D411yescharacteristic
40#107×C59—22no—
41C4⋊C8×C59—22no—
42#95×C59—41no—
43C22×C4×C59—41no—
44C22×C4×C59—41no—
45C22×C4×C59D811yescharacteristic
46C22×C4×C59D811yescharacteristic
47C22×C4×C59C2×D411yescharacteristic
48C2×C22⋊D59—42no—
49#11745685—82no—
50#11745685—82no—
51C2×D236—22no—
52C2×D236—41no—
53C2×D236—41no—
54C2×C4×D59—42no—
55C2×Dic118—41no—
56C2×C8×C59—21no—
57C2×C8×C59—41no—
58C2×C8×C59—41no—
59C2×C8×C59C2×D412yes—
60C4⋊C4×C59—22no—
61C4⋊C4×C59—41no—
62C4⋊C4×C59—41no—
63C4⋊C4×C59—41no—
64C4⋊C4×C59—82no—
65C42×C59—21no—
66C42×C59—21no—
67(C2×D59)⋊C4—42no—
68(C2×D59)⋊C4—42no—
69C4⋊Dic59—22no—
70C4⋊Dic59—41no—
71D472—42no—
72D472—81no—
73C59⋊C16—42no—
74C23×C59C2×D811yescharacteristic
75C22×D59—42no—
76C2×C4×C59—21no—
77C2×C4×C59—21no—
78C2×C4×C59—21no—
79C2×C4×C59—21no—
80C2×C4×C59—21no—
81C2×C4×C59—41no—
82C2×C4×C59—42no—
83C2×C4×C59—81no—
84C2×C4×C59—81no—
85C2×C4×C59—81no—
86C2×C4×C59—81no—
87C2×C4×C59—81no—
88C2×C4×C59D1612yes—
89C2×C4×C59#12011yescharacteristic
90C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
91C22⋊D59—82no—
92C22⋊D59—82no—
93C2×Dic59—42no—
94D236—42no—
95D236—42no—
96D236—82no—
97D236—82no—
98C4×D59—84no—
99Dic118—82no—
100C8×C59—22no—
101C8×C59—41no—
102#12288—591nomaximal, Sylow
103C22×C59—21no—
104C22×C59—21no—
105C22×C59#44711yescharacteristic
106C22×C59C2×D1611yescharacteristic
107C22×C59#50911yescharacteristic
108C2×D59—82no—
109C2×D59—82no—
110C2×D59—82no—
111C4×C59—21no—
112C4×C59—22no—
113C4×C59—41no—
114C4×C59—82no—
115C4×C59—161no—
116C4×C59#45611yescharacteristic
117C4×C59#45811yescharacteristic
118Dic59—82no—
119#3245—591no—
120#2232—592no—
121#2086D5911yescharacteristic, complemented
122#1347—592no—
123#1322—591no—
124C2×C59—21no—
125C2×C59—21no—
126C2×C59#161011yescharacteristic
127C2×C59#218411yescharacteristic
128C2×C59#219111yescharacteristic
129D59—162no—
130#532—1181no—
131#474—1182no—
132#466—592no—
133#466—1181no—
134#459—1182no—
135C2×#107C2×D5911yescharacteristic
136C2×C4⋊C8C2×D5911yescharacteristic
137#389—21no—
138#366—1182no—
139#357—1182no—
140C22⋊C16—592no—
141#340—21no—
142#340C2×D5911yescharacteristic
143C59#1228811yescharacteristic, complemented, Sylow
144C2×#41—2361no—
145C2×D8—592no—
146C2×D8—1181no—
147C22×C8C22⋊D5911yescharacteristic
148C22×C8C22×D5911yescharacteristic
149C4⋊Q8—1181no—
150#127—1181no—
151#124—2361no—
152#121—1182no—
153#121—2361no—
154C4×D4—1182no—
155C4×D4—2361no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—41no—
159C2×C42C22⋊D5911yescharacteristic
160C2×C16—1182no—
161#107—22no—
162#107—592no—
163#107—1181no—
164C4⋊C8—22no—
165#102—1182no—
166#102—1182no—
167#95—41no—
168#41—4722no—
169#41—4722no—
170C2×Q8—2361no—
171C2×D4—1182no—
172C2×D4—2361no—
173C2×D4—2361no—
174C2×D4—2362no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—2362no—
178C22×C4#1174566911yescharacteristic
179C22×C4#1174566911yescharacteristic
180C22×C4C2×C22⋊D5911yescharacteristic
181D8—4721no—
182D8—2362no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8D4×D5912yes—
187C4⋊C4—22no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—82no—
192C4⋊C4—1182no—
193C4⋊C4—2361no—
194C22⋊C4—2362no—
195C22⋊C4—2362no—
196C42—21no—
197C42—21no—
198C16—2362no—
199C23—2362no—
200C23C2×#1174566911yescharacteristic
201Q8—4722no—
202D4—4722no—
203D4—4722no—
204D4—4722no—
205D4—4722no—
206D4—2362no—
207D4—2362no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—4724no—
221C2×C4—2362no—
222C2×C4#1427366812yes—
223C2×C4#1427369511yescharacteristic
224C2×C4#1427375211yescharacteristic
225C8—22no—
226C8—41no—
227C22—21no—
228C22—21no—
229C22—4722no—
230C22—4722no—
231C22—4722no—
232C22#1683753511yescharacteristic
233C22C2×#1427366811yescharacteristic
234C22#1683783711yescharacteristic
235C4—22no—
236C4—21no—
237C4—41no—
238C4—82no—
239C4—161no—
240C4—4722no—
241C4#1683758411yescharacteristic
242C4#1683759111yescharacteristic
243C2—21no—
244C2—21no—
245C2—9442no—
246C2#2142309811yescharacteristic
247C2#2142600711yescharacteristic
248C2#2142604611yescharacteristic
subgroupG/H
centerC22#16837535
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC59×#2086C2
socleC22×C59#447
prime radical#21435406C2
hypercenter#2086D59
hypocenterC59#12288
LCSG ▷ C22×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #2086

Complements

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

NH ≅ G/Ntype
2. #21426276C2semidirect
3. #21422104C2semidirect
4. #21422036C2semidirect
5. C59×#2086C2semidirect
121. #2086D59semidirect
143. C59#12288semidirect

Maximal subgroups

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

subgroupindex
#214354062
#214262762
#214221042
#214220362
C59×#20862
#1228859