Finite Group Databasebeta

← #21425401

Subgroups — #21425401

Order 7552.

264 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 65 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#16838200C211yesmaximal, characteristic
2#16837429C211yesmaximal, characteristic, complemented
3#16837398C211yesmaximal, characteristic, complemented
4#16837396C211yesmaximal, characteristic, complemented
5#16837375C211yesmaximal, characteristic, complemented
6#16837158C211yesmaximal, characteristic, complemented
7C59×#410C211yesmaximal, characteristic, complemented
8C2×#11745685C2211yescharacteristic
9#14273758—21no—
10#14273757—21no—
11#14273753—21no—
12#14273752—21no—
13#14273747—22no—
14C2×#11745664C2211yescharacteristic
15C22×C59⋊C8C2211yescharacteristic
16#14273686C2211yescharacteristic
17C4⋊(C59⋊C8)—22no—
18#14273651—21no—
19#14273650—21no—
20#14273648—21no—
21#14273647—21no—
22#14273641—22no—
23C2×#41×C59C2211yescharacteristic
24#117×C59C2211yescharacteristic
25#117×C59C2211yescharacteristic
26#99×C59—22no—
27#41×C59—21no—
28#41×C59—21no—
29#41×C59—21no—
30#41×C59—21no—
31C2×Q8×C59D411yescharacteristic
32C2×D4×C59—21no—
33C2×D4×C59D411yescharacteristic
34C22×C4×C59—21no—
35C22×C4×C59C2311yescharacteristic
36C2×C22⋊D59—21no—
37#11745685—21no—
38#11745685—21no—
39#11745685—21no—
40#11745685—21no—
41C2×D236D411yescharacteristic
42C2×C4×D59—21no—
43C2×Dic118D411yescharacteristic
44C4⋊C4×C59—21no—
45C4⋊C4×C59—21no—
46C22⋊C4×C59—21no—
47C22⋊C4×C59D412yes—
48C42×C59—21no—
49C42×C59—21no—
50C22⋊Dic59—21no—
51#11745672—22no—
52#11745671—22no—
53#11745670—22no—
54#11745669—22no—
55C4⋊Dic59—21no—
56C4×Dic59—21no—
57#11745664—21no—
58#11745664—21no—
59C2×C59⋊C8—21no—
60C2×C59⋊C8—21no—
61C2×C59⋊C8—21no—
62C2×C59⋊C8D412yes—
63C23×C59—21no—
64C23×C59C2×D411yescharacteristic
65C22×D59—21no—
66Q8×C59—21no—
67Q8×C59—21no—
68D4×C59—21no—
69D4×C59—21no—
70D4×C59—41no—
71D4×C59—41no—
72C2×C4×C59—22no—
73C2×C4×C59—21no—
74C2×C4×C59—41no—
75C2×C4×C59—41no—
76C2×C4×C59—41no—
77C2×C4×C59C2×D411yescharacteristic
78C2×C4×C59C2×D411yescharacteristic
79C2×C4×C59C2×D411yescharacteristic
80C2×C4×C59#4111yescharacteristic
81C2×C4×C59#4111yescharacteristic
82C2×C4×C59#4111yescharacteristic
83C22⋊D59—41no—
84C22⋊D59—41no—
85C2×Dic59—21no—
86C2×Dic59—41no—
87D236—21no—
88D236—21no—
89C4×D59—41no—
90C4×D59—41no—
91Dic118—21no—
92Dic118—21no—
93C59⋊C8—22no—
94C59⋊C8—41no—
95C22×C59—41no—
96C22×C59—41no—
97C22×C59—81no—
98C22×C59#12111yescharacteristic
99C22×C59#12111yescharacteristic
100C22×C59#12111yescharacteristic
101C2×D59—41no—
102C2×D59—81no—
103C4×C59—42no—
104C4×C59—41no—
105C4×C59—81no—
106C4×C59#12011yescharacteristic
107C4×C59#12311yescharacteristic
108C4×C59#12311yescharacteristic
109C4×C59#12411yescharacteristic
110Dic59—41no—
111Dic59—81no—
112#2053—591nomaximal, Sylow
113C2×C59—21no—
114C2×C59—21no—
115C2×C59—21no—
116C2×C59—81no—
117C2×C59#39411yescharacteristic
118D59—81no—
119#575—591no—
120#424—591no—
121#416—591no—
122#416—591no—
123#410—591no—
124#410D5911yescharacteristic, complemented
125H(3,ℤ/4)—591no—
126C59#205311yescharacteristic, complemented, Sylow
127C2×#41—591no—
128C2×#41C2×D5911yescharacteristic
129#137—1181no—
130#137—1181no—
131AGL(1,ℤ/8)—1181no—
132AGL(1,ℤ/8)—1181no—
133#135—1182no—
134C2×M16—591no—
135C22×C8—591no—
136#117—591no—
137#117C2×D5911yescharacteristic
138#117C2×D5911yescharacteristic
139C4⋊C8—1182no—
140#103—1181no—
141#103—1181no—
142#102—1181no—
143#102—1181no—
144#99—22no—
145#99—1182no—
146#41—21no—
147#41—21no—
148#41—21no—
149#41—21no—
150#41—1181no—
151#41—1181no—
152#41—1181no—
153#41—1181no—
154C2×Q8—591no—
155C2×Q8D23611yescharacteristic
156C2×D4—21no—
157C2×D4—591no—
158C2×D4—1181no—
159C2×D4D23611yescharacteristic
160C22×C4—21no—
161C22×C4—1181no—
162C22×C4C22×D5911yescharacteristic
163Q16—1182no—
164SD16—1182no—
165SD16—1182no—
166D8—1182no—
167M16—1181no—
168M16—1181no—
169C2×C8—592no—
170C2×C8—1181no—
171C2×C8—1181no—
172C2×C8—1181no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—1181no—
176C22⋊C4—21no—
177C22⋊C4—1181no—
178C22⋊C4C22⋊D5912yes—
179C42—21no—
180C42—21no—
181C42—1181no—
182C23—21no—
183C23—1181no—
184C23C2×C22⋊D5911yescharacteristic
185Q8—21no—
186Q8—21no—
187Q8—1181no—
188Q8—1181no—
189D4—21no—
190D4—21no—
191D4—41no—
192D4—41no—
193D4—1181no—
194D4—1181no—
195D4—2361no—
196D4—2361no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—1181no—
203C2×C4—2361no—
204C2×C4—2361no—
205C2×C4—2361no—
206C2×C4C2×D23611yescharacteristic
207C2×C4#1174568511yescharacteristic
208C2×C4D4×D5911yescharacteristic
209C2×C4D4×D5911yescharacteristic
210C2×C4#1174568711yescharacteristic
211C2×C4#1174568711yescharacteristic
212C8—1182no—
213C8—2361no—
214C22—41no—
215C22—41no—
216C22—81no—
217C22—4721no—
218C22—2361no—
219C22#1427369811yescharacteristic
220C22#1427369911yescharacteristic
221C22#1427370111yescharacteristic
222C4—42no—
223C4—41no—
224C4—81no—
225C4—4721no—
226C4—2361no—
227C4#1427369711yescharacteristic
228C4#1427370911yescharacteristic
229C4#1427371111yescharacteristic
230C4#1427371311yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—81no—
235C2—4721no—
236C2#1683731811yescharacteristic
subgroupG/H
centerC4#14273713
commutatorC22×C4×C59C23
FrattiniC22×C4C22×D59
FittingC59×#410C2
socleC2×C59#394
hypercenter#410D59
hypocenterC59#2053
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #16837429C2semidirect
3. #16837398C2semidirect
4. #16837396C2semidirect
5. #16837375C2semidirect
6. #16837158C2semidirect
7. C59×#410C2semidirect
124. #410D59semidirect
126. C59#2053semidirect