Finite Group Databasebeta

← #21424455

Subgroups — #21424455

Order 7552.

310 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 79 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#16838168C211yesmaximal, characteristic, complemented
3#16837396C211yesmaximal, characteristic, complemented
4#16837369C211yesmaximal, characteristic, complemented
5#16837172C211yesmaximal, characteristic, complemented
6#16837166C211yesmaximal, characteristic, complemented
7C59×#421C211yesmaximal, characteristic, complemented
8C2×#11745685C2211yescharacteristic
9#14273758C412yescomplemented
10#14273757—21no—
11#14273753C412yescomplemented
12#14273752—21no—
13#14273747—22no—
14#14273735—22no—
15#14273733—22no—
16C2×#11745664C2211yescharacteristic
17C2×#11745664C2211yescharacteristic
18M16×D59—22no—
19C22×C59⋊C8C2211yescharacteristic
20#14273650—21no—
21#14273647—21no—
22C22⋊(C59⋊C8)—22no—
23C2×#41×C59C2211yescharacteristic
24C2×M16×C59C2211yescharacteristic
25#117×C59C2211yescharacteristic
26#104×C59—22no—
27#41×C59—21no—
28#41×C59—21no—
29#41×C59—21no—
30#41×C59—21no—
31C2×Q8×C59C2×C411yescharacteristic
32C2×D4×C59—21no—
33C2×D4×C59C2×C411yescharacteristic
34C22×C4×C59—21no—
35C22×C4×C59C2311yescharacteristic
36C2×C22⋊D59—21no—
37#11745685—21no—
38#11745685—21no—
39#11745685C2×C412yes—
40#11745685D412yes—
41C2×D236D411yescharacteristic
42C2×C4×D59—21no—
43C2×Dic118D411yescharacteristic
44M16×C59—21no—
45M16×C59D412yes—
46C2×C8×C59—21no—
47C4⋊C4×C59—21no—
48C22⋊C4×C59—21no—
49C42×C59—21no—
50#11745672—22no—
51#11745671—22no—
52#11745670—22no—
53#11745669—22no—
54#11745664—21no—
55#11745664—21no—
56#11745664—21no—
57#11745664C2×C412yes—
58C2×C59⋊C8—21no—
59C2×C59⋊C8—21no—
60C2×C59⋊C8—21no—
61C2×C59⋊C8—21no—
62C2×C59⋊C8D412yes—
63#11745659—24no—
64C8×D59—24no—
65C23×C59—21no—
66C23×C59#4111yescharacteristic
67C22×D59—21no—
68Q8×C59—21no—
69Q8×C59—21no—
70D4×C59—21no—
71D4×C59—21no—
72D4×C59—41no—
73D4×C59—41no—
74C2×C4×C59—21no—
75C2×C4×C59—41no—
76C2×C4×C59—41no—
77C2×C4×C59—41no—
78C2×C4×C59C22×C411yescharacteristic
79C2×C4×C59C2×D411yescharacteristic
80C2×C4×C59C2×D411yescharacteristic
81C2×C4×C59C2×D411yescharacteristic
82C2×C4×C59C2×D411yescharacteristic
83C2×C4×C59#4111yescharacteristic
84C22⋊D59—22no—
85C22⋊D59—22no—
86C2×Dic59—21no—
87D236—21no—
88D236C22⋊C412yes—
89C4×D59—22no—
90C4×D59—22no—
91Dic118—21no—
92Dic118C22⋊C412yes—
93C8×C59—22no—
94C59⋊C8—22no—
95C59⋊C8—22no—
96C59⋊C8—41no—
97C22×C59—41no—
98C22×C59—42no—
99C22×C59—41no—
100C22×C59C4×D411yescharacteristic
101C22×C59#12111yescharacteristic
102C22×C59#12311yescharacteristic
103C2×D59—22no—
104C2×D59—81no—
105C4×C59—41no—
106C4×C59—81no—
107C4×C59C2×C22⋊C411yescharacteristic
108C4×C59C4×D411yescharacteristic
109C4×C59#12011yescharacteristic
110C4×C59#12111yescharacteristic
111Dic59—22no—
112#1881—591nomaximal, Sylow
113C2×C59—21no—
114C2×C59—21no—
115C2×C59—21no—
116C2×C59—81no—
117C2×C59#38611yescharacteristic
118D59—42no—
119#575—591no—
120#568—591no—
121#421D5911yescharacteristic, complemented
122#416—591no—
123#408—591no—
124#344—591no—
125#341—591no—
126C59#188111yescharacteristic, complemented, Sylow
127C2×#41—591no—
128C2×#41C2×D5911yescharacteristic
129#137—592no—
130#137—1181no—
131AGL(1,ℤ/8)—592no—
132AGL(1,ℤ/8)—1181no—
133#135—1182no—
134#131—1182no—
135#131—1182no—
136C2×M16—591no—
137C2×M16—591no—
138C2×M16—1182no—
139C2×M16C2×D5911yescharacteristic
140C22×C8—591no—
141#117C2×D5911yescharacteristic
142#104—22no—
143#103—1181no—
144#102—1181no—
145C22⋊C8—1182no—
146#41—21no—
147#41—21no—
148#41—21no—
149#41—21no—
150#41—592no—
151#41—592no—
152#41—1181no—
153#41—1181no—
154C2×Q8—591no—
155C2×Q8C4×D5911yescharacteristic
156C2×D4—21no—
157C2×D4—591no—
158C2×D4—1181no—
159C2×D4C4×D5911yescharacteristic
160C22×C4—21no—
161C22×C4—1181no—
162C22×C4C22×D5911yescharacteristic
163Q16—1182no—
164SD16—1182no—
165SD16—1182no—
166D8—1182no—
167M16—21no—
168M16—592no—
169M16—1184no—
170M16—1181no—
171M16—1181no—
172M16—1181no—
173M16C22⋊D5912yes—
174C2×C8—21no—
175C2×C8—592no—
176C2×C8—1184no—
177C2×C8—1181no—
178C2×C8—1181no—
179C2×C8—1181no—
180C2×C8—1181no—
181C4⋊C4—21no—
182C22⋊C4—21no—
183C42—21no—
184C23—21no—
185C23—1181no—
186C23#1174568711yescharacteristic
187Q8—21no—
188Q8—21no—
189Q8—592no—
190Q8—1181no—
191D4—21no—
192D4—21no—
193D4—41no—
194D4—41no—
195D4—592no—
196D4—1182no—
197D4—1182no—
198D4—1181no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—1181no—
204C2×C4—1182no—
205C2×C4—1182no—
206C2×C4C2×C4×D5911yescharacteristic
207C2×C4#1174568511yescharacteristic
208C2×C4D4×D5911yescharacteristic
209C2×C4D4×D5911yescharacteristic
210C2×C4D4×D5911yescharacteristic
211C2×C4C2×C22⋊D5911yescharacteristic
212C8—22no—
213C8—1182no—
214C8—1182no—
215C8—2361no—
216C22—41no—
217C22—42no—
218C22—41no—
219C22—1182no—
220C22—4721no—
221C22(C22⋊D59)⋊C411yescharacteristic
222C22#1427369811yescharacteristic
223C22#1427370911yescharacteristic
224C4—41no—
225C4—81no—
226C4—1182no—
227C4C22⋊C4×D5911yescharacteristic
228C4C4×C22⋊D5911yescharacteristic
229C4#1427369511yescharacteristic
230C4#1427370011yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—81no—
235C2—2362no—
236C2#1683727811yescharacteristic
subgroupG/H
centerC4#14273695
commutatorC2×C4×C59C22×C4
FrattiniC22×C4C22×D59
FittingC59×#421C2
socleC2×C59#386
hypercenter#421D59
hypocenterC59#1881
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #16838168C2semidirect
3. #16837396C2semidirect
4. #16837369C2semidirect
5. #16837172C2semidirect
6. #16837166C2semidirect
7. C59×#421C2semidirect
9. #14273758C4semidirect
11. #14273753C4semidirect
121. #421D59semidirect
126. C59#1881semidirect