Finite Group Databasebeta

← #21426069

Subgroups — #21426069

Order 7552.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#16838207C211yesmaximal, characteristic
2#16837845C211yesmaximal, characteristic, complemented
3C2×#14273672C211yesmaximal, characteristic, complemented
4#16837677C211yesmaximal, characteristic, complemented
5#16837216C211yesmaximal, characteristic, complemented
6#16837210C211yesmaximal, characteristic, complemented
7C59×#350C211yesmaximal, characteristic, complemented
8#14273773—21no—
9#14273757—21no—
10#14273755—21no—
11#14273755—22no—
12#14273749—21no—
13#14273749C2211yescharacteristic
14C2×Dic236C2211yescharacteristic, complemented
15C2×D472—21no—
16C2×D472C2211yescharacteristic
17Dic472—21no—
18#14273672—22no—
19#14273672—21no—
20#14273663C2211yescharacteristic, complemented
21#14273645—21no—
22#131×C59C2211yescharacteristic
23M32×C59C2211yescharacteristic, complemented
24C2×C16×C59C2211yescharacteristic, complemented
25#41×C59D411yescharacteristic
26#11745689—41no—
27D4×D59—42no—
28D4×D59—41no—
29#11745685—22no—
30#11745685—21no—
31C2×D236—22no—
32C2×D236—21no—
33C2×Dic118—21no—
34M16×C59—21no—
35M16×C59D411yescharacteristic
36C2×C8×C59—21no—
37C2×C8×C59C2311yescharacteristic
38#11745672—41no—
39#11745670—41no—
40#11745664—21no—
41Dic236—21no—
42Dic236—21no—
43Dic236D411yescharacteristic
44D472—22no—
45D472—21no—
46D472D411yescharacteristic
47D472D412yes—
48#11745660—22no—
49#11745660—21no—
50C16×C59—21no—
51C16×C59—21no—
52C22×D59—42no—
53C22×D59—41no—
54Q8×C59D811yescharacteristic
55D4×C59—21no—
56D4×C59D811yescharacteristic
57C2×C4×C59—21no—
58C2×C4×C59C2×D411yescharacteristic
59C22⋊D59—42no—
60C22⋊D59—41no—
61C2×Dic59—41no—
62D236—22no—
63D236—21no—
64D236—42no—
65D236—41no—
66C4×D59—42no—
67C4×D59—41no—
68Dic118—21no—
69Dic118—41no—
70Dic118—41no—
71C8×C59—21no—
72C8×C59C2×D411yescharacteristic
73C8×C59C2×D411yescharacteristic
74C59⋊C8—41no—
75C22×C59—21no—
76C22×C59AGL(1,ℤ/8)11yescharacteristic
77C2×D59—42no—
78C2×D59—41no—
79C2×D59—161no—
80C2×D59—162no—
81C4×C59—21no—
82C4×C59#12011yescharacteristic
83C4×C59C2×D811yescharacteristic
84Dic59—41no—
85Dic59—81no—
86#2194—591nomaximal, Sylow
87C2×C59—21no—
88C2×C59—41no—
89C2×C59#44711yescharacteristic
90D59—82no—
91D59—81no—
92#576—591no—
93#510—591no—
94C2×SD32—591no—
95#473—591no—
96#361—591no—
97#359—591no—
98#350D5911yescharacteristic, complemented
99C59#219411yescharacteristic, complemented, Sylow
100H(5,𝔽2)—1181no—
101#137—1181no—
102AGL(1,ℤ/8)—1182no—
103AGL(1,ℤ/8)—1181no—
104#135—591no—
105#135—1181no—
106C2×Q16—591no—
107C2×D8—591no—
108C2×D8—1181no—
109#131C2×D5911yescharacteristic
110Q32—1181no—
111SD32—1182no—
112SD32—1181no—
113M32C2×D5911yescharacteristic, complemented
114C2×C16C2×D5911yescharacteristic, complemented
115#108—591no—
116#101—1181no—
117#41—1182no—
118#41—1181no—
119#41—2361no—
120#41D23611yescharacteristic
121C2×Q8—1181no—
122C2×D4—1182no—
123C2×D4—1181no—
124C2×D4—2362no—
125C2×D4—2361no—
126Q16—591no—
127Q16—1181no—
128Q16—1181no—
129Q16—2361no—
130SD16—1182no—
131SD16—1181no—
132SD16—2361no—
133D8—592no—
134D8—591no—
135D8—1182no—
136D8—1181no—
137M16—21no—
138M16—1181no—
139M16D23611yescharacteristic
140C2×C8—21no—
141C2×C8C22×D5911yescharacteristic
142C16—21no—
143C16—21no—
144C23—2362no—
145C23—2361no—
146Q8—1181no—
147Q8—2361no—
148Q8—2361no—
149Q8D47211yescharacteristic
150D4—21no—
151D4—1182no—
152D4—1181no—
153D4—2362no—
154D4—2362no—
155D4—2361no—
156D4—2361no—
157D4D47211yescharacteristic
158C2×C4—21no—
159C2×C4—2362no—
160C2×C4—2361no—
161C2×C4—2361no—
162C2×C4C2×D23611yescharacteristic
163C8—21no—
164C8—2361no—
165C8D4×D5911yescharacteristic
166C8D4×D5911yescharacteristic
167C22—21no—
168C22—9441no—
169C22—9442no—
170C22—2362no—
171C22—2361no—
172C22#1427375511yescharacteristic
173C4—21no—
174C4—4721no—
175C4—2361no—
176C4#1427369711yescharacteristic
177C4C2×D47211yescharacteristic
178C2—21no—
179C2—41no—
180C2—4722no—
181C2—4721no—
182C2#1683753811yescharacteristic
subgroupG/H
centerC2#16837538
commutatorC2×C8×C59C23
FrattiniC2×C8C22×D59
FittingC59×#350C2
socleC2×C59#447
hypercenter#350D59
hypocenterC59#2194
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #350

Complements

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

NH ≅ G/Ntype
2. #16837845C2semidirect
3. C2×#14273672C2semidirect
4. #16837677C2semidirect
5. #16837216C2semidirect
6. #16837210C2semidirect
7. C59×#350C2semidirect
14. C2×Dic236C22semidirect
20. #14273663C22semidirect
23. M32×C59C22semidirect
24. C2×C16×C59C22semidirect
98. #350D59semidirect
99. C59#2194semidirect
113. M32C2×D59semidirect
114. C2×C16C2×D59semidirect