Finite Group Databasebeta

← #30640701

Subgroups — #30640701

Order 16256.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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#22550316C211yesmaximal, characteristic, complemented
2C2×#16861153C211yesmaximal, characteristic, complemented
3#22550151C211yesmaximal, characteristic, complemented
4#22549689C211yesmaximal, characteristic, complemented
5#22549683C211yesmaximal, characteristic, complemented
6#22549660C211yesmaximal, characteristic, complemented
7#576×C127C211yesmaximal, characteristic
8#16861239—21no—
9C2×Dic508C2211yescharacteristic, complemented
10#16861155—21no—
11#16861153—22no—
12#16861153—21no—
13#16861149C2211yescharacteristic, complemented
14C2×C127⋊C16C2211yescharacteristic, complemented
15#16861145C2211yescharacteristic, complemented
16#16861127—21no—
17H(5,𝔽2)×C127—21no—
18AGL(1,ℤ/8)×C127—22no—
19AGL(1,ℤ/8)×C127—21no—
20#135×C127—21no—
21#135×C127C2211yescharacteristic
22C2×D8×C127—21no—
23C2×D8×C127C2211yescharacteristic
24#131×C127C2211yescharacteristic
25C2×Dic254—21no—
26Dic508—21no—
27Dic508—21no—
28#14523238—41no—
29#14523236—41no—
30#14523232—21no—
31C127⋊C16—21no—
32C127⋊C16—21no—
33#41×C127—22no—
34#41×C127—21no—
35#41×C127—41no—
36#41×C127D411yescharacteristic
37C2×D4×C127—22no—
38C2×D4×C127—21no—
39C2×D4×C127—42no—
40C2×D4×C127—41no—
41Q16×C127D411yescharacteristic
42SD16×C127—22no—
43SD16×C127—21no—
44D8×C127—22no—
45D8×C127—21no—
46D8×C127D411yescharacteristic
47D8×C127D412yes—
48M16×C127—21no—
49M16×C127D411yescharacteristic
50C2×C8×C127—21no—
51C2×C8×C127C2311yescharacteristic
52Dic254—41no—
53Dic254—41no—
54C2×Dic127—41no—
55C127⋊C8—41no—
56C23×C127—42no—
57C23×C127—41no—
58Q8×C127—21no—
59Q8×C127D811yescharacteristic
60D4×C127—21no—
61D4×C127—21no—
62D4×C127—22no—
63D4×C127—42no—
64D4×C127—42no—
65D4×C127—41no—
66D4×C127—41no—
67D4×C127D811yescharacteristic
68C2×C4×C127—21no—
69C2×C4×C127—42no—
70C2×C4×C127—41no—
71C2×C4×C127C2×D411yescharacteristic
72C8×C127—21no—
73C8×C127C2×D411yescharacteristic
74C8×C127C2×D411yescharacteristic
75C22×C127—21no—
76C22×C127—42no—
77C22×C127—41no—
78C22×C127—161no—
79C22×C127—162no—
80C22×C127AGL(1,ℤ/8)11yescharacteristic
81C4×C127—21no—
82C4×C127—41no—
83C4×C127#12011yescharacteristic
84C4×C127C2×D811yescharacteristic
85Dic127—81no—
86C2×C127—21no—
87C2×C127—41no—
88C2×C127—82no—
89C2×C127—81no—
90C2×C127#44711yescharacteristic
91#2194—1271nomaximal, Sylow
92C127#219411yescharacteristic, complemented, Sylow
93#576D12711yescharacteristic
94#510—1271no—
95C2×SD32—1271no—
96#473—1271no—
97#361—1271no—
98#359—1271no—
99#350—1271no—
100H(5,𝔽2)—21no—
101#137—2541no—
102AGL(1,ℤ/8)—22no—
103AGL(1,ℤ/8)—21no—
104#135—21no—
105#135C2×D12711yescharacteristic
106C2×Q16—1271no—
107C2×D8—21no—
108C2×D8C2×D12711yescharacteristic
109#131C2×D12711yescharacteristic
110Q32—2541no—
111SD32—2542no—
112SD32—2541no—
113M32—1271no—
114C2×C16—1271no—
115#108—1271no—
116#101—2541no—
117#41—22no—
118#41—21no—
119#41—41no—
120#41C22⋊D12711yescharacteristic
121C2×Q8—2541no—
122C2×D4—22no—
123C2×D4—21no—
124C2×D4—42no—
125C2×D4—41no—
126Q16—5081no—
127Q16—2541no—
128Q16—2541no—
129Q16C22⋊D12711yescharacteristic
130SD16—22no—
131SD16—21no—
132SD16—5081no—
133D8—22no—
134D8—21no—
135D8C22⋊D12711yescharacteristic
136D8C22⋊D12712yes—
137M16—21no—
138M16—2541no—
139M16C22⋊D12711yescharacteristic
140C2×C8—21no—
141C2×C8C22×D12711yescharacteristic
142C16—2541no—
143C16—2541no—
144C23—42no—
145C23—41no—
146Q8—21no—
147Q8—5081no—
148Q8—5081no—
149Q8#1452323311yescharacteristic
150D4—21no—
151D4—21no—
152D4—22no—
153D4—42no—
154D4—42no—
155D4—41no—
156D4—41no—
157D4#1452323311yescharacteristic
158C2×C4—21no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—5081no—
162C2×C4C2×C22⋊D12711yescharacteristic
163C8—21no—
164C8—5081no—
165C8C2×C22⋊D12711yescharacteristic
166C8C2×C22⋊D12711yescharacteristic
167C22—21no—
168C22—42no—
169C22—41no—
170C22—161no—
171C22—162no—
172C22#1686123511yescharacteristic
173C4—21no—
174C4—41no—
175C4—10161no—
176C4#1686117811yescharacteristic
177C4C2×#1452323311yescharacteristic
178C2—21no—
179C2—41no—
180C2—82no—
181C2—81no—
182C2#2255001011yescharacteristic
subgroupG/H
centerC2#22550010
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting, prime radical#576×C127C2
socleC2×C127#447
hypercenter#576D127
hypocenterC127#2194
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #22550316C2semidirect
2. C2×#16861153C2semidirect
3. #22550151C2semidirect
4. #22549689C2semidirect
5. #22549683C2semidirect
6. #22549660C2semidirect
9. C2×Dic508C22semidirect
13. #16861149C22semidirect
14. C2×C127⋊C16C22semidirect
15. #16861145C22semidirect
92. C127#2194semidirect

Maximal subgroups

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

subgroupindex
#225503162
C2×#168611532
#225501512
#225496892
#225496832
#225496602
#576×C1272
#2194127