Finite Group Databasebeta

← #30640699

Subgroups — #30640699

Order 16256.

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#22550681C211yesmaximal, characteristic
2#22550317C211yesmaximal, characteristic, complemented
3C2×#16861152C211yesmaximal, characteristic, complemented
4#22549688C211yesmaximal, characteristic, complemented
5#22549682C211yesmaximal, characteristic, complemented
6#22549660C211yesmaximal, characteristic, complemented
7#473×C127C211yesmaximal, characteristic, complemented
8#16861255—21no—
9#16861237—22no—
10#16861237—21no—
11#16861231—21no—
12#16861231C2211yescharacteristic
13C2×D1016—21no—
14C2×D1016C2211yescharacteristic
15#16861155—21no—
16#16861152—22no—
17#16861152—21no—
18#16861149C2211yescharacteristic, complemented
19C2×C127⋊C16C2211yescharacteristic, complemented
20#137×C127—21no—
21C2×Q16×C127C2211yescharacteristic, complemented
22#131×C127C2211yescharacteristic
23#108×C127C2211yescharacteristic, complemented
24#101×C127—21no—
25#14523249—41no—
26#14523248—22no—
27#14523248—21no—
28C2×D508—22no—
29C2×D508—21no—
30D4×D127—42no—
31D4×D127—41no—
32Dic508D411yescharacteristic
33#14523237—22no—
34#14523237—21no—
35D1016—22no—
36D1016—21no—
37D1016D411yescharacteristic
38D1016D412yes—
39C127⋊C16—21no—
40C127⋊C16—21no—
41#41×C127D411yescharacteristic
42C2×Q8×C127—21no—
43Q16×C127—21no—
44Q16×C127—21no—
45Q16×C127—41no—
46SD16×C127—41no—
47M16×C127—21no—
48M16×C127—21no—
49M16×C127D411yescharacteristic
50C2×C8×C127—21no—
51C2×C8×C127C2311yescharacteristic
52C22×D127—42no—
53C22×D127—41no—
54Dic254—21no—
55D508—21no—
56D508—22no—
57D508—42no—
58D508—41no—
59C22⋊D127—42no—
60C22⋊D127—41no—
61C4×D127—42no—
62C4×D127—41no—
63Q8×C127—41no—
64Q8×C127—41no—
65Q8×C127D811yescharacteristic
66D4×C127—21no—
67D4×C127D811yescharacteristic
68C2×C4×C127—21no—
69C2×C4×C127—41no—
70C2×C4×C127C2×D411yescharacteristic
71C8×C127—21no—
72C8×C127—41no—
73C8×C127C2×D411yescharacteristic
74C8×C127C2×D411yescharacteristic
75C22×C127—21no—
76C22×C127AGL(1,ℤ/8)11yescharacteristic
77C2×D127—42no—
78C2×D127—41no—
79C2×D127—161no—
80C2×D127—162no—
81C4×C127—21no—
82C4×C127—81no—
83C4×C127#12011yescharacteristic
84C4×C127C2×D811yescharacteristic
85Dic127—41no—
86C2×C127—21no—
87C2×C127—41no—
88C2×C127#44711yescharacteristic
89D127—82no—
90D127—81no—
91#2194—1271nomaximal, Sylow
92C127#219411yescharacteristic, complemented, Sylow
93#576—1271no—
94#510—1271no—
95C2×SD32—1271no—
96#473D12711yescharacteristic, complemented
97#361—1271no—
98#359—1271no—
99#350—1271no—
100H(5,𝔽2)—2541no—
101#137—21no—
102AGL(1,ℤ/8)—2542no—
103AGL(1,ℤ/8)—2541no—
104#135—1271no—
105#135—2541no—
106C2×Q16C2×D12711yescharacteristic, complemented
107C2×D8—1271no—
108C2×D8—2541no—
109#131C2×D12711yescharacteristic
110Q32—2541no—
111SD32—2542no—
112SD32—2541no—
113M32—1271no—
114C2×C16—1271no—
115#108C2×D12711yescharacteristic, complemented
116#101—21no—
117#41—5081no—
118#41—2542no—
119#41—2541no—
120#41C22⋊D12711yescharacteristic
121C2×Q8—21no—
122C2×D4—5082no—
123C2×D4—5081no—
124C2×D4—2542no—
125C2×D4—2541no—
126Q16—21no—
127Q16—21no—
128Q16—41no—
129Q16—1271no—
130SD16—41no—
131SD16—2542no—
132SD16—2541no—
133D8—1272no—
134D8—1271no—
135D8—2542no—
136D8—2541no—
137M16—21no—
138M16—21no—
139M16C22⋊D12711yescharacteristic
140C2×C8—21no—
141C2×C8C22×D12711yescharacteristic
142C16—2541no—
143C16—2541no—
144C23—5082no—
145C23—5081no—
146Q8—41no—
147Q8—41no—
148Q8—2541no—
149Q8#1452323311yescharacteristic
150D4—21no—
151D4—5082no—
152D4—5082no—
153D4—5081no—
154D4—5081no—
155D4—2542no—
156D4—2541no—
157D4#1452323311yescharacteristic
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—5082no—
161C2×C4—5081no—
162C2×C4C2×C22⋊D12711yescharacteristic
163C8—21no—
164C8—41no—
165C8D4×D12711yescharacteristic
166C8D4×D12711yescharacteristic
167C22—21no—
168C22—20321no—
169C22—20322no—
170C22—5082no—
171C22—5081no—
172C22#1686123411yescharacteristic
173C4—21no—
174C4—81no—
175C4—5081no—
176C4#1686117711yescharacteristic
177C4C2×#1452323311yescharacteristic
178C2—21no—
179C2—41no—
180C2—10162no—
181C2—10161no—
182C2#2255000911yescharacteristic
subgroupG/H
centerC2#22550009
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#473×C127C2
socleC2×C127#447
prime radical#22550681C2
hypercenter#473D127
hypocenterC127#2194
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #473

Complements

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

NH ≅ G/Ntype
2. #22550317C2semidirect
3. C2×#16861152C2semidirect
4. #22549688C2semidirect
5. #22549682C2semidirect
6. #22549660C2semidirect
7. #473×C127C2semidirect
18. #16861149C22semidirect
19. C2×C127⋊C16C22semidirect
21. C2×Q16×C127C22semidirect
23. #108×C127C22semidirect
92. C127#2194semidirect
96. #473D127semidirect
106. C2×Q16C2×D127semidirect
115. #108C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506812
#225503172
C2×#168611522
#225496882
#225496822
#225496602
#473×C1272
#2194127