Finite Group Databasebeta

← #30640846

Subgroups — #30640846

Order 16256.

160 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 47 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
1C2×#16861152C211yesmaximal, characteristic, complemented
2C2×#16861148C211yesmaximal, characteristic, complemented
3#22550114C211yesmaximal, characteristic, complemented
4#22550113C211yesmaximal, characteristic
5#22549701C211yesmaximal, characteristic, complemented
6#22549678C211yesmaximal, characteristic, complemented
7#358×C127C211yesmaximal, characteristic, complemented
8C2×D1016C2211yescharacteristic
9C2×C8×D127C2211yescharacteristic
10#16861185—21no—
11#16861184—21no—
12#16861152—22no—
13#16861148—22no—
14C2×C127⋊C16C2211yescharacteristic
15#16861143C2211yescharacteristic
16#16861133—21no—
17#16861129—21no—
18C2×Q16×C127C2211yescharacteristic
19C2×C16×C127C2211yescharacteristic
20#107×C127C2211yescharacteristic
21C2×D508—21no—
22C2×D508—41no—
23C2×C4×D127D411yescharacteristic
24D1016—21no—
25D1016—21no—
26C2×C127⋊C8D411yescharacteristic
27C8×D127—21no—
28C8×D127—21no—
29C4⋊Dic127—21no—
30Dic127⋊C4—41no—
31(C2×D127)⋊C4—41no—
32(C2×D127)⋊C4—41no—
33C127⋊C16D412yescomplemented
34C2×Q8×C127—21no—
35Q16×C127—21no—
36Q16×C127—21no—
37C2×C8×C127C2311yescharacteristic
38C4⋊C4×C127—21no—
39C16×C127—21no—
40C22×D127—41no—
41C22×D127D811yescharacteristic
42D508—41no—
43D508—41no—
44D508—82no—
45C2×Dic127—41no—
46C2×Dic127D811yescharacteristic
47C4×D127—21no—
48C4×D127—21no—
49C127⋊C8—21no—
50Q8×C127—41no—
51Q8×C127—41no—
52C2×C4×C127—41no—
53C2×C4×C127—41no—
54C2×C4×C127C2×D411yescharacteristic
55C8×C127C2×D411yescharacteristic
56C8×C127#4111yescharacteristic
57C22×C127C2×D811yescharacteristic
58C2×D127—21no—
59C2×D127—21no—
60C2×D127—21no—
61C2×D127—81no—
62C2×D127—81no—
63C2×D127—81no—
64C4×C127—81no—
65C4×C127—81no—
66C4×C127#12111yescharacteristic
67C4×C127#13511yescharacteristic
68Dic127—21no—
69Dic127—81no—
70C2×C127#46611yescharacteristic
71C2×C127#50911yescharacteristic
72C2×C127#51011yescharacteristic
73D127—41no—
74D127—161no—
75#2220—1271nomaximal, Sylow
76C127#222011yescharacteristic, complemented, Sylow
77C2×SD32—1271no—
78C2×M32—1271no—
79#467—1271no—
80#466—1271no—
81#366—1271no—
82#358D12711yescharacteristic, complemented
83#357—1271no—
84C2×Q16C2×D12711yescharacteristic
85C2×D8—1271no—
86C22×C8—1271no—
87C22⋊Q8—2541no—
88#121—2541no—
89SD32—2542no—
90M32—2542no—
91C2×C16—1271no—
92C2×C16C2×D12711yescharacteristic
93#107—1271no—
94#107C2×D12711yescharacteristic
95#103—2541no—
96#102—2541no—
97C2×Q8—21no—
98C2×D4—5081no—
99C2×D4—2541no—
100C22×C4—1271no—
101Q16—21no—
102Q16—21no—
103D8—2541no—
104D8—2541no—
105C2×C8—1271no—
106C2×C8—2541no—
107C2×C8—2541no—
108C2×C8C22×D12711yescharacteristic
109C4⋊C4—21no—
110C4⋊C4—5081no—
111C4⋊C4—2541no—
112C22⋊C4—5081no—
113C22⋊C4—5081no—
114C16—21no—
115C16—1272no—
116C23—1271no—
117C23—5081no—
118Q8—41no—
119Q8—41no—
120D4—10162no—
121D4—5081no—
122D4—5081no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—1271no—
126C2×C4—5081no—
127C2×C4—2541no—
128C2×C4—2541no—
129C2×C4D4×D12711yescharacteristic
130C8—2541no—
131C8D4×D12711yescharacteristic
132C8#1452324811yescharacteristic
133C22—10161no—
134C22—10161no—
135C22—10161no—
136C22—2541no—
137C22—2541no—
138C22—2541no—
139C22D8×D12711yescharacteristic
140C4—81no—
141C4—81no—
142C4—10161no—
143C4—2541no—
144C4#1686118211yescharacteristic
145C4#1686123011yescharacteristic
146C2—20321no—
147C2—5081no—
148C2#2255011111yescharacteristic
149C2#2255031311yescharacteristic
150C2#2255031811yescharacteristic
subgroupG/H
centerC22D8×D127
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#358×C127C2
socleC22×C127C2×D8
prime radical#22550113C2
hypercenter#358D127
hypocenterC127#2220
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ #358

Complements

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

NH ≅ G/Ntype
1. C2×#16861152C2semidirect
2. C2×#16861148C2semidirect
3. #22550114C2semidirect
5. #22549701C2semidirect
6. #22549678C2semidirect
7. #358×C127C2semidirect
33. C127⋊C16D4semidirect
76. C127#2220semidirect
82. #358D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611522
C2×#168611482
#225501142
#225501132
#225497012
#225496782
#358×C1272
#2220127