Finite Group Databasebeta

← #30640197

Subgroups — #30640197

Order 16256.

184 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 59 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#22549924C211yesmaximal, characteristic
2#22549885C211yesmaximal, characteristic
3#22549885C211yesmaximal, characteristic
4#22549648C211yesmaximal, characteristic
5#22549647C211yesmaximal, characteristic
6#22549633C211yesmaximal, characteristic
7C2×#101×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10#16861168C2211yescharacteristic
11#16861168C2211yescharacteristic
12#16861144—22no—
13#16861133—22no—
14#16861133—22no—
15C22×Q8×C127C2211yescharacteristic
16C2×M16×C127C2211yescharacteristic
17C2×M16×C127C2211yescharacteristic
18#101×C127—22no—
19#14523232—21no—
20#14523232—21no—
21#14523232—21no—
22#14523232—21no—
23C2×C127⋊C8—21no—
24C2×C127⋊C8D412yes—
25C4⋊Dic127—21no—
26C4⋊Dic127—21no—
27C22⋊Dic127—21no—
28C22⋊Dic127—21no—
29C4×Dic127—21no—
30C4×Dic127—21no—
31C2×Q8×C127—21no—
32C2×Q8×C127—21no—
33C2×Q8×C127—21no—
34C2×Q8×C127—21no—
35C2×Q8×C127—21no—
36C2×Q8×C127D412yes—
37C22×C4×C127—21no—
38C22×C4×C127C2311yescharacteristic
39M16×C127—21no—
40M16×C127—21no—
41M16×C127—21no—
42M16×C127—21no—
43C2×C8×C127—21no—
44C2×C8×C127—21no—
45C2×Dic127—41no—
46C2×Dic127—41no—
47C127⋊C8—22no—
48C127⋊C8—41no—
49C23×C127#4111yescharacteristic
50Q8×C127—22no—
51Q8×C127—22no—
52Q8×C127—41no—
53Q8×C127—41no—
54C2×C4×C127—22no—
55C2×C4×C127—41no—
56C2×C4×C127—41no—
57C2×C4×C127C2×D411yescharacteristic
58C2×C4×C127C2×D411yescharacteristic
59C2×C4×C127#4111yescharacteristic
60C2×C4×C127#4111yescharacteristic
61C2×C4×C127#4111yescharacteristic
62C2×C4×C127#4111yescharacteristic
63C8×C127—41no—
64C8×C127—41no—
65C22×C127—41no—
66C22×C127#12311yescharacteristic
67C22×C127#12411yescharacteristic
68C22×C127#12611yescharacteristic
69C4×C127—42no—
70C4×C127#12111yescharacteristic
71C4×C127#12311yescharacteristic
72C4×C127#12311yescharacteristic
73C4×C127#12611yescharacteristic
74Dic127—81no—
75Dic127—81no—
76C2×C127—21no—
77C2×C127—21no—
78C2×C127—21no—
79C2×C127#39711yescharacteristic
80#2082—1271nomaximal, Sylow
81C127#208211yescharacteristic, complemented, Sylow
82#430—1271no—
83#419—1271no—
84#419—1271no—
85C2×#101D12711yescharacteristic
86#344—1271no—
87#344—1271no—
88H(3,ℤ/4)—1271no—
89C22×Q8C2×D12711yescharacteristic
90C2×M16—1271no—
91C2×M16—1271no—
92C2×M16C2×D12711yescharacteristic
93C2×M16C2×D12711yescharacteristic
94#117—1271no—
95#117—1271no—
96#108—2542no—
97#103—2542no—
98#103—2542no—
99#101—22no—
100C2×Q8—21no—
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8C22⋊D12712yes—
106C22×C4—21no—
107C22×C4C22×D12711yescharacteristic
108M16—21no—
109M16—21no—
110M16—21no—
111M16—21no—
112M16—2541no—
113M16—2541no—
114M16—2541no—
115M16—2541no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—1272no—
119C2×C8—2541no—
120C4⋊C4—2541no—
121C4⋊C4—2541no—
122C22⋊C4—2541no—
123C22⋊C4—2541no—
124C42—2541no—
125C42—2541no—
126C23#1452324811yescharacteristic
127Q8—22no—
128Q8—22no—
129Q8—41no—
130Q8—41no—
131C2×C4—22no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—5081no—
135C2×C4—5081no—
136C2×C4D4×D12711yescharacteristic
137C2×C4C2×C22⋊D12711yescharacteristic
138C2×C4#1452324711yescharacteristic
139C2×C4#1452324711yescharacteristic
140C2×C4#1452324711yescharacteristic
141C2×C4#1452324811yescharacteristic
142C8—41no—
143C8—41no—
144C8—5081no—
145C8—2542no—
146C22—41no—
147C22#1686119211yescharacteristic
148C22#1686119511yescharacteristic
149C22#1686120211yescharacteristic
150C4—42no—
151C4—10161no—
152C4—10161no—
153C4#1686118111yescharacteristic
154C4#1686119011yescharacteristic
155C4#1686119111yescharacteristic
156C4#1686120211yescharacteristic
157C2—21no—
158C2—21no—
159C2—21no—
160C2#2254980411yescharacteristic
subgroupG/H
centerC2#22549804
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#101×C127C2
socleC2×C127#397
prime radicalC23×C127#41
hypercenterC2×#101D127
hypocenterC127#2082
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
81. C127#2082semidirect

Maximal subgroups

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

subgroupindex
#225499242
#225498852
#225496482
#225496472
#225496332
C2×#101×C1272
#2082127