Finite Group Databasebeta

← #30639265

Subgroups — #30639265

Order 16256.

314 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 85 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#22550450C211yesmaximal, characteristic
2#22550354C211yesmaximal, characteristic, complemented
3#22549892C212yesmaximal
4#22549633C212yesmaximal, complemented
5C2×#101×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C2211yescharacteristic
7#16861207C412yescomplemented
8#16861198C412yescomplemented
9#16861192—22no—
10#16861185—22no—
11Q8×Dic127—24no—
12#16861168—22no—
13#16861168C2211yescharacteristic
14#16861168C2212yes—
15#16861135—24no—
16C22×Q8×C127C2211yescharacteristic
17C2×M16×C127C2212yes—
18#101×C127—22no—
19#14523248—22no—
20#14523248—22no—
21C2×Dic254C2×C411yescharacteristic
22C2×D508C2×C411yescharacteristic
23C2×C22⋊D127—21no—
24C2×C4×D127—21no—
25C4⋊Dic127—24no—
26C4⋊Dic127—21no—
27C4⋊Dic127—22no—
28Dic127⋊C4—24no—
29C22⋊Dic127—21no—
30C22⋊Dic127—22no—
31(C2×D127)⋊C4—24no—
32C4×Dic127—24no—
33C4×Dic127C2×C412yes—
34C4×Dic127D414yes—
35C2×Q8×C127—22no—
36C2×Q8×C127—22no—
37C2×Q8×C127C2×C412yes—
38C2×Q8×C127D412yes—
39C22×C4×C127D412yes—
40C22×C4×C127C2311yescharacteristic
41M16×C127—22no—
42M16×C127—22no—
43C2×C8×C127—22no—
44C22×D127—21no—
45Dic254—22no—
46D508—22no—
47C22⋊D127—41no—
48C22⋊D127—41no—
49C2×Dic127—22no—
50C2×Dic127—21no—
51C2×Dic127—24no—
52C4×D127—42no—
53C23×C127C2×D411yescharacteristic
54Q8×C127—24no—
55Q8×C127—24no—
56C2×C4×C127—22no—
57C2×C4×C127—22no—
58C2×C4×C127C22⋊C414yes—
59C2×C4×C127C22×C411yescharacteristic
60C2×C4×C127C2×D412yes—
61C2×C4×C127C2×D411yescharacteristic
62C2×C4×C127#4112yes—
63C8×C127—42no—
64C22×C127—41no—
65C22×C127C2×C22⋊C411yescharacteristic
66C22×C127#12011yescharacteristic
67C22×C127#12311yescharacteristic
68C2×D127—42no—
69C2×D127—41no—
70C4×C127—24no—
71C4×C127C4×D412yes—
72C4×C127#12112yes—
73Dic127—44no—
74Dic127—42no—
75Dic127—41no—
76C2×C127—21no—
77C2×C127—21no—
78C2×C127—21no—
79C2×C127#38611yescharacteristic
80D127—81no—
81#1912—1271nomaximal, Sylow
82C127#191211yescharacteristic, complemented, Sylow
83#536—1271no—
84#519—1271no—
85#421—1272no—
86C2×#101D12711yescharacteristic, complemented
87H(3,ℤ/4)—1272no—
88C2×#41—1271no—
89C22×Q8C2×D12711yescharacteristic
90C2×M16C2×D12712yes—
91C4⋊Q8—1272no—
92#124—1272no—
93#123—2542no—
94C22⋊Q8—2542no—
95C4×Q8—2544no—
96#117—1272no—
97#117—1271no—
98#117—2542no—
99#104—2544no—
100#101—22no—
101#41—2542no—
102#41—2542no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—1271no—
106C2×Q8C4×D12712yes—
107C2×Q8C22⋊D12712yes—
108C2×D4—1271no—
109C2×D4—2541no—
110C22×C4—2541no—
111C22×C4D50812yes—
112C22×C4C22×D12711yescharacteristic
113M16—22no—
114M16—22no—
115C2×C8—22no—
116C4⋊C4—2544no—
117C4⋊C4—2544no—
118C4⋊C4—2542no—
119C4⋊C4—2541no—
120C22⋊C4—2542no—
121C22⋊C4—2541no—
122C22⋊C4—2544no—
123C42—1272no—
124C42—1274no—
125C42—2544no—
126C23—2541no—
127C23C2×D50811yescharacteristic
128Q8—24no—
129Q8—24no—
130Q8—2542no—
131D4—5081no—
132D4—5081no—
133D4—2542no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—5082no—
137C2×C4—2542no—
138C2×C4—2541no—
139C2×C4—2544no—
140C2×C4(C2×D127)⋊C414yes—
141C2×C4C2×C4×D12711yescharacteristic
142C2×C4D4×D12712yes—
143C2×C4C2×C22⋊D12711yescharacteristic
144C2×C4#1452324712yes—
145C8—42no—
146C22—41no—
147C22—5082no—
148C22—5081no—
149C22C2×(C2×D127)⋊C411yescharacteristic
150C22#1686117911yescharacteristic
151C22#1686119311yescharacteristic
152C4—24no—
153C4—5084no—
154C4—5082no—
155C4—5081no—
156C4(C22⋊D127)⋊C412yes—
157C4#1686118112yes—
158C2—21no—
159C2—21no—
160C2—21no—
161C2—10161no—
162C2#2254975311yescharacteristic
subgroupG/H
centerC2#22549753
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#101×C127C2
socleC2×C127#386
prime radicalC2×#14523248C22
cosocleC22×C4×C127C23
hypercenterC2×#101D127
hypocenterC127#1912
LCSG ▷ C2×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
2. #22550354C2semidirect
4. #22549633C2semidirect
5. C2×#101×C127C2semidirect
7. #16861207C4semidirect
8. #16861198C4semidirect
82. C127#1912semidirect
86. C2×#101D127semidirect

Maximal subgroups

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

subgroupindex
#225504502
#225503542
#225498922
#225496332
C2×#101×C1272
#1912127