Finite Group Databasebeta

← #30639388

Subgroups — #30639388

Order 16256.

212 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 75 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#22549919C211yesmaximal, characteristic
2C2×#16861133C211yesmaximal, characteristic
3C2×#16861128C211yesmaximal, characteristic
4#22549832C211yesmaximal, characteristic
5#22549644C211yesmaximal, characteristic
6#22549640C211yesmaximal, characteristic
7#419×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C22×C127⋊C8C2211yescharacteristic
11C2×C4⋊Dic127C2211yescharacteristic
12#16861142—21no—
13#16861140—21no—
14#16861133—21no—
15#16861133—21no—
16#16861128C414yescomplemented
17#16861120—21no—
18C4×C127⋊C8—21no—
19C22×Q8×C127C2211yescharacteristic
20C2×M16×C127C2211yescharacteristic
21#117×C127C2211yescharacteristic
22#103×C127—21no—
23#103×C127—21no—
24C22×Dic127—21no—
25#14523232—21no—
26#14523232—21no—
27#14523232C2×C412yes—
28#14523232C2×C412yes—
29C2×C127⋊C8—21no—
30C2×C127⋊C8—21no—
31C2×C127⋊C8—21no—
32C2×C127⋊C8—21no—
33C2×C127⋊C8D411yescharacteristic
34C2×C127⋊C8D411yescharacteristic
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37C2×Q8×C127—21no—
38C2×Q8×C127—21no—
39C2×Q8×C127—21no—
40C2×Q8×C127—21no—
41C2×Q8×C127—21no—
42C2×Q8×C127C2×C412yes—
43C22×C4×C127—21no—
44C22×C4×C127C2311yescharacteristic
45M16×C127—21no—
46M16×C127—21no—
47C2×C8×C127—21no—
48C4⋊C4×C127D411yescharacteristic
49C4⋊C4×C127D411yescharacteristic
50C22⋊C4×C127—21no—
51C42×C127—21no—
52C2×Dic127—41no—
53C2×Dic127—81no—
54C127⋊C8—22no—
55C127⋊C8—22no—
56C127⋊C8—21no—
57C127⋊C8—21no—
58C23×C127#4111yescharacteristic
59Q8×C127—22no—
60Q8×C127—22no—
61Q8×C127—41no—
62Q8×C127—41no—
63C2×C4×C127—22no—
64C2×C4×C127—21no—
65C2×C4×C127—21no—
66C2×C4×C127—41no—
67C2×C4×C127—41no—
68C2×C4×C127C22×C411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C2×C4×C127#4111yescharacteristic
72C2×C4×C127#4111yescharacteristic
73C2×C4×C127#4111yescharacteristic
74C8×C127—41no—
75C22×C127—21no—
76C22×C127—21no—
77C22×C127#11711yescharacteristic
78C22×C127#12111yescharacteristic
79C22×C127#12611yescharacteristic
80C4×C127—42no—
81C4×C127—41no—
82C4×C127—41no—
83C4×C127C4×D411yescharacteristic
84C4×C127C4×D411yescharacteristic
85C4×C127#12311yescharacteristic
86C4×C127#12411yescharacteristic
87Dic127—81no—
88C2×C127—21no—
89C2×C127—21no—
90C2×C127#38811yescharacteristic
91C2×C127#47111yescharacteristic
92C2×C127#47311yescharacteristic
93#1930—1271nomaximal, Sylow
94C127#193011yescharacteristic, complemented, Sylow
95#428—1271no—
96#419D12711yescharacteristic
97C2×#103—1271no—
98C2×#101—1271no—
99#405—1271no—
100#343—1271no—
101#341—1271no—
102C22×Q8C2×D12711yescharacteristic
103C2×M16—1271no—
104C2×M16—1271no—
105C2×M16C2×D12711yescharacteristic
106C22×C8—1271no—
107#117C2×D12711yescharacteristic
108C2×C4⋊C4—1271no—
109#107—2541no—
110#106—2541no—
111#103—21no—
112#103—21no—
113#103—2541no—
114#103—2541no—
115#101—1274no—
116#97—2541no—
117C4×C8—2541no—
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8C4×D12712yes—
124C22×C4—21no—
125C22×C4—2541no—
126C22×C4C22×D12711yescharacteristic
127M16—21no—
128M16—21no—
129M16—1272no—
130M16—1272no—
131M16—2541no—
132M16—2541no—
133C2×C8—21no—
134C2×C8—1271no—
135C2×C8—1271no—
136C2×C8—2541no—
137C2×C8—2541no—
138C2×C8—2541no—
139C2×C8—2541no—
140C4⋊C4—2541no—
141C4⋊C4—2541no—
142C4⋊C4C22⋊D12711yescharacteristic
143C4⋊C4C22⋊D12711yescharacteristic
144C22⋊C4—21no—
145C42—21no—
146C23#1452324711yescharacteristic
147Q8—22no—
148Q8—22no—
149Q8—41no—
150Q8—41no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—10161no—
157C2×C4—5081no—
158C2×C4C2×C4×D12711yescharacteristic
159C2×C4D4×D12711yescharacteristic
160C2×C4C2×C22⋊D12711yescharacteristic
161C2×C4#1452324711yescharacteristic
162C2×C4#1452324711yescharacteristic
163C2×C4#1452324811yescharacteristic
164C8—41no—
165C8—2541no—
166C8—2541no—
167C8—2542no—
168C8—2542no—
169C22—21no—
170C22—21no—
171C22#1686116511yescharacteristic
172C22#1686118111yescharacteristic
173C22#1686120211yescharacteristic
174C4—42no—
175C4—41no—
176C4—41no—
177C4—10161no—
178C4(C22⋊D127)⋊C411yescharacteristic
179C4C4×C22⋊D12711yescharacteristic
180C4#1686119111yescharacteristic
181C4#1686119611yescharacteristic
182C2—21no—
183C2—21no—
184C2#2254977011yescharacteristic
185C2#2255014111yescharacteristic
186C2#2255014911yescharacteristic
subgroupG/H
centerC22#16861181
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#419×C127C2
socleC22×C127#121
prime radicalC23×C127#41
cosocleC22×C4×C127C23
hypercenter#419D127
hypocenterC127#1930
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
16. #16861128C4semidirect
94. C127#1930semidirect

Maximal subgroups

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

subgroupindex
#225499192
C2×#168611332
C2×#168611282
#225498322
#225496442
#225496402
#419×C1272
#1930127