Finite Group Databasebeta

← #30638970

Subgroups — #30638970

Order 16256.

262 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 89 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, complemented
2#22549899C211yesmaximal, characteristic, complemented
3#22549893C211yesmaximal, characteristic
4C2×#16861136C211yesmaximal, characteristic
5#22549647C211yesmaximal, characteristic, complemented
6#22549637C211yesmaximal, characteristic, complemented
7#568×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10#16861168C2211yescharacteristic
11C2×C4×Dic127C2211yescharacteristic
12#16861144—22no—
13Dic127⋊C8—22no—
14#16861136—22no—
15#16861136C412yescomplemented
16#16861136C412yescomplemented
17C2×#41×C127C2211yescharacteristic
18#131×C127—22no—
19#131×C127—22no—
20C2×M16×C127—22no—
21C2×M16×C127C2211yescharacteristic
22C2×M16×C127C2211yescharacteristic
23C22×Dic127—21no—
24#14523232—21no—
25#14523232—21no—
26#14523232—21no—
27#14523232C2×C412yes—
28C2×C127⋊C8—21no—
29C2×C127⋊C8—21no—
30C4⋊Dic127—21no—
31C22⋊Dic127—21no—
32C4×Dic127—21no—
33C4×Dic127—21no—
34C4×Dic127C2×C411yescharacteristic
35C4×Dic127C2×C411yescharacteristic
36#41×C127—21no—
37#41×C127—21no—
38#41×C127C2×C412yes—
39#41×C127D412yes—
40C2×Q8×C127Q811yescharacteristic
41C2×D4×C127—21no—
42C2×D4×C127Q811yescharacteristic
43C22×C4×C127—21no—
44C22×C4×C127C2311yescharacteristic
45M16×C127—24no—
46M16×C127—21no—
47M16×C127—21no—
48M16×C127—21no—
49M16×C127D412yes—
50C2×C8×C127—24no—
51C2×C8×C127—21no—
52C2×C8×C127D412yes—
53C2×Dic127—21no—
54C2×Dic127—21no—
55C2×Dic127—42no—
56C2×Dic127—41no—
57C127⋊C8—22no—
58C127⋊C8—41no—
59C23×C127—21no—
60C23×C127#4111yescharacteristic
61Q8×C127—21no—
62Q8×C127C4⋊C412yes—
63D4×C127—22no—
64D4×C127—22no—
65D4×C127—21no—
66D4×C127C4⋊C412yes—
67C2×C4×C127—21no—
68C2×C4×C127—22no—
69C2×C4×C127—22no—
70C2×C4×C127C22×C411yescharacteristic
71C2×C4×C127C2×D411yescharacteristic
72C2×C4×C127C2×D411yescharacteristic
73C2×C4×C127C2×D411yescharacteristic
74C2×C4×C127C2×Q811yescharacteristic
75C2×C4×C127#4111yescharacteristic
76C8×C127—22no—
77C8×C127—22no—
78C22×C127—22no—
79C22×C127—41no—
80C22×C127—81no—
81C22×C127C4×D411yescharacteristic
82C22×C127C22⋊Q811yescharacteristic
83C22×C127#12311yescharacteristic
84C4×C127—22no—
85C4×C127C2×C4⋊C411yescharacteristic
86C4×C127C4×D411yescharacteristic
87C4×C127#12011yescharacteristic
88C4×C127C22⋊Q811yescharacteristic
89Dic127—22no—
90Dic127—41no—
91Dic127—81no—
92C2×C127—21no—
93C2×C127—21no—
94C2×C127—21no—
95C2×C127—42no—
96C2×C127#38511yescharacteristic
97#1861—1271nomaximal, Sylow
98C127#186111yescharacteristic, complemented, Sylow
99#568D12711yescharacteristic
100#430—1271no—
101#423—1271no—
102#421—1271no—
103C2×#104—1271no—
104#344—1271no—
105#339—1271no—
106C2×#41C2×D12711yescharacteristic
107#131—22no—
108#131—22no—
109C2×M16—22no—
110C2×M16—1271no—
111C2×M16—1271no—
112C2×M16C2×D12711yescharacteristic
113C2×M16C2×D12711yescharacteristic
114#117—1271no—
115C2×C42—1271no—
116#108—2542no—
117C4⋊C8—2542no—
118#104—1272no—
119#104—1272no—
120#104—2542no—
121#41—21no—
122#41—21no—
123#41C4×D12712yes—
124#41C22⋊D12712yes—
125C2×Q8Dic25411yescharacteristic
126C2×D4—21no—
127C2×D4Dic25411yescharacteristic
128C22×C4—21no—
129C22×C4—2541no—
130C22×C4C22×D12711yescharacteristic
131M16—24no—
132M16—21no—
133M16—21no—
134M16—21no—
135M16—1272no—
136M16—2541no—
137M16—2541no—
138M16—2541no—
139M16C22⋊D12712yes—
140C2×C8—24no—
141C2×C8—21no—
142C2×C8—2541no—
143C2×C8—2541no—
144C2×C8C22⋊D12712yes—
145C4⋊C4—2541no—
146C22⋊C4—2541no—
147C42—1271no—
148C42—1271no—
149C42—2541no—
150C42—2541no—
151C23—21no—
152C23#1452324811yescharacteristic
153Q8—21no—
154Q8Dic127⋊C412yes—
155D4—22no—
156D4—22no—
157D4—21no—
158D4Dic127⋊C412yes—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—5081no—
163C2×C4—5082no—
164C2×C4—2541no—
165C2×C4—2541no—
166C2×C4C2×C4×D12711yescharacteristic
167C2×C4C2×C22⋊D12711yescharacteristic
168C2×C4C2×C22⋊D12711yescharacteristic
169C2×C4C2×C22⋊D12711yescharacteristic
170C2×C4C2×Dic25411yescharacteristic
171C2×C4#1452324811yescharacteristic
172C8—22no—
173C8—22no—
174C8—5081no—
175C8—2542no—
176C22—22no—
177C22—41no—
178C22—81no—
179C22C4×C22⋊D12711yescharacteristic
180C22#1686118811yescharacteristic
181C22#1686119211yescharacteristic
182C4—22no—
183C4—10161no—
184C4—5081no—
185C4—2542no—
186C4C2×Dic127⋊C411yescharacteristic
187C4C4×C22⋊D12711yescharacteristic
188C4#1686117811yescharacteristic
189C4#1686118811yescharacteristic
190C2—21no—
191C2—21no—
192C2—21no—
193C2—42no—
194C2#2254974911yescharacteristic
subgroupG/H
centerC4#16861178
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#568×C127C2
socleC2×C127#385
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenter#568D127
hypocenterC127#1861
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
1. #22549924C2semidirect
2. #22549899C2semidirect
5. #22549647C2semidirect
6. #22549637C2semidirect
15. #16861136C4semidirect
16. #16861136C4semidirect
98. C127#1861semidirect

Maximal subgroups

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

subgroupindex
#225499242
#225498992
#225498932
C2×#168611362
#225496472
#225496372
#568×C1272
#1861127