Finite Group Databasebeta

← #30638207

Subgroups — #30638207

Order 16256.

198 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 53 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#22550252C211yesmaximal, characteristic, complemented
2#22550071C211yesmaximal, characteristic, complemented
3#22549954C211yesmaximal, characteristic, complemented
4#22549626C211yesmaximal, characteristic, complemented
5#22549620C211yesmaximal, characteristic, complemented
6#22549619C211yesmaximal, characteristic, complemented
7#459×C127C211yesmaximal, characteristic, complemented
8C2×#14523233—21no—
9C2×#14523233—21no—
10C2×#14523233—21no—
11C2×#14523232—21no—
12#16861206C2211yescharacteristic
13C4⋊(C127⋊C8)C2211yescharacteristic
14C4⋊(C127⋊C8)C2211yescharacteristic
15#16861129—21no—
16C22⋊(C127⋊C8)—21no—
17C4×C127⋊C8C2211yescharacteristic
18C2×D8×C127—21no—
19#127×C127C2211yescharacteristic
20C4×D4×C127C2211yescharacteristic
21C4⋊C8×C127C2211yescharacteristic
22#102×C127—21no—
23C2×D508—21no—
24C2×D508—21no—
25C2×D508—21no—
26#14523233—42no—
27#14523233—42no—
28#14523233—42no—
29#14523232—42no—
30C2×C127⋊C8—21no—
31C2×C127⋊C8—21no—
32C2×C127⋊C8D411yescharacteristic
33C2×C127⋊C8D411yescharacteristic
34C2×D4×C127—21no—
35C2×D4×C127—21no—
36C2×D4×C127—21no—
37C2×D4×C127D411yescharacteristic
38C22×C4×C127—21no—
39D8×C127—42no—
40C2×C8×C127—21no—
41C4⋊C4×C127D411yescharacteristic
42C22⋊C4×C127—21no—
43C42×C127C2311yescharacteristic
44C22×D127—41no—
45D508—41no—
46D508—41no—
47D508—41no—
48D508—41no—
49D508—81no—
50C127⋊C8—21no—
51C127⋊C8—41no—
52C127⋊C8—41no—
53C127⋊C8D812yescomplemented
54C23×C127—21no—
55C23×C127—41no—
56D4×C127—21no—
57D4×C127—21no—
58D4×C127—41no—
59D4×C127—41no—
60D4×C127—41no—
61D4×C127—41no—
62D4×C127—81no—
63C2×C4×C127—21no—
64C2×C4×C127—41no—
65C2×C4×C127—41no—
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127C2×D411yescharacteristic
68C2×C4×C127#4111yescharacteristic
69C8×C127—41no—
70C22×C127—41no—
71C22×C127—41no—
72C22×C127—41no—
73C22×C127—81no—
74C22×C127—81no—
75C22×C127—81no—
76C22×C127#12111yescharacteristic
77C2×D127—81no—
78C2×D127—81no—
79C2×D127—81no—
80C4×C127—21no—
81C4×C127—41no—
82C4×C127C2×D811yescharacteristic
83C4×C127AGL(1,ℤ/8)11yescharacteristic
84C4×C127AGL(1,ℤ/8)11yescharacteristic
85C4×C127AGL(1,ℤ/8)11yescharacteristic
86C2×C127—81no—
87C2×C127—161no—
88C2×C127#45911yescharacteristic
89C2×C127#46911yescharacteristic
90C2×C127#47211yescharacteristic
91D127—161no—
92#1687—1271nomaximal, Sylow
93C127#168711yescharacteristic, complemented, Sylow
94#493—1271no—
95#459—1271no—
96#459D12711yescharacteristic, complemented
97#436—1271no—
98#334—1271no—
99#331—1271no—
100#331—1271no—
101C2×D8—21no—
102C2×D8—2541no—
103C2×D8—2541no—
104C2×D8—2541no—
105C2×M16—2541no—
106#127—1271no—
107#127C2×D12711yescharacteristic
108C4×D4C2×D12711yescharacteristic
109C4⋊C8—1271no—
110C4⋊C8—1271no—
111C4⋊C8C2×D12711yescharacteristic
112#102—21no—
113#102—2541no—
114C22⋊C8—2541no—
115C4×C8—1271no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—2541no—
120C2×D4—2541no—
121C2×D4—2541no—
122C2×D4C22⋊D12711yescharacteristic
123C22×C4—21no—
124D8—42no—
125D8—5082no—
126D8—5082no—
127D8—5082no—
128M16—5082no—
129C2×C8—21no—
130C2×C8—1271no—
131C2×C8—1271no—
132C2×C8—2541no—
133C2×C8—2541no—
134C4⋊C4C22⋊D12711yescharacteristic
135C22⋊C4—21no—
136C42C22×D12711yescharacteristic
137C23—21no—
138C23—41no—
139C23—5081no—
140D4—21no—
141D4—21no—
142D4—41no—
143D4—41no—
144D4—41no—
145D4—41no—
146D4—81no—
147D4—10161no—
148D4—5081no—
149D4—5081no—
150D4—5081no—
151D4—5081no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4D4×D12711yescharacteristic
156C2×C4C2×C22⋊D12711yescharacteristic
157C2×C4#1452324711yescharacteristic
158C8—41no—
159C8—1272no—
160C8—5081no—
161C8—5081no—
162C8—2541no—
163C22—41no—
164C22—41no—
165C22—41no—
166C22—81no—
167C22—81no—
168C22—81no—
169C22—10161no—
170C22—10161no—
171C22—10161no—
172C22#1686118111yescharacteristic
173C4—21no—
174C4—41no—
175C4D8×D12711yescharacteristic
176C4#1686123311yescharacteristic
177C4#1686123411yescharacteristic
178C4#1686123511yescharacteristic
179C2—81no—
180C2—161no—
181C2—20321no—
182C2#2255006911yescharacteristic
183C2#2255012711yescharacteristic
184C2#2255014411yescharacteristic
subgroupG/H
centerC22#16861181
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#459×C127C2
socleC22×C127#121
hypercenter#459D127
hypocenterC127#1687
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #22550252C2semidirect
2. #22550071C2semidirect
3. #22549954C2semidirect
4. #22549626C2semidirect
5. #22549620C2semidirect
6. #22549619C2semidirect
7. #459×C127C2semidirect
53. C127⋊C8D8semidirect
93. C127#1687semidirect
96. #459D127semidirect

Maximal subgroups

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

subgroupindex
#225502522
#225500712
#225499542
#225496262
#225496202
#225496192
#459×C1272
#1687127