Finite Group Databasebeta

← #30638832

Subgroups — #30638832

Order 16256.

262 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 73 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#22550344C211yesmaximal, characteristic
2#22549858C211yesmaximal, characteristic
3#22549849C211yesmaximal, characteristic
4#22549637C211yesmaximal, characteristic, complemented
5#22549635C211yesmaximal, characteristic, complemented
6#22549632C211yesmaximal, characteristic, complemented
7H(3,ℤ/4)×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10#16861174—21no—
11#16861171—21no—
12(C22⋊D127)⋊C4—22no—
13#16861168C2211yescharacteristic
14#16861166C412yescomplemented
15C2×C4×Dic127C2211yescharacteristic
16C42×D127C412yescomplemented
17#16861127—21no—
18#16861125—21no—
19#16861123—22no—
20C2×M16×C127C2211yescharacteristic
21#117×C127C2211yescharacteristic
22#117×C127C2211yescharacteristic
23#14523248—21no—
24#14523248—21no—
25#14523248—21no—
26#14523248—21no—
27C2×Dic254D411yescharacteristic
28C2×D508D411yescharacteristic
29C2×C22⋊D127D412yes—
30C22×Dic127—21no—
31C2×C4×D127—22no—
32C2×C4×D127C2×C412yes—
33#14523232—21no—
34#14523232—21no—
35C2×C127⋊C8—21no—
36C4⋊Dic127—21no—
37Dic127⋊C4—22no—
38C22⋊Dic127—21no—
39(C2×D127)⋊C4—22no—
40C4×Dic127—22no—
41C4×Dic127—21no—
42C4×Dic127—21no—
43C4×Dic127C2×C411yescharacteristic
44C4×Dic127C2×C411yescharacteristic
45C22×C4×C127C2311yescharacteristic
46M16×C127—21no—
47M16×C127—21no—
48C2×C8×C127—21no—
49C4⋊C4×C127—21no—
50C4⋊C4×C127—21no—
51C22⋊C4×C127—21no—
52C22⋊C4×C127—21no—
53C42×C127—21no—
54C42×C127C2×C412yes—
55C22×D127C22⋊C412yes—
56Dic254—21no—
57Dic254—21no—
58D508—21no—
59D508—21no—
60C22⋊D127—22no—
61C22⋊D127—22no—
62C2×Dic127—21no—
63C2×Dic127—21no—
64C2×Dic127—42no—
65C2×Dic127—41no—
66C2×Dic127C22⋊C412yes—
67C4×D127—22no—
68C4×D127—22no—
69C4×D127—44no—
70C127⋊C8—41no—
71C23×C127C2×D411yescharacteristic
72C2×C4×C127—22no—
73C2×C4×C127—41no—
74C2×C4×C127C22×C411yescharacteristic
75C2×C4×C127C2×D411yescharacteristic
76C2×C4×C127#4111yescharacteristic
77C2×C4×C127#4111yescharacteristic
78C2×C4×C127#4111yescharacteristic
79C2×C4×C127#4111yescharacteristic
80C8×C127—41no—
81C22×C127—41no—
82C22×C127C2×C22⋊C411yescharacteristic
83C22×C127#12311yescharacteristic
84C22×C127#12311yescharacteristic
85C2×D127—22no—
86C2×D127—42no—
87C4×C127—42no—
88C4×C127—81no—
89C4×C127#11711yescharacteristic
90C4×C127#11711yescharacteristic
91C4×C127#12311yescharacteristic
92C4×C127#12311yescharacteristic
93Dic127—22no—
94Dic127—22no—
95Dic127—41no—
96Dic127—81no—
97C2×C127—21no—
98C2×C127—21no—
99C2×C127—21no—
100C2×C127#38111yescharacteristic
101D127—42no—
102#1827—1271nomaximal, Sylow
103C127#182711yescharacteristic, complemented, Sylow
104C4×#41—1271no—
105#413—1271no—
106#410—1271no—
107#339—1271no—
108#339—1271no—
109H(3,ℤ/4)—1271no—
110H(3,ℤ/4)D12711yescharacteristic, complemented
111C2×#41—1271no—
112C2×M16—1271no—
113C2×M16C2×D12711yescharacteristic
114C4×Q8—2541no—
115C4×D4—2542no—
116C4×D4—2541no—
117#117—1272no—
118#117—1271no—
119#117C2×D12711yescharacteristic
120#117C2×D12711yescharacteristic
121C2×C42—1271no—
122C2×C42—1272no—
123#101—2541no—
124#100—2541no—
125#99—2542no—
126#41—2541no—
127#41—2541no—
128#41—2541no—
129#41—2541no—
130C2×Q8—1271no—
131C2×D4—1272no—
132C2×D4—1271no—
133C22×C4—1272no—
134C22×C4—2542no—
135C22×C4—2541no—
136C22×C4C22×D12711yescharacteristic
137M16—21no—
138M16—21no—
139M16—2541no—
140M16—2541no—
141C2×C8—21no—
142C2×C8—2541no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—2542no—
146C4⋊C4—2541no—
147C22⋊C4—21no—
148C22⋊C4—21no—
149C22⋊C4—2542no—
150C22⋊C4—2541no—
151C42—21no—
152C42—1271no—
153C42—1271no—
154C42—2542no—
155C42—2541no—
156C42—2541no—
157C42C4×D12712yes—
158C23—1272no—
159C23D4×D12711yescharacteristic
160Q8—2541no—
161Q8—2541no—
162D4—2542no—
163D4—2542no—
164D4—2541no—
165D4—2541no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—1272no—
169C2×C4—5084no—
170C2×C4—5081no—
171C2×C4—5082no—
172C2×C4—2541no—
173C2×C4—2541no—
174C2×C4—2542no—
175C2×C4—2542no—
176C2×C4C2×C4×D12711yescharacteristic
177C2×C4D4×D12711yescharacteristic
178C2×C4#1452324711yescharacteristic
179C2×C4#1452324811yescharacteristic
180C2×C4#1452324811yescharacteristic
181C2×C4#1452324911yescharacteristic
182C8—41no—
183C8—5081no—
184C22—41no—
185C22—5082no—
186C22—2542no—
187C22C22⋊C4×D12711yescharacteristic
188C22#1686119111yescharacteristic
189C22#1686119411yescharacteristic
190C4—42no—
191C4—81no—
192C4—10161no—
193C4—5081no—
194C4—2542no—
195C4—2542no—
196C4#1686116611yescharacteristic
197C4#1686116711yescharacteristic
198C4#1686119111yescharacteristic
199C4#1686119411yescharacteristic
200C2—21no—
201C2—21no—
202C2—21no—
203C2—5082no—
204C2#2254973811yescharacteristic
subgroupG/H
centerC4#16861166
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingH(3,ℤ/4)×C127C2
socleC2×C127#381
prime radicalC2×#14523248C22
cosocleC22×C4×C127C23
hypercenterH(3,ℤ/4)D127
hypocenterC127#1827
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ #117 ◁ H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
4. #22549637C2semidirect
5. #22549635C2semidirect
6. #22549632C2semidirect
7. H(3,ℤ/4)×C127C2semidirect
14. #16861166C4semidirect
16. C42×D127C4semidirect
103. C127#1827semidirect
110. H(3,ℤ/4)D127semidirect

Maximal subgroups

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

subgroupindex
#225503442
#225498582
#225498492
#225496372
#225496352
#225496322
H(3,ℤ/4)×C1272
#1827127