Finite Group Databasebeta

← #30640674

Subgroups — #30640674

Order 16256.

226 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 47 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#22550673C211yesmaximal, characteristic, complemented
2#22550312C211yesmaximal, characteristic, complemented
3#22550126C211yesmaximal, characteristic, complemented
4#22549687C211yesmaximal, characteristic, complemented
5#22549686C211yesmaximal, characteristic, complemented
6#22549657C211yesmaximal, characteristic, complemented
7#509×C127C211yesmaximal, characteristic, complemented
8#41×D127—21no—
9#16861241—21no—
10#16861238—21no—
11#16861236—21no—
12#16861233—21no—
13#16861231C2211yescharacteristic
14#16861229—21no—
15C2×#14523231C2211yescharacteristic
16#16861180—21no—
17#16861153—21no—
18#16861150—21no—
19#16861149C2211yescharacteristic, complemented
20#16861141C2211yescharacteristic, complemented
21#16861130—21no—
22#135×C127C2211yescharacteristic
23C2×D8×C127C2211yescharacteristic, complemented
24SD32×C127—21no—
25D16×C127—21no—
26M32×C127C2211yescharacteristic, complemented
27#14523249—21no—
28#14523248—21no—
29#14523248—41no—
30#14523247—21no—
31#14523247—41no—
32Q8×D127—21no—
33C2×C22⋊D127—41no—
34D4×D127—21no—
35D4×D127—41no—
36C2×C4×D127—41no—
37C2×C4×D127D411yescharacteristic
38Dic508D411yescharacteristic, complemented
39#14523238—21no—
40#14523237—21no—
41#14523236—21no—
42#14523235—21no—
43D1016D411yescharacteristic, complemented
44#14523233—21no—
45#14523231—21no—
46#14523231—21no—
47C2×C127⋊C8D411yescharacteristic
48C4⋊Dic127—21no—
49C22⋊Dic127—41no—
50C127⋊C16—21no—
51#41×C127—21no—
52C2×D4×C127—21no—
53Q16×C127D411yescharacteristic, complemented
54SD16×C127—21no—
55D8×C127—21no—
56D8×C127—21no—
57D8×C127D411yescharacteristic, complemented
58C2×C8×C127C2311yescharacteristic
59C16×C127—21no—
60C22×D127—41no—
61C22×D127D811yescharacteristic
62Dic254—21no—
63Dic254—41no—
64D508—21no—
65D508—41no—
66C22⋊D127—41no—
67C22⋊D127—41no—
68C22⋊D127—41no—
69C22⋊D127—82no—
70C2×Dic127—41no—
71C2×Dic127—41no—
72C2×Dic127D811yescharacteristic
73C4×D127—21no—
74C4×D127—21no—
75C4×D127—41no—
76C4×D127—41no—
77C4×D127—41no—
78C127⋊C8—21no—
79C23×C127—41no—
80Q8×C127—21no—
81D4×C127—21no—
82D4×C127—41no—
83D4×C127—41no—
84D4×C127—41no—
85C2×C4×C127—41no—
86C2×C4×C127C2×D411yescharacteristic
87C8×C127C2×D411yescharacteristic
88C8×C127C2×D411yescharacteristic
89C22×C127—41no—
90C22×C127—81no—
91C22×C127—81no—
92C22×C127—81no—
93C22×C127C2×D811yescharacteristic
94C2×D127—21no—
95C2×D127—41no—
96C2×D127—41no—
97C2×D127—82no—
98C4×C127—41no—
99C4×C127#12011yescharacteristic
100C4×C127AGL(1,ℤ/8)11yescharacteristic
101Dic127—21no—
102Dic127—41no—
103Dic127—81no—
104C2×C127—21no—
105C2×C127—81no—
106C2×C127—161no—
107C2×C127#44711yescharacteristic
108D127—41no—
109D127—81no—
110#2190—1271nomaximal, Sylow
111C127#219011yescharacteristic, complemented, Sylow
112#575—1271no—
113#509—1271no—
114#509D12711yescharacteristic, complemented
115#469—1271no—
116#360—1271no—
117#360—1271no—
118#349—1271no—
119C2×#41—2541no—
120#137—2541no—
121#137—2541no—
122AGL(1,ℤ/8)—2541no—
123AGL(1,ℤ/8)—2541no—
124#135—1271no—
125#135—2541no—
126#135C2×D12711yescharacteristic
127C2×D8C2×D12711yescharacteristic, complemented
128C2×M16—1271no—
129#121—2541no—
130SD32—21no—
131SD32—2541no—
132D16—21no—
133D16—2541no—
134M32—1271no—
135M32C2×D12711yescharacteristic, complemented
136#106—1271no—
137#102—2541no—
138#41—21no—
139#41—5081no—
140#41—5081no—
141#41—2541no—
142#41—2541no—
143#41—2541no—
144C2×Q8—2541no—
145C2×D4—21no—
146C2×D4—5081no—
147C2×D4—5081no—
148C2×D4—2541no—
149C22×C4—1271no—
150C22×C4—5081no—
151Q16—1271no—
152Q16—2541no—
153Q16C22⋊D12711yescharacteristic, complemented
154SD16—21no—
155SD16—2541no—
156SD16—2541no—
157SD16—2541no—
158D8—21no—
159D8—21no—
160D8—1271no—
161D8—2541no—
162D8C22⋊D12711yescharacteristic, complemented
163M16—2541no—
164M16—2541no—
165C2×C8—1271no—
166C2×C8C22×D12711yescharacteristic
167C4⋊C4—2541no—
168C22⋊C4—5081no—
169C16—21no—
170C16—2541no—
171C23—41no—
172C23—1271no—
173C23—5081no—
174Q8—21no—
175Q8—5081no—
176Q8—2541no—
177D4—21no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—10162no—
182D4—5081no—
183D4—5081no—
184D4—5081no—
185D4—5081no—
186D4—2541no—
187C2×C4—41no—
188C2×C4—1271no—
189C2×C4—5081no—
190C2×C4—5081no—
191C2×C4—5081no—
192C2×C4—5081no—
193C2×C4—5081no—
194C2×C4—2541no—
195C2×C4—2541no—
196C2×C4D4×D12711yescharacteristic
197C8—2541no—
198C8D4×D12711yescharacteristic
199C8C2×C22⋊D12711yescharacteristic
200C22—41no—
201C22—81no—
202C22—81no—
203C22—81no—
204C22—10162no—
205C22—5081no—
206C22—5081no—
207C22—2541no—
208C22D8×D12711yescharacteristic
209C4—41no—
210C4—10161no—
211C4—5081no—
212C4—2541no—
213C4#1686117711yescharacteristic
214C4#1686123311yescharacteristic
215C2—21no—
216C2—81no—
217C2—161no—
218C2—10161no—
219C2—5081no—
220C2#2255000811yescharacteristic
subgroupG/H
centerC2#22550008
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#509×C127C2
socleC2×C127#447
hypercenter#509D127
hypocenterC127#2190
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #22550673C2semidirect
2. #22550312C2semidirect
3. #22550126C2semidirect
4. #22549687C2semidirect
5. #22549686C2semidirect
6. #22549657C2semidirect
7. #509×C127C2semidirect
19. #16861149C22semidirect
20. #16861141C22semidirect
23. C2×D8×C127C22semidirect
26. M32×C127C22semidirect
38. Dic508D4semidirect
43. D1016D4semidirect
53. Q16×C127D4semidirect
57. D8×C127D4semidirect
111. C127#2190semidirect
114. #509D127semidirect
127. C2×D8C2×D127semidirect
135. M32C2×D127semidirect
153. Q16C22⋊D127semidirect
162. D8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225506732
#225503122
#225501262
#225496872
#225496862
#225496572
#509×C1272
#2190127