Finite Group Databasebeta

← #30639772

Subgroups — #30639772

Order 16256.

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

normal subgroups only

This group has 406 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
1C2×#16861234C212yesmaximal, complemented
2C2×#16861136C212yesmaximal, complemented
3#22549777C211yesmaximal, characteristic, complemented
4#22549645C211yesmaximal, characteristic, complemented
5#537×C127C211yesmaximal, characteristic, complemented
6C22×D508C2211yescharacteristic
7#16861234—24no—
8#16861234—24no—
9C2×#14523235—22no—
10C2×#14523233—22no—
11C2×#14523232C2212yes—
12C2×(C2×D127)⋊C4—22no—
13C2×C4×Dic127C2211yescharacteristic
14#16861136—24no—
15C2×#41×C127C2212yes—
16C22⋊Q8×C127—22no—
17#121×C127—22no—
18#121×C127—22no—
19C2×C4⋊C4×C127C2211yescharacteristic
20C23×D127—21no—
21C2×D508—22no—
22C2×D508—22no—
23C2×D508—21no—
24C2×D508D412yescomplemented
25C22×Dic127—21no—
26#14523235—28no—
27#14523233—28no—
28#14523232—22no—
29#14523232D414yescomplemented
30C2×C127⋊C8—22no—
31(C2×D127)⋊C4—44no—
32C4×Dic127—22no—
33#41×C127—22no—
34#41×C127—22no—
35#41×C127—22no—
36#41×C127—22no—
37C2×Q8×C127D412yes—
38C2×D4×C127—22no—
39C2×D4×C127—22no—
40C2×D4×C127D412yes—
41C22×C4×C127—22no—
42C22×C4×C127D412yes—
43C22×C4×C127C2311yescharacteristic
44C4⋊C4×C127—22no—
45C4⋊C4×C127—22no—
46C22⋊C4×C127—24no—
47C22×D127—22no—
48C22×D127—42no—
49C22×D127—42no—
50C22×D127—41no—
51C22×D127—41no—
52D508—24no—
53D508—42no—
54C2×Dic127—22no—
55C2×Dic127—41no—
56C2×Dic127—41no—
57C127⋊C8—24no—
58C23×C127—22no—
59C23×C127C2×D411yescharacteristic
60Q8×C127—22no—
61Q8×C127—22no—
62D4×C127—22no—
63D4×C127—22no—
64D4×C127—42no—
65D4×C127—42no—
66D4×C127—82no—
67C2×C4×C127—22no—
68C2×C4×C127—22no—
69C2×C4×C127—42no—
70C2×C4×C127—42no—
71C2×C4×C127—42no—
72C2×C4×C127C2×D411yescharacteristic
73C2×C4×C127C2×D412yes—
74C2×C4×C127C2×D412yes—
75C2×C4×C127#4111yescharacteristic
76C22×C127—21no—
77C22×C127—21no—
78C22×C127—42no—
79C22×C127—42no—
80C22×C127—42no—
81C22×C127#12011yescharacteristic
82C22×C127#12111yescharacteristic
83C22×C127#12711yescharacteristic
84C2×D127—44no—
85C2×D127—42no—
86C2×D127—82no—
87C2×D127—81no—
88C2×D127—81no—
89C4×C127—42no—
90C4×C127—42no—
91C4×C127#12012yes—
92C4×C127#12112yes—
93Dic127—42no—
94C2×C127—21no—
95C2×C127—21no—
96C2×C127—82no—
97C2×C127#39211yescharacteristic
98C2×C127#45311yescharacteristic
99C2×C127#45411yescharacteristic
100D127—82no—
101#2006—1271nomaximal, Sylow
102C127#200611yescharacteristic, complemented, Sylow
103C2×AGL(1,ℤ/8)—1272no—
104#537D12711yescharacteristic, complemented
105C2×#104—1272no—
106#390—1271no—
107#343—1271no—
108C2×#41C2×D12712yes—
109C22×D4—1271no—
110AGL(1,ℤ/8)—2544no—
111AGL(1,ℤ/8)—2544no—
112C2×SD16—2542no—
113C2×D8—2542no—
114C2×M16—1272no—
115C22⋊Q8—22no—
116#121—22no—
117#121—22no—
118C2×C4⋊C4C2×D12711yescharacteristic
119C2×C22⋊C4—2542no—
120C2×C42—1271no—
121#104—2544no—
122C24—2541no—
123#41—22no—
124#41—22no—
125#41—22no—
126#41—22no—
127C2×Q8C22⋊D12712yes—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—1272no—
131C2×D4—2542no—
132C2×D4—2542no—
133C2×D4—2541no—
134C2×D4C22⋊D12712yes—
135C22×C4—22no—
136C22×C4—2541no—
137C22×C4C22⋊D12712yes—
138C22×C4C22×D12711yescharacteristic
139SD16—2548no—
140D8—2548no—
141M16—1274no—
142M16—2542no—
143C2×C8—2542no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C22⋊C4—24no—
147C22⋊C4—5084no—
148C42—2542no—
149C23—22no—
150C23—5082no—
151C23—5082no—
152C23—5081no—
153C23—5081no—
154C23—2542no—
155C23C2×C22⋊D12711yescharacteristic
156Q8—22no—
157Q8—22no—
158D4—22no—
159D4—22no—
160D4—42no—
161D4—42no—
162D4—82no—
163D4—5082no—
164D4—2544no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—5081no—
171C2×C4—5081no—
172C2×C4—2542no—
173C2×C4D4×D12711yescharacteristic
174C2×C4D4×D12712yes—
175C2×C4C2×C22⋊D12712yes—
176C2×C4#1452324711yescharacteristic
177C8—2544no—
178C22—21no—
179C22—21no—
180C22—42no—
181C22—42no—
182C22—42no—
183C22—10162no—
184C22—10161no—
185C22—10161no—
186C22—5084no—
187C22—5082no—
188C22#1686117811yescharacteristic
189C22#1686118011yescharacteristic
190C22#1686120511yescharacteristic
191C4—42no—
192C4—42no—
193C4—5082no—
194C4#1686117712yes—
195C4#1686118112yes—
196C2—21no—
197C2—21no—
198C2—82no—
199C2—10162no—
200C2#2254978411yescharacteristic
201C2#2255004411yescharacteristic
202C2#2255004911yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#537×C127C2
socleC22×C127#120
hypercenter#537D127
hypocenterC127#2006
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#16861234C2semidirect
2. C2×#16861136C2semidirect
3. #22549777C2semidirect
4. #22549645C2semidirect
5. #537×C127C2semidirect
24. C2×D508D4semidirect
29. #14523232D4semidirect
102. C127#2006semidirect
104. #537D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
C2×#168611362
#225497772
#225496452
#537×C1272
#2006127