Finite Group Databasebeta

← #30639794

Subgroups — #30639794

Order 16256.

394 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 71 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#22550383C212yesmaximal, complemented
2#22549893C212yesmaximal, complemented
3#22549839C211yesmaximal, characteristic, complemented
4#22549633C211yesmaximal, characteristic, complemented
5#575×C127C211yesmaximal, characteristic, complemented
6C22×C4×D127C2211yescharacteristic
7C2×#14523232C2211yescharacteristic
8#16861190—22no—
9#16861185—22no—
10#16861180—22no—
11#16861177—22no—
12C4×C22⋊D127—24no—
13#16861168C2212yes—
14#16861136—24no—
15(C2×D127)⋊C8—22no—
16C2×#41×C127C2212yes—
17#137×C127—22no—
18AGL(1,ℤ/8)×C127—22no—
19#135×C127—22no—
20C2×M16×C127C2211yescharacteristic
21C23×D127D411yescharacteristic
22C2×C22⋊D127—24no—
23C22×Dic127D411yescharacteristic
24C2×C4×D127—22no—
25C2×C4×D127—21no—
26C2×C4×D127—22no—
27C2×C4×D127—21no—
28#14523232—22no—
29C2×C127⋊C8—21no—
30C4⋊Dic127—22no—
31Dic127⋊C4—24no—
32C22⋊Dic127—24no—
33C22⋊Dic127—22no—
34(C2×D127)⋊C4—24no—
35C4×Dic127D414yescomplemented
36#41×C127—22no—
37#41×C127—22no—
38#41×C127—22no—
39#41×C127—22no—
40C2×Q8×C127D412yescomplemented
41C2×D4×C127—22no—
42C2×D4×C127D412yescomplemented
43C22×C4×C127—22no—
44C22×C4×C127C2311yescharacteristic
45Q16×C127—22no—
46SD16×C127—24no—
47D8×C127—22no—
48M16×C127—22no—
49C2×C8×C127D412yescomplemented
50C22×D127—22no—
51C22×D127—21no—
52C22×D127—42no—
53C22⋊D127—48no—
54C2×Dic127—24no—
55C2×Dic127—22no—
56C2×Dic127—21no—
57C4×D127—41no—
58C4×D127—42no—
59C4×D127—41no—
60C127⋊C8—41no—
61C23×C127—22no—
62C23×C127C2×D411yescharacteristic
63Q8×C127—22no—
64Q8×C127—22no—
65D4×C127—22no—
66D4×C127—22no—
67D4×C127—42no—
68D4×C127—42no—
69C2×C4×C127—22no—
70C2×C4×C127—42no—
71C2×C4×C127—42no—
72C2×C4×C127C2×D412yes—
73C2×C4×C127C2×D412yes—
74C2×C4×C127C2×D411yescharacteristic
75C2×C4×C127#4111yescharacteristic
76C8×C127—22no—
77C22×C127—41no—
78C22×C127—44no—
79C22×C127—42no—
80C22×C127#12012yes—
81C22×C127#12111yescharacteristic
82C2×D127—24no—
83C2×D127—22no—
84C2×D127—42no—
85C2×D127—41no—
86C2×D127—41no—
87C4×C127—42no—
88C4×C127#12011yescharacteristic
89C4×C127#12112yes—
90C4×C127#12711yescharacteristic
91Dic127—44no—
92Dic127—41no—
93C2×C127—22no—
94C2×C127—21no—
95C2×C127—82no—
96C2×C127#39211yescharacteristic
97D127—42no—
98#2010—1271nomaximal, Sylow
99C127#201011yescharacteristic, complemented, Sylow
100#575D12711yescharacteristic, complemented
101#525—1272no—
102#421—1272no—
103#407—1271no—
104H(3,ℤ/4)—1271no—
105C2×#41C2×D12712yes—
106C23×C4—1271no—
107#137—22no—
108AGL(1,ℤ/8)—22no—
109#135—22no—
110C2×M16—1271no—
111C2×M16C2×D12711yescharacteristic
112#123—2542no—
113C22⋊Q8—2542no—
114#121—2542no—
115#120—2542no—
116C4×D4—2544no—
117#117—1272no—
118#104—2544no—
119C22⋊C8—2542no—
120C24—1271no—
121#41—22no—
122#41—22no—
123#41—22no—
124#41—22no—
125C2×Q8C22⋊D12712yescomplemented
126C2×D4—22no—
127C2×D4—2544no—
128C2×D4C22⋊D12712yescomplemented
129C22×C4—22no—
130C22×C4—1271no—
131C22×C4—2542no—
132C22×C4—2541no—
133C22×C4—2542no—
134C22×C4—2541no—
135C22×C4C22×D12711yescharacteristic
136Q16—22no—
137SD16—24no—
138D8—22no—
139M16—22no—
140M16—2542no—
141C2×C8—2541no—
142C2×C8C22⋊D12712yescomplemented
143C4⋊C4—2544no—
144C4⋊C4—2542no—
145C22⋊C4—2544no—
146C22⋊C4—2544no—
147C22⋊C4—2542no—
148C42—1274no—
149C23—22no—
150C23—5082no—
151C23—2542no—
152C23—2541no—
153C23D4×D12711yescharacteristic
154Q8—22no—
155Q8—22no—
156D4—22no—
157D4—22no—
158D4—42no—
159D4—42no—
160D4—5088no—
161C2×C4—22no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—5081no—
165C2×C4—5081no—
166C2×C4—5082no—
167C2×C4—2544no—
168C2×C4—2542no—
169C2×C4—2541no—
170C2×C4D4×D12712yes—
171C2×C4C2×C22⋊D12711yescharacteristic
172C2×C4C2×C22⋊D12712yes—
173C2×C4#1452324711yescharacteristic
174C8—22no—
175C8—5081no—
176C22—41no—
177C22—44no—
178C22—42no—
179C22—5082no—
180C22—5081no—
181C22—5081no—
182C22—2544no—
183C22—2542no—
184C22#1686117712yes—
185C22#1686118011yescharacteristic
186C4—42no—
187C4—5084no—
188C4—5081no—
189C4#1686117811yescharacteristic
190C4#1686118112yes—
191C4#1686120511yescharacteristic
192C2—22no—
193C2—21no—
194C2—82no—
195C2—5082no—
196C2#2254978411yescharacteristic
subgroupG/H
centerC4#16861205
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#575×C127C2
socleC2×C127#392
hypercenter#575D127
hypocenterC127#2010
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #22550383C2semidirect
2. #22549893C2semidirect
3. #22549839C2semidirect
4. #22549633C2semidirect
5. #575×C127C2semidirect
35. C4×Dic127D4semidirect
40. C2×Q8×C127D4semidirect
42. C2×D4×C127D4semidirect
49. C2×C8×C127D4semidirect
99. C127#2010semidirect
100. #575D127semidirect
125. C2×Q8C22⋊D127semidirect
128. C2×D4C22⋊D127semidirect
142. C2×C8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225503832
#225498932
#225498392
#225496332
#575×C1272
#2010127