Finite Group Databasebeta

← #30639776

Subgroups — #30639776

Order 16256.

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

normal subgroups only

This group has 370 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
1C2×#16861240C212yesmaximal, complemented
2#22550457C211yesmaximal, characteristic
3C2×#16861137C212yesmaximal, complemented
4#22549645C211yesmaximal, characteristic, complemented
5#391×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C2212yes—
7#16861240—24no—
8#16861240—24no—
9C2×#14523238—22no—
10C2×#14523235—22no—
11C2×#14523232C2212yes—
12#16861189—22no—
13#16861184—22no—
14#16861183—22no—
15#16861137—24no—
16C22×Q8×C127C2211yescharacteristic
17C2×C4⋊C4×C127C2211yescharacteristic
18C2×C42×C127C2211yescharacteristic
19#95×C127—22no—
20#14523248—22no—
21#14523248—22no—
22#14523248—22no—
23#14523248—22no—
24C2×Dic254D412yes—
25C2×D508—22no—
26C2×D508D412yes—
27C2×C22⋊D127—22no—
28C2×C4×D127—22no—
29#14523238—28no—
30#14523235—28no—
31#14523232—22no—
32#14523232D414yescomplemented
33C2×C127⋊C8—22no—
34C4⋊Dic127—22no—
35(C2×D127)⋊C4—24no—
36C2×Q8×C127—21no—
37C2×Q8×C127—22no—
38C2×Q8×C127—22no—
39C2×Q8×C127D412yescomplemented
40C22×C4×C127—21no—
41C22×C4×C127—21no—
42C22×C4×C127D412yes—
43C22×C4×C127C2311yescharacteristic
44C4⋊C4×C127—22no—
45C42×C127—22no—
46C22×D127—22no—
47Dic254—22no—
48Dic254—22no—
49D508—22no—
50D508—22no—
51D508—82no—
52C22⋊D127—42no—
53C22⋊D127—42no—
54C2×Dic127—22no—
55C4×D127—42no—
56C4×D127—42no—
57C127⋊C8—24no—
58C23×C127C2×D411yescharacteristic
59Q8×C127—24no—
60Q8×C127—42no—
61C2×C4×C127—22no—
62C2×C4×C127—22no—
63C2×C4×C127—22no—
64C2×C4×C127—41no—
65C2×C4×C127—41no—
66C2×C4×C127—41no—
67C2×C4×C127—41no—
68C2×C4×C127—42no—
69C2×C4×C127C2×D412yes—
70C2×C4×C127C2×D412yes—
71C2×C4×C127C2×D411yescharacteristic
72C2×C4×C127#4111yescharacteristic
73C22×C127—21no—
74C22×C127—21no—
75C22×C127#12011yescharacteristic
76C22×C127#12111yescharacteristic
77C22×C127#12711yescharacteristic
78C2×D127—42no—
79C2×D127—42no—
80C2×D127—42no—
81C4×C127—42no—
82C4×C127—42no—
83C4×C127—42no—
84C4×C127#12012yes—
85C4×C127#12112yes—
86Dic127—42no—
87C2×C127—21no—
88C2×C127—21no—
89C2×C127#39211yescharacteristic
90C2×C127#45511yescharacteristic
91C2×C127#45611yescharacteristic
92D127—82no—
93#2007—1271nomaximal, Sylow
94C127#200711yescharacteristic, complemented, Sylow
95C2×#137—1272no—
96#537—1271no—
97C2×#104—1272no—
98#391D12711yescharacteristic, complemented
99#343—1271no—
100C2×#41—1272no—
101C22×Q8C2×D12711yescharacteristic
102#137—2544no—
103#137—2544no—
104C2×Q16—2542no—
105C2×SD16—2542no—
106C2×M16—1272no—
107C22⋊Q8—2542no—
108#121—2542no—
109#121—2542no—
110C2×C4⋊C4C2×D12711yescharacteristic
111C2×C42C2×D12711yescharacteristic
112#104—2544no—
113#95—22no—
114#41—2542no—
115#41—2542no—
116#41—2542no—
117#41—2542no—
118C2×Q8—21no—
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—1272no—
122C2×Q8D50812yescomplemented
123C2×D4—1272no—
124C2×D4—2542no—
125C2×D4—2542no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—2542no—
129C22×C4C22⋊D12712yes—
130C22×C4C22×D12711yescharacteristic
131Q16—2548no—
132SD16—2548no—
133M16—1274no—
134M16—2542no—
135C2×C8—2542no—
136C4⋊C4—22no—
137C4⋊C4—2542no—
138C22⋊C4—2544no—
139C42—22no—
140C23—2542no—
141C23C2×C22⋊D12711yescharacteristic
142Q8—24no—
143Q8—42no—
144Q8—2542no—
145Q8—2542no—
146D4—10162no—
147D4—5082no—
148D4—5082no—
149D4—2542no—
150D4—2542no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—5082no—
160C2×C4—5082no—
161C2×C4—2542no—
162C2×C4D4×D12712yes—
163C2×C4D4×D12712yes—
164C2×C4C2×D50811yescharacteristic
165C2×C4#1452324811yescharacteristic
166C8—2544no—
167C22—21no—
168C22—21no—
169C22—5082no—
170C22—5082no—
171C22—5082no—
172C22#1686117711yescharacteristic
173C22#1686118311yescharacteristic
174C22#1686120511yescharacteristic
175C4—42no—
176C4—42no—
177C4—42no—
178C4—5082no—
179C4#1686117912yes—
180C4#1686118212yes—
181C2—21no—
182C2—21no—
183C2—10162no—
184C2#2254978511yescharacteristic
185C2#2255005311yescharacteristic
186C2#2255005811yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#391×C127C2
socleC22×C127#120
prime radical#22550457C2
hypercenter#391D127
hypocenterC127#2007
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#16861240C2semidirect
3. C2×#16861137C2semidirect
4. #22549645C2semidirect
5. #391×C127C2semidirect
32. #14523232D4semidirect
39. C2×Q8×C127D4semidirect
94. C127#2007semidirect
98. #391D127semidirect
122. C2×Q8D508semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
#225504572
C2×#168611372
#225496452
#391×C1272
#2007127