Finite Group Databasebeta

← #30639833

Subgroups — #30639833

Order 16256.

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

normal subgroups only

This group has 350 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#22550674C212yesmaximal, complemented
2#22549844C211yesmaximal, characteristic, complemented
3#22549843C212yesmaximal, complemented
4C127⋊#338C211yesmaximal, characteristic, complemented
5#575×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C2211yescharacteristic
7#16861240—22no—
8#16861239—22no—
9#16861235—22no—
10#16861234—22no—
11#16861229—24no—
12C2×#14523232C2212yes—
13C22×C127⋊C8C2211yescharacteristic
14C22⋊(C127⋊C8)—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
21#14523248—22no—
22#14523248—22no—
23C2×Dic254D411yescharacteristic
24C2×D508D411yescharacteristic
25C2×C22⋊D127—21no—
26C2×C4×D127—21no—
27#14523238—24no—
28#14523236—24no—
29#14523235—24no—
30#14523233—24no—
31#14523232—22no—
32#14523232—22no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8—22no—
35C2×C127⋊C8D414yes—
36#41×C127—22no—
37#41×C127—22no—
38#41×C127—22no—
39#41×C127—22no—
40C2×Q8×C127D412yes—
41C2×D4×C127—22no—
42C2×D4×C127D412yes—
43C22×C4×C127—22no—
44C22×C4×C127C2311yescharacteristic
45Q16×C127—22no—
46SD16×C127—24no—
47D8×C127—22no—
48M16×C127—22no—
49C2×C8×C127D412yes—
50C22×D127—21no—
51Dic254—22no—
52D508—22no—
53C22⋊D127—42no—
54C2×Dic127—21no—
55C4×D127—41no—
56C4×D127—41no—
57C127⋊C8—24no—
58C127⋊C8—41no—
59C23×C127—22no—
60C23×C127#4111yescharacteristic
61Q8×C127—22no—
62Q8×C127—22no—
63D4×C127—22no—
64D4×C127—22no—
65D4×C127—42no—
66D4×C127—42no—
67C2×C4×C127—22no—
68C2×C4×C127—42no—
69C2×C4×C127—42no—
70C2×C4×C127C2×D411yescharacteristic
71C2×C4×C127C2×D412yes—
72C2×C4×C127C2×D412yes—
73C2×C4×C127C2×D411yescharacteristic
74C8×C127—22no—
75C22×C127—41no—
76C22×C127—44no—
77C22×C127—42no—
78C22×C127#12111yescharacteristic
79C22×C127#12112yes—
80C2×D127—42no—
81C2×D127—41no—
82C4×C127—42no—
83C4×C127#12012yes—
84C4×C127#12011yescharacteristic
85C4×C127#12711yescharacteristic
86Dic127—41no—
87C2×C127—21no—
88C2×C127—22no—
89C2×C127—82no—
90C2×C127#39211yescharacteristic
91D127—81no—
92#2017—1271nomaximal, Sylow
93C127#201711yescharacteristic, complemented, Sylow
94#575—1272no—
95#575D12711yescharacteristic, complemented
96#408—1272no—
97#408—1271no—
98#338—1271no—
99C2×#41—1271no—
100C2×#41C2×D12712yes—
101#137—22no—
102#137—2542no—
103#137—2542no—
104AGL(1,ℤ/8)—22no—
105AGL(1,ℤ/8)—2542no—
106AGL(1,ℤ/8)—2542no—
107#135—22no—
108#135—2544no—
109C2×M16—1272no—
110C2×M16C2×D12711yescharacteristic
111C22×C8—1271no—
112C22⋊C8—2544no—
113C22⋊C8—2542no—
114#41—22no—
115#41—22no—
116#41—22no—
117#41—22no—
118#41—2542no—
119#41—2542no—
120C2×Q8—1271no—
121C2×Q8C22⋊D12712yes—
122C2×D4—22no—
123C2×D4—1271no—
124C2×D4—2541no—
125C2×D4C22⋊D12712yes—
126C22×C4—22no—
127C22×C4—2541no—
128C22×C4C22×D12711yescharacteristic
129Q16—22no—
130Q16—2544no—
131SD16—24no—
132SD16—2544no—
133SD16—2544no—
134D8—22no—
135D8—2544no—
136M16—22no—
137M16—2542no—
138M16—2542no—
139C2×C8—1274no—
140C2×C8—2542no—
141C2×C8—2541no—
142C2×C8C22⋊D12712yes—
143C23—22no—
144C23—2541no—
145C23#1452324711yescharacteristic
146Q8—22no—
147Q8—22no—
148Q8—2542no—
149D4—22no—
150D4—22no—
151D4—42no—
152D4—42no—
153D4—5082no—
154D4—2542no—
155C2×C4—22no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—5081no—
159C2×C4—5081no—
160C2×C4—2541no—
161C2×C4D4×D12712yes—
162C2×C4D4×D12711yescharacteristic
163C2×C4C2×C22⋊D12711yescharacteristic
164C2×C4C2×C22⋊D12712yes—
165C8—22no—
166C8—5081no—
167C8—2544no—
168C22—41no—
169C22—44no—
170C22—42no—
171C22—5082no—
172C22—5081no—
173C22#1686118011yescharacteristic
174C22#1686118112yes—
175C4—42no—
176C4—5081no—
177C4#1686117712yes—
178C4#1686117811yescharacteristic
179C4#1686120511yescharacteristic
180C2—22no—
181C2—21no—
182C2—82no—
183C2—10161no—
184C2#2254978411yescharacteristic
subgroupG/H
centerC4#16861205
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#575×C127C2
socleC2×C127#392
hypercenter#575D127
hypocenterC127#2017
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. #22550674C2semidirect
2. #22549844C2semidirect
3. #22549843C2semidirect
4. C127⋊#338C2semidirect
5. #575×C127C2semidirect
93. C127#2017semidirect
95. #575D127semidirect

Maximal subgroups

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

subgroupindex
#225506742
#225498442
#225498432
C127⋊#3382
#575×C1272
#2017127