Finite Group Databasebeta

← #30665214

Subgroups — #30665214

Order 16768.

200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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#22554036C211yesmaximal, characteristic
2#22553594C211yesmaximal, characteristic, complemented
3#22553514C211yesmaximal, characteristic
4#22553058C211yesmaximal, characteristic, complemented
5#22553044C211yesmaximal, characteristic, complemented
6#22553041C211yesmaximal, characteristic
7#500×C131C211yesmaximal, characteristic, complemented
8C2×#14523500—21no—
9#16862241—22no—
10#16862235—21no—
11#16862226—21no—
12Q8×Dic131—21no—
13C4×Dic262C2211yescharacteristic
14C4×D524C2211yescharacteristic
15#16862204—21no—
16#16862177—21no—
17C4⋊(C131⋊C8)C2211yescharacteristic
18C4⋊(C131⋊C8)C2211yescharacteristic
19#16862170—21no—
20#16862169—21no—
21C4⋊Q8×C131C2211yescharacteristic
22C4⋊Q8×C131C2211yescharacteristic
23#107×C131—22no—
24C4×C8×C131C2211yescharacteristic
25C2×Dic262D411yescharacteristic
26C2×D524D411yescharacteristic
27C2×C4×D131—21no—
28#14523500—42no—
29C2×C131⋊C8—21no—
30C2×C131⋊C8—21no—
31C4⋊Dic131—22no—
32C4⋊Dic131D411yescharacteristic
33C4⋊Dic131D411yescharacteristic
34Dic131⋊C4—21no—
35(C2×D131)⋊C4—22no—
36(C2×D131)⋊C4—22no—
37(C2×D131)⋊C4—21no—
38C4×Dic131—22no—
39C4×Dic131—21no—
40C2×Q8×C131—21no—
41C2×Q8×C131—21no—
42C2×C8×C131—21no—
43C4⋊C4×C131—21no—
44C4⋊C4×C131—21no—
45C4⋊C4×C131—21no—
46C4⋊C4×C131D412yes—
47C42×C131C2311yescharacteristic
48C22×D131—21no—
49Dic262—21no—
50Dic262—21no—
51D524—21no—
52D524—21no—
53C2×Dic131—21no—
54C2×Dic131—21no—
55C2×Dic131—21no—
56C4×D131—41no—
57C4×D131—41no—
58C131⋊C8—41no—
59C131⋊C8—41no—
60Q8×C131—41no—
61Q8×C131—41no—
62Q8×C131—41no—
63Q8×C131—41no—
64C2×C4×C131—22no—
65C2×C4×C131—41no—
66C2×C4×C131C2×D411yescharacteristic
67C2×C4×C131C2×D411yescharacteristic
68C2×C4×C131C2×D411yescharacteristic
69C8×C131—22no—
70C22×C131#12011yescharacteristic
71C2×D131—41no—
72C2×D131—41no—
73C2×D131—41no—
74C4×C131—21no—
75C4×C131—42no—
76C4×C131—81no—
77C4×C131#13511yescharacteristic
78C4×C131#13511yescharacteristic
79C4×C131AGL(1,ℤ/8)11yescharacteristic
80C4×C131#13711yescharacteristic
81Dic131—41no—
82Dic131—41no—
83Dic131—41no—
84C2×C131#44911yescharacteristic
85C2×C131#45211yescharacteristic
86C2×C131#45611yescharacteristic
87D131—81no—
88C131#165011yescharacteristic, complemented, Sylow
89#1650—1311nomaximal, Sylow
90#556—1311no—
91#500D13111yescharacteristic, complemented
92#475—1311no—
93#461—1311no—
94#332—1311no—
95Q8⋊C8—1311no—
96D4⋊C8—1311no—
97C2×SD16—2621no—
98C4⋊Q8C2×D13111yescharacteristic
99C4⋊Q8C2×D13111yescharacteristic
100#126—2622no—
101#124—2621no—
102C22⋊Q8—2621no—
103C4×Q8—2621no—
104C4×Q8—1311no—
105C4×D4—1311no—
106#117—2621no—
107#107—22no—
108#106—2621no—
109C4⋊C8—1311no—
110C4⋊C8—1311no—
111#103—2621no—
112#103—2621no—
113C4×C8C2×D13111yescharacteristic
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—1311no—
117C2×D4—1311no—
118C22×C4—2621no—
119SD16—5242no—
120C2×C8—21no—
121C2×C8—2621no—
122C2×C8—2621no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—2622no—
127C4⋊C4—2621no—
128C4⋊C4—1311no—
129C4⋊C4—1311no—
130C4⋊C4C22⋊D13112yes—
131C22⋊C4—2622no—
132C22⋊C4—2622no—
133C22⋊C4—2621no—
134C42—2622no—
135C42—2621no—
136C42C22×D13111yescharacteristic
137C23—2621no—
138Q8—41no—
139Q8—41no—
140Q8—41no—
141Q8—41no—
142Q8—2621no—
143Q8—2621no—
144D4—2621no—
145D4—2621no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—2621no—
149C2×C4—2621no—
150C2×C4—2621no—
151C2×C4—5241no—
152C2×C4—5241no—
153C2×C4D4×D13111yescharacteristic
154C2×C4D4×D13111yescharacteristic
155C2×C4C2×C22⋊D13111yescharacteristic
156C8—22no—
157C8—5241no—
158C8—5241no—
159C22—5241no—
160C22—5241no—
161C22—5241no—
162C22#1686221411yescharacteristic
163C4—21no—
164C4—42no—
165C4—81no—
166C4—5241no—
167C4—5241no—
168C4—5241no—
169C4#1686226511yescharacteristic
170C4#1686226911yescharacteristic
171C4#1686227011yescharacteristic
172C4#1686227811yescharacteristic
173C2—10481no—
174C2#2255345411yescharacteristic
175C2#2255347311yescharacteristic
176C2#2255349411yescharacteristic
subgroupG/H
centerC22#16862214
commutator, cosocleC42×C131C23
FrattiniC42C22×D131
Fitting#500×C131C2
socleC22×C131#120
prime radicalC2×D524D4
hypercenter#500D131
hypocenterC131#1650
LCSG ▷ C42×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
2. #22553594C2semidirect
4. #22553058C2semidirect
5. #22553044C2semidirect
7. #500×C131C2semidirect
88. C131#1650semidirect
91. #500D131semidirect

Maximal subgroups

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

subgroupindex
#225540362
#225535942
#225535142
#225530582
#225530442
#225530412
#500×C1312
#1650131