Finite Group Databasebeta

← #30664997

Subgroups — #30664997

Order 16768.

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

normal subgroups only

This group has 246 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#22553893C211yesmaximal, characteristic
2#22553630C212yesmaximal, complemented
3#22553045C212yesmaximal, complemented
4#22553038C211yesmaximal, characteristic, complemented
5#400×C131C211yesmaximal, characteristic, complemented
6C2×#14523513—21no—
7#16862226—22no—
8#16862221—22no—
9#16862220—21no—
10#16862220C2212yes—
11#16862177—22no—
12#16862166—22no—
13C22⋊(C131⋊C8)C2212yes—
14C2×C4⋊C4×C131C2211yescharacteristic
15C2×C4⋊C4×C131C2211yescharacteristic
16#95×C131—21no—
17#95×C131—21no—
18#95×C131C2211yescharacteristic
19#14523513—42no—
20#14523513—42no—
21C2×Dic262—21no—
22C2×D524—21no—
23C2×D524—21no—
24C2×D524D412yes—
25C2×C22⋊D131—22no—
26C2×C4×D131—22no—
27C2×C131⋊C8—22no—
28C4⋊Dic131—21no—
29C4⋊Dic131D412yes—
30(C2×D131)⋊C4—22no—
31(C2×D131)⋊C4—22no—
32C22×C4×C131—21no—
33C22×C4×C131—21no—
34C22×C4×C131D411yescharacteristic
35C22×C4×C131D411yescharacteristic
36C22×C4×C131C2311yescharacteristic
37C4⋊C4×C131—22no—
38C4⋊C4×C131—22no—
39C22×D131—22no—
40Dic262—42no—
41D524—42no—
42D524—42no—
43D524—81no—
44C22⋊D131—42no—
45C22⋊D131—42no—
46C2×Dic131—22no—
47C4×D131—44no—
48C131⋊C8—42no—
49C23×C131C2×D411yescharacteristic
50C2×C4×C131—21no—
51C2×C4×C131—21no—
52C2×C4×C131—21no—
53C2×C4×C131—41no—
54C2×C4×C131—41no—
55C2×C4×C131—41no—
56C2×C4×C131—41no—
57C2×C4×C131—42no—
58C2×C4×C131—41no—
59C2×C4×C131—41no—
60C2×C4×C131SD1612yes—
61C2×C4×C131C2×D412yes—
62C22×C131—21no—
63C22×C131—21no—
64C22×C131#12011yescharacteristic
65C22×C131C2×SD1611yescharacteristic
66C22×C131AGL(1,ℤ/8)11yescharacteristic
67C2×D131—42no—
68C2×D131—42no—
69C2×D131—42no—
70C4×C131—22no—
71C4×C131—22no—
72C4×C131—41no—
73C4×C131—81no—
74C4×C131—81no—
75Dic131—42no—
76C2×C131—21no—
77C2×C131—21no—
78C2×C131#45011yescharacteristic
79C2×C131#45411yescharacteristic
80C2×C131#45811yescharacteristic
81D131—82no—
82C131#161611yescharacteristic, complemented, Sylow
83#1616—1311nomaximal, Sylow
84#537—1311no—
85#481—1312no—
86#400D13111yescharacteristic, complemented
87#327—1312no—
88#324—1311no—
89C2×#41—2621no—
90C22⋊Q8—2622no—
91#121—2622no—
92#121—2621no—
93#121—1312no—
94C2×C4⋊C4C2×D13111yescharacteristic
95C2×C4⋊C4C2×D13111yescharacteristic
96#106—2622no—
97#102—2622no—
98C22⋊C8—1312no—
99#95—21no—
100#95—21no—
101#95C2×D13111yescharacteristic
102#41—5242no—
103#41—5242no—
104C2×Q8—2621no—
105C2×D4—2621no—
106C2×D4—2621no—
107C2×D4—2622no—
108C2×D4—1312no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—2622no—
112C22×C4C22⋊D13111yescharacteristic
113C22×C4C22⋊D13111yescharacteristic
114C22×C4C22×D13111yescharacteristic
115C2×C8—2622no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—2621no—
119C4⋊C4—1312no—
120C22⋊C4—2622no—
121C22⋊C4—2622no—
122C23—2622no—
123C23C2×C22⋊D13111yescharacteristic
124Q8—5242no—
125D4—5242no—
126D4—5242no—
127D4—5242no—
128D4—5242no—
129D4—10481no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—2622no—
141C2×C4—5244no—
142C2×C4#1452350012yes—
143C2×C4D4×D13112yes—
144C8—5242no—
145C22—21no—
146C22—21no—
147C22—5242no—
148C22—5242no—
149C22—5242no—
150C22#1686221411yescharacteristic
151C22C2×#1452350011yescharacteristic
152C22#1686227111yescharacteristic
153C4—22no—
154C4—22no—
155C4—41no—
156C4—81no—
157C4—81no—
158C4—5242no—
159C2—21no—
160C2—21no—
161C2—10482no—
162C2#2255346211yescharacteristic
163C2#2255348411yescharacteristic
164C2#2255350111yescharacteristic
subgroupG/H
centerC22#16862214
commutator, cosocleC22×C4×C131C23
FrattiniC22×C4C22×D131
Fitting#400×C131C2
socleC22×C131#120
prime radical#22553893C2
hypercenter#400D131
hypocenterC131#1616
LCSG ▷ C22×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #400

Complements

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

NH ≅ G/Ntype
2. #22553630C2semidirect
3. #22553045C2semidirect
4. #22553038C2semidirect
5. #400×C131C2semidirect
82. C131#1616semidirect
86. #400D131semidirect

Maximal subgroups

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

subgroupindex
#225538932
#225536302
#225530452
#225530382
#400×C1312
#1616131