Finite Group Databasebeta

← #30676252

Subgroups — #30676252

Order 16768.

382 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 125 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#22553917C211yesmaximal, characteristic
2#22553636C212yesmaximal, complemented
3#22553619C212yesmaximal, complemented
4#22553512C212yesmaximal, complemented
5#22553451C212yesmaximal, complemented
6#22553410C212yesmaximal, complemented
7C4×#14523500C212yesmaximal, complemented
8#22553373C211yesmaximal, characteristic, complemented
9#519×C131C211yesmaximal, characteristic, complemented
10C2×#14523500C2212yes—
11C2×#14523500C2212yes—
12#16862243C2211yescharacteristic
13#16862238C2211yescharacteristic
14#16862231—22no—
15#16862220C2212yes—
16#16862219—22no—
17C4×D524C2212yes—
18#16862179C2212yes—
19#16862177C2212yes—
20C4⋊(C131⋊C8)C2212yes—
21#16862170C2212yes—
22#16862170C2212yes—
23#16862166C2212yes—
24#16862166C2212yes—
25C22⋊(C131⋊C8)C2212yes—
26#16862157C2211yescharacteristic
27C4×C131⋊C8C2211yescharacteristic
28C22×Q8×C131C2211yescharacteristic
29C4×Q8×C131—22no—
30C4×Q8×C131C2212yes—
31C4×Q8×C131C2212yes—
32#117×C131—22no—
33#117×C131C2211yescharacteristic
34#117×C131C2211yescharacteristic
35C2×D524—22no—
36C2×D524C2312yes—
37C2×C22⋊D131—22no—
38C2×C4×D131—22no—
39#14523500—24no—
40#14523500—24no—
41C2×C131⋊C8C2312yes—
42C2×C131⋊C8C2312yes—
43C4⋊Dic131C2312yes—
44Dic131⋊C4—22no—
45(C2×D131)⋊C4—22no—
46(C2×D131)⋊C4—22no—
47C2×Q8×C131—24no—
48C2×Q8×C131—21no—
49C2×Q8×C131C2312yes—
50C22×C4×C131—21no—
51C22×C4×C131C2311yescharacteristic
52C4⋊C4×C131—22no—
53C4⋊C4×C131—22no—
54C4⋊C4×C131D412yes—
55C4⋊C4×C131C2312yes—
56C22⋊C4×C131—21no—
57C22⋊C4×C131D412yes—
58C42×C131—22no—
59C42×C131—22no—
60C42×C131—21no—
61C42×C131C2311yescharacteristic
62C42×C131C2311yescharacteristic
63C22×D131—22no—
64D524—22no—
65D524—22no—
66D524—82no—
67C22⋊D131—82no—
68C2×Dic131—22no—
69C4×D131—82no—
70C131⋊C8—22no—
71C131⋊C8—22no—
72C23×C131C2×D411yescharacteristic
73Q8×C131—24no—
74Q8×C131—22no—
75Q8×C131#4114yes—
76C2×C4×C131—24no—
77C2×C4×C131—22no—
78C2×C4×C131—22no—
79C2×C4×C131—22no—
80C2×C4×C131C2×D412yes—
81C2×C4×C131C2×D412yes—
82C2×C4×C131C2×D411yescharacteristic
83C2×C4×C131C2411yescharacteristic
84C22×C131—21no—
85C22×C131—22no—
86C22×C131C22×D411yescharacteristic
87C2×D131—42no—
88C2×D131—42no—
89C2×D131—42no—
90C4×C131—24no—
91C4×C131—21no—
92C4×C131—22no—
93C4×C131—22no—
94C4×C131—42no—
95C4×C131C2×#4112yes—
96Dic131—42no—
97C2×C131—41no—
98C2×C131#52511yescharacteristic
99C2×C131#57712yes—
100D131—82no—
101C131#312611yescharacteristic, complemented, Sylow
102#3126—1311nomaximal, Sylow
103#540—1311no—
104#519D13111yescharacteristic, complemented
105#482—1312no—
106#479—1312no—
107#460—1312no—
108#448—1312no—
109#440—1312no—
110C4×SD16—1312no—
111#433—1311no—
112C22×Q8C2×D13111yescharacteristic
113C2×SD16—1312no—
114C2×SD16—1312no—
115#127—1311no—
116#125—1311no—
117#123—2622no—
118#121—2622no—
119#121—1312no—
120C4×Q8—22no—
121C4×Q8C2×D13112yes—
122C4×Q8C2×D13112yes—
123C4×D4—1312no—
124#117—22no—
125#117C2×D13111yescharacteristic
126#117C2×D13111yescharacteristic
127#107—1312no—
128#106—1312no—
129C4⋊C8—1312no—
130#103—1312no—
131#103—1312no—
132#102—1312no—
133#102—1312no—
134C22⋊C8—1312no—
135#97—1311no—
136C4×C8—1311no—
137C2×Q8—24no—
138C2×Q8—21no—
139C2×Q8C22×D13112yes—
140C2×D4—2622no—
141C2×D4—2622no—
142C2×D4—1312no—
143C22×C4—21no—
144C22×C4—2622no—
145C22×C4C22×D13111yescharacteristic
146SD16—2624no—
147SD16—2624no—
148C2×C8—1312no—
149C2×C8—1312no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—2622no—
153C4⋊C4—1312no—
154C4⋊C4C22⋊D13112yes—
155C4⋊C4C22×D13112yes—
156C22⋊C4—21no—
157C22⋊C4—2622no—
158C22⋊C4—2622no—
159C22⋊C4C22⋊D13112yes—
160C42—22no—
161C42—22no—
162C42—21no—
163C42C22×D13111yescharacteristic
164C42C22×D13111yescharacteristic
165C23—2622no—
166C23C2×C22⋊D13111yescharacteristic
167Q8—24no—
168Q8—22no—
169Q8#1452351314yes—
170D4—2622no—
171D4—2622no—
172D4—10482no—
173D4—10482no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—2622no—
179C2×C4—10482no—
180C2×C4C2×C22⋊D13112yes—
181C2×C4C2×C22⋊D13112yes—
182C2×C4C2×C22⋊D13111yescharacteristic
183C2×C4C23×D13111yescharacteristic
184C8—2622no—
185C8—2622no—
186C22—22no—
187C22—21no—
188C22—5242no—
189C22—5242no—
190C22—5242no—
191C22C22×C22⋊D13111yescharacteristic
192C4—22no—
193C4—22no—
194C4—24no—
195C4—21no—
196C4—42no—
197C4—5242no—
198C4C2×#1452351312yes—
199C2—41no—
200C2—10482no—
201C2#2255382311yescharacteristic
202C2#2255412012yes—
subgroupG/H
centerC22C22×C22⋊D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#519×C131C2
socleC22×C131C22×D4
prime radical#22553917C2
hypercenter#519D131
hypocenterC131#3126
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. #22553636C2semidirect
3. #22553619C2semidirect
4. #22553512C2semidirect
5. #22553451C2semidirect
6. #22553410C2semidirect
7. C4×#14523500C2semidirect
8. #22553373C2semidirect
9. #519×C131C2semidirect
101. C131#3126semidirect
104. #519D131semidirect

Maximal subgroups

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

subgroupindex
#225539172
#225536362
#225536192
#225535122
#225534512
#225534102
C4×#145235002
#225533732
#519×C1312
#3126131