Finite Group Databasebeta

← #30676247

Subgroups — #30676247

Order 16768.

382 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 121 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#22553789C211yesmaximal, characteristic, complemented
2#22553633C212yesmaximal, complemented
3#22553616C212yesmaximal, complemented
4#22553509C212yesmaximal, complemented
5#22553450C212yesmaximal, complemented
6#22553412C212yesmaximal, complemented
7C4×#14523501C212yesmaximal, complemented
8#22553373C211yesmaximal, characteristic, complemented
9#540×C131C211yesmaximal, characteristic
10C22×Dic262C2211yescharacteristic
11C2×#14523501C2212yes—
12C2×#14523501C2212yes—
13C4×Dic262C2212yes—
14C4×Dic262C2212yes—
15#16862211—22no—
16#16862205C2211yescharacteristic
17#16862202—22no—
18#16862179C2212yes—
19#16862177C2212yes—
20C4⋊(C131⋊C8)C2212yes—
21#16862169C2212yes—
22#16862169C2212yes—
23#16862167C2212yes—
24#16862167C2212yes—
25C22⋊(C131⋊C8)C2212yes—
26#16862157C2211yescharacteristic
27C4×C131⋊C8C2211yescharacteristic
28#127×C131C2211yescharacteristic
29#125×C131C2211yescharacteristic
30#123×C131—22no—
31#121×C131—22no—
32#121×C131C2212yes—
33C4×D4×C131C2212yes—
34#117×C131C2211yescharacteristic
35C2×Dic262—24no—
36C2×Dic262—21no—
37C2×Dic262C2312yes—
38C22×Dic131—21no—
39#14523501—24no—
40#14523501—24no—
41C2×C131⋊C8C2312yes—
42C2×C131⋊C8C2312yes—
43C4⋊Dic131C2312yes—
44Dic131⋊C4—22no—
45Dic131⋊C4—22no—
46C22⋊Dic131—21no—
47C4×Dic131—21no—
48C4×Dic131—22no—
49C4×Dic131—22no—
50C2×D4×C131—22no—
51C2×D4×C131—22no—
52C2×D4×C131C2312yes—
53C22×C4×C131—22no—
54C22×C4×C131C2311yescharacteristic
55C4⋊C4×C131—22no—
56C4⋊C4×C131D412yes—
57C4⋊C4×C131C2312yes—
58C22⋊C4×C131—22no—
59C22⋊C4×C131—22no—
60C22⋊C4×C131D412yes—
61C42×C131C2311yescharacteristic
62C42×C131C2311yescharacteristic
63Dic262—24no—
64Dic262—22no—
65Dic262#4114yes—
66C2×Dic131—24no—
67C2×Dic131—22no—
68C2×Dic131—22no—
69C131⋊C8—22no—
70C131⋊C8—22no—
71C23×C131—22no—
72C23×C131C2×D411yescharacteristic
73D4×C131—22no—
74D4×C131—22no—
75D4×C131—82no—
76D4×C131—82no—
77C2×C4×C131—22no—
78C2×C4×C131—22no—
79C2×C4×C131—82no—
80C2×C4×C131C2×D412yes—
81C2×C4×C131C2×D412yes—
82C2×C4×C131C2×D411yescharacteristic
83C2×C4×C131C2411yescharacteristic
84C22×C131—21no—
85C22×C131—22no—
86C22×C131—42no—
87C22×C131—42no—
88C22×C131—42no—
89C22×C131C22×D411yescharacteristic
90C4×C131—21no—
91C4×C131—22no—
92C4×C131—22no—
93C4×C131—42no—
94C4×C131C2×#4112yes—
95Dic131—24no—
96Dic131—42no—
97C2×C131—41no—
98C2×C131—82no—
99C2×C131#52511yescharacteristic
100C2×C131#57712yes—
101C131#312611yescharacteristic, complemented, Sylow
102#3126—1311nomaximal, Sylow
103#540D13111yescharacteristic
104#519—1311no—
105#482—1312no—
106#479—1312no—
107#460—1312no—
108#448—1312no—
109#440—1312no—
110C4×SD16—1312no—
111#433—1311no—
112C22×Q8—1311no—
113C2×SD16—1312no—
114C2×SD16—1312no—
115#127C2×D13111yescharacteristic
116#125C2×D13111yescharacteristic
117#123—22no—
118#121—22no—
119#121C2×D13112yes—
120C4×Q8—2622no—
121C4×Q8—1312no—
122C4×Q8—1312no—
123C4×D4C2×D13112yes—
124#117—2622no—
125#117—1311no—
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—2624no—
138C2×Q8—2621no—
139C2×Q8—1312no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4C22×D13112yes—
143C22×C4—22no—
144C22×C4—2621no—
145C22×C4C22×D13111yescharacteristic
146SD16—2624no—
147SD16—2624no—
148C2×C8—1312no—
149C2×C8—1312no—
150C4⋊C4—22no—
151C4⋊C4—2622no—
152C4⋊C4—2622no—
153C4⋊C4—1312no—
154C4⋊C4C22⋊D13112yes—
155C4⋊C4C22×D13112yes—
156C22⋊C4—22no—
157C22⋊C4—22no—
158C22⋊C4—2621no—
159C22⋊C4C22⋊D13112yes—
160C42—2622no—
161C42—2622no—
162C42—2621no—
163C42C22×D13111yescharacteristic
164C42C22×D13111yescharacteristic
165C23—22no—
166C23C2×C22⋊D13111yescharacteristic
167Q8—2624no—
168Q8—2622no—
169Q8—1314no—
170D4—22no—
171D4—22no—
172D4—82no—
173D4—82no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—2624no—
177C2×C4—2622no—
178C2×C4—2622no—
179C2×C4—82no—
180C2×C4C2×C22⋊D13112yes—
181C2×C4C2×C22⋊D13112yes—
182C2×C4C2×C22⋊D13111yescharacteristic
183C2×C4C23×D13111yescharacteristic
184C8—2622no—
185C8—2622no—
186C22—22no—
187C22—21no—
188C22—42no—
189C22—42no—
190C22—42no—
191C22C22×C22⋊D13111yescharacteristic
192C4—22no—
193C4—22no—
194C4—21no—
195C4—42no—
196C4—2624no—
197C4—5242no—
198C4#41×D13112yes—
199C2—41no—
200C2—82no—
201C2#2255381911yescharacteristic
202C2#2255412312yes—
subgroupG/H
centerC22C22×C22⋊D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting, prime radical#540×C131C2
socleC22×C131C22×D4
hypercenter#540D131
hypocenterC131#3126
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #22553789C2semidirect
2. #22553633C2semidirect
3. #22553616C2semidirect
4. #22553509C2semidirect
5. #22553450C2semidirect
6. #22553412C2semidirect
7. C4×#14523501C2semidirect
8. #22553373C2semidirect
101. C131#3126semidirect

Maximal subgroups

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

subgroupindex
#225537892
#225536332
#225536162
#225535092
#225534502
#225534122
C4×#145235012
#225533732
#540×C1312
#3126131