Finite Group Databasebeta

← #20275405

Subgroups — #20275405

Order 6784.

248 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C2×#13133142C211yesmaximal, characteristic
2C2×#13133142C211yesmaximal, characteristic
3#16810529C211yesmaximal, characteristic
4#16810521C211yesmaximal, characteristic
5#16810386C211yesmaximal, characteristic
6#16810386C211yesmaximal, characteristic
7C2×#97×C53C211yesmaximal, characteristic
8C22×Dic106C2211yescharacteristic
9C2×C4⋊Dic53C2211yescharacteristic
10C2×C4⋊Dic53C2211yescharacteristic
11C2×C4⋊Dic53C2211yescharacteristic
12C2×#11719919—21no—
13#13133142—22no—
14#13133142C414yescomplemented
15#13133124—21no—
16#13133124—21no—
17#13133124—21no—
18C22×C8×C53C2211yescharacteristic
19C22×C8×C53C2211yescharacteristic
20C2×C42×C53C2211yescharacteristic
21#97×C53—22no—
22C22×C4×C53D411yescharacteristic
23C22×C4×C53D411yescharacteristic
24C22×C4×C53C2311yescharacteristic
25C22×Dic53—21no—
26C22×Dic53—21no—
27C22×Dic53—21no—
28C22×Dic53—21no—
29C2×Dic106—21no—
30C2×Dic106—21no—
31C2×Dic106—21no—
32C2×Dic106—21no—
33C2×Dic106—21no—
34C2×Dic106C2×C412yes—
35C2×C8×C53—21no—
36C2×C8×C53—21no—
37C2×C8×C53—21no—
38C2×C8×C53—21no—
39C2×C8×C53C2×C412yes—
40C2×C8×C53D412yes—
41C42×C53—21no—
42C42×C53—21no—
43C4⋊Dic53—21no—
44C4⋊Dic53—21no—
45C4⋊Dic53—21no—
46C4⋊Dic53—21no—
47C4⋊Dic53—21no—
48C4⋊Dic53C2×C412yes—
49#11719919—42no—
50C23×C53C2×D411yescharacteristic
51C2×C4×C53—21no—
52C2×C4×C53—21no—
53C2×C4×C53—21no—
54C2×C4×C53—21no—
55C2×C4×C53—21no—
56C2×C4×C53—21no—
57C2×C4×C53C22×C411yescharacteristic
58C2×C4×C53C2×D411yescharacteristic
59C2×C4×C53#4111yescharacteristic
60C2×C4×C53#4111yescharacteristic
61C2×C4×C53#4111yescharacteristic
62C2×C4×C53#4111yescharacteristic
63C2×Dic53—22no—
64C2×Dic53—22no—
65C2×Dic53—41no—
66C2×Dic53—41no—
67C2×Dic53—41no—
68C2×Dic53—41no—
69C2×Dic53—41no—
70C2×Dic53—41no—
71C2×Dic53—41no—
72C2×Dic53—41no—
73C2×Dic53—41no—
74C2×Dic53—41no—
75Dic106—22no—
76Dic106—22no—
77Dic106—41no—
78Dic106—41no—
79C8×C53—22no—
80C8×C53—22no—
81C22×C53C4×D411yescharacteristic
82C22×C53#12111yescharacteristic
83C22×C53#12411yescharacteristic
84C22×C53AGL(1,ℤ/8)11yescharacteristic
85C22×C53#13711yescharacteristic
86C22×C53#13711yescharacteristic
87C22×C53#13711yescharacteristic
88C4×C53—41no—
89C4×C53—41no—
90C4×C53#11711yescharacteristic
91C4×C53C4×D411yescharacteristic
92C4×C53#12311yescharacteristic
93C4×C53#12611yescharacteristic
94Dic53—42no—
95Dic53—42no—
96Dic53—81no—
97Dic53—81no—
98#1935—531nomaximal, Sylow
99C2×C53#38811yescharacteristic
100C2×C53#44011yescharacteristic
101C2×C53#44111yescharacteristic
102C2×C53#46811yescharacteristic
103C2×C53#47011yescharacteristic
104C2×C53#48911yescharacteristic
105C2×C53#49111yescharacteristic
106C2×#103—531no—
107C2×#103—531no—
108C2×#97D5311yescharacteristic
109#391—531no—
110#389—531no—
111#340—531no—
112#340—531no—
113C53#193511yescharacteristic, complemented, Sylow
114C22×Q8—531no—
115C22×C8C2×D5311yescharacteristic
116C22×C8C2×D5311yescharacteristic
117C2×C4⋊C4—531no—
118C2×C4⋊C4—531no—
119C2×C4⋊C4—531no—
120C2×C4⋊C4—1061no—
121C2×C42C2×D5311yescharacteristic
122#103—534no—
123#103—1062no—
124#97—22no—
125#95—1061no—
126#95—1061no—
127#95—1061no—
128C2×Q8—532no—
129C2×Q8—1061no—
130C2×Q8—1061no—
131C2×Q8—1061no—
132C2×Q8—1061no—
133C2×Q8—1061no—
134C22×C4—1061no—
135C22×C4—1061no—
136C22×C4—1061no—
137C22×C4—1061no—
138C22×C4D21211yescharacteristic
139C22×C4D21211yescharacteristic
140C22×C4C22×D5311yescharacteristic
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8C4×D5312yes—
146C2×C8C22⋊D5312yes—
147C4⋊C4—532no—
148C4⋊C4—1061no—
149C4⋊C4—1061no—
150C4⋊C4—1061no—
151C4⋊C4—1061no—
152C4⋊C4—1061no—
153C4⋊C4—2122no—
154C42—21no—
155C42—21no—
156C23C2×D21211yescharacteristic
157Q8—1062no—
158Q8—1062no—
159Q8—2121no—
160Q8—2121no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—1062no—
168C2×C4—1062no—
169C2×C4—2121no—
170C2×C4—2121no—
171C2×C4—2121no—
172C2×C4—2121no—
173C2×C4—2121no—
174C2×C4—2121no—
175C2×C4—2121no—
176C2×C4—2121no—
177C2×C4—2121no—
178C2×C4—2121no—
179C2×C4C2×C4×D5311yescharacteristic
180C2×C4#1171994511yescharacteristic
181C2×C4#1171994511yescharacteristic
182C2×C4#1171994511yescharacteristic
183C2×C4#1171994511yescharacteristic
184C2×C4C2×C22⋊D5311yescharacteristic
185C8—22no—
186C8—22no—
187C22C4×D21211yescharacteristic
188C22#1313319111yescharacteristic
189C22#1313320511yescharacteristic
190C22#1313324511yescharacteristic
191C22#1313325011yescharacteristic
192C22#1313325011yescharacteristic
193C22#1313325011yescharacteristic
194C4—41no—
195C4—41no—
196C4—4241no—
197C4—4241no—
198C4—2122no—
199C4—2122no—
200C4#1313317411yescharacteristic
201C4C4×C22⋊D5311yescharacteristic
202C4#1313320011yescharacteristic
203C4#1313321111yescharacteristic
204C2#1681051411yescharacteristic
205C2#1681072511yescharacteristic
206C2#1681073111yescharacteristic
207C2#1681086911yescharacteristic
208C2#1681088111yescharacteristic
209C2#1681098311yescharacteristic
210C2#1681099211yescharacteristic
subgroupG/H
centerC23C2×D212
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC2×#97×C53C2
socleC23×C53C2×D4
hypercenterC2×#97D53
hypocenterC53#1935
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#97

Complements

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

NH ≅ G/Ntype
14. #13133142C4semidirect
113. C53#1935semidirect