Finite Group Databasebeta

← #354×D53 #20276374

Subgroups — #354×D53 #20276374

Order 6784.

284 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 57 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#124×D53C211yesmaximal, characteristic
2#99×D53C212yesmaximal
3#16810418C211yesmaximal, characteristic, complemented
4#16810417C212yesmaximal, complemented
5C53×#354C211yesmaximal, characteristic, complemented
6C2×Q8×D53C411yescharacteristic, complemented
7C2×D4×D53C2211yescharacteristic
8#13133206C411yescharacteristic, complemented
9#13133205C411yescharacteristic, complemented
10#13133204C2211yescharacteristic
11#13133203—22no—
12C22⋊C4×D53—22no—
13C22⋊C4×D53—22no—
14C42×D53C411yescharacteristic, complemented
15#13133132C2212yes—
16#13133131—22no—
17#124×C53C2211yescharacteristic
18#99×C53C2212yes—
19C23×D53D412yes—
20C2×Q8×C53C2×C411yescharacteristic, complemented
21C2×D4×C53C2311yescharacteristic
22C2×C22⋊D53D412yes—
23Q8×D53—41no—
24Q8×D53—41no—
25D4×D53—41no—
26D4×D53—41no—
27C2×D212C2×C411yescharacteristic
28C2×C4×D53—21no—
29C2×C4×D53—21no—
30C2×C4×D53—42no—
31C2×C4×D53C2×C411yescharacteristic
32C2×Dic106—21no—
33C2×Dic106C2×C411yescharacteristic, complemented
34C22⋊C4×C53—22no—
35C22⋊C4×C53—22no—
36C42×C53C2×C411yescharacteristic, complemented
37C22⋊Dic53—22no—
38C22⋊Dic53—22no—
39#11719921—22no—
40#11719921—22no—
41#11719921—42no—
42C4×Dic53—21no—
43C4×Dic53C2×C411yescharacteristic, complemented
44C23×C53C2×D412yes—
45C22×D53—22no—
46C22×D53—22no—
47C22×D53—44no—
48C22×D53C22⋊C411yescharacteristic
49C22×D53C22⋊C412yes—
50Q8×C53—41no—
51D4×C53—41no—
52C2×C4×C53—21no—
53C2×C4×C53—21no—
54C2×C4×C53—42no—
55C2×C4×C53C22×C411yescharacteristic
56C22⋊D53—42no—
57C2×Dic53—21no—
58C2×Dic53—21no—
59C2×Dic53—42no—
60C2×Dic53C22⋊C411yescharacteristic
61D212—41no—
62C4×D53—21no—
63C4×D53—21no—
64C4×D53—42no—
65C4×D53—41no—
66C4×D53—41no—
67C4×D53—84no—
68Dic106—42no—
69Dic106—41no—
70C22×C53—22no—
71C22×C53—42no—
72C22×C53C2×C22⋊C411yescharacteristic
73C2×D53—22no—
74C2×D53—22no—
75C2×D53—44no—
76C2×D53—42no—
77C2×D53—82no—
78C2×D53#9911yescharacteristic
79C2×D53#9911yescharacteristic
80C4×C53—21no—
81C4×C53—41no—
82C4×C53—41no—
83C4×C53—82no—
84Dic53—21no—
85Dic53—41no—
86Dic53—41no—
87Dic53—82no—
88C2×#354—531nomaximal, Sylow
89C2×C53—21no—
90C2×C53—42no—
91C2×C53C2×#9911yescharacteristic
92D53—21no—
93D53—42no—
94D53#35412yesdirect product factor
95C2×#124—531no—
96C2×#99—532no—
97#354—531no—
98#354—532no—
99#354D5311yescharacteristic, direct product factor
100C53C2×#35411yescharacteristic, complemented, Sylow
101C22×Q8—531no—
102C22×D4—531no—
103#124—531no—
104#124—531no—
105#124—531no—
106#124—1062no—
107#124C2×D5311yescharacteristic
108C2×C22⋊C4—1062no—
109C2×C22⋊C4—1062no—
110C2×C42—531no—
111#99—532no—
112#99—1062no—
113#99C2×D5312yes—
114C24—532no—
115C2×Q8—531no—
116C2×Q8—1061no—
117C2×Q8—2121no—
118C2×Q8—2121no—
119C2×Q8C4×D5311yescharacteristic, complemented
120C2×D4—532no—
121C2×D4—531no—
122C2×D4—2121no—
123C2×D4—2121no—
124C2×D4C22×D5311yescharacteristic
125C22×C4—531no—
126C22×C4—1061no—
127C22×C4—1061no—
128C22×C4—2122no—
129C22⋊C4—22no—
130C22⋊C4—22no—
131C22⋊C4—1062no—
132C22⋊C4—1062no—
133C22⋊C4—1062no—
134C22⋊C4—1062no—
135C22⋊C4—2122no—
136C42—531no—
137C42—1061no—
138C42C4×D5311yescharacteristic, complemented
139C23—531no—
140C23—532no—
141C23—1062no—
142C23—1062no—
143C23—2124no—
144C23D4×D5312yes—
145Q8—41no—
146Q8—2121no—
147Q8—2122no—
148D4—41no—
149D4—2121no—
150D4—2122no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—42no—
154C2×C4—531no—
155C2×C4—1061no—
156C2×C4—1061no—
157C2×C4—1061no—
158C2×C4—1061no—
159C2×C4—4244no—
160C2×C4—2121no—
161C2×C4—2121no—
162C2×C4—2122no—
163C2×C4—2122no—
164C2×C4C2×C4×D5311yescharacteristic
165C22—22no—
166C22—42no—
167C22—531no—
168C22—531no—
169C22—1062no—
170C22—1062no—
171C22—4242no—
172C22—2122no—
173C22—2124no—
174C22C22⋊C4×D5311yescharacteristic
175C4—21no—
176C4—41no—
177C4—41no—
178C4—82no—
179C4—1061no—
180C4—4242no—
181C4—2121no—
182C4—2121no—
183C2—21no—
184C2—42no—
185C2—532no—
186C2—1061no—
187C2—2122no—
188C2#99×D5311yescharacteristic
subgroupG/H
centerC2#99×D53
commutatorC2×C4×C53C22×C4
FrattiniC2×D4C22×D53
FittingC53×#354C2
socleC2×C53C2×#99
hypercenter#354D53
hypocenterC53C2×#354
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. #16810418C2semidirect
4. #16810417C2semidirect
5. C53×#354C2semidirect
6. C2×Q8×D53C4semidirect
8. #13133206C4semidirect
9. #13133205C4semidirect
14. C42×D53C4semidirect
20. C2×Q8×C53C2×C4semidirect
33. C2×Dic106C2×C4semidirect
36. C42×C53C2×C4semidirect
43. C4×Dic53C2×C4semidirect
94. D53#354direct (N × H)
99. #354D53direct (N × H)
100. C53C2×#354semidirect
119. C2×Q8C4×D53semidirect
138. C42C4×D53semidirect