Finite Group Databasebeta

← #20290153

Subgroups — #20290153

Order 6784.

288 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 68 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×#13133274C211yesmaximal, characteristic
2#16811565C211yesmaximal, characteristic
3#16811477C212yesmaximal, complemented
4#16811476C212yesmaximal, complemented
5C22⋊Q8×D53C211yesmaximal, characteristic
6C22×#88324—21no—
7C2×C4⋊#6157—21no—
8C2×C22⋊#6157—21no—
9#13133274—21no—
10#13133274C2212yes—
11(C2×#6157)⋊C4C2212yes—
12C2×Q8×D53C411yescharacteristic, complemented
13C22×C4×D53C2211yescharacteristic
14#13133197C2211yescharacteristic
15#13133196C411yescharacteristic, complemented
16#13133195—21no—
17#13133194—21no—
18#13133193C2211yescharacteristic
19C4⋊C4×D53—21no—
20C4⋊C4×D53C411yescharacteristic, complemented
21C22⋊C4×D53—21no—
22C22⋊Q8×C53C411yescharacteristic, complemented
23C23×D53D411yescharacteristic
24C22×#6157—22no—
25C2×Q8×C53C2×C411yescharacteristic
26C22×C4×C53C2×C411yescharacteristic
27C22×Dic53D411yescharacteristic
28Q8×D53—42no—
29C2×C4×D53—21no—
30C2×C4×D53—21no—
31C2×C4×D53—42no—
32C2×C4×D53C2×C411yescharacteristic
33C2×C4×D53D411yescharacteristic
34C2×C4×D53D411yescharacteristic
35C2×C4×D53C2311yescharacteristic
36C2×Dic106—21no—
37C2×Dic106C2×C411yescharacteristic
38C22⋊#6157—42no—
39C2×#88324—22no—
40C2×#88324—42no—
41C4⋊#6157—42no—
42C4⋊C4×C53—21no—
43C4⋊C4×C53C2×C411yescharacteristic
44C22⋊C4×C53—21no—
45C22⋊Dic53—21no—
46#11719921—21no—
47#11719921—21no—
48C4⋊Dic53—21no—
49C4⋊Dic53C2×C411yescharacteristic
50#11719919—21no—
51#11719919—21no—
52#11719919—21no—
53C23×C53C22⋊C411yescharacteristic
54C22×D53—41no—
55C22×D53—41no—
56C22×D53C22⋊C411yescharacteristic
57C22×D53SD1612yes—
58C22×D53Q1612yes—
59C22×D53C2×D411yescharacteristic
60C2×#6157—42no—
61C2×#6157—42no—
62C2×#6157—42no—
63Q8×C53—41no—
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53—41no—
67C2×C4×C53C22⋊C411yescharacteristic
68C2×C4×C53C22×C411yescharacteristic
69C2×Dic53—22no—
70C2×Dic53—21no—
71C2×Dic53—21no—
72C2×Dic53C22⋊C411yescharacteristic
73C2×Dic53C2×D411yescharacteristic
74C4×D53—22no—
75C4×D53—22no—
76C4×D53—41no—
77C4×D53—41no—
78C4×D53—42no—
79C4×D53—42no—
80Dic106—42no—
81Dic106—41no—
82#88324—42no—
83C22×C53—21no—
84C22×C53—21no—
85C22×C53#10312yes—
86C22×C53C2×C22⋊C411yescharacteristic
87C2×D53—21no—
88C2×D53—21no—
89C2×D53—21no—
90C2×D53—21no—
91C2×D53—42no—
92C2×D53—42no—
93C2×D53#10312yes—
94C2×D53C2×SD1611yescharacteristic
95C2×D53C2×Q1611yescharacteristic
96#6157—82no—
97C4×C53—21no—
98C4×C53—21no—
99C4×C53—41no—
100C4×C53—41no—
101Dic53—21no—
102Dic53—41no—
103Dic53—41no—
104Dic53#9912yes—
105#1520—531nomaximal, Sylow
106C2×C53—22no—
107C2×C53C2×#9911yescharacteristic
108C2×C53C2×#10311yescharacteristic
109C2×C53#42111yescharacteristic
110D53—21no—
111D53—22no—
112D53—21no—
113C2×C22⋊Q8—531no—
114C2×C22⋊C8—531no—
115#380—531no—
116#328—532no—
117#328—532no—
118C53#152011yescharacteristic, complemented, Sylow
119C22×Q8—531no—
120C23×C4—531no—
121C22×C8—1061no—
122C22⋊Q8—531no—
123C22⋊Q8—531no—
124C22⋊Q8—531no—
125C22⋊Q8—1061no—
126C22⋊Q8—1061no—
127C22⋊Q8#615711yescharacteristic, complemented
128C2×C4⋊C4—531no—
129C2×C4⋊C4—1061no—
130C2×C4⋊C4—1061no—
131C2×C22⋊C4—1061no—
132C2×C22⋊C4—1061no—
133C22⋊C8—532no—
134C22⋊C8—1061no—
135#95—532no—
136C24—531no—
137C2×Q8—531no—
138C2×Q8—1061no—
139C2×Q8—2122no—
140C2×Q8C2×#615711yescharacteristic
141C22×C4—531no—
142C22×C4—531no—
143C22×C4—531no—
144C22×C4—531no—
145C22×C4—531no—
146C22×C4—1061no—
147C22×C4—1061no—
148C22×C4—1062no—
149C22×C4—2122no—
150C22×C4C2×#615711yescharacteristic
151C2×C8—1062no—
152C2×C8—2122no—
153C4⋊C4—21no—
154C4⋊C4—531no—
155C4⋊C4—1061no—
156C4⋊C4—1061no—
157C4⋊C4—1061no—
158C4⋊C4—1061no—
159C4⋊C4—2122no—
160C4⋊C4C2×#615711yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—1061no—
163C22⋊C4—1061no—
164C22⋊C4—1061no—
165C22⋊C4—2122no—
166C23—531no—
167C23—531no—
168C23—532no—
169C23—532no—
170C23—2121no—
171C23—2121no—
172C23C22⋊#615711yescharacteristic
173Q8—41no—
174Q8—2121no—
175Q8—2122no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—531no—
180C2×C4—531no—
181C2×C4—1062no—
182C2×C4—1062no—
183C2×C4—1062no—
184C2×C4—1061no—
185C2×C4—1061no—
186C2×C4—2121no—
187C2×C4—2121no—
188C2×C4—2122no—
189C2×C4—2122no—
190C2×C4—2122no—
191C2×C4—2122no—
192C2×C4—2122no—
193C2×C4C22⋊#615711yescharacteristic
194C2×C4C22×#615711yescharacteristic
195C8—2122no—
196C22—21no—
197C22—21no—
198C22—531no—
199C22—532no—
200C22—531no—
201C22—1061no—
202C22—1061no—
203C22—1061no—
204C22—1061no—
205C22—2122no—
206C22—2122no—
207C22#1313328012yes—
208C22C2×C22⋊#615711yescharacteristic
209C4—21no—
210C4—21no—
211C4—41no—
212C4—41no—
213C4—532no—
214C4—1061no—
215C4—4242no—
216C4—2121no—
217C4—2121no—
218C2—22no—
219C2—1061no—
220C2—1062no—
221C2—1061no—
222C2D53⋊#9911yescharacteristic
223C2C2×#1313328011yescharacteristic
224C2#1681162011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC22⋊Q8×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC22⋊Q8#6157
hypocenterC53#1520
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #16811477C2semidirect
4. #16811476C2semidirect
12. C2×Q8×D53C4semidirect
15. #13133196C4semidirect
20. C4⋊C4×D53C4semidirect
22. C22⋊Q8×C53C4semidirect
118. C53#1520semidirect
127. C22⋊Q8#6157semidirect