Finite Group Databasebeta

← #20290586

Subgroups — #20290586

Order 6784.

292 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 84 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
1D53⋊(C4⋊C8)C211yesmaximal, characteristic
2C2×#13133280C212yesmaximal
3#16811567C211yesmaximal, characteristic
4#16811496C212yesmaximal
5C4×Q8×D53C211yesmaximal, characteristic
6C2×D53⋊C8C2212yes—
7C2×C4⋊#6157C2212yes—
8#13133284—21no—
9C4⋊#88324—21no—
10#13133280—24no—
11(C2×#6157)⋊C4—21no—
12(C2×#6157)⋊C4—21no—
13C2×Q8×D53C2211yescharacteristic
14Q8×Dic53C411yescharacteristic, complemented
15C4×Dic106—21no—
16C4×Dic106C411yescharacteristic, complemented
17#13133182—21no—
18#13133182C411yescharacteristic, complemented
19C4⋊C4×D53—21no—
20C4⋊C4×D53C2211yescharacteristic
21C42×D53—21no—
22C42×D53C2211yescharacteristic
23C4×Q8×C53C411yescharacteristic, complemented
24C22×#6157—22no—
25C2×Q8×C53C2×C411yescharacteristic
26Q8×D53—21no—
27Q8×D53—21no—
28Q8×D53D412yes—
29Q8×D53Q812yes—
30C2×C4×D53—21no—
31C2×C4×D53—21no—
32C2×C4×D53D411yescharacteristic
33C2×C4×D53D411yescharacteristic
34C2×C4×D53C2311yescharacteristic
35C2×Dic106—21no—
36C2×Dic106C2×C411yescharacteristic
37C2×#88324—22no—
38C4⋊#6157—22no—
39C4⋊#6157—22no—
40D53⋊C8—22no—
41D53⋊C8—22no—
42C4⋊C4×C53—21no—
43C4⋊C4×C53C2×C411yescharacteristic
44C42×C53—21no—
45C42×C53C2×C411yescharacteristic
46C4⋊Dic53—21no—
47C4⋊Dic53C2×C411yescharacteristic
48#11719919—21no—
49#11719919—21no—
50#11719919—21no—
51C4×Dic53—21no—
52C4×Dic53—21no—
53C4×Dic53—21no—
54C4×Dic53C2×C411yescharacteristic
55C4×Dic53D411yescharacteristic
56C4×Dic53D411yescharacteristic
57C22×D53C2×D411yescharacteristic
58C2×#6157—42no—
59C2×#6157—42no—
60C2×#6157—42no—
61Q8×C53—21no—
62Q8×C53C4⋊C412yes—
63C2×C4×C53—21no—
64C2×C4×C53—21no—
65C2×C4×C53C22⋊C411yescharacteristic
66C2×C4×C53C22⋊C411yescharacteristic
67C2×C4×C53C22×C411yescharacteristic
68C2×Dic53—21no—
69C2×Dic53—21no—
70C2×Dic53C22⋊C411yescharacteristic
71C2×Dic53C22⋊C411yescharacteristic
72C2×Dic53C2×D411yescharacteristic
73C4×D53—22no—
74C4×D53—22no—
75C4×D53—22no—
76C4×D53—22no—
77C4×D53—42no—
78C4×D53C2×D411yescharacteristic
79C4×D53C2×Q811yescharacteristic
80C4×D53#4111yescharacteristic
81C4×D53#4111yescharacteristic
82Dic106—21no—
83Dic106—22no—
84Dic106—22no—
85Dic106C4⋊C412yes—
86#88324—42no—
87C22×C53C2×C22⋊C411yescharacteristic
88C2×D53#12111yescharacteristic
89C2×D53C22⋊Q811yescharacteristic
90C2×D53#13512yes—
91C2×D53#13712yes—
92#6157—82no—
93C4×C53—22no—
94C4×C53—21no—
95C4×C53—21no—
96C4×C53—41no—
97C4×C53C2×C4⋊C411yescharacteristic
98C4×C53#11711yescharacteristic
99Dic53—22no—
100Dic53—21no—
101Dic53—21no—
102Dic53—41no—
103Dic53#12111yescharacteristic
104Dic53C22⋊Q811yescharacteristic
105#1805—531nomaximal, Sylow
106C2×C53#38011yescharacteristic
107C2×C53#41611yescharacteristic
108C2×C53#41911yescharacteristic
109D53#46412yes—
110D53#47912yes—
111C2×C4×Q8—531no—
112C2×C4⋊C8—531no—
113C2×#103—532no—
114#382—531no—
115#340—532no—
116C53#180511yescharacteristic, complemented, Sylow
117C22×Q8—531no—
118C22×C8—532no—
119C4×Q8—531no—
120C4×Q8—531no—
121C4×Q8—531no—
122C4×Q8—1061no—
123C4×Q8—1061no—
124C4×Q8#615711yescharacteristic, complemented
125C2×C4⋊C4—531no—
126C2×C4⋊C4—532no—
127C2×C4⋊C4—1061no—
128C2×C42—531no—
129C2×C42—1061no—
130C4⋊C8—1061no—
131C4⋊C8—1061no—
132#103—1064no—
133#95—1061no—
134#95—1061no—
135C2×Q8—532no—
136C2×Q8—532no—
137C2×Q8—531no—
138C2×Q8—1061no—
139C2×Q8—1061no—
140C2×Q8—1061no—
141C2×Q8C2×#615711yescharacteristic
142C22×C4—531no—
143C22×C4—531no—
144C22×C4—531no—
145C22×C4—1061no—
146C22×C4—1061no—
147C22×C4—1062no—
148C2×C8—1062no—
149C2×C8—1062no—
150C2×C8—1062no—
151C4⋊C4—21no—
152C4⋊C4—531no—
153C4⋊C4—1061no—
154C4⋊C4—1061no—
155C4⋊C4—1061no—
156C4⋊C4—1062no—
157C4⋊C4—1062no—
158C4⋊C4—1061no—
159C4⋊C4C2×#615711yescharacteristic
160C42—21no—
161C42—531no—
162C42—531no—
163C42—531no—
164C42—1061no—
165C42—1061no—
166C42—1061no—
167C42C2×#615711yescharacteristic
168C23—531no—
169Q8—21no—
170Q8—532no—
171Q8—1062no—
172Q8—1062no—
173Q8—1061no—
174Q8C4⋊#615712yes—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—531no—
178C2×C4—531no—
179C2×C4—531no—
180C2×C4—531no—
181C2×C4—531no—
182C2×C4—531no—
183C2×C4—531no—
184C2×C4—1062no—
185C2×C4—1062no—
186C2×C4—1062no—
187C2×C4—1062no—
188C2×C4—1061no—
189C2×C4—1061no—
190C2×C4—2122no—
191C2×C4—2122no—
192C2×C4—2122no—
193C2×C4—2122no—
194C2×C4C22⋊#615711yescharacteristic
195C2×C4C22⋊#615711yescharacteristic
196C2×C4C22×#615711yescharacteristic
197C8—2122no—
198C22—531no—
199C22—531no—
200C22—532no—
201C22—532no—
202C22C2×C22⋊#615711yescharacteristic
203C4—22no—
204C4—21no—
205C4—21no—
206C4—41no—
207C4—531no—
208C4—531no—
209C4—1062no—
210C4—1061no—
211C4—1061no—
212C4—4242no—
213C4—2121no—
214C4C2×C4⋊#615711yescharacteristic
215C4#1313329611yescharacteristic
216C2—532no—
217C2—532no—
218C2#1681156511yescharacteristic
219C2#1681161311yescharacteristic
220C2#1681161611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC4×Q8×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC4×Q8#6157
hypocenterC53#1805
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
14. Q8×Dic53C4semidirect
16. C4×Dic106C4semidirect
18. #13133182C4semidirect
23. C4×Q8×C53C4semidirect
116. C53#1805semidirect
124. C4×Q8#6157semidirect