Finite Group Databasebeta

← #20287594

Subgroups — #20287594

Order 6784.

338 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 125 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#16811348C211yesmaximal, characteristic, complemented
2#16811330C211yesmaximal, characteristic
3#16811024C212yesmaximal, complemented
4#16811019C211yesmaximal, characteristic, complemented
5#16810929C212yesmaximal, complemented
6#16810921C212yesmaximal
7#16810919C211yesmaximal, characteristic, complemented
8#16810747C211yesmaximal, characteristic, complemented
9#16810714C212yesmaximal
10C4×#11719923C211yesmaximal, characteristic
11C53×#474C211yesmaximal, characteristic
12C2×#11719923C2211yescharacteristic
13#13133217C2211yescharacteristic
14#13133215—21no—
15#13133208C2212yes—
16#13133208C2212yes—
17#13133207—22no—
18#13133196—21no—
19#13133194—22no—
20Q8×Dic53—21no—
21C4×Dic106C2211yescharacteristic
22C4×Dic106C2211yescharacteristic
23#13133182C2212yes—
24D4×Dic53C2212yes—
25C2×C4⋊Dic53—21no—
26#13133151C2212yes—
27#13133148C2212yes—
28#13133148C2212yes—
29#13133148C2211yescharacteristic
30#13133146C2212yes—
31#13133142C2212yes—
32#13133140C2211yescharacteristic
33#13133138C2211yescharacteristic
34C4×#88322C2211yescharacteristic
35C8×Dic53C2212yes—
36C4⋊Q8×C53C2211yescharacteristic
37C4×D4×C53C2211yescharacteristic
38#107×C53C2212yes—
39C4⋊C8×C53C2211yescharacteristic
40#102×C53C2212yes—
41C2×Q8×C53—21no—
42C2×D4×C53C2311yescharacteristic
43C22×C4×C53—21no—
44C22×Dic53—22no—
45C2×Dic106—21no—
46C2×Dic106D411yescharacteristic
47C2×Dic106C2311yescharacteristic
48C2×C8×C53C2312yes—
49C4⋊C4×C53—21no—
50C4⋊C4×C53C2312yes—
51C4⋊C4×C53C2311yescharacteristic
52C22⋊C4×C53—21no—
53C42×C53C2311yescharacteristic
54C22⋊Dic53—22no—
55#11719923Q814yescomplemented
56C4⋊Dic53—21no—
57C4⋊Dic53—22no—
58C4⋊Dic53—22no—
59C4⋊Dic53D411yescharacteristic
60C4⋊Dic53C2311yescharacteristic
61C4⋊Dic53C2312yes—
62#11719919—21no—
63#11719919—22no—
64#11719919—22no—
65#11719919—22no—
66#11719919D412yes—
67C4×Dic53—21no—
68C4×Dic53—22no—
69C4×Dic53C2312yes—
70C2×#88322C2311yescharacteristic
71C2×#88322C2311yescharacteristic
72C23×C53—21no—
73Q8×C53—42no—
74D4×C53—21no—
75D4×C53C2×Q812yes—
76C2×C4×C53—21no—
77C2×C4×C53—22no—
78C2×C4×C53—42no—
79C2×C4×C53C2×D411yescharacteristic
80C2×C4×C53C2×D411yescharacteristic
81C2×C4×C53C2411yescharacteristic
82C2×Dic53—21no—
83C2×Dic53—22no—
84C2×Dic53—21no—
85C2×Dic53—82no—
86C2×Dic53C2×D412yes—
87C2×Dic53C2×D412yes—
88Dic106—22no—
89Dic106—21no—
90Dic106—81no—
91Dic106C2×Q812yes—
92C8×C53—22no—
93#88322—21no—
94#88322C2×Q812yes—
95C22×C53—22no—
96C22×C53—41no—
97C22×C53—41no—
98C22×C53C22×D411yescharacteristic
99C4×C53—21no—
100C4×C53—41no—
101C4×C53—42no—
102C4×C53#13512yes—
103C4×C53C22×Q811yescharacteristic
104C4×C532−1+411yescharacteristic
105Dic53—22no—
106Dic53—22no—
107Dic53—22no—
108Dic53—42no—
109Dic53—41no—
110#3381—531nomaximal, Sylow
111C2×C53—42no—
112C2×C53D4×Q811yescharacteristic
113C2×C53C2×#13511yescharacteristic
114C2×C53#57711yescharacteristic
115#557—531no—
116#554—531no—
117#499—532no—
118#498—531no—
119#479—532no—
120#478—532no—
121#477—531no—
122#474D5311yescharacteristic
123C8×Q8—531no—
124C4×SD16—531no—
125C4×SD16—532no—
126C53#338111yescharacteristic, complemented, Sylow
127C2×SD16—531no—
128C4⋊Q8—531no—
129C4⋊Q8—1061no—
130C4⋊Q8C2×D5311yescharacteristic
131#125—532no—
132#125—532no—
133#125—1062no—
134C22⋊Q8—1062no—
135C22⋊Q8—1061no—
136C4×Q8—531no—
137C4×Q8—532no—
138C4×Q8—531no—
139C4×Q8—1061no—
140C4×D4—532no—
141C4×D4C2×D5311yescharacteristic
142C2×C4⋊C4—1061no—
143#107—532no—
144#107C2×D5312yes—
145#106—532no—
146#106—531no—
147#106—532no—
148C4⋊C8—532no—
149C4⋊C8C2×D5311yescharacteristic
150#103—532no—
151#103—531no—
152#102—531no—
153#102C2×D5312yes—
154C4×C8—532no—
155C4×C8—531no—
156C2×Q8—21no—
157C2×Q8—531no—
158C2×Q8—531no—
159C2×Q8—1061no—
160C2×D4C22×D5311yescharacteristic
161C22×C4—21no—
162C22×C4—1062no—
163SD16—534no—
164C2×C8—531no—
165C2×C8—531no—
166C2×C8C22×D5312yes—
167C4⋊C4—21no—
168C4⋊C4—531no—
169C4⋊C4—532no—
170C4⋊C4—531no—
171C4⋊C4—532no—
172C4⋊C4—1062no—
173C4⋊C4—1062no—
174C4⋊C4—1062no—
175C4⋊C4—1062no—
176C4⋊C4—1062no—
177C4⋊C4—1061no—
178C4⋊C4—1061no—
179C4⋊C4C22×D5312yes—
180C4⋊C4C22×D5311yescharacteristic
181C22⋊C4—21no—
182C22⋊C4—1062no—
183C42—532no—
184C42—1062no—
185C42—1061no—
186C42C22×D5311yescharacteristic
187C23—21no—
188Q8—42no—
189Q8—532no—
190Q8—1062no—
191Q8—1061no—
192Q8—4241no—
193D4—21no—
194D4Q8×D5312yes—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—42no—
198C2×C4—532no—
199C2×C4—532no—
200C2×C4—1061no—
201C2×C4—1061no—
202C2×C4—1062no—
203C2×C4—4242no—
204C2×C4D4×D5311yescharacteristic
205C2×C4D4×D5311yescharacteristic
206C2×C4C23×D5311yescharacteristic
207C8—22no—
208C8—532no—
209C8—1061no—
210C22—22no—
211C22—41no—
212C22—41no—
213C22C2×D4×D5311yescharacteristic
214C4—21no—
215C4—41no—
216C4—42no—
217C4—1062no—
218C4—1062no—
219C4—1062no—
220C4—2122no—
221C4—2121no—
222C4#1313323612yes—
223C4C2×Q8×D5311yescharacteristic
224C4#1313326411yescharacteristic
225C2—42no—
226C2#1681128611yescharacteristic
227C2C2×#1313323611yescharacteristic
228C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#474C2
socleC22×C53C22×D4
hypercenter#474D53
hypocenterC53#3381
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #16811348C2semidirect
3. #16811024C2semidirect
4. #16811019C2semidirect
5. #16810929C2semidirect
7. #16810919C2semidirect
8. #16810747C2semidirect
55. #11719923Q8semidirect
126. C53#3381semidirect