Finite Group Databasebeta

← #20284685

Subgroups — #20284685

Order 6784.

386 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 133 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#16811194C211yesmaximal, characteristic
2#16811106C211yesmaximal, characteristic
3#16810965C212yesmaximal, complemented
4#16810965C212yesmaximal, complemented
5#16810959C212yesmaximal, complemented
6#16810954C212yesmaximal, complemented
7#16810665C211yesmaximal, characteristic, complemented
8#16810661C211yesmaximal, characteristic, complemented
9#16810633C211yesmaximal, characteristic, complemented
10C2×#13133142C211yesmaximal, characteristic, complemented
11C53×#408C211yesmaximal, characteristic
12C22×Dic106C2211yescharacteristic
13#13133215—21no—
14#13133204—21no—
15#13133198—22no—
16#13133196C2212yes—
17#13133196C2212yes—
18Q8×Dic53—21no—
19D4×Dic53—22no—
20D4×Dic53—21no—
21#13133176C2211yescharacteristic
22#13133176C2212yes—
23#13133176C2211yescharacteristic
24C2×C4⋊Dic53C2211yescharacteristic
25C2×C4⋊Dic53C2212yes—
26#13133151C2211yescharacteristic
27#13133151C2211yescharacteristic
28#13133151C2212yes—
29#13133149C2211yescharacteristic
30#13133149C2211yescharacteristic
31#13133149C2212yes—
32#13133142C2212yes—
33#13133142C2212yes—
34#13133142C2212yes—
35#13133142C2212yes—
36C2×#41×C53C2211yescharacteristic
37C2×M16×C53C2211yescharacteristic
38C22×C8×C53C2211yescharacteristic
39C22⋊C8×C53C2212yes—
40C22⋊C8×C53C2212yes—
41#41×C53—22no—
42#41×C53—22no—
43C2×Q8×C53D411yescharacteristic
44C2×D4×C53D412yes—
45C2×D4×C53D411yescharacteristic
46C22×C4×C53C2312yes—
47C22×C4×C53C2311yescharacteristic
48C22×Dic53—21no—
49C22×Dic53—22no—
50C22×Dic53—21no—
51C2×Dic106—22no—
52C2×Dic106—22no—
53C2×Dic106—21no—
54C2×Dic106C2312yes—
55M16×C53—22no—
56C2×C8×C53—22no—
57C2×C8×C53C2311yescharacteristic
58C2×C8×C53C2311yescharacteristic
59C2×C8×C53C2312yes—
60C22⋊Dic53—22no—
61C22⋊Dic53—21no—
62C22⋊Dic53—21no—
63C22⋊Dic53—22no—
64C22⋊Dic53—22no—
65C4⋊Dic53—22no—
66C4⋊Dic53—21no—
67C4⋊Dic53C2312yes—
68C4⋊Dic53C2311yescharacteristic
69C4⋊Dic53C2311yescharacteristic
70C4⋊Dic53C2311yescharacteristic
71C4⋊Dic53C2311yescharacteristic
72#11719919—22no—
73#11719919—22no—
74C4×Dic53—22no—
75C4×Dic53—21no—
76C4×Dic53—21no—
77C23×C53C2×D412yes—
78C23×C53C2×D411yescharacteristic
79Q8×C53—22no—
80D4×C53—24no—
81D4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53C2×D412yes—
85C2×C4×C53C2×D411yescharacteristic
86C2×C4×C53#4112yes—
87C2×C4×C53#4112yes—
88C2×C4×C53C2411yescharacteristic
89C2×Dic53—22no—
90C2×Dic53—21no—
91C2×Dic53—22no—
92C2×Dic53—21no—
93C2×Dic53—21no—
94C2×Dic53—21no—
95C2×Dic53—42no—
96C2×Dic53—81no—
97C2×Dic53—82no—
98Dic106—22no—
99Dic106—22no—
100Dic106—42no—
101C8×C53—21no—
102C8×C53—21no—
103C8×C53—22no—
104C22×C53—21no—
105C22×C53—21no—
106C22×C53—22no—
107C22×C53—22no—
108C22×C53—22no—
109C22×C53#12312yes—
110C22×C53C22×D411yescharacteristic
111C4×C53—22no—
112C4×C53#12312yes—
113C4×C53C2×#4111yescharacteristic
114C4×C53C2×#4111yescharacteristic
115Dic53—42no—
116Dic53—41no—
117Dic53—42no—
118Dic53—41no—
119Dic53—41no—
120Dic53—41no—
121#3097—531nomaximal, Sylow
122C2×C53—22no—
123C2×C53—42no—
124C2×C53C2×#12311yescharacteristic
125C2×C53#57711yescharacteristic
126C2×C53#57811yescharacteristic
127#536—531no—
128#520—531no—
129#485—532no—
130#485—532no—
131#484—532no—
132#483—532no—
133#428—531no—
134#427—531no—
135#419—531no—
136C2×#103—531no—
137#408D5311yescharacteristic
138C53#309711yescharacteristic, complemented, Sylow
139C2×#41C2×D5311yescharacteristic
140C22×Q8—531no—
141C2×M16C2×D5311yescharacteristic
142C22×C8C2×D5311yescharacteristic
143C4⋊Q8—1061no—
144#124—1061no—
145#123—1062no—
146C22⋊Q8—532no—
147C22⋊Q8—532no—
148C4×Q8—1061no—
149C4×D4—1062no—
150C4×D4—1061no—
151#117—531no—
152#117—532no—
153#117—531no—
154C2×C4⋊C4—531no—
155C2×C4⋊C4—532no—
156#107—531no—
157#107—531no—
158#107—532no—
159#106—531no—
160#106—531no—
161#106—532no—
162#103—532no—
163#103—532no—
164#103—532no—
165#103—532no—
166C22⋊C8C2×D5312yes—
167C22⋊C8C2×D5312yes—
168#41—22no—
169#41—22no—
170C2×Q8—532no—
171C2×Q8—1061no—
172C2×Q8—1062no—
173C2×Q8—1062no—
174C2×Q8D21211yescharacteristic
175C2×D4D21212yes—
176C2×D4D21211yescharacteristic
177C22×C4—1062no—
178C22×C4—1061no—
179C22×C4—1061no—
180C22×C4C22×D5312yes—
181C22×C4C22×D5311yescharacteristic
182M16—22no—
183C2×C8—22no—
184C2×C8C22×D5311yescharacteristic
185C2×C8C22×D5312yes—
186C2×C8C22×D5311yescharacteristic
187C4⋊C4—532no—
188C4⋊C4—531no—
189C4⋊C4—531no—
190C4⋊C4—531no—
191C4⋊C4—531no—
192C4⋊C4—1062no—
193C4⋊C4—1061no—
194C4⋊C4—1062no—
195C4⋊C4—1062no—
196C22⋊C4—1062no—
197C22⋊C4—1061no—
198C22⋊C4—1061no—
199C22⋊C4—1062no—
200C22⋊C4—1062no—
201C42—1062no—
202C42—1061no—
203C42—1061no—
204C23C2×D21212yes—
205C23C2×D21211yescharacteristic
206Q8—22no—
207Q8—1062no—
208Q8—1062no—
209Q8—2122no—
210D4—24no—
211D4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—1061no—
215C2×C4—1061no—
216C2×C4—1061no—
217C2×C4—1062no—
218C2×C4—1061no—
219C2×C4—1062no—
220C2×C4—4241no—
221C2×C4—4242no—
222C2×C4—2122no—
223C2×C4C2×D21212yes—
224C2×C4C2×D21211yescharacteristic
225C2×C4#1171994712yes—
226C2×C4#1171994712yes—
227C2×C4C23×D5311yescharacteristic
228C8—21no—
229C8—21no—
230C8—22no—
231C22—21no—
232C22—21no—
233C22—22no—
234C22—22no—
235C22—22no—
236C22#1313320112yes—
237C22C22×D21211yescharacteristic
238C4—22no—
239C4—2122no—
240C4—2122no—
241C4—2121no—
242C4—2121no—
243C4—2121no—
244C4—2121no—
245C4#1313320112yes—
246C4C2×#1171994711yescharacteristic
247C4C2×#1171994711yescharacteristic
248C2—22no—
249C2—42no—
250C2C2×#1313320111yescharacteristic
251C2#1681143511yescharacteristic
252C2#1681144011yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#408C2
socleC22×C53C22×D4
hypercenter#408D53
hypocenterC53#3097
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
3. #16810965C2semidirect
4. #16810965C2semidirect
5. #16810959C2semidirect
6. #16810954C2semidirect
7. #16810665C2semidirect
8. #16810661C2semidirect
9. #16810633C2semidirect
10. C2×#13133142C2semidirect
138. C53#3097semidirect