Finite Group Databasebeta

← #20274665

Subgroups — #20274665

Order 6784.

548 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 97 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#16811459C211yesmaximal, characteristic
2#16810607C211yesmaximal, characteristic, complemented
3#16810605C212yesmaximal, complemented
4#16810589C212yesmaximal, complemented
5C53×#408C211yesmaximal, characteristic, complemented
6#13133265—21no—
7#13133264—22no—
8#13133263—22no—
9#13133262—21no—
10#13133262C414yes—
11C2×#11719945—22no—
12C2×#11719945C2212yes—
13#41×D53—24no—
14C2×#11719917C2212yescomplemented
15#13133136—21no—
16#13133135—22no—
17#13133134—21no—
18#13133133—22no—
19#13133129—24no—
20C2×#41×C53C2211yescharacteristic
21C2×M16×C53C2211yescharacteristic, complemented
22C22×C8×C53C2211yescharacteristic, complemented
23C22⋊C8×C53—22no—
24#41×C53—22no—
25#41×C53—22no—
26C2×Q8×C53D411yescharacteristic
27C2×D4×C53C2×C412yes—
28C2×D4×C53D411yescharacteristic
29C22×C4×C53D412yes—
30C22×C4×C53C2311yescharacteristic
31C2×C22⋊D53—24no—
32C2×C22⋊D53C2×C414yes—
33#11719949—24no—
34Q8×D53—24no—
35#11719947—24no—
36#11719947—28no—
37D4×D53—24no—
38D4×D53—28no—
39#11719945—24no—
40#11719945—24no—
41#11719945—24no—
42#11719945—22no—
43#11719945—22no—
44#11719945—22no—
45#11719945—22no—
46C2×D212—22no—
47C2×D212D412yes—
48C2×C4×D53—24no—
49C2×C4×D53D414yes—
50C2×Dic106—22no—
51C2×Dic106D412yes—
52M16×C53—22no—
53C2×C8×C53—21no—
54C2×C8×C53—21no—
55C2×C8×C53—22no—
56#11719917—22no—
57#11719917—22no—
58C2×#88322—22no—
59C23×C53C22⋊C412yes—
60C23×C53C22×C411yescharacteristic
61C22×D53—24no—
62C22×D53C22⋊C414yes—
63Q8×C53—22no—
64D4×C53—24no—
65D4×C53—22no—
66C2×C4×C53—22no—
67C2×C4×C53—22no—
68C2×C4×C53C22⋊C412yes—
69C2×C4×C53C2×D411yescharacteristic
70C2×C4×C53C2×D412yes—
71C2×C4×C53C2×D412yes—
72C2×C4×C53C2×D411yescharacteristic
73C22⋊D53—28no—
74C22⋊D53—24no—
75C22⋊D53—24no—
76C2×Dic53—24no—
77C2×Dic53C22⋊C414yes—
78D212—24no—
79D212—22no—
80D212—22no—
81C4×D53—28no—
82C4×D53—24no—
83C4×D53—24no—
84Dic106—24no—
85Dic106—22no—
86Dic106—22no—
87C8×C53—41no—
88C8×C53—41no—
89#88322—42no—
90C22×C53—22no—
91C22×C53—41no—
92C22×C53—42no—
93C22×C53C2×C22⋊C412yes—
94C22×C53C2×C22⋊C411yescharacteristic
95C2×D53—24no—
96C2×D53—44no—
97C2×D53—84no—
98C4×C53—22no—
99C4×C53#12011yescharacteristic
100C4×C53#12011yescharacteristic
101C4×C53#12012yes—
102Dic53—24no—
103#1791—531nomaximal, Sylow
104C2×C53—22no—
105C2×C53—21no—
106C2×C53—42no—
107C2×C53#37911yescharacteristic
108D53—44no—
109#585—531no—
110#413—531no—
111#413—532no—
112#408—532no—
113#408D5311yescharacteristic, complemented
114C53#179111yescharacteristic, complemented, Sylow
1152−1+4—1062no—
1162−1+4—1061no—
117H(5,𝔽2)—534no—
118H(5,𝔽2)—1062no—
119H(5,𝔽2)—1061no—
120C2×#41—532no—
121C2×#41—1064no—
122C2×#41—1062no—
123C2×#41C2×D5311yescharacteristic
124C2×M16—532no—
125C2×M16C2×D5311yescharacteristic, complemented
126C22×C8C2×D5311yescharacteristic, complemented
127#101—1062no—
128#101—1061no—
129#100—1062no—
130#100—1061no—
131C22⋊C8—22no—
132C22⋊C8—1064no—
133#41—22no—
134#41—22no—
135#41—1064no—
136#41—1064no—
137#41—1068no—
138#41—1064no—
139#41—1064no—
140#41—1062no—
141#41—1062no—
142#41—1062no—
143#41—1062no—
144#41—1064no—
145C2×Q8—532no—
146C2×Q8—1062no—
147C2×Q8—1064no—
148C2×Q8D21211yescharacteristic
149C2×D4—534no—
150C2×D4—532no—
151C2×D4—1062no—
152C2×D4—1068no—
153C2×D4—1064no—
154C2×D4—1064no—
155C2×D4C4×D5312yes—
156C2×D4D21211yescharacteristic
157C22×C4—534no—
158C22×C4—1064no—
159C22×C4C22⋊D5312yes—
160C22×C4C22×D5311yescharacteristic
161M16—22no—
162M16—1062no—
163M16—1062no—
164C2×C8—22no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—1062no—
168C23—534no—
169C23—1064no—
170C23#1171992112yes—
171C23C2×C4×D5311yescharacteristic
172Q8—22no—
173Q8—1064no—
174Q8—1062no—
175Q8—1062no—
176D4—24no—
177D4—22no—
178D4—1064no—
179D4—1068no—
180D4—1064no—
181D4—1064no—
182D4—1062no—
183D4—1062no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—534no—
187C2×C4—1068no—
188C2×C4—1064no—
189C2×C4—1064no—
190C2×C4—1064no—
191C2×C4#1171992112yes—
192C2×C4C2×D21211yescharacteristic
193C2×C4D4×D5312yes—
194C2×C4D4×D5312yes—
195C2×C4C2×C22⋊D5311yescharacteristic
196C8—41no—
197C8—41no—
198C8—2122no—
199C22—22no—
200C22—41no—
201C22—42no—
202C22—1064no—
203C22—4244no—
204C22—2124no—
205C22C22⋊C4×D5312yes—
206C22C2×#1171992111yescharacteristic
207C4—22no—
208C4—1064no—
209C4#1313318511yescharacteristic
210C4#1313318511yescharacteristic
211C4#1313318712yes—
212C2—21no—
213C2—22no—
214C2—42no—
215C2—2124no—
216C2#1681047811yescharacteristic
subgroupG/H
centerC4#13133185
commutatorC23×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#408C2
socleC2×C53#379
hypercenter#408D53
hypocenterC53#1791
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #16810607C2semidirect
3. #16810605C2semidirect
4. #16810589C2semidirect
5. C53×#408C2semidirect
14. C2×#11719917C22semidirect
21. C2×M16×C53C22semidirect
22. C22×C8×C53C22semidirect
113. #408D53semidirect
114. C53#1791semidirect
125. C2×M16C2×D53semidirect
126. C22×C8C2×D53semidirect