Finite Group Databasebeta

← #20275261

Subgroups — #20275261

Order 6784.

340 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 79 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#16811389C211yesmaximal, characteristic, complemented
2#16811192C211yesmaximal, characteristic
3#16810605C211yesmaximal, characteristic, complemented
4C2×#13133133C211yesmaximal, characteristic, complemented
5C2×#13133133C211yesmaximal, characteristic, complemented
6#16810393C211yesmaximal, characteristic, complemented
7C53×#344C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9C22×D212C2211yescharacteristic
10#13133225—22no—
11#13133223—22no—
12C2×#11719917C2211yescharacteristic
13C2×#11719917C2211yescharacteristic
14M16×D53—22no—
15#13133214—21no—
16#13133205—21no—
17#13133192—22no—
18#13133187C414yes—
19#13133135—21no—
20#13133133—22no—
21#13133133—22no—
22#13133133—21no—
23C2×M16×C53C2211yescharacteristic
24C2×M16×C53C2211yescharacteristic
25#117×C53C2211yescharacteristic
26C23×D53C2×C412yes—
27C22×C4×C53C2311yescharacteristic
28C2×C22⋊D53C2×C412yes—
29#11719945—21no—
30#11719945—21no—
31#11719945—21no—
32#11719945—21no—
33C2×D212—21no—
34C2×D212—21no—
35C2×D212—21no—
36C2×D212—21no—
37C2×D212—21no—
38C2×D212—21no—
39C2×D212—24no—
40C2×D212D411yescharacteristic
41C2×C4×D53D412yes—
42C2×Dic106D411yescharacteristic
43M16×C53—21no—
44M16×C53—21no—
45M16×C53—21no—
46M16×C53D412yescomplemented
47C2×C8×C53—21no—
48C2×C8×C53—21no—
49C4⋊C4×C53—21no—
50C22⋊C4×C53C2×C412yes—
51C42×C53—21no—
52#11719921—24no—
53#11719917—21no—
54#11719917—21no—
55#11719917—21no—
56#11719917D412yescomplemented
57C2×#88322—21no—
58C2×#88322—21no—
59#11719912—24no—
60C8×D53—24no—
61C23×C53C22×C411yescharacteristic
62C22×D53—22no—
63C22×D53—22no—
64C22×D53—22no—
65C22×D53—82no—
66C22×D53C22⋊C412yes—
67C2×C4×C53—22no—
68C2×C4×C53C2×D411yescharacteristic
69C2×C4×C53C2×D411yescharacteristic
70C2×C4×C53C2×D411yescharacteristic
71C2×C4×C53C2×D411yescharacteristic
72C2×C4×C53#4111yescharacteristic
73C2×C4×C53#4111yescharacteristic
74C22⋊D53—22no—
75C22⋊D53—22no—
76C2×Dic53C22⋊C412yes—
77D212—21no—
78D212—21no—
79D212—41no—
80D212—41no—
81D212—41no—
82D212—41no—
83D212—84no—
84C4×D53—22no—
85C4×D53—22no—
86Dic106—21no—
87Dic106—21no—
88C8×C53—22no—
89C8×C53—41no—
90#88322—22no—
91#88322—41no—
92C22×C53—41no—
93C22×C53C2×C22⋊C411yescharacteristic
94C22×C53C4×D411yescharacteristic
95C22×C53C4×D411yescharacteristic
96C2×D53—22no—
97C2×D53—42no—
98C2×D53—44no—
99C2×D53—42no—
100C2×D53—82no—
101C2×D53—82no—
102C4×C53—42no—
103C4×C53#12011yescharacteristic
104C4×C53#12111yescharacteristic
105C4×C53#12111yescharacteristic
106C4×C53#12311yescharacteristic
107Dic53—22no—
108#1914—531nomaximal, Sylow
109C2×C53—21no—
110C2×C53—21no—
111C2×C53—21no—
112C2×C53#38611yescharacteristic
113D53—42no—
114D53—82no—
115#568—531no—
116#535—531no—
117#413—531no—
118C2×#100—531no—
119C2×#100—531no—
120#344—531no—
121#344D5311yescharacteristic, complemented
122C53#191411yescharacteristic, complemented, Sylow
123C2×#41—531no—
124C22×D4—531no—
125#131—1062no—
126#131—1062no—
127C2×M16—531no—
128C2×M16—531no—
129C2×M16—1062no—
130C2×M16C2×D5311yescharacteristic
131C2×M16C2×D5311yescharacteristic
132#127—1061no—
133#124—1061no—
134#121—1062no—
135#120—534no—
136#117C2×D5311yescharacteristic
137#101—1061no—
138#100—1062no—
139#100—1061no—
140#100—1062no—
141C24—532no—
142#41—1061no—
143#41—1061no—
144#41—1061no—
145#41—1061no—
146C2×Q8—531no—
147C2×D4—532no—
148C2×D4—531no—
149C2×D4—1064no—
150C2×D4—1061no—
151C2×D4—1061no—
152C2×D4—1061no—
153C2×D4—1061no—
154C2×D4—1061no—
155C2×D4—1061no—
156C22×C4—532no—
157C22×C4C22×D5311yescharacteristic
158M16—21no—
159M16—21no—
160M16—21no—
161M16—532no—
162M16—1064no—
163M16—1061no—
164M16—1061no—
165M16—1061no—
166M16D21212yescomplemented
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—1064no—
170C2×C8—1061no—
171C2×C8—1061no—
172C4⋊C4—21no—
173C22⋊C4—1064no—
174C22⋊C4C4×D5312yes—
175C42—21no—
176C23—532no—
177C23—1062no—
178C23—1062no—
179C23—1062no—
180C23—4242no—
181C23C2×C4×D5311yescharacteristic
182Q8—1061no—
183Q8—1061no—
184D4—1062no—
185D4—1062no—
186D4—1061no—
187D4—1061no—
188D4—4244no—
189D4—2121no—
190D4—2121no—
191D4—2121no—
192D4—2121no—
193C2×C4—22no—
194C2×C4—532no—
195C2×C4—1062no—
196C2×C4—1062no—
197C2×C4C2×D21211yescharacteristic
198C2×C4#1171994511yescharacteristic
199C2×C4D4×D5311yescharacteristic
200C2×C4D4×D5311yescharacteristic
201C2×C4D4×D5311yescharacteristic
202C2×C4#1171994911yescharacteristic
203C8—22no—
204C8—41no—
205C8—1062no—
206C8—2121no—
207C22—41no—
208C22—1062no—
209C22—4242no—
210C22—4242no—
211C22—2124no—
212C22—2122no—
213C22—2122no—
214C22C22⋊C4×D5311yescharacteristic
215C22#1313317911yescharacteristic
216C22C4×D21211yescharacteristic
217C4—42no—
218C4—1062no—
219C4#1313318711yescharacteristic
220C4#1313319011yescharacteristic
221C4#1313319211yescharacteristic
222C4#1313320211yescharacteristic
223C2—21no—
224C2—21no—
225C2—21no—
226C2—4242no—
227C2—2122no—
228C2#1681050111yescharacteristic
subgroupG/H
centerC2#16810501
commutatorC23×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#344C2
socleC2×C53#386
hypercenter#344D53
hypocenterC53#1914
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
1. #16811389C2semidirect
3. #16810605C2semidirect
4. C2×#13133133C2semidirect
5. C2×#13133133C2semidirect
6. #16810393C2semidirect
7. C53×#344C2semidirect
46. M16×C53D4semidirect
56. #11719917D4semidirect
121. #344D53semidirect
122. C53#1914semidirect
166. M16D212semidirect