Finite Group Databasebeta

← #20274768

Subgroups — #20274768

Order 6784.

256 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C2×#13133225C211yesmaximal, characteristic
2C2×#13133152C211yesmaximal, characteristic, complemented
3#16810659C211yesmaximal, characteristic, complemented
4#16810641C211yesmaximal, characteristic
5#16810639C211yesmaximal, characteristic
6#16810395C211yesmaximal, characteristic, complemented
7C53×#344C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9#13133225—21no—
10#13133225—21no—
11#13133225C412yescomplemented
12#13133225C412yescomplemented
13C2×#11719917C2211yescharacteristic
14C2×#11719912—21no—
15C22×#88322C2211yescharacteristic
16C2×C8×D53—21no—
17#13133176C2211yescharacteristic
18#13133152—21no—
19#13133152—21no—
20#13133150—21no—
21#13133148—21no—
22#13133145—22no—
23#13133143—22no—
24C2×M16×C53C2211yescharacteristic
25C2×M16×C53C2211yescharacteristic
26#117×C53C2211yescharacteristic
27C22×C4×C53C2311yescharacteristic
28C2×C22⋊D53—21no—
29#11719945—21no—
30#11719945—21no—
31#11719945C2×C412yes—
32#11719945D412yes—
33C2×D212Q811yescharacteristic
34C2×C4×D53—21no—
35C2×Dic106Q811yescharacteristic
36M16×C53—21no—
37M16×C53—21no—
38M16×C53—21no—
39M16×C53C2×C412yes—
40C2×C8×C53—21no—
41C2×C8×C53—21no—
42C4⋊C4×C53—21no—
43C22⋊C4×C53—21no—
44C42×C53—21no—
45C22⋊Dic53—21no—
46C4⋊Dic53—21no—
47C4×Dic53—21no—
48#11719917—21no—
49#11719917—21no—
50C2×#88322—21no—
51C2×#88322C2×C411yescharacteristic
52C2×#88322C2×C411yescharacteristic
53C2×#88322D411yescharacteristic
54C2×#88322D411yescharacteristic
55C2×#88322D411yescharacteristic
56C2×#88322D411yescharacteristic
57#11719912—22no—
58#11719912—22no—
59C8×D53—22no—
60C8×D53—22no—
61C23×C53#4111yescharacteristic
62C22×D53—21no—
63C2×C4×C53—41no—
64C2×C4×C53C22×C411yescharacteristic
65C2×C4×C53C2×D411yescharacteristic
66C2×C4×C53C2×D411yescharacteristic
67C2×C4×C53C2×D411yescharacteristic
68C2×C4×C53C2×Q811yescharacteristic
69C2×C4×C53#4111yescharacteristic
70C22⋊D53—22no—
71C22⋊D53—22no—
72C2×Dic53—21no—
73C2×Dic53—41no—
74D212—21no—
75D212C4⋊C412yes—
76C4×D53—22no—
77C4×D53—22no—
78Dic106—21no—
79Dic106C4⋊C412yes—
80C8×C53—22no—
81C8×C53—41no—
82#88322—41no—
83#88322C22⋊C411yescharacteristic
84#88322C22⋊C411yescharacteristic
85#88322C22⋊C411yescharacteristic
86#88322C22⋊C411yescharacteristic
87C22×C53—41no—
88C22×C53#11711yescharacteristic
89C22×C53#12111yescharacteristic
90C22×C53C22⋊Q811yescharacteristic
91C2×D53—22no—
92C2×D53—42no—
93C4×C53—81no—
94C4×C53C2×C22⋊C411yescharacteristic
95C4×C53C2×C4⋊C411yescharacteristic
96C4×C53#12111yescharacteristic
97C4×C53C22⋊Q811yescharacteristic
98Dic53—22no—
99Dic53—81no—
100#1814—531nomaximal, Sylow
101C2×C53—21no—
102C2×C53—21no—
103C2×C53—21no—
104C2×C53#38011yescharacteristic
105D53—42no—
106C2×#131—531no—
107C2×#108—531no—
108#427—531no—
109#421—531no—
110#421—531no—
111#344—531no—
112#344D5311yescharacteristic, complemented
113C53#181411yescharacteristic, complemented, Sylow
114C2×#41—531no—
115#131—532no—
116#131—532no—
117#131—1061no—
118#131—1061no—
119C2×M16—531no—
120C2×M16—1061no—
121C2×M16C2×D5311yescharacteristic
122C2×M16C2×D5311yescharacteristic
123C22×C8—531no—
124C22×C8—1061no—
125#117—531no—
126#117C2×D5311yescharacteristic
127#108—1061no—
128#108—1061no—
129#107—1061no—
130#106—1061no—
131#104—1062no—
132#104—1062no—
133#41—532no—
134#41—532no—
135#41—1061no—
136#41—1061no—
137C2×Q8—531no—
138C2×D4—531no—
139C2×D4—1061no—
140C22×C4—1061no—
141C22×C4C22×D5311yescharacteristic
142M16—21no—
143M16—21no—
144M16—21no—
145M16—1062no—
146M16—1062no—
147M16—1061no—
148M16—1061no—
149M16C4×D5312yes—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—531no—
153C2×C8—531no—
154C2×C8—531no—
155C2×C8—531no—
156C2×C8—531no—
157C2×C8—531no—
158C2×C8—1062no—
159C2×C8—1062no—
160C2×C8—1061no—
161C4⋊C4—21no—
162C4⋊C4—1061no—
163C22⋊C4—21no—
164C22⋊C4—1061no—
165C42—21no—
166C42—1061no—
167C23—1061no—
168C23#1171994511yescharacteristic
169Q8—532no—
170Q8—1061no—
171D4—532no—
172D4—1062no—
173D4—1062no—
174D4—1061no—
175C2×C4—41no—
176C2×C4—1061no—
177C2×C4—1062no—
178C2×C4—1062no—
179C2×C4—2121no—
180C2×C4C2×C4×D5311yescharacteristic
181C2×C4#1171994511yescharacteristic
182C2×C4D4×D5311yescharacteristic
183C2×C4D4×D5311yescharacteristic
184C2×C4D4×D5311yescharacteristic
185C2×C4Q8×D5311yescharacteristic
186C8—22no—
187C8—41no—
188C8—531no—
189C8—531no—
190C8—531no—
191C8—531no—
192C8—2121no—
193C22—41no—
194C22—1062no—
195C22—2122no—
196C22#1313317411yescharacteristic
197C22#1313319011yescharacteristic
198C22#1313319511yescharacteristic
199C4—81no—
200C4—1062no—
201C4—4241no—
202C4C22⋊C4×D5311yescharacteristic
203C4C4⋊C4×D5311yescharacteristic
204C4#1313319011yescharacteristic
205C4#1313319511yescharacteristic
206C2—21no—
207C2—21no—
208C2—21no—
209C2—2122no—
210C2#1681048211yescharacteristic
subgroupG/H
centerC4#13133190
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#344C2
socleC2×C53#380
hypercenter#344D53
hypocenterC53#1814
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
2. C2×#13133152C2semidirect
3. #16810659C2semidirect
6. #16810395C2semidirect
7. C53×#344C2semidirect
11. #13133225C4semidirect
12. #13133225C4semidirect
112. #344D53semidirect
113. C53#1814semidirect