Finite Group Databasebeta

← #28298481

Subgroups — #28298481

Order 13568.

284 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C4×#13133278C211yesmaximal, characteristic
2#20289957C211yesmaximal, characteristic
3#20275650C211yesmaximal, characteristic
4#16811640—21no—
5C2×#13133278C412yes—
6C4×#11719940—21no—
7#16811503—21no—
8#16811469—21no—
9#16811468C412yes—
10C2×C4×Dic106C2211yescharacteristic
11C2×C4×#88322C411yescharacteristic
12#16810377—21no—
13#16810377—21no—
14C2×C4⋊C8×C53C411yescharacteristic
15C2×#11719940—22no—
16#13133284—41no—
17#13133278—22no—
18#13133278C814yescomplemented
19C22⋊#88324—22no—
20#13133272—41no—
21C4×#88324—41no—
22C4×#88324—41no—
23C22×Dic106C2×C411yescharacteristic
24C22×#88322—21no—
25C4×Dic106—41no—
26C4×Dic106—41no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919D411yescharacteristic
29C2×#11719919Q811yescharacteristic
30C2×C4×Dic53D411yescharacteristic
31C2×C4×Dic53D411yescharacteristic
32C4×#88322—41no—
33C22×C8×C53—21no—
34C2×C42×C53C2×C411yescharacteristic
35C4⋊C8×C53—41no—
36C22×C4×C53C4211yescharacteristic
37C22×C4×C53C22⋊C411yescharacteristic
38C22×C4×C53C4⋊C411yescharacteristic
39C22×Dic53C22⋊C411yescharacteristic
40C22×Dic53C22⋊C411yescharacteristic
41C22×Dic53C4⋊C411yescharacteristic
42C22×Dic53C4⋊C411yescharacteristic
43C2×Dic106—41no—
44C2×Dic106—41no—
45C2×Dic106C2×C812yes—
46C2×Dic106M1612yes—
47#11719940—24no—
48#11719940—42no—
49C2×#88324—42no—
50C2×#88324—42no—
51C2×C8×C53—42no—
52C2×C8×C53—41no—
53C42×C53—21no—
54C42×C53—21no—
55C4⋊Dic53—41no—
56#11719919—41no—
57#11719919—41no—
58C4×Dic53—21no—
59C4×Dic53—21no—
60C4×Dic53—21no—
61C4×Dic53—21no—
62C2×#88322—22no—
63C2×#88322—41no—
64C2×#88322—41no—
65C23×C53#9511yescharacteristic
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—41no—
72C2×C4×C53C4×C811yescharacteristic
73C2×C4×C53#9711yescharacteristic
74C2×C4×C53#9911yescharacteristic
75C2×C4×C53#9911yescharacteristic
76C2×Dic53—21no—
77C2×Dic53—21no—
78C2×Dic53—21no—
79C2×Dic53—21no—
80C2×Dic53—21no—
81C2×Dic53—21no—
82C2×Dic53—41no—
83C2×Dic53—41no—
84C2×Dic53C22⋊C812yes—
85C2×Dic53C4⋊C812yes—
86Dic106—42no—
87Dic106—42no—
88#88324—44no—
89#88324—82no—
90C8×C53—81no—
91#88322—42no—
92#7206—531nomaximal, Sylow
93C22×C53—21no—
94C22×C53—21no—
95C22×C53#33611yescharacteristic
96C22×C53#34211yescharacteristic
97C22×C53#34411yescharacteristic
98C4×C53—21no—
99C4×C53—21no—
100C4×C53—21no—
101C4×C53—41no—
102C4×C53#35511yescharacteristic
103C4×C53#35611yescharacteristic
104Dic53—22no—
105Dic53—22no—
106Dic53—41no—
107Dic53—41no—
108#1984—531no—
109C4×#101—531no—
110#1461—531no—
111C2×C53—21no—
112C2×C53—21no—
113C2×C53#128211yescharacteristic
114C2×C53#139411yescharacteristic
115C2×C53#139511yescharacteristic
116C2×C4×Q8—531no—
117#431—1061no—
118C2×C4⋊C8#615711yescharacteristic
119C2×#101—532no—
120C4×M16—1061no—
121C2×C4×C8—531no—
122#343—1061no—
123#336—1061no—
124#336—1061no—
125#324—532no—
126#324—1061no—
127C53#720611yescharacteristic, complemented, Sylow
128C22×Q8—531no—
129C2×M16—1062no—
130C22×C8—21no—
131C22×C8—1061no—
132C4×Q8—2121no—
133C4×Q8—2121no—
134C2×C4⋊C4—531no—
135C2×C4⋊C4—531no—
136C2×C4⋊C4—531no—
137C2×C42—531no—
138C2×C42—531no—
139C2×C42C2×#615711yescharacteristic
140C4⋊C8—41no—
141C4⋊C8—2121no—
142#101—534no—
143#101—1062no—
144C22⋊C8—1062no—
145#97—2121no—
146C4×C8—2121no—
147C4×C8—2121no—
148C4×C8—2121no—
149C2×Q8—532no—
150C2×Q8—532no—
151C2×Q8—2121no—
152C2×Q8—2121no—
153C22×C4—531no—
154C22×C4—531no—
155C22×C4—531no—
156C22×C4—531no—
157C22×C4C4×#615711yescharacteristic
158C22×C4C4⋊#615711yescharacteristic
159C22×C4C22⋊#615711yescharacteristic
160M16—1064no—
161M16—2122no—
162C2×C8—42no—
163C2×C8—41no—
164C2×C8—1062no—
165C2×C8—2122no—
166C2×C8—2121no—
167C2×C8—2122no—
168C2×C8—2121no—
169C4⋊C4—2121no—
170C4⋊C4—2121no—
171C4⋊C4—2121no—
172C42—21no—
173C42—21no—
174C42—1061no—
175C42—1061no—
176C42—1061no—
177C42—1061no—
178C23(C2×#6157)⋊C411yescharacteristic
179Q8—2122no—
180Q8—2122no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—532no—
188C2×C4—532no—
189C2×C4—1061no—
190C2×C4—1061no—
191C2×C4—1061no—
192C2×C4—1061no—
193C2×C4—1061no—
194C2×C4—1061no—
195C2×C4—2121no—
196C2×C4—2121no—
197C2×C4C8×#615711yescharacteristic
198C2×C4#1313327111yescharacteristic
199C2×C4#1313327511yescharacteristic
200C2×C4#1313327611yescharacteristic
201C8—81no—
202C8—4242no—
203C8—2122no—
204C8—2124no—
205C22—21no—
206C22—21no—
207C22(C2×#6157)⋊C811yescharacteristic
208C22#1681150111yescharacteristic
209C22#1681150511yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—41no—
214C4—1062no—
215C4—1062no—
216C4—2121no—
217C4—2121no—
218C4#1681153111yescharacteristic
219C4#1681153311yescharacteristic
220C2—21no—
221C2—21no—
222C2#2028955711yescharacteristic
223C2#2028981311yescharacteristic
224C2#2028981611yescharacteristic
subgroupG/H
centerC2×C4#13133275
commutatorC2×C4×C53C4×C8
FrattiniC2×C42C2×#6157
FittingC2×C4⋊C8×C53C4
socleC22×C53#342
prime radicalC23×C53#95
cosocleC2×C4×Dic106C22
hypercenterC2×C4⋊C8#6157
hypocenterC53#7206
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4⋊C8

Complements

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

NH ≅ G/Ntype
18. #13133278C8semidirect
127. C53#7206semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C4×#131332782
#202899572
#202756502
#720653