Finite Group Databasebeta

← #27296066

Subgroups — #27296066

Order 13568.

218 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 69 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#20276581C211yesmaximal, characteristic
2#20273630C211yesmaximal, characteristic
3#20273535C211yesmaximal, characteristic, complemented
4#20272781C211yesmaximal, characteristic, complemented
5#20272686C211yesmaximal, characteristic, complemented
6#20272683C211yesmaximal, characteristic
7C53×#1313C211yesmaximal, characteristic, complemented
8C2×#13133162C411yescharacteristic
9C8×Dic106—21no—
10C4×Dic212C2211yescharacteristic
11C4×D424C2211yescharacteristic
12#16810702—21no—
13#16810449C411yescharacteristic
14#16810428C411yescharacteristic
15#16810425C411yescharacteristic
16#16810374C2211yescharacteristic
17#16810374C2211yescharacteristic
18#16810355—21no—
19#16810352—21no—
20C53×#346C2211yescharacteristic
21C53×#346C2211yescharacteristic
22C53×#322C2211yescharacteristic
23C2×Dic212C2×C411yescharacteristic
24C2×D424C2×C411yescharacteristic
25C2×#11719912—41no—
26C4×Dic106—21no—
27C4×D212—21no—
28#13133162—22no—
29#13133151C2×C411yescharacteristic
30#13133151C2×C411yescharacteristic
31C4⋊#88322—21no—
32C4⋊#88322—21no—
33#13133146—41no—
34#13133142—21no—
35#13133139—21no—
36#13133129—41no—
37C8×Dic53—41no—
38C2×C16×C53—21no—
39C2×C16×C53C2×C411yescharacteristic
40C2×C16×C53C2×C411yescharacteristic
41C4×C8×C53D411yescharacteristic
42C4×C8×C53D411yescharacteristic
43C4×C8×C53C2311yescharacteristic
44C2×D212—21no—
45C2×C4×D53—41no—
46C2×Dic106—21no—
47C2×C8×C53—21no—
48C2×C8×C53—21no—
49C2×C8×C53C22×C411yescharacteristic
50C2×C8×C53#4111yescharacteristic
51C42×C53C2×D411yescharacteristic
52#11719921—41no—
53C4⋊Dic53—21no—
54C4⋊Dic53—21no—
55#11719919—41no—
56C4×Dic53—41no—
57C2×#88322—41no—
58C2×#88322—41no—
59Dic212—21no—
60Dic212M1612yescomplemented
61D424—21no—
62D424—21no—
63#11719912—82no—
64C16×C53—22no—
65C16×C53—21no—
66C16×C53—21no—
67C22×D53—41no—
68C2×C4×C53C4×D411yescharacteristic
69C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
70C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
71C2×Dic53—41no—
72C2×Dic53—41no—
73C2×Dic53—41no—
74D212—41no—
75D212—41no—
76C4×D53—81no—
77C4×D53—81no—
78Dic106—22no—
79Dic106—41no—
80C8×C53—21no—
81C8×C53—41no—
82C8×C53—41no—
83C8×C53C2×M1611yescharacteristic
84C8×C53#13111yescharacteristic
85#88322—81no—
86#88322—81no—
87#12080—531nomaximal, Sylow
88C22×C53#44211yescharacteristic
89C2×D53—81no—
90C2×D53—81no—
91C2×D53—81no—
92C4×C53—21no—
93C4×C53—21no—
94C4×C53#43511yescharacteristic
95C4×C53#44311yescharacteristic
96Dic53—42no—
97Dic53—81no—
98Dic53—81no—
99#2175—531no—
100#1594—531no—
101#1579—531no—
102#1368—531no—
103#1334—531no—
104#1333—531no—
105#1313D5311yescharacteristic, complemented
106C2×C53#158611yescharacteristic
107C2×C53#218011yescharacteristic
108C2×C53AGL(1,ℤ/16)11yescharacteristic
109D53—161no—
110C2×SD32—531no—
111C8×Q8—1061no—
112C4×Q16—531no—
113C4×D8—531no—
114#436—1061no—
115#366—531no—
116#358—531no—
117#357—531no—
118#346C2×D5311yescharacteristic
119#346C2×D5311yescharacteristic
120#334—531no—
121#334—531no—
122Q8⋊C8—1061no—
123D4⋊C8—1061no—
124#322C2×D5311yescharacteristic
125C53#1208011yescharacteristic, complemented, Sylow
126C2×Q16—531no—
127C2×D8—531no—
128C2×M16—2121no—
129C4×Q8—1061no—
130C4×D4—1061no—
131SD32—1062no—
132C2×C16—21no—
133C2×C16C4×D5311yescharacteristic
134C2×C16C4×D5311yescharacteristic
135#107—531no—
136#107—531no—
137C4⋊C8—1061no—
138C4⋊C8—1061no—
139C4⋊C8—2121no—
140#103—1061no—
141#102—1061no—
142C22⋊C8—2121no—
143C4×C8—2121no—
144C4×C8D21211yescharacteristic
145C4×C8D21211yescharacteristic
146C4×C8C22×D5311yescharacteristic
147C2×Q8—1061no—
148C2×D4—1061no—
149C22×C4—2121no—
150Q16—532no—
151Q16—1061no—
152D8—1061no—
153D8—1061no—
154M16—4242no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—2121no—
158C2×C8—2121no—
159C2×C8C2×C4×D5311yescharacteristic
160C2×C8#1171994511yescharacteristic
161C4⋊C4—1061no—
162C4⋊C4—1061no—
163C4⋊C4—2121no—
164C22⋊C4—2121no—
165C42—2121no—
166C42C2×D21211yescharacteristic
167C16—22no—
168C16—21no—
169C16—21no—
170C23—2121no—
171Q8—1062no—
172Q8—2121no—
173D4—2121no—
174D4—2121no—
175C2×C4—4241no—
176C2×C4—4241no—
177C2×C4—2121no—
178C2×C4—2121no—
179C2×C4—2121no—
180C2×C4C4×D21211yescharacteristic
181C2×C4#1313324511yescharacteristic
182C2×C4#1313324511yescharacteristic
183C8—21no—
184C8—41no—
185C8—41no—
186C8—4241no—
187C8—4241no—
188C8M16×D5311yescharacteristic
189C8#1313322511yescharacteristic
190C22—4241no—
191C22—4241no—
192C22—4241no—
193C22#1681073611yescharacteristic
194C4—21no—
195C4—21no—
196C4—4241no—
197C4—4241no—
198C4—2122no—
199C4#1681069611yescharacteristic
200C4#1681074111yescharacteristic
201C2—8481no—
202C2#2027358111yescharacteristic
203C2#2027660811yescharacteristic
204C2#2027661511yescharacteristic
subgroupG/H
centerC22#16810736
commutatorC2×C8×C53C22×C4
FrattiniC4×C8C22×D53
FittingC53×#1313C2
socleC22×C53#442
prime radicalC2×D424C2×C4
cosocleC4×C8×C53C23
hypercenter#1313D53
hypocenterC53#12080
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #322 ◁ #1313

Complements

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

NH ≅ G/Ntype
3. #20273535C2semidirect
4. #20272781C2semidirect
5. #20272686C2semidirect
7. C53×#1313C2semidirect
60. Dic212M16semidirect
105. #1313D53semidirect
125. C53#12080semidirect

Maximal subgroups

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

subgroupindex
#202765812
#202736302
#202735352
#202727812
#202726862
#202726832
C53×#13132
#1208053