Finite Group Databasebeta

← #27271476

Subgroups — #27271476

Order 13568.

216 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 43 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#20275516C211yesmaximal, characteristic
2#20272661C211yesmaximal, characteristic
3C53×#1325C211yesmaximal, characteristic
4C2×#13133217C411yescharacteristic
5C2×#13133140—21no—
6C2×#13133140—21no—
7C2×C4×#88322C411yescharacteristic
8#16810349—21no—
9C53×#384C2211yescharacteristic
10C53×#324—21no—
11C22×Dic106—21no—
12C22×#88322—21no—
13#13133217—21no—
14#13133217—41no—
15#13133217C812yescomplemented
16C2×C4⋊Dic53—41no—
17#13133140—42no—
18#13133140—42no—
19C22⋊#88322—41no—
20C4×#88322—41no—
21C2×C4⋊C4×C53—41no—
22C2×C4⋊C4×C53D411yescharacteristic
23C2×C4⋊C4×C53D411yescharacteristic
24C2×C42×C53C2×C411yescharacteristic
25C22⋊C8×C53—41no—
26C22×C4×C53—21no—
27C22×C4×C53—21no—
28C22×C4×C53—21no—
29C22×C4×C53C22⋊C411yescharacteristic
30C22×Dic53—41no—
31C2×Dic106—22no—
32C2×Dic106—41no—
33C2×Dic106—41no—
34C2×Dic106—41no—
35C2×Dic106—41no—
36C2×Dic106—41no—
37C2×C8×C53—81no—
38C4⋊C4×C53—41no—
39C4⋊C4×C53—41no—
40C4⋊C4×C53—82no—
41C42×C53—21no—
42C42×C53C2×C811yescharacteristic
43C42×C53M1611yescharacteristic
44C4⋊Dic53—42no—
45C4⋊Dic53—42no—
46C2×#88322—22no—
47C2×#88322—41no—
48C2×#88322—41no—
49C2×#88322—81no—
50C23×C53#9911yescharacteristic
51C2×C4×C53—22no—
52C2×C4×C53—22no—
53C2×C4×C53—21no—
54C2×C4×C53—21no—
55C2×C4×C53—21no—
56C2×C4×C53—21no—
57C2×C4×C53—41no—
58C2×C4×C53—41no—
59C2×C4×C53—81no—
60C2×C4×C53—81no—
61C2×C4×C53C22⋊C811yescharacteristic
62C2×C4×C53#10011yescharacteristic
63C2×Dic53—42no—
64C2×Dic53—81no—
65C2×Dic53—81no—
66Dic106—42no—
67Dic106—42no—
68Dic106—81no—
69Dic106—81no—
70C8×C53—161no—
71#88322—42no—
72#88322—161no—
73#7355—531nomaximal, Sylow
74C22×C53—21no—
75C22×C53—21no—
76C22×C53#32311yescharacteristic
77C22×C53#35311yescharacteristic
78C22×C53#35511yescharacteristic
79C4×C53—22no—
80C4×C53—22no—
81C4×C53—21no—
82C4×C53—21no—
83C4×C53—81no—
84C4×C53—81no—
85Dic53—82no—
86#1956—531no—
87#1325—531no—
88#1325D5311yescharacteristic
89C2×C53—21no—
90C2×C53—21no—
91C2×C53#131811yescharacteristic
92C2×C53#140911yescharacteristic
93C2×C53#141111yescharacteristic
94C2×C4⋊Q8—531no—
95C2×#103—1061no—
96C2×#103—1061no—
97C2×C4×C8—531no—
98#384C2×D5311yescharacteristic
99#324—21no—
100#324—1061no—
101C53#735511yescharacteristic, complemented, Sylow
102C22×Q8—1061no—
103C22×C8—1061no—
104C4⋊Q8—532no—
105C4⋊Q8—1061no—
106C4⋊Q8—2121no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—2121no—
109C2×C4⋊C4D21211yescharacteristic
110C2×C4⋊C4C22⋊D5311yescharacteristic
111C2×C42C4×D5311yescharacteristic
112#103—2122no—
113#103—2122no—
114C22⋊C8—41no—
115C22⋊C8—2121no—
116C4×C8—2121no—
117C2×Q8—1062no—
118C2×Q8—2121no—
119C2×Q8—2121no—
120C2×Q8—2121no—
121C2×Q8—2121no—
122C2×Q8—2121no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—2121no—
127C22×C4#1171992111yescharacteristic
128C2×C8—81no—
129C2×C8—1062no—
130C2×C8—4241no—
131C2×C8—2121no—
132C2×C8—2121no—
133C4⋊C4—41no—
134C4⋊C4—41no—
135C4⋊C4—82no—
136C4⋊C4—2122no—
137C4⋊C4—2122no—
138C42—21no—
139C42C8×D5311yescharacteristic
140C42#1171991211yescharacteristic
141C23#1313313111yescharacteristic
142Q8—4241no—
143Q8—4241no—
144Q8—2122no—
145Q8—2122no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—81no—
155C2×C4—81no—
156C2×C4—4241no—
157C2×C4—4241no—
158C2×C4—2122no—
159C2×C4#1313312911yescharacteristic
160C2×C4#1313313311yescharacteristic
161C8—161no—
162C8—8481no—
163C8—2122no—
164C22—21no—
165C22—21no—
166C22#1681034611yescharacteristic
167C22#1681041511yescharacteristic
168C22#1681041911yescharacteristic
169C4—22no—
170C4—22no—
171C4—21no—
172C4—21no—
173C4—81no—
174C4—81no—
175C4—4242no—
176C2—21no—
177C2—21no—
178C2#2027264611yescharacteristic
179C2#2027287911yescharacteristic
180C2#2027288311yescharacteristic
subgroupG/H
centerC22#16810415
commutatorC42×C53C2×C8
Frattini#384C2×D53
FittingC53×#1325C2
socleC22×C53#353
prime radicalC23×C53#99
cosocleC53×#384C22
hypercenter#1325D53
hypocenterC53#7355
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #384 ◁ #1325

Complements

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

NH ≅ G/Ntype
15. #13133217C8semidirect
101. C53#7355semidirect

Maximal subgroups

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

subgroupindex
#202755162
#202726612
C53×#13252
#735553