Finite Group Databasebeta

← #27304434

Subgroups — #27304434

Order 13568.

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

normal subgroups only

This group has 398 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#20276400C212yesmaximal
2#20272884C214yesmaximal, complemented
3C53×#3152C211yesmaximal, characteristic
4#16810607—22no—
5#16810422C2214yes—
6#16810420—22no—
7C2×#137×C53—22no—
8C53×#532C2211yescharacteristic
9C2×C4⋊Q8×C53C411yescharacteristic
10C53×#497—22no—
11C53×#494C2212yes—
12C53×#492C412yes—
13C4×M16×C53C411yescharacteristic
14C2×#11719917—42no—
15#13133136—24no—
16#13133134—42no—
17C2×#41×C53D412yes—
18C22×Q8×C53—21no—
19#137×C53—44no—
20#137×C53—44no—
21C2×Q16×C53—24no—
22C2×SD16×C53—24no—
23C2×M16×C53—21no—
24C4⋊Q8×C53—21no—
25C4⋊Q8×C53—41no—
26C4⋊Q8×C53C2×C412yes—
27C4⋊Q8×C53C2311yescharacteristic
28#127×C53C2×C411yescharacteristic
29#124×C53D412yes—
30#121×C53—41no—
31C4×D4×C53—41no—
32C2×C4⋊C4×C53—41no—
33C2×C42×C53C2×C411yescharacteristic
34#97×C53—21no—
35C4×C8×C53C2×C412yes—
36#41×C53—42no—
37#41×C53—42no—
38C2×Q8×C53—22no—
39C2×Q8×C53—42no—
40C2×Q8×C53—42no—
41C2×Q8×C53—41no—
42C2×Q8×C53C2×D412yes—
43C2×D4×C53—22no—
44C2×D4×C53—41no—
45C2×D4×C53C22⋊C412yes—
46C22×C4×C53—21no—
47C22×C4×C53—22no—
48C22×C4×C53—41no—
49C22×C4×C53C22⋊C411yescharacteristic
50Q16×C53—48no—
51SD16×C53—48no—
52M16×C53—24no—
53C2×C8×C53—22no—
54C4⋊C4×C53—41no—
55C4⋊C4×C53—42no—
56C4⋊C4×C53—42no—
57C22⋊C4×C53—42no—
58C42×C53—21no—
59C42×C53C22⋊C411yescharacteristic
60C42×C53C22×C411yescharacteristic
61#11719917—44no—
62#11719917—82no—
63C2×#88322—82no—
64C23×C53—22no—
65C23×C53#9911yescharacteristic
66Q8×C53—42no—
67Q8×C53—44no—
68Q8×C53—82no—
69D4×C53—42no—
70D4×C53—82no—
71D4×C53—82no—
72C2×C4×C53—21no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—21no—
76C2×C4×C53—42no—
77C2×C4×C53—42no—
78C2×C4×C53—42no—
79C2×C4×C53—42no—
80C2×C4×C53—81no—
81C2×C4×C53—81no—
82C2×C4×C53#9911yescharacteristic
83C2×C4×C53C2×C22⋊C411yescharacteristic
84C8×C53—24no—
85#88322—84no—
86#13592—531nomaximal, Sylow
87C22×C53—21no—
88C22×C53—41no—
89C22×C53—42no—
90C22×C53—82no—
91C22×C53C2×#9911yescharacteristic
92C4×C53—22no—
93C4×C53—21no—
94C4×C53—41no—
95C4×C53—42no—
96C4×C53—82no—
97C4×C53#35312yes—
98#3152D5311yescharacteristic
99#2132—532no—
100#1411—534no—
101C2×C53—21no—
102C2×C53—41no—
103C2×C53—82no—
104C2×C53C2×#35311yescharacteristic
105C2×#137—22no—
106#532C2×D5311yescharacteristic
107C2×C4⋊Q8Dic5311yescharacteristic
108#497—22no—
109#494C2×D5312yes—
110#492Dic5312yes—
111#413—1062no—
112C4×M16Dic5311yescharacteristic
113#356—534no—
114#355—1062no—
115C53#1359211yescharacteristic, complemented, Sylow
116C2×#41C22⋊D5312yes—
117C22×Q8—21no—
118#137—44no—
119#137—44no—
120C2×Q16—24no—
121C2×SD16—24no—
122C2×M16—21no—
123C2×M16—2122no—
124C4⋊Q8—21no—
125C4⋊Q8—41no—
126C4⋊Q8C2×Dic5312yes—
127C4⋊Q8C22×D5311yescharacteristic
128#127C2×Dic5311yescharacteristic
129#124C22⋊D5312yes—
130#121—41no—
131C4×D4—41no—
132C2×C4⋊C4—41no—
133C2×C42C2×Dic5311yescharacteristic
134#101—1064no—
135#100—2122no—
136#97—21no—
137C4×C8C2×Dic5312yes—
138#41—42no—
139#41—42no—
140C2×Q8—22no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—41no—
144C2×Q8C2×C22⋊D5312yes—
145C2×D4—22no—
146C2×D4—41no—
147C2×D4C22⋊Dic5312yes—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—41no—
151C22×C4C22⋊Dic5311yescharacteristic
152Q16—48no—
153SD16—48no—
154M16—24no—
155M16—4242no—
156M16—2124no—
157C2×C8—22no—
158C2×C8—4242no—
159C4⋊C4—41no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C22⋊C4—42no—
163C42—21no—
164C42C22⋊Dic5311yescharacteristic
165C42C22×Dic5311yescharacteristic
166C23—22no—
167C23#1313313211yescharacteristic
168Q8—42no—
169Q8—44no—
170Q8—82no—
171D4—42no—
172D4—82no—
173D4—82no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4#1313313211yescharacteristic
185C2×C4C2×C22⋊Dic5311yescharacteristic
186C8—24no—
187C8—4244no—
188C22—21no—
189C22—41no—
190C22—42no—
191C22—82no—
192C22C2×#1313313211yescharacteristic
193C4—22no—
194C4—21no—
195C4—41no—
196C4—42no—
197C4—82no—
198C4#1681041612yes—
199C2—21no—
200C2—41no—
201C2—82no—
202C2C2×#1681041611yescharacteristic
subgroupG/H
centerC2C2×#16810416
commutatorC42×C53C22×C4
FrattiniC4⋊Q8C22×D53
FittingC53×#3152C2
socleC2×C53C2×#353
prime radicalC53×#532C22
cosocleC4⋊Q8×C53C23
hypercenter#3152D53
hypocenterC53#13592
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #3152

Complements

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

NH ≅ G/Ntype
2. #20272884C2semidirect
115. C53#13592semidirect

Maximal subgroups

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

subgroupindex
#202764002
#202728842
C53×#31522
#1359253