Finite Group Databasebeta

← #27304452

Subgroups — #27304452

Order 13568.

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

normal subgroups only

This group has 318 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#20272888C214yesmaximal, complemented
3C53×#3135C211yesmaximal, characteristic
4#16810607—22no—
5#16810422C2214yes—
6#16810420—22no—
7C53×#533C411yescharacteristic
8C53×#532C2211yescharacteristic
9C53×#491C2212yes—
10C53×#490C412yes—
11C53×#489—22no—
12C53×#416—22no—
13C2×#97×C53C411yescharacteristic
14C2×#11719917—42no—
15#13133136—24no—
16#13133134—42no—
17C2×#41×C53D412yes—
18C22×C8×C53—21no—
19C4⋊Q8×C53—21no—
20C4⋊Q8×C53C2311yescharacteristic
21#127×C53C2×C411yescharacteristic
22#125×C53C2×C412yes—
23#124×C53D412yes—
24C22⋊Q8×C53—41no—
25#121×C53—41no—
26C4×Q8×C53—41no—
27C4×D4×C53—41no—
28#117×C53—21no—
29C2×C42×C53C2×C411yescharacteristic
30#103×C53—24no—
31#102×C53—24no—
32#97×C53—21no—
33#97×C53C2×C412yes—
34#41×C53—42no—
35#41×C53—42no—
36C2×Q8×C53—41no—
37C2×Q8×C53C2×D412yes—
38C2×D4×C53—22no—
39C2×D4×C53—41no—
40C2×D4×C53C22⋊C412yes—
41C22×C4×C53—21no—
42C22×C4×C53—22no—
43C22×C4×C53C22⋊C411yescharacteristic
44C2×C8×C53—22no—
45C2×C8×C53—41no—
46C2×C8×C53—41no—
47C4⋊C4×C53—22no—
48C4⋊C4×C53—41no—
49C4⋊C4×C53—41no—
50C4⋊C4×C53—42no—
51C22⋊C4×C53—42no—
52C22⋊C4×C53—41no—
53C42×C53—21no—
54C42×C53—41no—
55C42×C53C22⋊C411yescharacteristic
56C42×C53C22×C411yescharacteristic
57#11719917—44no—
58#11719917—82no—
59C2×#88322—82no—
60C23×C53—22no—
61C23×C53#9911yescharacteristic
62Q8×C53—42no—
63Q8×C53—82no—
64D4×C53—42no—
65D4×C53—82no—
66D4×C53—82no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—22no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—42no—
73C2×C4×C53—42no—
74C2×C4×C53—42no—
75C2×C4×C53—81no—
76C2×C4×C53#9911yescharacteristic
77C2×C4×C53C2×C22⋊C411yescharacteristic
78C8×C53—42no—
79#88322—84no—
80#13598—531nomaximal, Sylow
81C22×C53—21no—
82C22×C53—41no—
83C22×C53—42no—
84C22×C53—82no—
85C22×C53C2×#9911yescharacteristic
86C4×C53—21no—
87C4×C53—41no—
88C4×C53—41no—
89C4×C53—42no—
90C4×C53—82no—
91C4×C53#35411yescharacteristic
92C4×C53#35411yescharacteristic
93#3135D5311yescharacteristic
94#2132—532no—
95#1413—534no—
96C2×C53—21no—
97C2×C53—41no—
98C2×C53—82no—
99C2×C53C2×#35411yescharacteristic
100#533Dic5311yescharacteristic
101#532C2×D5311yescharacteristic
102#491C2×D5312yes—
103#490Dic5312yes—
104#489—22no—
105#416—22no—
106#413—1062no—
107C2×#97Dic5311yescharacteristic
108#356—534no—
109#355—1062no—
110C53#1359811yescharacteristic, complemented, Sylow
111C2×#41C22⋊D5312yes—
112C2×M16—2122no—
113C22×C8—21no—
114C4⋊Q8—21no—
115C4⋊Q8C22×D5311yescharacteristic
116#127C2×Dic5311yescharacteristic
117#125C2×Dic5312yes—
118#124C22⋊D5312yes—
119C22⋊Q8—41no—
120#121—41no—
121C4×Q8—41no—
122C4×D4—41no—
123#117—21no—
124C2×C42C2×Dic5311yescharacteristic
125#103—24no—
126#102—24no—
127#101—1064no—
128#100—2122no—
129#97—21no—
130#97C2×Dic5312yes—
131#41—42no—
132#41—42no—
133C2×Q8—41no—
134C2×Q8C2×C22⋊D5312yes—
135C2×D4—22no—
136C2×D4—41no—
137C2×D4C22⋊Dic5312yes—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4C22⋊Dic5311yescharacteristic
141M16—4242no—
142M16—2124no—
143C2×C8—22no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—4242no—
147C4⋊C4—22no—
148C4⋊C4—41no—
149C4⋊C4—42no—
150C4⋊C4—41no—
151C22⋊C4—42no—
152C22⋊C4—41no—
153C42—21no—
154C42—41no—
155C42C22⋊Dic5311yescharacteristic
156C42C22×Dic5311yescharacteristic
157C23—22no—
158C23#1313313211yescharacteristic
159Q8—42no—
160Q8—82no—
161D4—42no—
162D4—82no—
163D4—82no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—81no—
173C2×C4#1313313211yescharacteristic
174C2×C4C2×C22⋊Dic5311yescharacteristic
175C8—42no—
176C8—4244no—
177C22—21no—
178C22—41no—
179C22—42no—
180C22—82no—
181C22C2×#1313313211yescharacteristic
182C4—21no—
183C4—41no—
184C4—41no—
185C4—42no—
186C4—82no—
187C4#1681041811yescharacteristic
188C4#1681041811yescharacteristic
189C2—21no—
190C2—41no—
191C2—82no—
192C2C2×#1681041811yescharacteristic
subgroupG/H
centerC4#16810418
commutatorC42×C53C22×C4
FrattiniC4⋊Q8C22×D53
FittingC53×#3135C2
socleC2×C53C2×#354
prime radicalC53×#532C22
cosocleC4⋊Q8×C53C23
hypercenter#3135D53
hypocenterC53#13598
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #3135

Complements

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

NH ≅ G/Ntype
2. #20272888C2semidirect
110. C53#13598semidirect

Maximal subgroups

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

subgroupindex
#202764002
#202728882
C53×#31352
#1359853