Finite Group Databasebeta

← #28310089

Subgroups — #28310089

Order 13568.

224 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 66 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#20290348C211yesmaximal, characteristic
2#20290303C211yesmaximal, characteristic, complemented
3#20289748C211yesmaximal, characteristic, complemented
4#20289672C211yesmaximal, characteristic, complemented
5#20289669C211yesmaximal, characteristic
6#88324⋊C16C211yesmaximal, characteristic, complemented
7#20276570C211yesmaximal, characteristic
8#16811672—21no—
9#16811658—21no—
10#16811509C2211yescharacteristic
11#16811486C2211yescharacteristic
12#16811486C2211yescharacteristic
13#16811473—21no—
14D4⋊#88324—21no—
15C8×#88324C2211yescharacteristic
16C2×#13133161C411yescharacteristic
17#16810717C2211yescharacteristic
18D8×Dic53C2211yescharacteristic
19#16810450C411yescharacteristic
20#16810428C411yescharacteristic
21C16×Dic53C2211yescharacteristic
22C53×#357C411yescharacteristic
23C2×#11719940—41no—
24C2×#11719910—21no—
25#13133284—41no—
26C4⋊#88324—21no—
27C4⋊#88324—21no—
28C22⋊#88324—41no—
29C4×#88324—41no—
30C4×#88324D411yescharacteristic
31C4×#88324D411yescharacteristic
32C2×Dic212C2×C411yescharacteristic
33#13133182—21no—
34D4×Dic53—21no—
35#13133161—22no—
36C2×#11719908C2×C411yescharacteristic
37#13133151C2×C411yescharacteristic
38#13133140—21no—
39#13133138—21no—
40C8×Dic53C2311yescharacteristic
41C2×D8×C53C2×C411yescharacteristic
42C2×C16×C53C2×C411yescharacteristic
43#107×C53C2×C411yescharacteristic
44C2×D4×C53—21no—
45C22×Dic53—41no—
46C2×Dic106—21no—
47#11719940—82no—
48C2×#88324—21no—
49C2×#88324—41no—
50C2×#88324—41no—
51C2×#88324D811yescharacteristic
52C2×#88324D811yescharacteristic
53D8×C53—21no—
54D8×C53—21no—
55C2×C8×C53C22×C411yescharacteristic
56C4⋊C4×C53—21no—
57C22⋊Dic53—41no—
58C4⋊Dic53—21no—
59#11719919—41no—
60C4×Dic53—41no—
61C4×Dic53C2×D411yescharacteristic
62C2×#88322#4111yescharacteristic
63Dic212—21no—
64Dic212M1612yescomplemented
65#11719910—41no—
66C16×C53—21no—
67#11719908—21no—
68C23×C53—41no—
69D4×C53—41no—
70D4×C53—41no—
71C2×C4×C53—41no—
72C2×C4×C53C4×D411yescharacteristic
73C2×Dic53—41no—
74C2×Dic53—41no—
75C2×Dic53—81no—
76C2×Dic53—81no—
77C2×Dic53C2×D811yescharacteristic
78C2×Dic53#13511yescharacteristic
79Dic106—22no—
80Dic106—41no—
81#88324—21no—
82#88324—41no—
83#88324—81no—
84#88324—81no—
85#88324SD3212yescomplemented
86C8×C53C2×M1611yescharacteristic
87C8×C53#13111yescharacteristic
88#88322—21no—
89#12071—531nomaximal, Sylow
90C22×C53—81no—
91C22×C53—81no—
92C22×C53—81no—
93C22×C53C4×D811yescharacteristic
94C4×C53—81no—
95C4×C53#43511yescharacteristic
96C4×C53#44411yescharacteristic
97Dic53—21no—
98Dic53—42no—
99Dic53—81no—
100Dic53C2×SD3211yescharacteristic
101Dic53#50811yescharacteristic
102C4×SD32—531no—
103#1589—531no—
104C8×Q16—531no—
105#1372—531no—
106#1332—531no—
107#1331—531no—
108#1312—531no—
109C2×C53—161no—
110C2×C53#158111yescharacteristic
111C2×C53C4×SD3211yescharacteristic
112C2×C53#217911yescharacteristic
113C2×SD32—531no—
114C8×Q8—1061no—
115C4×Q16—531no—
116C4×D8—531no—
117#436—1061no—
118#367—531no—
119#358—531no—
120#357#615711yescharacteristic
121C4×C16—531no—
122C4×C16—531no—
123#335—531no—
124#335—531no—
125Q8⋊C8—1061no—
126D4⋊C8—1061no—
127C82—531no—
128C53#1207111yescharacteristic, complemented, Sylow
129C2×Q16—531no—
130C2×D8C2×#615711yescharacteristic
131C2×M16—2121no—
132C4×Q8—1061no—
133C4×D4—1061no—
134SD32—1062no—
135C2×C16—531no—
136C2×C16—1061no—
137C2×C16C2×#615711yescharacteristic
138#107—531no—
139#107C2×#615711yescharacteristic
140C4⋊C8—1061no—
141C4⋊C8—1061no—
142C4⋊C8—2121no—
143#103—1061no—
144#102—1061no—
145C22⋊C8—2121no—
146C4×C8—531no—
147C4×C8—531no—
148C4×C8—531no—
149C4×C8—2121no—
150C2×Q8—1061no—
151C2×D4—21no—
152C22×C4—2121no—
153Q16—532no—
154Q16—1061no—
155D8—21no—
156D8—21no—
157M16—4242no—
158C2×C8—531no—
159C2×C8—531no—
160C2×C8—531no—
161C2×C8—1061no—
162C2×C8—2121no—
163C2×C8—2121no—
164C2×C8C22×#615711yescharacteristic
165C4⋊C4—21no—
166C4⋊C4—1061no—
167C4⋊C4—2121no—
168C22⋊C4—2121no—
169C42—531no—
170C42—2121no—
171C16—21no—
172C16—1061no—
173C16—2121no—
174C23—41no—
175Q8—1062no—
176Q8—2121no—
177D4—41no—
178D4—41no—
179C2×C4—41no—
180C2×C4—531no—
181C2×C4—531no—
182C2×C4—4241no—
183C2×C4—4241no—
184C2×C4—2121no—
185C2×C4—2121no—
186C2×C4D4×#615711yescharacteristic
187C8—532no—
188C8—1061no—
189C8—1061no—
190C8—4241no—
191C8—4241no—
192C8—2121no—
193C8D53⋊M1611yescharacteristic
194C8#1313330411yescharacteristic
195C22—81no—
196C22—81no—
197C22—81no—
198C22D8×#615711yescharacteristic
199C4—81no—
200C4—531no—
201C4—531no—
202C4—1061no—
203C4—4241no—
204C4—2122no—
205C4#1681165511yescharacteristic
206C4#1681166811yescharacteristic
207C2—161no—
208C2#2029031711yescharacteristic
209C2SD32×#615711yescharacteristic
210C2#2029111911yescharacteristic
subgroupG/H
centerC22D8×#6157
commutatorC2×C8×C53C22×C4
FrattiniC2×C8C22×#6157
FittingC53×#357C4
socleC22×C53C4×D8
prime radicalC2×D8×C53C2×C4
cosocleC8×Dic53C23
hypercenter#357#6157
hypocenterC53#12071
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ #357

Complements

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

NH ≅ G/Ntype
2. #20290303C2semidirect
3. #20289748C2semidirect
4. #20289672C2semidirect
6. #88324⋊C16C2semidirect
64. Dic212M16semidirect
85. #88324SD32semidirect
128. C53#12071semidirect

Maximal subgroups

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

subgroupindex
#202903482
#202903032
#202897482
#202896722
#202896692
#88324⋊C162
#202765702
#1207153