Finite Group Databasebeta

← #28299735

Subgroups — #28299735

Order 13568.

456 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 118 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#20290631C211yesmaximal, characteristic
2#20290631C211yesmaximal, characteristic
3#20289533C214yesmaximal
4C4⋊Q8×Dic53C211yesmaximal, characteristic
5C2×#13133284—21no—
6C2×#13133284—21no—
7#16811489—21no—
8#16811489—21no—
9#16811489C2212yes—
10#16811489C2212yes—
11C2×Q8×Dic53—21no—
12#16810530—21no—
13#16810522—21no—
14#16810522C2212yes—
15#16810493C411yescharacteristic
16C4⋊C4×Dic53—21no—
17C4⋊C4×Dic53C412yes—
18C42×Dic53C2211yescharacteristic
19C2×C4⋊Q8×C53C411yescharacteristic
20C22×#88324—22no—
21C22×#88324—22no—
22#13133284—42no—
23#13133284—42no—
24Q8×Dic53—42no—
25Q8×Dic53—42no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—22no—
29C2×C4⋊Dic53C2×C412yes—
30C2×C4×Dic53—22no—
31C2×C4×Dic53C2×C411yescharacteristic
32C2×C4×Dic53D411yescharacteristic
33C2×C4×Dic53D411yescharacteristic
34C2×C4×Dic53D411yescharacteristic
35C2×C4×Dic53D411yescharacteristic
36C2×C4×Dic53C2311yescharacteristic
37#13133124—22no—
38#13133124—22no—
39C22×Q8×C53—21no—
40C4⋊Q8×C53—24no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53C2×C412yes—
43C2×C42×C53C2×C411yescharacteristic
44C2×Q8×C53—24no—
45C2×Q8×C53—42no—
46C2×Q8×C53—42no—
47C22×C4×C53—22no—
48C22×C4×C53C22⋊C411yescharacteristic
49C22×C4×C53C22⋊C411yescharacteristic
50C22×C4×C53C22×C411yescharacteristic
51C22×Dic53—22no—
52C22×Dic53C22⋊C411yescharacteristic
53C22×Dic53C22⋊C411yescharacteristic
54C22×Dic53C2×D411yescharacteristic
55C22×Dic53C2×D411yescharacteristic
56C2×#88324—42no—
57C2×#88324—42no—
58C2×#88324—42no—
59C2×#88324—42no—
60C2×#88324—42no—
61C2×#88324—42no—
62C4⋊C4×C53—24no—
63C4⋊C4×C53—22no—
64C4⋊C4×C53M1614yes—
65C42×C53—22no—
66C4⋊Dic53—42no—
67C4⋊Dic53—42no—
68C4⋊Dic53—42no—
69C4⋊Dic53—42no—
70C4⋊Dic53—42no—
71C4×Dic53—22no—
72C4×Dic53—41no—
73C4×Dic53—41no—
74C4×Dic53—41no—
75C4×Dic53—44no—
76C4×Dic53D812yes—
77C4×Dic53SD1612yes—
78C4×Dic53SD1612yes—
79C4×Dic53Q1612yes—
80C23×C53C2×C22⋊C411yescharacteristic
81Q8×C53—48no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—24no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53#10212yes—
92C2×C4×C53#10312yes—
93C2×C4×C53C2×M1612yes—
94C2×Dic53—21no—
95C2×Dic53—21no—
96C2×Dic53—21no—
97C2×Dic53—21no—
98C2×Dic53—21no—
99C2×Dic53—21no—
100C2×Dic53—21no—
101C2×Dic53—42no—
102C2×Dic53—42no—
103C2×Dic53—42no—
104C2×Dic53#10112yes—
105C2×Dic53#10212yes—
106C2×Dic53#10312yes—
107C2×Dic53C2×D811yescharacteristic
108C2×Dic53C2×SD1611yescharacteristic
109C2×Dic53C2×SD1611yescharacteristic
110C2×Dic53C2×Q1611yescharacteristic
111#88324—82no—
112#88324—82no—
113#7652—531nomaximal, Sylow
114C22×C53#40711yescharacteristic
115C22×C53#41011yescharacteristic
116C22×C53C2×#10111yescharacteristic
117C22×C53C2×#10211yescharacteristic
118C22×C53C2×#10311yescharacteristic
119C22×C53#42111yescharacteristic
120C22×C53#42111yescharacteristic
121C4×C53—22no—
122C4×C53—22no—
123C4×C53—22no—
124C4×C53—44no—
125Dic53—21no—
126Dic53—21no—
127Dic53—21no—
128Dic53—21no—
129Dic53—41no—
130Dic53—82no—
131Dic53#33214yes—
132C4×C4⋊Q8—531no—
133#1831—531no—
134#1831—531no—
135#1271—534no—
136C2×C53#146711yescharacteristic
137C2×C53#148611yescharacteristic
138C2×C53#148711yescharacteristic
139C2×C53#151711yescharacteristic
140C2×C53#151811yescharacteristic
141C2×C53#153711yescharacteristic
142C2×C53C2×#33211yescharacteristic
143C2×C4⋊Q8#615711yescharacteristic
144C2×C4×Q8—1061no—
145C2×C4⋊C8—1061no—
146C2×C4⋊C8—1061no—
147#391—1061no—
148#389—532no—
149#389—1061no—
150#384—531no—
151C4×C4⋊C4—532no—
152C4×C4⋊C4—1061no—
153C43—531no—
154#336—532no—
155#336—532no—
156#336—1061no—
157#336—1061no—
158C53#765211yescharacteristic, complemented, Sylow
159C22×Q8—21no—
160C22×C8—1062no—
161C22×C8—1062no—
162C4⋊Q8—24no—
163C4×Q8—2122no—
164C4×Q8—2122no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—532no—
167C2×C4⋊C4—1062no—
168C2×C4⋊C4—1061no—
169C2×C4⋊C4—1061no—
170C2×C4⋊C4C2×#615712yes—
171C2×C42—531no—
172C2×C42—531no—
173C2×C42—531no—
174C2×C42—531no—
175C2×C42—531no—
176C2×C42—531no—
177C2×C42—1062no—
178C2×C42C2×#615711yescharacteristic
179C4⋊C8—2122no—
180C4⋊C8—2122no—
181#95—1062no—
182#95—1062no—
183C2×Q8—24no—
184C2×Q8—42no—
185C2×Q8—42no—
186C22×C4—22no—
187C22×C4—531no—
188C22×C4—531no—
189C22×C4—531no—
190C22×C4—531no—
191C22×C4—1062no—
192C22×C4C22⋊#615711yescharacteristic
193C22×C4C22⋊#615711yescharacteristic
194C22×C4C22×#615711yescharacteristic
195C2×C8—2122no—
196C2×C8—2122no—
197C2×C8—2122no—
198C2×C8—2122no—
199C2×C8—2122no—
200C2×C8—2122no—
201C4⋊C4—24no—
202C4⋊C4—22no—
203C4⋊C4—2122no—
204C4⋊C4—2122no—
205C4⋊C4—2122no—
206C4⋊C4—2122no—
207C4⋊C4—2122no—
208C4⋊C4#1171994014yes—
209C42—22no—
210C42—532no—
211C42—532no—
212C42—532no—
213C42—532no—
214C42—1062no—
215C42—2121no—
216C42—2121no—
217C42—2121no—
218C42—2124no—
219C23C2×C22⋊#615711yescharacteristic
220Q8—48no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—24no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—532no—
231C2×C4—532no—
232C2×C4—532no—
233C2×C4—531no—
234C2×C4—531no—
235C2×C4—531no—
236C2×C4—531no—
237C2×C4—1061no—
238C2×C4—1061no—
239C2×C4—1061no—
240C2×C4—1061no—
241C2×C4—1061no—
242C2×C4—1061no—
243C2×C4—1061no—
244C2×C4—2122no—
245C2×C4—2122no—
246C2×C4—2122no—
247C2×C4#1313327912yes—
248C2×C4#1313328012yes—
249C2×C4C2×#1171994012yes—
250C8—4242no—
251C8—4242no—
252C22#1681159311yescharacteristic
253C22#1681160411yescharacteristic
254C22D53⋊#10111yescharacteristic
255C22C2×#1313327911yescharacteristic
256C22C2×#1313328011yescharacteristic
257C22#1681162111yescharacteristic
258C22#1681162211yescharacteristic
259C4—22no—
260C4—22no—
261C4—22no—
262C4—44no—
263C4—534no—
264C4—1061no—
265C4—1061no—
266C4—1061no—
267C4—1061no—
268C4—4242no—
269C4—2121no—
270C2#2028997811yescharacteristic
271C2#2029004011yescharacteristic
272C2#2029004311yescharacteristic
273C2#2029014611yescharacteristic
274C2#2029014711yescharacteristic
275C2#2029020711yescharacteristic
276C2D53⋊#33211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4⋊Q8×C53C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4⋊Q8#6157
hypocenterC53#7652
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
158. C53#7652semidirect

Maximal subgroups

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

subgroupindex
#202906312
#202895332
C4⋊Q8×Dic532
#765253