Finite Group Databasebeta

← #49122282

Subgroups — #49122282

Order 36224.

382 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C2×#22583635C211yesmaximal, characteristic
2#31827036C211yesmaximal, characteristic, complemented
3C2×#22583598C211yesmaximal, characteristic, complemented
4#31826471C211yesmaximal, characteristic, complemented
5C2×#22583528C211yesmaximal, characteristic, complemented
6#31826225C211yesmaximal, characteristic, complemented
7C2×#104×C283C211yesmaximal, characteristic, complemented
8C22×Dic566C2211yescharacteristic
9C22×D1132C2211yescharacteristic
10#22583635—22no—
11#22583635—22no—
12C2×#17985379—21no—
13C2×#17985377—21no—
14C2×#17985376C2211yescharacteristic
15#22583608—21no—
16#22583598—22no—
17#22583598—22no—
18#22583597—21no—
19#22583591—22no—
20#22583589—22no—
21#22583569C2211yescharacteristic
22C2×(C2×D283)⋊C4—22no—
23#22583532—22no—
24#22583528—22no—
25C2×#41×C283C2211yescharacteristic
26C2×M16×C283C2211yescharacteristic
27C2×C42×C283C2211yescharacteristic
28#104×C283—22no—
29C23×D283—21no—
30C2×Dic566—21no—
31C2×Dic566—21no—
32C2×Dic566—21no—
33C2×Dic566—21no—
34C2×Dic566—21no—
35C2×Dic566—21no—
36C2×D1132—21no—
37C2×D1132—21no—
38C2×D1132—21no—
39C2×D1132—21no—
40C2×D1132—21no—
41C2×D1132D412yes—
42C22×Dic283D412yes—
43#17985379—24no—
44#17985377—24no—
45#17985376—21no—
46#17985376D412yes—
47C2×C283⋊C8—21no—
48C4⋊Dic283D412yes—
49Dic283⋊C4—24no—
50C22⋊Dic283—24no—
51C22⋊Dic283—21no—
52(C2×D283)⋊C4—28no—
53C4×Dic283—21no—
54#41×C283—21no—
55#41×C283—21no—
56#41×C283—21no—
57#41×C283—21no—
58C2×Q8×C283D411yescharacteristic
59C2×D4×C283—21no—
60C2×D4×C283D411yescharacteristic
61C22×C4×C283—21no—
62C22×C4×C283—21no—
63C22×C4×C283C2311yescharacteristic
64M16×C283—21no—
65M16×C283—21no—
66C2×C8×C283—21no—
67C42×C283—21no—
68C42×C283D412yes—
69C22×D283—22no—
70C22×D283—44no—
71C22×D283—41no—
72C22×D283—41no—
73Dic566—41no—
74Dic566—41no—
75Dic566—41no—
76Dic566—41no—
77D1132—22no—
78D1132—22no—
79D1132—41no—
80D1132—41no—
81C2×Dic283—22no—
82C2×Dic283—22no—
83C2×Dic283—22no—
84C2×Dic283—22no—
85C283⋊C8—22no—
86C23×C283—21no—
87C23×C283C2×D411yescharacteristic
88Q8×C283—21no—
89Q8×C283—21no—
90D4×C283—21no—
91D4×C283—21no—
92D4×C283—41no—
93D4×C283—41no—
94C2×C4×C283—22no—
95C2×C4×C283—21no—
96C2×C4×C283—42no—
97C2×C4×C283—41no—
98C2×C4×C283—41no—
99C2×C4×C283C2×D411yescharacteristic
100C2×C4×C283C2×D411yescharacteristic
101C2×C4×C283C2×D411yescharacteristic
102C2×C4×C283C2×D411yescharacteristic
103C2×C4×C283C2×D411yescharacteristic
104C2×C4×C283#4111yescharacteristic
105C8×C283—41no—
106C22×C283—22no—
107C22×C283—41no—
108C22×C283—42no—
109C22×C283#12011yescharacteristic
110C22×C283#12011yescharacteristic
111C22×C283#12111yescharacteristic
112C2×D283—44no—
113C2×D283—42no—
114C2×D283—82no—
115C2×D283—82no—
116C4×C283—42no—
117C4×C283—41no—
118C4×C283#12011yescharacteristic
119C4×C283#12111yescharacteristic
120C4×C283#12111yescharacteristic
121C4×C283#12711yescharacteristic
122Dic283—42no—
123Dic283—42no—
124C2×C283—21no—
125C2×C283—21no—
126C2×C283—81no—
127C2×C283#39211yescharacteristic
128C2×C283#45412yes—
129D283—82no—
130C283#200311yescharacteristic, complemented, Sylow
131#2003—2831nomaximal, Sylow
132C2×AGL(1,ℤ/8)—2831no—
133#536—2831no—
134C2×#124—2831no—
135C2×#104D28311yescharacteristic, complemented
136#418—2831no—
137C2×#101—2831no—
138#339—2831no—
139C2×#41C2×D28311yescharacteristic
140C22×Q8—2831no—
141C22×D4—2831no—
142AGL(1,ℤ/8)—5662no—
143AGL(1,ℤ/8)—5662no—
144C2×SD16—5661no—
145C2×D8—5661no—
146C2×M16—2831no—
147C2×M16C2×D28311yescharacteristic
148C4⋊Q8—5661no—
149#124—5662no—
150#124—5661no—
151#124—5662no—
152#123—5662no—
153C22⋊Q8—5662no—
154#117—2831no—
155C2×C22⋊C4—5662no—
156C2×C42C2×D28311yescharacteristic
157#104—22no—
158#102—5662no—
159#101—5662no—
160C24—5661no—
161#41—21no—
162#41—21no—
163#41—21no—
164#41—21no—
165C2×Q8—5661no—
166C2×Q8—5661no—
167C2×Q8—5661no—
168C2×Q8—5661no—
169C2×Q8—5661no—
170C2×Q8—5661no—
171C2×Q8D113211yescharacteristic
172C2×D4—21no—
173C2×D4—2832no—
174C2×D4—5661no—
175C2×D4—5661no—
176C2×D4—5661no—
177C2×D4—5661no—
178C2×D4—5661no—
179C2×D4D113211yescharacteristic
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—2832no—
183C22×C4C22×D28311yescharacteristic
184SD16—5664no—
185D8—5664no—
186M16—21no—
187M16—21no—
188M16—2832no—
189M16—5661no—
190C2×C8—21no—
191C2×C8—5661no—
192C4⋊C4—2832no—
193C4⋊C4—5664no—
194C22⋊C4—5668no—
195C22⋊C4—5664no—
196C22⋊C4—5661no—
197C42—21no—
198C42—5661no—
199C42C22⋊D28312yes—
200C23—21no—
201C23—5662no—
202C23—11324no—
203C23—11321no—
204C23—11321no—
205C23D4×D28311yescharacteristic
206Q8—21no—
207Q8—21no—
208Q8—11321no—
209Q8—11321no—
210Q8—11321no—
211Q8—11321no—
212D4—21no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—5662no—
217D4—5662no—
218D4—11321no—
219D4—11321no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—5662no—
226C2×C4—5662no—
227C2×C4—5662no—
228C2×C4—5662no—
229C2×C4D4×D28311yescharacteristic
230C2×C4D4×D28311yescharacteristic
231C2×C4D4×D28311yescharacteristic
232C2×C4C2×C22⋊D28311yescharacteristic
233C2×C4C2×D113211yescharacteristic
234C2×C4#1798539211yescharacteristic
235C8—41no—
236C8—5662no—
237C22—22no—
238C22—41no—
239C22—42no—
240C22—11324no—
241C22—11322no—
242C22—22642no—
243C22—22642no—
244C22#2258357811yescharacteristic
245C22#2258358011yescharacteristic
246C22#2258358311yescharacteristic
247C4—42no—
248C4—41no—
249C4—11322no—
250C4—11322no—
251C4#2258358011yescharacteristic
252C4#2258358311yescharacteristic
253C4#2258358411yescharacteristic
254C4#2258360611yescharacteristic
255C2—21no—
256C2—21no—
257C2—81no—
258C2—22642no—
259C2#3182637411yescharacteristic
260C2#3182663912yes—
subgroupG/H
centerC22#22583580
commutator, cosocleC22×C4×C283C23
FrattiniC22×C4C22×D283
FittingC2×#104×C283C2
socleC22×C283#120
prime radicalC2×#22583635C2
hypercenterC2×#104D283
hypocenterC283#2003
LCSG ▷ C22×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. #31827036C2semidirect
3. C2×#22583598C2semidirect
4. #31826471C2semidirect
5. C2×#22583528C2semidirect
6. #31826225C2semidirect
7. C2×#104×C283C2semidirect
130. C283#2003semidirect
135. C2×#104D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836352
#318270362
C2×#225835982
#318264712
C2×#225835282
#318262252
C2×#104×C2832
#2003283