Finite Group Databasebeta

← #49121227

Subgroups — #49121227

Order 36224.

548 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 97 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#31827301C211yesmaximal, characteristic
2#31826483C212yesmaximal, complemented
3#31826459C212yesmaximal, complemented
4#31826432C211yesmaximal, characteristic, complemented
5#410×C283C211yesmaximal, characteristic, complemented
6#22583658C412yescomplemented
7#22583657—22no—
8#22583656—22no—
9#22583655—22no—
10#22583655C412yescomplemented
11C2×#17985392—22no—
12C2×#17985392C2212yes—
13#41×D283—24no—
14C2×#17985376C2211yescharacteristic, complemented
15C22×C283⋊C8C2211yescharacteristic, complemented
16#22583538—24no—
17#22583533—22no—
18#22583530—22no—
19C22⋊(C283⋊C8)—22no—
20C2×#41×C283C2211yescharacteristic
21#117×C283C2212yescomplemented
22#99×C283—22no—
23#17985393—24no—
24#17985392—24no—
25#17985392—24no—
26#17985392—24no—
27#17985392—22no—
28#17985392—22no—
29#17985392C2×C414yes—
30#17985392D414yes—
31#17985391—28no—
32#17985391—24no—
33C2×Dic566—22no—
34C2×Dic566D412yes—
35Q8×D283—24no—
36C2×D1132—22no—
37C2×D1132D412yes—
38C2×C22⋊D283—24no—
39C2×C22⋊D283—22no—
40D4×D283—28no—
41D4×D283—24no—
42C2×C4×D283—24no—
43C2×C4×D283—22no—
44#17985376—22no—
45C2×C283⋊C8—21no—
46C2×C283⋊C8—22no—
47C2×C283⋊C8—21no—
48#41×C283—22no—
49#41×C283—22no—
50C2×Q8×C283C2×C411yescharacteristic
51C2×D4×C283C2×C411yescharacteristic
52C2×D4×C283D412yes—
53C22×C4×C283D412yes—
54C22×C4×C283C2311yescharacteristic
55C4⋊C4×C283—22no—
56C22⋊C4×C283—22no—
57C42×C283—22no—
58C22×D283—24no—
59C22×D283—22no—
60Dic566—24no—
61Dic566—22no—
62Dic566C22⋊C414yes—
63D1132—24no—
64D1132—22no—
65D1132C22⋊C414yes—
66C22⋊D283—28no—
67C22⋊D283—24no—
68C22⋊D283—24no—
69C2×Dic283—24no—
70C2×Dic283—22no—
71C4×D283—28no—
72C4×D283—24no—
73C4×D283—24no—
74C283⋊C8—41no—
75C283⋊C8—41no—
76C23×C283C22⋊C412yes—
77C23×C283C2×D411yescharacteristic
78Q8×C283—22no—
79D4×C283—24no—
80D4×C283—22no—
81C2×C4×C283—22no—
82C2×C4×C283—22no—
83C2×C4×C283—42no—
84C2×C4×C283C22⋊C412yes—
85C2×C4×C283C22×C411yescharacteristic
86C2×C4×C283C2×D412yes—
87C2×C4×C283C2×D412yes—
88C2×C4×C283C2×D411yescharacteristic
89C22×C283—22no—
90C22×C283—41no—
91C22×C283—42no—
92C22×C283C2×C22⋊C411yescharacteristic
93C22×C283#12012yes—
94C2×D283—24no—
95C2×D283—82no—
96C2×D283—84no—
97C4×C283—22no—
98C4×C283—82no—
99C4×C283C2×C22⋊C412yes—
100C4×C283#12011yescharacteristic
101C4×C283#12011yescharacteristic
102Dic283—24no—
103C2×C283—22no—
104C2×C283—21no—
105C2×C283—42no—
106C2×C283#37911yescharacteristic
107D283—44no—
108C283#179411yescharacteristic, complemented, Sylow
109#1794—2831nomaximal, Sylow
110#585—2831no—
111#421—2832no—
112#416—2832no—
113#410D28311yescharacteristic, complemented
114#408—2831no—
1152−1+4—2832no—
1162−1+4—5662no—
117H(5,𝔽2)—2832no—
118H(5,𝔽2)—5662no—
119H(5,𝔽2)—5662no—
120C2×#41—2832no—
121C2×#41—5664no—
122C2×#41—5662no—
123C2×#41C2×D28311yescharacteristic
124C2×M16—2831no—
125C22×C8—2831no—
126#117C2×D28312yescomplemented
127#104—5664no—
128#103—5662no—
129#102—5662no—
130#99—22no—
131C22⋊C8—5662no—
132#41—22no—
133#41—22no—
134#41—2834no—
135#41—2834no—
136#41—5664no—
137#41—5664no—
138#41—5664no—
139#41—5668no—
140#41—5664no—
141#41—5664no—
142#41—5662no—
143#41—5662no—
144C2×Q8—2832no—
145C2×Q8—5662no—
146C2×Q8—5664no—
147C2×Q8C4×D28311yescharacteristic
148C2×D4—2832no—
149C2×D4—5662no—
150C2×D4—5664no—
151C2×D4—5668no—
152C2×D4—5664no—
153C2×D4—5662no—
154C2×D4C4×D28311yescharacteristic
155C2×D4C22⋊D28312yes—
156C22×C4—5664no—
157C22×C4—5662no—
158C22×C4D113212yes—
159C22×C4C22×D28311yescharacteristic
160M16—5662no—
161C2×C8—5662no—
162C2×C8—5661no—
163C2×C8—5661no—
164C4⋊C4—22no—
165C22⋊C4—22no—
166C42—22no—
167C23—5664no—
168C23—5662no—
169C23(C2×D283)⋊C412yes—
170C23C2×C22⋊D28311yescharacteristic
171Q8—22no—
172Q8—2834no—
173Q8—5664no—
174Q8—5662no—
175D4—24no—
176D4—22no—
177D4—2834no—
178D4—5664no—
179D4—5668no—
180D4—5664no—
181D4—5664no—
182D4—5662no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—42no—
186C2×C4—5668no—
187C2×C4—5664no—
188C2×C4—5662no—
189C2×C4—5664no—
190C2×C4—5664no—
191C2×C4(C2×D283)⋊C412yes—
192C2×C4C2×C4×D28311yescharacteristic
193C2×C4D4×D28312yes—
194C2×C4D4×D28312yes—
195C2×C4C2×D113211yescharacteristic
196C8—11321no—
197C8—11321no—
198C22—22no—
199C22—41no—
200C22—42no—
201C22—5664no—
202C22—22642no—
203C22—22644no—
204C22C2×(C2×D283)⋊C411yescharacteristic
205C22#2258357812yes—
206C4—22no—
207C4—82no—
208C4—5664no—
209C4C22⋊C4×D28312yes—
210C4#2258358011yescharacteristic
211C4#2258358011yescharacteristic
212C2—22no—
213C2—21no—
214C2—42no—
215C2—11324no—
216C2#3182632011yescharacteristic
subgroupG/H
centerC4#22583580
commutatorC2×C4×C283C22×C4
FrattiniC22×C4C22×D283
Fitting#410×C283C2
socleC2×C283#379
prime radical#31827301C2
cosocleC22×C4×C283C23
hypercenter#410D283
hypocenterC283#1794
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #31826483C2semidirect
3. #31826459C2semidirect
4. #31826432C2semidirect
5. #410×C283C2semidirect
6. #22583658C4semidirect
10. #22583655C4semidirect
14. C2×#17985376C22semidirect
15. C22×C283⋊C8C22semidirect
21. #117×C283C22semidirect
108. C283#1794semidirect
113. #410D283semidirect
126. #117C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318273012
#318264832
#318264592
#318264322
#410×C2832
#1794283