Finite Group Databasebeta

← #49131649

Subgroups — #49131649

Order 36224.

400 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 125 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#31827018C211yesmaximal, characteristic, complemented
2C2×#22583599C211yesmaximal, characteristic
3#31826826C214yesmaximal, complemented
4#31826821C214yesmaximal, complemented
5#31826474C211yesmaximal, characteristic, complemented
6#31826471C211yesmaximal, characteristic
7#31826469C211yesmaximal, characteristic
8M16×Dic283C211yesmaximal, characteristic, complemented
9#419×C283C211yesmaximal, characteristic, complemented
10C22×D1132C2211yescharacteristic
11C2×#17985376C2211yescharacteristic
12#22583609C2212yescomplemented
13#22583601C2212yescomplemented
14#22583599—22no—
15#22583599C2212yes—
16#22583599C2212yes—
17#22583587—22no—
18C4×Dic566—22no—
19#22583569C2211yescharacteristic
20C2×(C2×D283)⋊C4—22no—
21C2×C4×Dic283C2211yescharacteristic
22#22583534C2214yescomplemented
23#22583532C2214yes—
24#22583530C2214yes—
25#22583520C2212yescomplemented
26C8×Dic283C2212yescomplemented
27C22×Q8×C283C2211yescharacteristic
28C2×M16×C283C2211yescharacteristic
29#117×C283C2211yescharacteristic
30#103×C283C2214yescomplemented
31C23×D283—21no—
32C2×Dic566—22no—
33C2×D1132—21no—
34C2×D1132—22no—
35C2×D1132—22no—
36C2×D1132C2312yes—
37C22×Dic283D412yes—
38#17985376—22no—
39C2×C283⋊C8C2312yes—
40C4⋊Dic283—22no—
41C4⋊Dic283C2312yes—
42Dic283⋊C4—22no—
43Dic283⋊C4—22no—
44C22⋊Dic283—21no—
45(C2×D283)⋊C4—24no—
46(C2×D283)⋊C4—24no—
47C4×Dic283—21no—
48C4×Dic283D412yes—
49C4×Dic283C2311yescharacteristic
50C4×Dic283C2311yescharacteristic
51C2×Q8×C283—21no—
52C2×Q8×C283—22no—
53C2×Q8×C283—22no—
54C2×Q8×C283C2312yes—
55C22×C4×C283—21no—
56C22×C4×C283C2311yescharacteristic
57M16×C283—22no—
58C2×C8×C283C2312yes—
59C4⋊C4×C283C2312yes—
60C22⋊C4×C283—21no—
61C42×C283—21no—
62C22×D283—22no—
63C22×D283—44no—
64C22×D283—42no—
65Dic566—44no—
66D1132—22no—
67D1132—22no—
68D1132—42no—
69C2×Dic283—22no—
70C2×Dic283—42no—
71C2×Dic283C2×D412yes—
72C2×Dic283C2×D412yes—
73C283⋊C8—22no—
74C23×C283C2×D411yescharacteristic
75Q8×C283—22no—
76Q8×C283—22no—
77Q8×C283—42no—
78C2×C4×C283—22no—
79C2×C4×C283—22no—
80C2×C4×C283—42no—
81C2×C4×C283C2×D411yescharacteristic
82C2×C4×C283#4112yes—
83C2×C4×C283#4112yes—
84C2×C4×C283C2411yescharacteristic
85C8×C283—22no—
86C22×C283—22no—
87C22×C283#12412yes—
88C22×C283C22×D411yescharacteristic
89C2×D283—42no—
90C2×D283—44no—
91C2×D283—84no—
92C4×C283—42no—
93C4×C283—42no—
94C4×C283#12412yes—
95C4×C283C2×#4111yescharacteristic
96C4×C283C2×#4111yescharacteristic
97Dic283—22no—
98Dic283—22no—
99Dic283—42no—
100C2×C283—22no—
101C2×C283C2×#12411yescharacteristic
102C2×C283#57512yes—
103D283—82no—
104C283#314011yescharacteristic, complemented, Sylow
105#3140—2831nomaximal, Sylow
106#533—2831no—
107C2×#124—2831no—
108#489—2834no—
109#488—2834no—
110#419—2831no—
111#419D28311yescharacteristic, complemented
112#418—2831no—
113#418—2831no—
114C4×M16—2831no—
115C22×Q8C2×D28311yescharacteristic
116C22×D4—2831no—
117C2×M16—2831no—
118C2×M16C2×D28311yescharacteristic
119C4⋊Q8—2832no—
120#125—2832no—
121#124—2832no—
122#124—2832no—
123#124—5662no—
124C22⋊Q8—5662no—
125C4×Q8—5662no—
126#117—2831no—
127#117C2×D28311yescharacteristic
128C2×C22⋊C4—5662no—
129C2×C42—2831no—
130#103—2834no—
131#103C2×D28314yescomplemented
132#102—2834no—
133#102—2834no—
134#97—2832no—
135C4×C8—2832no—
136C24—5661no—
137C2×Q8—21no—
138C2×Q8—22no—
139C2×Q8—22no—
140C2×Q8—5662no—
141C2×Q8C22×D28312yes—
142C2×D4—2832no—
143C2×D4—5661no—
144C2×D4—5662no—
145C2×D4—5662no—
146C22×C4—21no—
147C22×C4—2832no—
148C22×C4C22×D28311yescharacteristic
149M16—22no—
150M16—5662no—
151C2×C8—2832no—
152C2×C8C22×D28312yes—
153C4⋊C4—2832no—
154C4⋊C4—5662no—
155C4⋊C4—5662no—
156C4⋊C4—5662no—
157C4⋊C4C22×D28312yes—
158C22⋊C4—21no—
159C22⋊C4—5664no—
160C22⋊C4—5664no—
161C22⋊C4—5661no—
162C42—21no—
163C42—2832no—
164C42—2831no—
165C42—2831no—
166C42—5661no—
167C23—5662no—
168C23—11324no—
169C23—11322no—
170C23D4×D28311yescharacteristic
171Q8—22no—
172Q8—22no—
173Q8—42no—
174Q8—11324no—
175D4—5662no—
176D4—5662no—
177D4—11322no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—2832no—
182C2×C4—2832no—
183C2×C4—5662no—
184C2×C4—11322no—
185C2×C4D4×D28311yescharacteristic
186C2×C4#1798539112yes—
187C2×C4#1798539212yes—
188C2×C4C23×D28311yescharacteristic
189C8—22no—
190C8—5662no—
191C22—22no—
192C22—11322no—
193C22—11324no—
194C22—22644no—
195C22#2258359612yes—
196C22C2×D4×D28311yescharacteristic
197C4—42no—
198C4—42no—
199C4—5662no—
200C4—5662no—
201C4—11322no—
202C4#2258359612yes—
203C4C2×#1798539111yescharacteristic
204C4C2×#1798539211yescharacteristic
205C2—22no—
206C2—22642no—
207C2C2×#2258359611yescharacteristic
208C2#3182726412yes—
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#419×C283C2
socleC22×C283C22×D4
prime radicalC22×D1132C22
hypercenter#419D283
hypocenterC283#3140
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. #31827018C2semidirect
3. #31826826C2semidirect
4. #31826821C2semidirect
5. #31826474C2semidirect
8. M16×Dic283C2semidirect
9. #419×C283C2semidirect
12. #22583609C22semidirect
13. #22583601C22semidirect
22. #22583534C22semidirect
25. #22583520C22semidirect
26. C8×Dic283C22semidirect
30. #103×C283C22semidirect
104. C283#3140semidirect
111. #419D283semidirect
131. #103C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318270182
C2×#225835992
#318268262
#318268212
#318264742
#318264712
#318264692
M16×Dic2832
#419×C2832
#3140283