Finite Group Databasebeta

← #49122414

Subgroups — #49122414

Order 36224.

282 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 71 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#31826469C212yesmaximal
2#31826366C211yesmaximal, characteristic
3#31826234C211yesmaximal, characteristic, complemented
4#31826224C212yesmaximal, complemented
5#544×C283C211yesmaximal, characteristic, complemented
6C22×D1132C2211yescharacteristic
7C2×#17985376C2212yes—
8C2×(C2×D283)⋊C4—22no—
9C2×C4×Dic283C2211yescharacteristic
10#22583530—22no—
11#22583530—22no—
12C4⋊Q8×C283—22no—
13#125×C283—22no—
14C22⋊Q8×C283—22no—
15#117×C283C2212yes—
16C2×C4⋊C4×C283C2211yescharacteristic
17C23×D283—21no—
18C2×D1132—22no—
19C2×D1132—22no—
20C2×D1132—21no—
21C2×D1132Q812yescomplemented
22C22×Dic283—21no—
23#17985376—22no—
24#17985376—22no—
25C2×C283⋊C8—22no—
26(C2×D283)⋊C4—44no—
27C4×Dic283—22no—
28C2×Q8×C283—22no—
29C22×C4×C283D412yes—
30C22×C4×C283C2311yescharacteristic
31C4⋊C4×C283—22no—
32C4⋊C4×C283—24no—
33C4⋊C4×C283—22no—
34C4⋊C4×C283D412yes—
35C4⋊C4×C283D412yes—
36C22⋊C4×C283—22no—
37C42×C283—22no—
38C22×D283—22no—
39C22×D283—42no—
40C22×D283—42no—
41C22×D283—41no—
42C22×D283—41no—
43D1132—24no—
44D1132—42no—
45C2×Dic283—22no—
46C2×Dic283—41no—
47C2×Dic283—41no—
48C283⋊C8—42no—
49C23×C283C2×D411yescharacteristic
50Q8×C283—82no—
51C2×C4×C283—22no—
52C2×C4×C283—22no—
53C2×C4×C283—22no—
54C2×C4×C283—42no—
55C2×C4×C283C2×D412yes—
56C2×C4×C283C2×Q811yescharacteristic
57C2×C4×C283#4112yes—
58C2×C4×C283#4111yescharacteristic
59C22×C283—21no—
60C22×C283—21no—
61C22×C283#12011yescharacteristic
62C22×C283C22⋊Q811yescharacteristic
63C22×C283#12411yescharacteristic
64C2×D283—44no—
65C2×D283—42no—
66C2×D283—82no—
67C2×D283—81no—
68C2×D283—81no—
69C4×C283—42no—
70C4×C283—42no—
71C4×C283—42no—
72C4×C283C22⋊Q812yes—
73C4×C283#12412yes—
74Dic283—42no—
75C2×C283—21no—
76C2×C283—21no—
77C2×C283#39311yescharacteristic
78C2×C283#45311yescharacteristic
79C2×C283#45411yescharacteristic
80D283—82no—
81C283#202711yescharacteristic, complemented, Sylow
82#2027—2831nomaximal, Sylow
83#544D28311yescharacteristic, complemented
84#418—2832no—
85#390—2831no—
86#343—2831no—
87#339—2832no—
88C22×D4—2831no—
89C2×M16—2832no—
90C4⋊Q8—22no—
91#125—22no—
92C22⋊Q8—22no—
93#117C2×D28312yes—
94C2×C4⋊C4C2×D28311yescharacteristic
95C2×C22⋊C4—5662no—
96C2×C42—2831no—
97#102—5662no—
98#102—5662no—
99C24—5661no—
100C2×Q8—22no—
101C2×D4—2832no—
102C2×D4—5662no—
103C2×D4—5662no—
104C2×D4—5661no—
105C22×C4—5661no—
106C22×C4C22⋊D28312yes—
107C22×C4C22×D28311yescharacteristic
108M16—5662no—
109M16—5662no—
110C2×C8—5662no—
111C4⋊C4—22no—
112C4⋊C4—24no—
113C4⋊C4—22no—
114C4⋊C4C22⋊D28312yes—
115C4⋊C4C22⋊D28312yes—
116C22⋊C4—22no—
117C22⋊C4—11324no—
118C42—22no—
119C42—5662no—
120C23—5662no—
121C23—11322no—
122C23—11322no—
123C23—11321no—
124C23—11321no—
125C23C2×C22⋊D28311yescharacteristic
126Q8—82no—
127D4—5664no—
128D4—11322no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—42no—
133C2×C4—5662no—
134C2×C4—11321no—
135C2×C4—11321no—
136C2×C4C2×C22⋊D28312yes—
137C2×C4Q8×D28311yescharacteristic
138C2×C4#1798539312yes—
139C2×C4#1798539311yescharacteristic
140C8—11322no—
141C22—21no—
142C22—21no—
143C22—11324no—
144C22—11322no—
145C22—22642no—
146C22—22641no—
147C22—22641no—
148C22#2258357911yescharacteristic
149C22#2258358611yescharacteristic
150C22#2258359911yescharacteristic
151C4—42no—
152C4—42no—
153C4—42no—
154C4—11322no—
155C4#2258358612yes—
156C4#2258359912yes—
157C2—21no—
158C2—21no—
159C2—22642no—
160C2#3182637611yescharacteristic
161C2#3182663311yescharacteristic
162C2#3182663811yescharacteristic
subgroupG/H
centerC22#22583579
commutator, cosocleC22×C4×C283C23
FrattiniC22×C4C22×D283
Fitting#544×C283C2
socleC22×C283#120
prime radicalC22×D1132C22
hypercenter#544D283
hypocenterC283#2027
LCSG ▷ C22×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
3. #31826234C2semidirect
4. #31826224C2semidirect
5. #544×C283C2semidirect
21. C2×D1132Q8semidirect
81. C283#2027semidirect
83. #544D283semidirect

Maximal subgroups

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

subgroupindex
#318264692
#318263662
#318262342
#318262242
#544×C2832
#2027283