Finite Group Databasebeta

← #49123274

Subgroups — #49123274

Order 36224.

226 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 45 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#31826654C212yesmaximal
2#31826642C211yesmaximal, characteristic
3#31826260C211yesmaximal, characteristic, complemented
4#31826255C212yesmaximal, complemented
5#563×C283C211yesmaximal, characteristic, complemented
6#22583655C2211yescharacteristic
7#22583641—21no—
8#22583593—22no—
9#22583538—21no—
10#22583527C2211yescharacteristic
11#22583525C2212yes—
12#22583524—22no—
13C4⋊Q8×C283—21no—
14#126×C283—22no—
15#125×C283—21no—
16#124×C283C2211yescharacteristic
17#123×C283C2212yes—
18C22⋊Q8×C283—22no—
19#17985392—21no—
20#17985391—41no—
21C2×C22⋊D283—22no—
22D4×D283—41no—
23#17985382—41no—
24#17985379—41no—
25#17985376—21no—
26Dic283⋊C4—42no—
27C22⋊Dic283—22no—
28(C2×D283)⋊C4—42no—
29C2×Q8×C283—21no—
30C2×Q8×C283D411yescharacteristic
31C2×D4×C283C2311yescharacteristic
32C22×C4×C283D412yes—
33C4⋊C4×C283—21no—
34C4⋊C4×C283—22no—
35C4⋊C4×C283—22no—
36C4⋊C4×C283—21no—
37C4⋊C4×C283—22no—
38C22⋊C4×C283—22no—
39C22⋊C4×C283—22no—
40C22⋊C4×C283D412yes—
41C42×C283—21no—
42C42×C283D411yescharacteristic
43C22×D283—22no—
44Dic566—21no—
45D1132—21no—
46C22⋊D283—42no—
47C22⋊D283—41no—
48C2×Dic283—22no—
49C2×Dic283—42no—
50C4×D283—41no—
51C283⋊C8—41no—
52C23×C283C2×D412yes—
53Q8×C283—41no—
54Q8×C283—81no—
55D4×C283—41no—
56C2×C4×C283—21no—
57C2×C4×C283—22no—
58C2×C4×C283—21no—
59C2×C4×C283—22no—
60C2×C4×C283—42no—
61C2×C4×C283C2×D411yescharacteristic
62C22×C283—22no—
63C22×C283—42no—
64C22×C283#12011yescharacteristic
65C2×D283—41no—
66C2×D283—82no—
67C4×C283—21no—
68C4×C283—42no—
69C4×C283—41no—
70C4×C283—42no—
71C4×C283—41no—
72Dic283—41no—
73Dic283—82no—
74C2×C283—21no—
75C2×C283—42no—
76C2×C283B(4,𝔽2)11yescharacteristic
77D283—81no—
78C283#220111yescharacteristic, complemented, Sylow
79#2201—2831nomaximal, Sylow
80#563D28311yescharacteristic, complemented
81#458—2832no—
82#455—2831no—
83#354—2831no—
84#352—2832no—
85H(5,𝔽2)—2831no—
86#137—5661no—
87C4⋊Q8—21no—
88#126—22no—
89#125—21no—
90#124C2×D28311yescharacteristic
91#123—5662no—
92#123C2×D28312yes—
93C22⋊Q8—22no—
94#104—5661no—
95#100—2831no—
96#99—2832no—
97#99—5662no—
98#41—5661no—
99#41—11321no—
100C2×Q8—21no—
101C2×Q8C22⋊D28311yescharacteristic
102C2×D4—5662no—
103C2×D4—11321no—
104C2×D4C22×D28311yescharacteristic
105C22×C4C22⋊D28312yes—
106Q16—11321no—
107SD16—11321no—
108M16—5661no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—11322no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—5662no—
118C22⋊C4—11322no—
119C22⋊C4C22⋊D28312yes—
120C42—21no—
121C42C22⋊D28311yescharacteristic
122C23—5662no—
123C23C2×C22⋊D28312yes—
124Q8—41no—
125Q8—81no—
126Q8—5661no—
127D4—41no—
128D4—5661no—
129D4—11322no—
130D4—11321no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—5662no—
137C2×C4—11321no—
138C2×C4—11322no—
139C2×C4C2×C22⋊D28311yescharacteristic
140C8—11321no—
141C22—22no—
142C22—42no—
143C22—11321no—
144C22—22642no—
145C22#2258357911yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—41no—
151C4—11321no—
152C4—22642no—
153C2—21no—
154C2—42no—
155C2—22641no—
156C2#3182665211yescharacteristic
subgroupG/H
centerC2#31826652
commutator, cosocleC2×D4×C283C23
FrattiniC2×D4C22×D283
Fitting#563×C283C2
socleC2×C283B(4,𝔽2)
prime radical#22583655C22
hypercenter#563D283
hypocenterC283#2201
LCSG ▷ C2×D4×C283 ▷ C22×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #563

Complements

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

NH ≅ G/Ntype
3. #31826260C2semidirect
4. #31826255C2semidirect
5. #563×C283C2semidirect
78. C283#2201semidirect
80. #563D283semidirect

Maximal subgroups

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

subgroupindex
#318266542
#318266422
#318262602
#318262552
#563×C2832
#2201283