Finite Group Databasebeta

← #49120507

Subgroups — #49120507

Order 36224.

234 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 57 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#31827130C211yesmaximal, characteristic
2#31826742C211yesmaximal, characteristic
3#31826672C211yesmaximal, characteristic, complemented
4#31826211C211yesmaximal, characteristic, complemented
5#31826197C211yesmaximal, characteristic, complemented
6#31826194C211yesmaximal, characteristic
7#500×C283C211yesmaximal, characteristic, complemented
8C2×Q8×D283—21no—
9C2×#17985382—21no—
10#22583609—21no—
11#22583588—22no—
12#22583586—21no—
13C4×Dic566C2211yescharacteristic
14C4×D1132C2211yescharacteristic
15#22583572—21no—
16#22583543—21no—
17C4⋊(C283⋊C8)C2211yescharacteristic
18C4⋊(C283⋊C8)C2211yescharacteristic
19#22583534—21no—
20#22583530—21no—
21C4⋊Q8×C283C2211yescharacteristic
22C4⋊Q8×C283C2211yescharacteristic
23#107×C283—22no—
24C4×C8×C283C2211yescharacteristic
25C2×Dic566—22no—
26C2×Dic566D411yescharacteristic
27Q8×D283—42no—
28Q8×D283—42no—
29C2×D1132D411yescharacteristic
30C2×C4×D283—22no—
31C2×C4×D283—21no—
32#17985382—42no—
33C2×C283⋊C8—21no—
34C2×C283⋊C8—21no—
35C4⋊Dic283D411yescharacteristic
36C4⋊Dic283D411yescharacteristic
37Dic283⋊C4—22no—
38Dic283⋊C4—21no—
39(C2×D283)⋊C4—22no—
40(C2×D283)⋊C4—21no—
41C4×Dic283—21no—
42C2×Q8×C283—21no—
43C2×Q8×C283—21no—
44C2×C8×C283—21no—
45C4⋊C4×C283—21no—
46C4⋊C4×C283—21no—
47C4⋊C4×C283—21no—
48C4⋊C4×C283D412yes—
49C42×C283C2311yescharacteristic
50C22×D283—21no—
51Dic566—21no—
52Dic566—44no—
53Dic566D812yes—
54D1132—21no—
55D1132Q1612yescomplemented
56C2×Dic283—21no—
57C2×Dic283—21no—
58C2×Dic283—21no—
59C4×D283—44no—
60C4×D283—41no—
61C4×D283—41no—
62C283⋊C8—41no—
63C283⋊C8—41no—
64Q8×C283—41no—
65Q8×C283—41no—
66Q8×C283—41no—
67Q8×C283—41no—
68C2×C4×C283—22no—
69C2×C4×C283—41no—
70C2×C4×C283C2×D411yescharacteristic
71C2×C4×C283C2×D411yescharacteristic
72C2×C4×C283C2×D411yescharacteristic
73C8×C283—22no—
74C22×C283#12011yescharacteristic
75C2×D283—22no—
76C2×D283—41no—
77C2×D283—41no—
78C4×C283—21no—
79C4×C283—42no—
80C4×C283—81no—
81C4×C283C2×D811yescharacteristic
82C4×C283C2×Q1611yescharacteristic
83C4×C283AGL(1,ℤ/8)11yescharacteristic
84C4×C283#13711yescharacteristic
85Dic283—22no—
86Dic283—41no—
87Dic283—41no—
88C2×C283#44711yescharacteristic
89C2×C283#45111yescharacteristic
90C2×C283#45611yescharacteristic
91D283—42no—
92C283#164911yescharacteristic, complemented, Sylow
93#1649—2831nomaximal, Sylow
94D4×Q8—2831no—
95#500D28311yescharacteristic, complemented
96#474—2831no—
97#462—2831no—
98#332—2831no—
99Q8⋊C8—2831no—
100D4⋊C8—2831no—
101C22×Q8—5661no—
102C2×Q16—5661no—
103C4⋊Q8—5661no—
104C4⋊Q8C2×D28311yescharacteristic
105C4⋊Q8C2×D28311yescharacteristic
106C22⋊Q8—5661no—
107C22⋊Q8—5662no—
108C4×Q8—2831no—
109C4×D4—2831no—
110C4×D4—5661no—
111#107—22no—
112#107—5661no—
113C4⋊C8—2831no—
114C4⋊C8—2831no—
115#103—5661no—
116#102—5661no—
117C4×C8C2×D28311yescharacteristic
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—2831no—
121C2×Q8—5662no—
122C2×Q8—11322no—
123C2×Q8—11322no—
124C2×D4—2831no—
125C22×C4—5662no—
126C22×C4—5661no—
127Q16—11322no—
128C2×C8—21no—
129C2×C8—5661no—
130C2×C8—5661no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—2831no—
135C4⋊C4—2831no—
136C4⋊C4—5662no—
137C4⋊C4—5661no—
138C4⋊C4C22⋊D28312yes—
139C22⋊C4—5662no—
140C22⋊C4—5661no—
141C42—5661no—
142C42C22×D28311yescharacteristic
143C23—5661no—
144Q8—41no—
145Q8—41no—
146Q8—41no—
147Q8—41no—
148Q8—2832no—
149Q8—5661no—
150Q8—11324no—
151D4—2832no—
152D4—5661no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—5661no—
156C2×C4—5661no—
157C2×C4—5661no—
158C2×C4—11324no—
159C2×C4—11321no—
160C2×C4—11321no—
161C2×C4D4×D28311yescharacteristic
162C2×C4D4×D28311yescharacteristic
163C2×C4C2×C22⋊D28311yescharacteristic
164C8—22no—
165C8—11321no—
166C8—11321no—
167C22—5662no—
168C22—11321no—
169C22—11321no—
170C22#2258357811yescharacteristic
171C4—21no—
172C4—42no—
173C4—81no—
174C4—5662no—
175C4—11321no—
176C4—11321no—
177C4D8×D28311yescharacteristic
178C4Q16×D28311yescharacteristic
179C4#2258363411yescharacteristic
180C4#2258364211yescharacteristic
181C2—11322no—
182C2#3182659711yescharacteristic
183C2#3182661911yescharacteristic
184C2#3182664711yescharacteristic
subgroupG/H
centerC22#22583578
commutator, cosocleC42×C283C23
FrattiniC42C22×D283
Fitting#500×C283C2
socleC22×C283#120
prime radicalC2×D1132D4
hypercenter#500D283
hypocenterC283#1649
LCSG ▷ C42×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
3. #31826672C2semidirect
4. #31826211C2semidirect
5. #31826197C2semidirect
7. #500×C283C2semidirect
55. D1132Q16semidirect
92. C283#1649semidirect
95. #500D283semidirect

Maximal subgroups

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

subgroupindex
#318271302
#318267422
#318266722
#318262112
#318261972
#318261942
#500×C2832
#1649283