Finite Group Databasebeta

← #49123230

Subgroups — #49123230

Order 36224.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#31827270C211yesmaximal, characteristic
2#31826906C211yesmaximal, characteristic, complemented
3C2×#22583553C211yesmaximal, characteristic, complemented
4#31826277C211yesmaximal, characteristic, complemented
5#31826271C211yesmaximal, characteristic, complemented
6#31826249C211yesmaximal, characteristic, complemented
7#473×C283C211yesmaximal, characteristic, complemented
8#22583656—21no—
9#22583638—22no—
10#22583638—21no—
11#22583632—21no—
12#22583632C2211yescharacteristic
13C2×D2264—21no—
14C2×D2264C2211yescharacteristic
15#22583556—21no—
16#22583553—22no—
17#22583553—21no—
18#22583550C2211yescharacteristic, complemented
19C2×C283⋊C16C2211yescharacteristic, complemented
20#137×C283—21no—
21C2×Q16×C283C2211yescharacteristic, complemented
22#131×C283C2211yescharacteristic
23#108×C283C2211yescharacteristic, complemented
24#101×C283—21no—
25#17985393—41no—
26#17985392—22no—
27#17985392—21no—
28C2×D1132—22no—
29C2×D1132—21no—
30D4×D283—42no—
31D4×D283—41no—
32Dic1132D411yescharacteristic
33#17985381—22no—
34#17985381—21no—
35D2264—22no—
36D2264—21no—
37D2264D411yescharacteristic
38D2264D412yes—
39C283⋊C16—21no—
40C283⋊C16—21no—
41#41×C283D411yescharacteristic
42C2×Q8×C283—21no—
43Q16×C283—21no—
44Q16×C283—21no—
45Q16×C283—41no—
46SD16×C283—41no—
47M16×C283—21no—
48M16×C283—21no—
49M16×C283D411yescharacteristic
50C2×C8×C283—21no—
51C2×C8×C283C2311yescharacteristic
52C22×D283—42no—
53C22×D283—41no—
54Dic566—21no—
55D1132—21no—
56D1132—22no—
57D1132—42no—
58D1132—41no—
59C22⋊D283—42no—
60C22⋊D283—41no—
61C4×D283—42no—
62C4×D283—41no—
63Q8×C283—41no—
64Q8×C283—41no—
65Q8×C283D811yescharacteristic
66D4×C283—21no—
67D4×C283D811yescharacteristic
68C2×C4×C283—21no—
69C2×C4×C283—41no—
70C2×C4×C283C2×D411yescharacteristic
71C8×C283—21no—
72C8×C283—41no—
73C8×C283C2×D411yescharacteristic
74C8×C283C2×D411yescharacteristic
75C22×C283—21no—
76C22×C283AGL(1,ℤ/8)11yescharacteristic
77C2×D283—42no—
78C2×D283—41no—
79C2×D283—161no—
80C2×D283—162no—
81C4×C283—21no—
82C4×C283—81no—
83C4×C283#12011yescharacteristic
84C4×C283C2×D811yescharacteristic
85Dic283—41no—
86C2×C283—21no—
87C2×C283—41no—
88C2×C283#44711yescharacteristic
89D283—82no—
90D283—81no—
91C283#219411yescharacteristic, complemented, Sylow
92#2194—2831nomaximal, Sylow
93#576—2831no—
94#510—2831no—
95C2×SD32—2831no—
96#473D28311yescharacteristic, complemented
97#361—2831no—
98#359—2831no—
99#350—2831no—
100H(5,𝔽2)—5661no—
101#137—21no—
102AGL(1,ℤ/8)—5662no—
103AGL(1,ℤ/8)—5661no—
104#135—2831no—
105#135—5661no—
106C2×Q16C2×D28311yescharacteristic, complemented
107C2×D8—2831no—
108C2×D8—5661no—
109#131C2×D28311yescharacteristic
110Q32—5661no—
111SD32—5662no—
112SD32—5661no—
113M32—2831no—
114C2×C16—2831no—
115#108C2×D28311yescharacteristic, complemented
116#101—21no—
117#41—5662no—
118#41—5661no—
119#41—11321no—
120#41C22⋊D28311yescharacteristic
121C2×Q8—21no—
122C2×D4—5662no—
123C2×D4—5661no—
124C2×D4—11322no—
125C2×D4—11321no—
126Q16—21no—
127Q16—21no—
128Q16—41no—
129Q16—2831no—
130SD16—41no—
131SD16—5662no—
132SD16—5661no—
133D8—2832no—
134D8—2831no—
135D8—5662no—
136D8—5661no—
137M16—21no—
138M16—21no—
139M16C22⋊D28311yescharacteristic
140C2×C8—21no—
141C2×C8C22×D28311yescharacteristic
142C16—5661no—
143C16—5661no—
144C23—11322no—
145C23—11321no—
146Q8—41no—
147Q8—41no—
148Q8—5661no—
149Q8#1798537711yescharacteristic
150D4—21no—
151D4—5662no—
152D4—5661no—
153D4—11322no—
154D4—11322no—
155D4—11321no—
156D4—11321no—
157D4#1798537711yescharacteristic
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—11322no—
161C2×C4—11321no—
162C2×C4C2×C22⋊D28311yescharacteristic
163C8—21no—
164C8—41no—
165C8D4×D28311yescharacteristic
166C8D4×D28311yescharacteristic
167C22—21no—
168C22—11322no—
169C22—11321no—
170C22—45281no—
171C22—45282no—
172C22#2258363511yescharacteristic
173C4—21no—
174C4—81no—
175C4—11321no—
176C4#2258357811yescharacteristic
177C4C2×#1798537711yescharacteristic
178C2—21no—
179C2—41no—
180C2—22642no—
181C2—22641no—
182C2#3182659811yescharacteristic
subgroupG/H
centerC2#31826598
commutator, cosocleC2×C8×C283C23
FrattiniC2×C8C22×D283
Fitting#473×C283C2
socleC2×C283#447
prime radical#31827270C2
hypercenter#473D283
hypocenterC283#2194
LCSG ▷ C2×C8×C283 ▷ C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #473

Complements

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

NH ≅ G/Ntype
2. #31826906C2semidirect
3. C2×#22583553C2semidirect
4. #31826277C2semidirect
5. #31826271C2semidirect
6. #31826249C2semidirect
7. #473×C283C2semidirect
18. #22583550C22semidirect
19. C2×C283⋊C16C22semidirect
21. C2×Q16×C283C22semidirect
23. #108×C283C22semidirect
91. C283#2194semidirect
96. #473D283semidirect
106. C2×Q16C2×D283semidirect
115. #108C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272702
#318269062
C2×#225835532
#318262772
#318262712
#318262492
#473×C2832
#2194283