Finite Group Databasebeta

← #49123234

Subgroups — #49123234

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#31826908C211yesmaximal, characteristic, complemented
3C2×#22583555C211yesmaximal, characteristic, complemented
4#31826740C211yesmaximal, characteristic, complemented
5#31826279C211yesmaximal, characteristic, complemented
6#31826273C211yesmaximal, characteristic, complemented
7#350×C283C211yesmaximal, characteristic, complemented
8#22583656—21no—
9#22583640—21no—
10#22583638—22no—
11#22583638—21no—
12#22583632—21no—
13#22583632C2211yescharacteristic
14C2×Dic1132C2211yescharacteristic, complemented
15C2×D2264—21no—
16C2×D2264C2211yescharacteristic
17Dic2264—21no—
18#22583555—22no—
19#22583555—21no—
20#22583546C2211yescharacteristic, complemented
21#22583528—21no—
22#131×C283C2211yescharacteristic
23M32×C283C2211yescharacteristic, complemented
24C2×C16×C283C2211yescharacteristic, complemented
25#17985393—41no—
26#17985392—22no—
27#17985392—21no—
28C2×Dic566—21no—
29C2×D1132—22no—
30C2×D1132—21no—
31D4×D283—42no—
32D4×D283—41no—
33Dic1132—21no—
34Dic1132—21no—
35Dic1132D411yescharacteristic
36#17985382—41no—
37#17985381—22no—
38#17985381—21no—
39#17985380—41no—
40D2264—22no—
41D2264—21no—
42D2264D411yescharacteristic
43D2264D412yes—
44#17985376—21no—
45#41×C283D411yescharacteristic
46M16×C283—21no—
47M16×C283D411yescharacteristic
48C2×C8×C283—21no—
49C2×C8×C283C2311yescharacteristic
50C16×C283—21no—
51C16×C283—21no—
52C22×D283—42no—
53C22×D283—41no—
54Dic566—21no—
55Dic566—41no—
56Dic566—41no—
57D1132—21no—
58D1132—22no—
59D1132—42no—
60D1132—41no—
61C22⋊D283—42no—
62C22⋊D283—41no—
63C2×Dic283—41no—
64C4×D283—42no—
65C4×D283—41no—
66C283⋊C8—41no—
67Q8×C283D811yescharacteristic
68D4×C283—21no—
69D4×C283D811yescharacteristic
70C2×C4×C283—21no—
71C2×C4×C283C2×D411yescharacteristic
72C8×C283—21no—
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#12011yescharacteristic
83C4×C283C2×D811yescharacteristic
84Dic283—41no—
85Dic283—81no—
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#473—2831no—
97#361—2831no—
98#359—2831no—
99#350D28311yescharacteristic, complemented
100H(5,𝔽2)—5661no—
101#137—5661no—
102AGL(1,ℤ/8)—5662no—
103AGL(1,ℤ/8)—5661no—
104#135—2831no—
105#135—5661no—
106C2×Q16—2831no—
107C2×D8—2831no—
108C2×D8—5661no—
109#131C2×D28311yescharacteristic
110Q32—5661no—
111SD32—5662no—
112SD32—5661no—
113M32C2×D28311yescharacteristic, complemented
114C2×C16C2×D28311yescharacteristic, complemented
115#108—2831no—
116#101—5661no—
117#41—5662no—
118#41—5661no—
119#41—11321no—
120#41D113211yescharacteristic
121C2×Q8—5661no—
122C2×D4—5662no—
123C2×D4—5661no—
124C2×D4—11322no—
125C2×D4—11321no—
126Q16—2831no—
127Q16—5661no—
128Q16—5661no—
129Q16—11321no—
130SD16—5662no—
131SD16—5661no—
132SD16—11321no—
133D8—2832no—
134D8—2831no—
135D8—5662no—
136D8—5661no—
137M16—21no—
138M16—5661no—
139M16D113211yescharacteristic
140C2×C8—21no—
141C2×C8C22×D28311yescharacteristic
142C16—21no—
143C16—21no—
144C23—11322no—
145C23—11321no—
146Q8—5661no—
147Q8—11321no—
148Q8—11321no—
149Q8D226411yescharacteristic
150D4—21no—
151D4—5662no—
152D4—5661no—
153D4—11322no—
154D4—11322no—
155D4—11321no—
156D4—11321no—
157D4D226411yescharacteristic
158C2×C4—21no—
159C2×C4—11322no—
160C2×C4—11321no—
161C2×C4—11321no—
162C2×C4C2×D113211yescharacteristic
163C8—21no—
164C8—11321no—
165C8D4×D28311yescharacteristic
166C8D4×D28311yescharacteristic
167C22—21no—
168C22—11322no—
169C22—11321no—
170C22—45281no—
171C22—45282no—
172C22#2258363811yescharacteristic
173C4—21no—
174C4—11321no—
175C4—22641no—
176C4#2258358011yescharacteristic
177C4C2×D226411yescharacteristic
178C2—21no—
179C2—41no—
180C2—22642no—
181C2—22641no—
182C2#3182660111yescharacteristic
subgroupG/H
centerC2#31826601
commutator, cosocleC2×C8×C283C23
FrattiniC2×C8C22×D283
Fitting#350×C283C2
socleC2×C283#447
prime radical#31827270C2
hypercenter#350D283
hypocenterC283#2194
LCSG ▷ C2×C8×C283 ▷ C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #350

Complements

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

NH ≅ G/Ntype
2. #31826908C2semidirect
3. C2×#22583555C2semidirect
4. #31826740C2semidirect
5. #31826279C2semidirect
6. #31826273C2semidirect
7. #350×C283C2semidirect
14. C2×Dic1132C22semidirect
20. #22583546C22semidirect
23. M32×C283C22semidirect
24. C2×C16×C283C22semidirect
91. C283#2194semidirect
99. #350D283semidirect
113. M32C2×D283semidirect
114. C2×C16C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272702
#318269082
C2×#225835552
#318267402
#318262792
#318262732
#350×C2832
#2194283