Finite Group Databasebeta

← #49169237

Subgroups — #49169237

Order 37504.

656 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×#22627652C214yesmaximal, complemented
2#34036679C211yesmaximal, characteristic, complemented
3C2×#22627621C211yesmaximal, characteristic, complemented
4#34036518C214yesmaximal, complemented
5#34036506C212yesmaximal, complemented
6#34036502C212yesmaximal, complemented
7C4×M16×C293C211yesmaximal, characteristic, complemented
8C2×#17989802C2212yes—
9C22×D1172—22no—
10C22×D1172C2212yes—
11#22627652—216no—
12C2×#17989791C2218yescomplemented
13C2×D2344C2218yescomplemented
14#22627624C2211yescharacteristic, complemented
15#22627621—22no—
16#22627621C2212yes—
17#22627621C2212yes—
18#22627621C2211yescharacteristic, complemented
19#22627612C2212yescomplemented
20#22627598—22no—
21C4×D1172—22no—
22C2×M16×C293C2212yes—
23C2×C42×C293C2211yescharacteristic
24#97×C293C2212yescomplemented
25C4×C8×C293C2212yescomplemented
26C23×D293—22no—
27#17989802—28no—
28C2×Dic586C2312yes—
29C2×D1172—28no—
30C2×D1172—22no—
31C2×D1172—22no—
32C2×D1172—24no—
33C2×D1172—24no—
34C2×D1172—48no—
35C2×D1172C2312yes—
36C2×D1172C2314yes—
37C2×C22⋊D293—22no—
38C2×C4×D293—22no—
39#17989791—216no—
40D2344—216no—
41C4⋊Dic293—22no—
42#17989779—24no—
43C22×C4×C293D412yes—
44C22×C4×C293C2311yescharacteristic
45M16×C293D418yescomplemented
46C2×C8×C293C2314yes—
47C42×C293D412yes—
48C42×C293C2311yescharacteristic
49C42×C293C2311yescharacteristic
50C22×D293—22no—
51C22×D293—24no—
52C22×D293—48no—
53C22×D293—44no—
54Dic586—24no—
55D1172—24no—
56D1172—28no—
57D1172—48no—
58D1172—44no—
59D1172—44no—
60D1172—48no—
61C22⋊D293—44no—
62C2×Dic293—22no—
63C4×D293—44no—
64C23×C293C2×D411yescharacteristic
65C2×C4×C293—24no—
66C2×C4×C293C2×D412yes—
67C2×C4×C293C2×D411yescharacteristic
68C2×C4×C293C2×D414yes—
69C2×C4×C293C2×D412yes—
70C2×C4×C293C2411yescharacteristic
71C8×C293C2×D418yescomplemented
72C22×C293—22no—
73C22×C293#12712yes—
74C22×C293C22×D411yescharacteristic
75C2×D293—42no—
76C2×D293—44no—
77C2×D293—48no—
78C2×D293—44no—
79C2×D293—88no—
80C4×C293—22no—
81C4×C293#12712yes—
82C4×C293AGL(1,ℤ/8)14yescomplemented
83C4×C293C22×D412yes—
84Dic293—42no—
85C2×C293—22no—
86C2×C293C2×#12711yescharacteristic
87C2×C293C2×AGL(1,ℤ/8)12yes—
88D293—82no—
89D293—84no—
90C293#315111yescharacteristic, complemented, Sylow
91#3151—2931nomaximal, Sylow
92C2×AGL(1,ℤ/8)—2934no—
93#532—2931no—
94C2×#127—2931no—
95#496—2934no—
96#493—2932no—
97#492—2932no—
98C4×M16D29311yescharacteristic, complemented
99C2×#41—2932no—
100C22×D4—2932no—
101C22×D4—5862no—
102AGL(1,ℤ/8)—58616no—
103C2×SD16—2938no—
104C2×D8—2938no—
105C2×M16C2×D29312yes—
106C4⋊Q8—2931no—
107#127—2932no—
108#127—2932no—
109#127—2931no—
110#127—5862no—
111#124—2932no—
112#121—5862no—
113C4×D4—5862no—
114C2×C42C2×D29311yescharacteristic
115#97C2×D29312yescomplemented
116C4×C8C2×D29312yescomplemented
117C24—5862no—
118#41—5868no—
119C2×Q8—2932no—
120C2×D4—2932no—
121C2×D4—2934no—
122C2×D4—5862no—
123C2×D4—5868no—
124C2×D4—5862no—
125C2×D4—5864no—
126C2×D4—5864no—
127C2×D4—5862no—
128C2×D4—11728no—
129C22×C4—5862no—
130C22×C4D117212yes—
131C22×C4C22×D29311yescharacteristic
132SD16—58616no—
133D8—58616no—
134M16D117218yescomplemented
135C2×C8C22×D29314yes—
136C4⋊C4—5862no—
137C22⋊C4—5864no—
138C42D117212yes—
139C42C22×D29311yescharacteristic
140C42C22×D29311yescharacteristic
141C23—5862no—
142C23—5864no—
143C23—11728no—
144C23—11724no—
145C23C2×D117211yescharacteristic
146Q8—5864no—
147D4—5864no—
148D4—5868no—
149D4—11728no—
150D4—11724no—
151D4—11724no—
152D4—11728no—
153D4—11724no—
154C2×C4—24no—
155C2×C4—5862no—
156C2×C4—11724no—
157C2×C4C2×D117212yes—
158C2×C4C2×D117211yescharacteristic
159C2×C4C2×D117214yes—
160C2×C4C2×D117212yes—
161C2×C4C23×D29311yescharacteristic
162C8C2×D117218yescomplemented
163C22—22no—
164C22—23448no—
165C22—11728no—
166C22—11724no—
167C22—11722no—
168C22—11724no—
169C22#2262762112yes—
170C22C22×D117211yescharacteristic
171C4—22no—
172C4—11722no—
173C4#2262762112yes—
174C4#2262765214yescomplemented
175C4C22×D117212yes—
176C2—22no—
177C2—23444no—
178C2—23442no—
179C2C2×#2262762111yescharacteristic
180C2C2×#2262765212yes—
subgroupG/H
centerC22C22×D1172
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
FittingC4×M16×C293C2
socleC22×C293C22×D4
hypercenterC4×M16D293
hypocenterC293#3151
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
1. C2×#22627652C2semidirect
2. #34036679C2semidirect
3. C2×#22627621C2semidirect
4. #34036518C2semidirect
5. #34036506C2semidirect
6. #34036502C2semidirect
7. C4×M16×C293C2semidirect
12. C2×#17989791C22semidirect
13. C2×D2344C22semidirect
14. #22627624C22semidirect
18. #22627621C22semidirect
19. #22627612C22semidirect
24. #97×C293C22semidirect
25. C4×C8×C293C22semidirect
45. M16×C293D4semidirect
71. C8×C293C2×D4semidirect
82. C4×C293AGL(1,ℤ/8)semidirect
90. C293#3151semidirect
98. C4×M16D293semidirect
115. #97C2×D293semidirect
116. C4×C8C2×D293semidirect
134. M16D1172semidirect
162. C8C2×D1172semidirect
174. C4#22627652semidirect

Maximal subgroups

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

subgroupindex
C2×#226276522
#340366792
C2×#226276212
#340365182
#340365062
#340365022
C4×M16×C2932
#3151293