Finite Group Databasebeta

← #33006607

Subgroups — #33006607

Order 18688.

194 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 63 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#22612244C211yesmaximal, characteristic
2#22611818C211yesmaximal, characteristic, complemented
3#22608136C211yesmaximal, characteristic, complemented
4#22608121C211yesmaximal, characteristic, complemented
5#22608117C211yesmaximal, characteristic, complemented
6#22608067C211yesmaximal, characteristic, complemented
7C73×#2177C211yesmaximal, characteristic, complemented
8#17989019C2211yescharacteristic
9#17988705—21no—
10#17988425C412yescomplemented
11#17988422C412yescomplemented
12#17988421C2211yescharacteristic
13#17988421C2211yescharacteristic
14#17988417C2211yescharacteristic
15#17988395—21no—
16C2×D16×C73C2211yescharacteristic
17C4×D8×C73—21no—
18#367×C73C2211yescharacteristic
19#357×C73—21no—
20#346×C73C2211yescharacteristic
21#14556613—21no—
22#14556597—41no—
23Dic584C2×C411yescharacteristic, complemented
24#14556536—21no—
25D1168C2×C411yescharacteristic, complemented
26#14556527—21no—
27#14556519—41no—
28#14556497—21no—
29#14556497C2×C412yescomplemented
30C2×D8×C73—21no—
31C4×D4×C73—41no—
32D16×C73—21no—
33D16×C73C2×C412yes—
34C2×C16×C73D411yescharacteristic
35C2×C16×C73C2311yescharacteristic
36#107×C73—21no—
37#102×C73—41no—
38C4×C8×C73D411yescharacteristic
39#11924028—41no—
40Dic292—21no—
41#11924017—41no—
42D584—21no—
43#11924012—41no—
44#11924011—81no—
45C8×D73—81no—
46C2×D4×C73—41no—
47C22×C4×C73—81no—
48D8×C73—22no—
49D8×C73—41no—
50C2×C8×C73D811yescharacteristic
51C2×C8×C73C2×D411yescharacteristic
52C4⋊C4×C73—41no—
53C22⋊C4×C73—81no—
54C42×C73D811yescharacteristic
55C16×C73—21no—
56C16×C73C22×C411yescharacteristic
57C16×C73#4111yescharacteristic
58C23×C73—81no—
59D4×C73—42no—
60D4×C73—81no—
61C2×C4×C73—21no—
62C2×C4×C73—81no—
63C2×C4×C73—162no—
64C2×C4×C73C2×D811yescharacteristic
65C22⋊D73—81no—
66D292—41no—
67C4×D73—81no—
68Dic146—41no—
69C8×C73—21no—
70C8×C73C4×D411yescharacteristic
71C8×C73#13511yescharacteristic
72#10596831—81no—
73C22×C73—82no—
74C22×C73—161no—
75C22×C73—161no—
76C22×C73#50911yescharacteristic
77C2×D73—81no—
78C4×C73—41no—
79C4×C73—161no—
80C4×C73C4×D811yescharacteristic
81C4×C73#50911yescharacteristic
82Dic73—81no—
83#13658—731nomaximal, Sylow
84C2×C73—21no—
85C2×C73—162no—
86C2×C73#217711yescharacteristic
87D73—161no—
88#2263—731no—
89#2179—731no—
90#2177D7311yescharacteristic, complemented
91#1426—731no—
92#1419—731no—
93#1418—731no—
94#1398—731no—
95C73#1365811yescharacteristic, complemented, Sylow
96#508—731no—
97C2×D16C2×D7311yescharacteristic
98#443—1461no—
99C4×D8—21no—
100SD64—732no—
101D32—732no—
102M64—731no—
103M64—731no—
104#368—731no—
105#367C2×D7311yescharacteristic
106#359—1461no—
107#357—21no—
108#346C2×D7311yescharacteristic
109#135—1461no—
110C2×D8—21no—
111#131—2921no—
112C4×D4—41no—
113Q32—731no—
114SD32—1461no—
115D16—21no—
116D16—731no—
117D16C4×D7312yes—
118C2×C16C22⋊D7311yescharacteristic
119C2×C16C22×D7311yescharacteristic
120#108—1461no—
121#107—21no—
122#104—2921no—
123#102—41no—
124C4×C8C22⋊D7311yescharacteristic
125C32—732no—
126C32—1461no—
127#41—2921no—
128C2×D4—41no—
129C22×C4—81no—
130Q16—1461no—
131SD16—2921no—
132D8—22no—
133D8—41no—
134D8—1461no—
135M16—2921no—
136M16—5841no—
137C2×C8—5841no—
138C2×C8#1192401311yescharacteristic
139C2×C8C2×C22⋊D7311yescharacteristic
140C4⋊C4—41no—
141C22⋊C4—81no—
142C42#1192401311yescharacteristic
143C16—21no—
144C16C2×C4×D7311yescharacteristic
145C16#1192402811yescharacteristic
146C23—81no—
147Q8—2921no—
148D4—42no—
149D4—81no—
150D4—2921no—
151D4—5841no—
152C2×C4—21no—
153C2×C4—81no—
154C2×C4—162no—
155C2×C4—5841no—
156C2×C4C2×#1192401311yescharacteristic
157C8—21no—
158C8—5841no—
159C8C4×C22⋊D7311yescharacteristic
160C8#1455661111yescharacteristic
161C22—82no—
162C22—161no—
163C22—161no—
164C22—5841no—
165C22#1798902311yescharacteristic
166C4—41no—
167C4—161no—
168C4—5841no—
169C4C4×#1192401311yescharacteristic
170C4#1798902211yescharacteristic
171C2—21no—
172C2—162no—
173C2—11681no—
174C2#2261180511yescharacteristic
subgroupG/H
centerC2#22611805
commutatorC16×C73C22×C4
FrattiniC2×C16C22×D73
FittingC73×#2177C2
socleC2×C73#2177
prime radical#22612244C2
cosocleC2×C16×C73C23
hypercenter#2177D73
hypocenterC73#13658
LCSG ▷ C16×C73 ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #346 ◁ #2177

Complements

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

NH ≅ G/Ntype
2. #22611818C2semidirect
3. #22608136C2semidirect
4. #22608121C2semidirect
5. #22608117C2semidirect
6. #22608067C2semidirect
7. C73×#2177C2semidirect
10. #17988425C4semidirect
11. #17988422C4semidirect
23. Dic584C2×C4semidirect
25. D1168C2×C4semidirect
29. #14556497C2×C4semidirect
90. #2177D73semidirect
95. C73#13658semidirect

Maximal subgroups

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

subgroupindex
#226122442
#226118182
#226081362
#226081212
#226081172
#226080672
C73×#21772
#1365873