Finite Group Databasebeta

← #31918075

Subgroups — #31918075

Order 18176.

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#22590653C211yesmaximal, characteristic
2#22590227C211yesmaximal, characteristic, complemented
3#22586545C211yesmaximal, characteristic, complemented
4#22586530C211yesmaximal, characteristic, complemented
5#22586526C211yesmaximal, characteristic, complemented
6#22586476C211yesmaximal, characteristic, complemented
7C71×#2177C211yesmaximal, characteristic, complemented
8#17986431C2211yescharacteristic
9#17986117—21no—
10#17985837C412yescomplemented
11#17985834C412yescomplemented
12#17985833C2211yescharacteristic
13#17985833C2211yescharacteristic
14#17985829C2211yescharacteristic
15#17985807—21no—
16C2×D16×C71C2211yescharacteristic
17C4×D8×C71—21no—
18#367×C71C2211yescharacteristic
19#357×C71—21no—
20#346×C71C2211yescharacteristic
21#14556150—21no—
22#14556134—41no—
23Dic568C2×C411yescharacteristic, complemented
24#14556073—21no—
25D1136C2×C411yescharacteristic, complemented
26#14556064—21no—
27#14556056—41no—
28C71⋊C32—21no—
29C71⋊C32C2×C412yescomplemented
30C2×D8×C71—21no—
31C4×D4×C71—41no—
32D16×C71—21no—
33D16×C71C2×C412yes—
34C2×C16×C71D411yescharacteristic
35C2×C16×C71C2311yescharacteristic
36#107×C71—21no—
37#102×C71—41no—
38C4×C8×C71D411yescharacteristic
39#11765865—41no—
40Dic284—21no—
41#11765854—41no—
42D568—21no—
43#11765849—41no—
44#11765848—81no—
45C8×D71—81no—
46C2×D4×C71—41no—
47C22×C4×C71—81no—
48D8×C71—22no—
49D8×C71—41no—
50C2×C8×C71D811yescharacteristic
51C2×C8×C71C2×D411yescharacteristic
52C4⋊C4×C71—41no—
53C22⋊C4×C71—81no—
54C42×C71D811yescharacteristic
55C16×C71—21no—
56C16×C71C22×C411yescharacteristic
57C16×C71#4111yescharacteristic
58C23×C71—81no—
59D4×C71—42no—
60D4×C71—81no—
61C2×C4×C71—21no—
62C2×C4×C71—81no—
63C2×C4×C71—162no—
64C2×C4×C71C2×D811yescharacteristic
65C22⋊D71—81no—
66D284—41no—
67C4×D71—81no—
68Dic142—41no—
69C8×C71—21no—
70C8×C71C4×D411yescharacteristic
71C8×C71#13511yescharacteristic
72C71⋊C8—81no—
73C22×C71—82no—
74C22×C71—161no—
75C22×C71—161no—
76C22×C71#50911yescharacteristic
77C2×D71—81no—
78C4×C71—41no—
79C4×C71—161no—
80C4×C71C4×D811yescharacteristic
81C4×C71#50911yescharacteristic
82Dic71—81no—
83#13658—711nomaximal, Sylow
84C2×C71—21no—
85C2×C71—162no—
86C2×C71#217711yescharacteristic
87D71—161no—
88#2263—711no—
89#2179—711no—
90#2177D7111yescharacteristic, complemented
91#1426—711no—
92#1419—711no—
93#1418—711no—
94#1398—711no—
95C71#1365811yescharacteristic, complemented, Sylow
96#508—711no—
97C2×D16C2×D7111yescharacteristic
98#443—1421no—
99C4×D8—21no—
100SD64—712no—
101D32—712no—
102M64—711no—
103M64—711no—
104#368—711no—
105#367C2×D7111yescharacteristic
106#359—1421no—
107#357—21no—
108#346C2×D7111yescharacteristic
109#135—1421no—
110C2×D8—21no—
111#131—2841no—
112C4×D4—41no—
113Q32—711no—
114SD32—1421no—
115D16—21no—
116D16—711no—
117D16C4×D7112yes—
118C2×C16C22⋊D7111yescharacteristic
119C2×C16C22×D7111yescharacteristic
120#108—1421no—
121#107—21no—
122#104—2841no—
123#102—41no—
124C4×C8C22⋊D7111yescharacteristic
125C32—712no—
126C32—1421no—
127#41—2841no—
128C2×D4—41no—
129C22×C4—81no—
130Q16—1421no—
131SD16—2841no—
132D8—22no—
133D8—41no—
134D8—1421no—
135M16—2841no—
136M16—5681no—
137C2×C8—5681no—
138C2×C8#1176585011yescharacteristic
139C2×C8C2×C22⋊D7111yescharacteristic
140C4⋊C4—41no—
141C22⋊C4—81no—
142C42#1176585011yescharacteristic
143C16—21no—
144C16C2×C4×D7111yescharacteristic
145C16#1176586511yescharacteristic
146C23—81no—
147Q8—2841no—
148D4—42no—
149D4—81no—
150D4—2841no—
151D4—5681no—
152C2×C4—21no—
153C2×C4—81no—
154C2×C4—162no—
155C2×C4—5681no—
156C2×C4C2×#1176585011yescharacteristic
157C8—21no—
158C8—5681no—
159C8C4×C22⋊D7111yescharacteristic
160C8#1455614811yescharacteristic
161C22—82no—
162C22—161no—
163C22—161no—
164C22—5681no—
165C22#1798643511yescharacteristic
166C4—41no—
167C4—161no—
168C4—5681no—
169C4C4×#1176585011yescharacteristic
170C4#1798643411yescharacteristic
171C2—21no—
172C2—162no—
173C2—11361no—
174C2#2259021411yescharacteristic
subgroupG/H
centerC2#22590214
commutatorC16×C71C22×C4
FrattiniC2×C16C22×D71
FittingC71×#2177C2
socleC2×C71#2177
prime radical#22590653C2
cosocleC2×C16×C71C23
hypercenter#2177D71
hypocenterC71#13658
LCSG ▷ C16×C71 ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #346 ◁ #2177

Complements

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

NH ≅ G/Ntype
2. #22590227C2semidirect
3. #22586545C2semidirect
4. #22586530C2semidirect
5. #22586526C2semidirect
6. #22586476C2semidirect
7. C71×#2177C2semidirect
10. #17985837C4semidirect
11. #17985834C4semidirect
23. Dic568C2×C4semidirect
25. D1136C2×C4semidirect
29. C71⋊C32C2×C4semidirect
90. #2177D71semidirect
95. C71#13658semidirect

Maximal subgroups

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

subgroupindex
#225906532
#225902272
#225865452
#225865302
#225865262
#225864762
C71×#21772
#1365871