Finite Group Databasebeta

← #30770236

Subgroups — #30770236

Order 17152.

196 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 57 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#22562452C211yesmaximal, characteristic, complemented
2#22562285C211yesmaximal, characteristic
3#22562051C211yesmaximal, characteristic, complemented
4#22558276C211yesmaximal, characteristic, complemented
5#22558270C211yesmaximal, characteristic, complemented
6#22558245C211yesmaximal, characteristic, complemented
7C67×#2231C211yesmaximal, characteristic, complemented
8#16863831—21no—
9#16863813—21no—
10#16863773C2211yescharacteristic
11#16863217C2211yescharacteristic
12#16863217C2211yescharacteristic
13#16863208—21no—
14#16863196—21no—
15#16863196—21no—
16#16863180—21no—
17#16863172C2211yescharacteristic
18#499×C67C2211yescharacteristic
19#443×C67C2211yescharacteristic
20#365×C67—21no—
21#364×C67C2211yescharacteristic
22#360×C67—21no—
23C2×D536—21no—
24C2×D536—21no—
25#14524040—21no—
26#14523986—41no—
27#14523984—41no—
28#14523983—21no—
29#14523983—21no—
30#14523983—41no—
31C2×C67⋊C16—41no—
32C2×C67⋊C16D411yescharacteristic
33C2×C67⋊C16D411yescharacteristic
34#135×C67D411yescharacteristic
35#131×C67—21no—
36#125×C67—21no—
37M32×C67—21no—
38#108×C67D411yescharacteristic
39#107×C67—21no—
40#106×C67—21no—
41#104×C67—21no—
42C4×C8×C67C2311yescharacteristic
43C2×D268—41no—
44C2×D268—81no—
45D536—41no—
46D536—41no—
47D536—41no—
48D536—41no—
49C67⋊C16—21no—
50C67⋊C16—21no—
51C67⋊C16—41no—
52C67⋊C16—41no—
53#41×C67—21no—
54Q16×C67SD1611yescharacteristic
55SD16×C67—21no—
56D8×C67SD1611yescharacteristic
57M16×C67—21no—
58M16×C67—41no—
59C2×C8×C67—41no—
60C2×C8×C67C2×D411yescharacteristic
61C2×C8×C67C2×D411yescharacteristic
62C4⋊C4×C67—41no—
63C4⋊C4×C67—81no—
64C42×C67#4111yescharacteristic
65C16×C67—41no—
66C22×D67—81no—
67Q8×C67—21no—
68D4×C67—21no—
69D4×C67—41no—
70C2×C4×C67—21no—
71C2×C4×C67—41no—
72C2×C4×C67—81no—
73C2×C4×C67#12111yescharacteristic
74D268—81no—
75D268—81no—
76D268—162no—
77C8×C67—41no—
78C8×C67C2×SD1611yescharacteristic
79C8×C67#13511yescharacteristic
80C8×C67#13511yescharacteristic
81C8×C67AGL(1,ℤ/8)11yescharacteristic
82C22×C67—41no—
83C22×C67#47211yescharacteristic
84C2×D67—162no—
85C2×D67—161no—
86C4×C67—41no—
87C4×C67—41no—
88C4×C67—161no—
89C4×C67#46311yescharacteristic
90C4×C67#46511yescharacteristic
91#12465—671nomaximal, Sylow
92C2×C67—21no—
93C2×C67—81no—
94C2×C67#167711yescharacteristic
95D67—321no—
96#2244—671no—
97#2231D6711yescharacteristic, complemented
98#2213—671no—
99#2171—671no—
100#1369—671no—
101#1366—671no—
102#1357—671no—
103C67#1246511yescharacteristic, complemented, Sylow
104#509—1341no—
105#504—1341no—
106#499C2×D6711yescharacteristic
107#493—671no—
108#443C2×D6711yescharacteristic
109#365—21no—
110#364—671no—
111#364—671no—
112#364C2×D6711yescharacteristic
113#361—1341no—
114#360—21no—
115#357—1341no—
116#357—1341no—
117#350—1341no—
118#346—671no—
119#135C22⋊D6711yescharacteristic
120C2×D8—1341no—
121C2×D8—1341no—
122#131—21no—
123#127—1341no—
124#125—21no—
125SD32—2681no—
126D16—2681no—
127M32—21no—
128M32—2681no—
129M32—1341no—
130M32—1341no—
131C2×C16—2681no—
132C2×C16—671no—
133C2×C16—671no—
134#108C22⋊D6711yescharacteristic
135#107—21no—
136#106—21no—
137#104—21no—
138C4×C8C22×D6711yescharacteristic
139#41—21no—
140C2×D4—2681no—
141C2×D4—5361no—
142Q16#1176161511yescharacteristic
143SD16—21no—
144D8—2681no—
145D8—2681no—
146D8—2681no—
147D8—2681no—
148D8#1176161511yescharacteristic
149M16—21no—
150M16—41no—
151C2×C8—41no—
152C2×C8D4×D6711yescharacteristic
153C2×C8C2×C22⋊D6711yescharacteristic
154C4⋊C4—41no—
155C4⋊C4—81no—
156C42#1176162711yescharacteristic
157C16—41no—
158C16—2681no—
159C16—2681no—
160C16—1341no—
161C16—1341no—
162C23—5361no—
163Q8—21no—
164D4—21no—
165D4—41no—
166D4—5361no—
167D4—5361no—
168D4—10722no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—81no—
172C2×C4#1452401511yescharacteristic
173C8—41no—
174C8C2×#1176161511yescharacteristic
175C8#1452406311yescharacteristic
176C8#1452406411yescharacteristic
177C8#1452407011yescharacteristic
178C22—41no—
179C22—10722no—
180C22—10721no—
181C22#1686366411yescharacteristic
182C4—41no—
183C4—41no—
184C4—161no—
185C4#1686361311yescharacteristic
186C4#1686362711yescharacteristic
187C2—21no—
188C2—81no—
189C2—21441no—
190C2#2255962011yescharacteristic
subgroupG/H
centerC2#22559620
commutator, cosocleC4×C8×C67C23
FrattiniC4×C8C22×D67
FittingC67×#2231C2
socleC2×C67#1677
prime radical#22562285C2
hypercenter#2231D67
hypocenterC67#12465
LCSG ▷ C4×C8×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2231

Complements

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

NH ≅ G/Ntype
1. #22562452C2semidirect
3. #22562051C2semidirect
4. #22558276C2semidirect
5. #22558270C2semidirect
6. #22558245C2semidirect
7. C67×#2231C2semidirect
97. #2231D67semidirect
103. C67#12465semidirect

Maximal subgroups

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

subgroupindex
#225624522
#225622852
#225620512
#225582762
#225582702
#225582452
C67×#22312
#1246567