Finite Group Databasebeta

← #30770231

Subgroups — #30770231

Order 17152.

196 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 55 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#22562382C211yesmaximal, characteristic, complemented
2#22562282C211yesmaximal, characteristic
3#22562052C211yesmaximal, characteristic, complemented
4#22558277C211yesmaximal, characteristic, complemented
5#22558270C211yesmaximal, characteristic, complemented
6#22558246C211yesmaximal, characteristic, complemented
7C67×#2244C211yesmaximal, characteristic, complemented
8#16863832—21no—
9#16863812—21no—
10#16863514C2211yescharacteristic
11#16863218—21no—
12#16863217C2211yescharacteristic
13#16863217C2211yescharacteristic
14#16863217C2211yescharacteristic
15#16863209—21no—
16#16863205—21no—
17#16863179—21no—
18#499×C67C2211yescharacteristic
19#493×C67C2211yescharacteristic
20#357×C67—21no—
21#357×C67—21no—
22#346×C67C2211yescharacteristic
23#14524065D411yescharacteristic
24#14524049—21no—
25#14523987—41no—
26#14523984—41no—
27#14523983—21no—
28#14523983—21no—
29#14523983—21no—
30#14523982—41no—
31C16×D67—41no—
32#14523979D411yescharacteristic
33#14523971—21no—
34C2×D8×C67—21no—
35C2×D8×C67—21no—
36#127×C67—21no—
37#125×C67—21no—
38C2×C16×C67D411yescharacteristic
39C2×C16×C67D411yescharacteristic
40#107×C67—21no—
41#106×C67—21no—
42C4×C8×C67C2311yescharacteristic
43#11761628—21no—
44Dic268SD1611yescharacteristic
45#11761617—21no—
46D536SD1611yescharacteristic
47#11761612—21no—
48#11761611—41no—
49C8×D67—41no—
50C67⋊C16—41no—
51C67⋊C16—41no—
52C67⋊C16—41no—
53C2×D4×C67—41no—
54C2×D4×C67—81no—
55D8×C67—41no—
56D8×C67—41no—
57D8×C67—41no—
58D8×C67—41no—
59C2×C8×C67C2×D411yescharacteristic
60C2×C8×C67C2×D411yescharacteristic
61C4⋊C4×C67—41no—
62C4⋊C4×C67—81no—
63C42×C67#4111yescharacteristic
64C16×C67—21no—
65C16×C67—21no—
66C23×C67—81no—
67D4×C67—81no—
68D4×C67—81no—
69D4×C67—162no—
70C2×C4×C67—21no—
71C2×C4×C67—81no—
72C2×C4×C67#12111yescharacteristic
73C22⋊D67—41no—
74D268—21no—
75C4×D67—41no—
76Dic134—21no—
77C8×C67C2×SD1611yescharacteristic
78C8×C67#13511yescharacteristic
79C8×C67#13511yescharacteristic
80C8×C67AGL(1,ℤ/8)11yescharacteristic
81C67⋊C8—41no—
82C22×C67—162no—
83C22×C67—161no—
84C22×C67#47211yescharacteristic
85C2×D67—41no—
86C4×C67—41no—
87C4×C67—161no—
88C4×C67#46311yescharacteristic
89C4×C67#46511yescharacteristic
90Dic67—41no—
91#12465—671nomaximal, Sylow
92C2×C67—21no—
93C2×C67—321no—
94C2×C67#167711yescharacteristic
95D67—81no—
96#2244D6711yescharacteristic, complemented
97#2231—671no—
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#493C2×D6711yescharacteristic
108#443—671no—
109#365—1341no—
110#364—671no—
111#364—671no—
112#364—671no—
113#361—1341no—
114#360—1341no—
115#357—21no—
116#357—21no—
117#350—1341no—
118#346C2×D6711yescharacteristic
119#135—671no—
120C2×D8—21no—
121C2×D8—21no—
122#131—1341no—
123#127—21no—
124#125—21no—
125SD32—2681no—
126D16—2681no—
127M32—2681no—
128M32—1341no—
129M32—1341no—
130M32—1341no—
131C2×C16—2681no—
132C2×C16C22⋊D6711yescharacteristic
133C2×C16C22⋊D6711yescharacteristic
134#108—671no—
135#107—21no—
136#106—21no—
137#104—1341no—
138C4×C8C22×D6711yescharacteristic
139#41—1341no—
140C2×D4—41no—
141C2×D4—81no—
142Q16—671no—
143SD16—1341no—
144D8—41no—
145D8—41no—
146D8—41no—
147D8—41no—
148D8—671no—
149M16—2681no—
150M16—1341no—
151C2×C8—2681no—
152C2×C8D4×D6711yescharacteristic
153C2×C8C2×C22⋊D6711yescharacteristic
154C4⋊C4—41no—
155C4⋊C4—81no—
156C42#1176162711yescharacteristic
157C16—21no—
158C16—21no—
159C16—2681no—
160C16—2681no—
161C16—2681no—
162C23—81no—
163Q8—1341no—
164D4—81no—
165D4—81no—
166D4—2681no—
167D4—162no—
168D4—1341no—
169C2×C4—21no—
170C2×C4—81no—
171C2×C4—2681no—
172C2×C4#1452401411yescharacteristic
173C8—2681no—
174C8SD16×D6711yescharacteristic
175C8#1452406311yescharacteristic
176C8#1452406411yescharacteristic
177C8#1452406811yescharacteristic
178C22—2681no—
179C22—162no—
180C22—161no—
181C22#1686366311yescharacteristic
182C4—41no—
183C4—2681no—
184C4—161no—
185C4#1686360811yescharacteristic
186C4#1686362211yescharacteristic
187C2—21no—
188C2—5361no—
189C2—321no—
190C2#2255961511yescharacteristic
subgroupG/H
centerC2#22559615
commutator, cosocleC4×C8×C67C23
FrattiniC4×C8C22×D67
FittingC67×#2244C2
socleC2×C67#1677
prime radical#22562282C2
hypercenter#2244D67
hypocenterC67#12465
LCSG ▷ C4×C8×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2244

Complements

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

NH ≅ G/Ntype
1. #22562382C2semidirect
3. #22562052C2semidirect
4. #22558277C2semidirect
5. #22558270C2semidirect
6. #22558246C2semidirect
7. C67×#2244C2semidirect
96. #2244D67semidirect
103. C67#12465semidirect

Maximal subgroups

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

subgroupindex
#225623822
#225622822
#225620522
#225582772
#225582702
#225582462
C67×#22442
#1246567