Finite Group Databasebeta

← #30770232

Subgroups — #30770232

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#22562450C211yesmaximal, characteristic, complemented
2#22562385C211yesmaximal, characteristic, complemented
3#22562051C211yesmaximal, characteristic, complemented
4#22558276C211yesmaximal, characteristic, complemented
5#22558269C211yesmaximal, characteristic, complemented
6#22558246C211yesmaximal, characteristic, complemented
7C67×#2213C211yesmaximal, characteristic
8#16863813—21no—
9#16863798C2211yescharacteristic
10#16863220—21no—
11#16863217C2211yescharacteristic
12#16863217C2211yescharacteristic
13#16863206—21no—
14#16863197—21no—
15#16863197—21no—
16#16863180—21no—
17#16863172C2211yescharacteristic
18#509×C67—21no—
19#493×C67C2211yescharacteristic
20#443×C67C2211yescharacteristic
21#364×C67C2211yescharacteristic
22#361×C67—21no—
23#14524035—21no—
24#14523983—21no—
25#14523983—21no—
26#14523983—41no—
27C2×C67⋊C16—41no—
28C2×C67⋊C16D411yescharacteristic
29C2×C67⋊C16D411yescharacteristic
30#14523977—21no—
31#14523975—21no—
32#135×C67D411yescharacteristic
33C2×D8×C67—21no—
34C2×D8×C67—21no—
35#131×C67—21no—
36#127×C67—21no—
37SD32×C67—41no—
38D16×C67—41no—
39M32×C67—21no—
40#108×C67D411yescharacteristic
41#104×C67—21no—
42C4×C8×C67C2311yescharacteristic
43C4⋊Dic67—41no—
44Dic67⋊C4—81no—
45C67⋊C16—21no—
46C67⋊C16—21no—
47C67⋊C16—41no—
48C67⋊C16—41no—
49#41×C67—21no—
50C2×D4×C67—41no—
51C2×D4×C67—81no—
52Q16×C67SD1611yescharacteristic
53SD16×C67—21no—
54D8×C67—41no—
55D8×C67—41no—
56D8×C67—41no—
57D8×C67—41no—
58D8×C67SD1611yescharacteristic
59M16×C67—21no—
60M16×C67—41no—
61C2×C8×C67—41no—
62C2×C8×C67C2×D411yescharacteristic
63C2×C8×C67C2×D411yescharacteristic
64C42×C67#4111yescharacteristic
65C16×C67—41no—
66C23×C67—81no—
67Q8×C67—21no—
68D4×C67—21no—
69D4×C67—41no—
70D4×C67—81no—
71D4×C67—81no—
72D4×C67—162no—
73C2×C4×C67—21no—
74C2×C4×C67—41no—
75C2×C4×C67#12111yescharacteristic
76C2×Dic67—81no—
77C8×C67—41no—
78C8×C67C2×SD1611yescharacteristic
79C8×C67#13511yescharacteristic
80C8×C67#13511yescharacteristic
81C8×C67AGL(1,ℤ/8)11yescharacteristic
82C22×C67—41no—
83C22×C67—162no—
84C22×C67—161no—
85C22×C67#47211yescharacteristic
86C4×C67—41no—
87C4×C67—41no—
88C4×C67#46311yescharacteristic
89C4×C67#46511yescharacteristic
90Dic67—161no—
91#12465—671nomaximal, Sylow
92C2×C67—21no—
93C2×C67—81no—
94C2×C67—321no—
95C2×C67#167711yescharacteristic
96#2244—671no—
97#2231—671no—
98#2213D6711yescharacteristic
99#2171—671no—
100#1369—671no—
101#1366—671no—
102#1357—671no—
103C67#1246511yescharacteristic, complemented, Sylow
104#509—21no—
105#504—1341no—
106#499—671no—
107#493C2×D6711yescharacteristic
108#443C2×D6711yescharacteristic
109#365—1341no—
110#364—671no—
111#364—671no—
112#364C2×D6711yescharacteristic
113#361—21no—
114#360—1341no—
115#357—1341no—
116#357—1341no—
117#350—1341no—
118#346—671no—
119#135C22⋊D6711yescharacteristic
120C2×D8—21no—
121C2×D8—21no—
122#131—21no—
123#127—21no—
124#125—1341no—
125SD32—41no—
126D16—41no—
127M32—21no—
128M32—2681no—
129M32—1341no—
130M32—1341no—
131C2×C16—2681no—
132C2×C16—671no—
133C2×C16—671no—
134#108C22⋊D6711yescharacteristic
135#107—1341no—
136#106—1341no—
137#104—21no—
138C4×C8C22×D6711yescharacteristic
139#41—21no—
140C2×D4—41no—
141C2×D4—81no—
142Q16#1176161611yescharacteristic
143SD16—21no—
144D8—41no—
145D8—41no—
146D8—41no—
147D8—41no—
148D8#1176161611yescharacteristic
149M16—21no—
150M16—41no—
151C2×C8—41no—
152C2×C8D4×D6711yescharacteristic
153C2×C8C2×C22⋊D6711yescharacteristic
154C4⋊C4—2681no—
155C4⋊C4—5361no—
156C42#1176162711yescharacteristic
157C16—41no—
158C16—2681no—
159C16—2681no—
160C16—1341no—
161C16—1341no—
162C23—81no—
163Q8—21no—
164D4—21no—
165D4—41no—
166D4—81no—
167D4—81no—
168D4—162no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—5361no—
172C2×C4#1452401511yescharacteristic
173C8—41no—
174C8C2×#1176161611yescharacteristic
175C8#1452406211yescharacteristic
176C8#1452406311yescharacteristic
177C8#1452406711yescharacteristic
178C22—41no—
179C22—162no—
180C22—161no—
181C22#1686366411yescharacteristic
182C4—41no—
183C4—41no—
184C4—10721no—
185C4#1686360911yescharacteristic
186C4#1686362311yescharacteristic
187C2—21no—
188C2—81no—
189C2—321no—
190C2#2255961611yescharacteristic
subgroupG/H
centerC2#22559616
commutator, cosocleC4×C8×C67C23
FrattiniC4×C8C22×D67
Fitting, prime radicalC67×#2213C2
socleC2×C67#1677
hypercenter#2213D67
hypocenterC67#12465
LCSG ▷ C4×C8×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2213

Complements

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

NH ≅ G/Ntype
1. #22562450C2semidirect
2. #22562385C2semidirect
3. #22562051C2semidirect
4. #22558276C2semidirect
5. #22558269C2semidirect
6. #22558246C2semidirect
103. C67#12465semidirect

Maximal subgroups

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

subgroupindex
#225624502
#225623852
#225620512
#225582762
#225582692
#225582462
C67×#22132
#1246567