Finite Group Databasebeta

← #30770289

Subgroups — #30770289

Order 17152.

212 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 61 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#22562436C211yesmaximal, characteristic, complemented
2#22562040C211yesmaximal, characteristic
3#22559609C211yesmaximal, characteristic
4#22559571C211yesmaximal, characteristic
5#22558282C211yesmaximal, characteristic, complemented
6#22558266C211yesmaximal, characteristic, complemented
7C67×#1362C211yesmaximal, characteristic, complemented
8C2×#14523982—21no—
9#16863801C2211yescharacteristic
10#16863687—21no—
11#16863673—21no—
12C8×D268C2211yescharacteristic
13#16863200—21no—
14#16863200—21no—
15(C2×D67)⋊C16—21no—
16C4×C67⋊C16C2211yescharacteristic
17#16863149C2211yescharacteristic
18#16863144—21no—
19#16863142—21no—
20#494×C67C2211yescharacteristic
21C4⋊C16×C67C2211yescharacteristic
22#335×C67C2211yescharacteristic
23C2×C8×D67—21no—
24#14524043—21no—
25C4×D268D411yescharacteristic
26#14523982—42no—
27C2×C67⋊C16D411yescharacteristic
28C2×C67⋊C16D411yescharacteristic
29#14523977—21no—
30#14523977—21no—
31#14523976—41no—
32#14523975—41no—
33C4⋊(C67⋊C8)—21no—
34C4⋊(C67⋊C8)D411yescharacteristic
35#14523965—41no—
36#14523963—41no—
37(C2×D67)⋊C8—21no—
38C2×Q16×C67—21no—
39C2×Q16×C67—21no—
40C4⋊Q8×C67—21no—
41C2×C16×C67—21no—
42C4⋊C8×C67—21no—
43C4×C8×C67C2311yescharacteristic
44C2×Dic134—41no—
45C2×D268D811yescharacteristic
46C2×C4×D67—21no—
47C2×C67⋊C8—21no—
48C2×C67⋊C8—41no—
49C8×D67—41no—
50C8×D67—41no—
51C4⋊Dic67—41no—
52C4⋊Dic67—41no—
53C4⋊Dic67D811yescharacteristic
54(C2×D67)⋊C4—21no—
55C67⋊C16—21no—
56C67⋊C16—21no—
57C2×Q8×C67—41no—
58Q16×C67—41no—
59Q16×C67—41no—
60Q16×C67—81no—
61C2×C8×C67—41no—
62C2×C8×C67C2×D411yescharacteristic
63C2×C8×C67#4111yescharacteristic
64C4⋊C4×C67—41no—
65C4⋊C4×C67—41no—
66C42×C67C2×D411yescharacteristic
67C16×C67—41no—
68C22×D67—21no—
69Q8×C67—81no—
70Q8×C67—81no—
71C2×C4×C67—81no—
72C2×C4×C67#12111yescharacteristic
73C2×C4×C67C2×D811yescharacteristic
74C2×C4×C67#13511yescharacteristic
75C2×Dic67—21no—
76C2×Dic67—81no—
77D268—21no—
78D268—21no—
79C4×D67—41no—
80C4×D67—41no—
81Dic134—161no—
82C8×C67—21no—
83C8×C67—81no—
84C8×C67#13511yescharacteristic
85C8×C67AGL(1,ℤ/8)11yescharacteristic
86C67⋊C8—41no—
87C67⋊C8—81no—
88C22×C67#46611yescharacteristic
89C2×D67—41no—
90C2×D67—41no—
91C2×D67—41no—
92C4×C67—21no—
93C4×C67—161no—
94C4×C67#46311yescharacteristic
95C4×C67#47111yescharacteristic
96C4×C67#51011yescharacteristic
97C4×C67#51011yescharacteristic
98Dic67—41no—
99Dic67—161no—
100#12473—671nomaximal, Sylow
101C2×C67#168111yescharacteristic
102C2×C67#221911yescharacteristic
103C2×C67#222311yescharacteristic
104D67—81no—
105#2241—671no—
106#2169—671no—
107#1675—671no—
108#1669—671no—
109#1372—671no—
110#1365—671no—
111#1362D6711yescharacteristic, complemented
112C67#1247311yescharacteristic, complemented, Sylow
113C2×M32—1341no—
114#500—671no—
115#494C2×D6711yescharacteristic
116#476—1341no—
117#474—1341no—
118C8×D4—671no—
119C4⋊C16C2×D6711yescharacteristic
120#358—1341no—
121#358—1341no—
122C22⋊C16—1341no—
123C4×C16—671no—
124#335—671no—
125#335C2×D6711yescharacteristic
126#333—1341no—
127#332—1341no—
128C2×Q16—21no—
129C2×Q16—21no—
130C22×C8—1341no—
131C4⋊Q8—21no—
132C4⋊Q8—1341no—
133C4×D4—671no—
134M32—2682no—
135C2×C16—21no—
136C2×C16—671no—
137C2×C16—671no—
138#107—2681no—
139#107—1341no—
140#107—1341no—
141#106—2681no—
142C4⋊C8—21no—
143C4⋊C8—671no—
144C4⋊C8—1341no—
145#102—2681no—
146#102—2681no—
147C22⋊C8—1341no—
148C4×C8C22×D6711yescharacteristic
149C2×Q8—41no—
150C2×Q8—2681no—
151C2×D4—671no—
152C22×C4—1341no—
153Q16—41no—
154Q16—41no—
155Q16—81no—
156C2×C8—41no—
157C2×C8—2681no—
158C2×C8—2681no—
159C2×C8—2681no—
160C2×C8—1341no—
161C2×C8D4×D6711yescharacteristic
162C2×C8#1176162811yescharacteristic
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—2681no—
166C4⋊C4—2681no—
167C4⋊C4—671no—
168C22⋊C4—1341no—
169C42D4×D6711yescharacteristic
170C16—41no—
171C16—1341no—
172C16—1341no—
173C23—1341no—
174Q8—81no—
175Q8—81no—
176Q8—10721no—
177D4—1341no—
178D4—1341no—
179C2×C4—81no—
180C2×C4—2681no—
181C2×C4—2681no—
182C2×C4—5361no—
183C2×C4—1341no—
184C2×C4#1452401611yescharacteristic
185C2×C4D8×D6711yescharacteristic
186C2×C4#1452406411yescharacteristic
187C8—21no—
188C8—81no—
189C8—2681no—
190C8—5361no—
191C8#1452406411yescharacteristic
192C8#1452406711yescharacteristic
193C22—2681no—
194C22—2681no—
195C22—2681no—
196C22#1686363111yescharacteristic
197C4—21no—
198C4—2681no—
199C4—161no—
200C4—10721no—
201C4#1686361011yescharacteristic
202C4#1686365811yescharacteristic
203C4#1686383511yescharacteristic
204C4#1686383811yescharacteristic
205C2—5361no—
206C2#2255964511yescharacteristic
207C2#2256231611yescharacteristic
208C2#2256234211yescharacteristic
subgroupG/H
centerC22#16863631
commutator, cosocleC4×C8×C67C23
FrattiniC4×C8C22×D67
FittingC67×#1362C2
socleC22×C67#466
prime radicalC2×D268D8
hypercenter#1362D67
hypocenterC67#12473
LCSG ▷ C4×C8×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
1. #22562436C2semidirect
5. #22558282C2semidirect
6. #22558266C2semidirect
7. C67×#1362C2semidirect
111. #1362D67semidirect
112. C67#12473semidirect

Maximal subgroups

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

subgroupindex
#225624362
#225620402
#225596092
#225595712
#225582822
#225582662
C67×#13622
#1247367