Finite Group Databasebeta

← #30767346

Subgroups — #30767346

Order 17152.

292 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 79 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#22570859C211yesmaximal, characteristic
2#22562025C211yesmaximal, characteristic, complemented
3#22558271C212yesmaximal, complemented
4#22558245C212yesmaximal, complemented
5C67×#2153C211yesmaximal, characteristic, complemented
6#16864001C412yes—
7#16863798C2211yescharacteristic
8#16863773C2211yescharacteristic
9#16863761C412yes—
10#16863649—21no—
11#16863632—21no—
12#16863509—21no—
13C4×D536—21no—
14#16863217C2212yes—
15#16863178—21no—
16#16863174—21no—
17C2×M32×C67—21no—
18#405×C67C2211yescharacteristic
19#364×C67—21no—
20#364×C67C2212yes—
21C2×D536—21no—
22C2×D536—21no—
23#14524040C2×C412yes—
24#14524035C2×C412yes—
25#14524028—42no—
26#14524017—22no—
27#14524016—42no—
28C4×D268—22no—
29#14523983—22no—
30#14523977—21no—
31#14523975—21no—
32#14523965—22no—
33#14523965—22no—
34C2×M16×C67D411yescharacteristic
35C22×C8×C67D411yescharacteristic
36#117×C67C2×C411yescharacteristic
37M32×C67—22no—
38M32×C67—41no—
39C2×C16×C67—41no—
40#97×C67C2×C411yescharacteristic
41#97×C67D411yescharacteristic
42C4×C8×C67D411yescharacteristic
43C4×C8×C67C2311yescharacteristic
44C2×D268—22no—
45C2×D268—42no—
46C2×C22⋊D67—42no—
47C2×C4×D67—42no—
48D536—42no—
49D536—41no—
50D536—41no—
51C4⋊Dic67—22no—
52Dic67⋊C4—42no—
53(C2×D67)⋊C4—42no—
54(C2×D67)⋊C4—42no—
55C67⋊C16—42no—
56C22×C4×C67C22⋊C411yescharacteristic
57M16×C67—22no—
58C2×C8×C67—21no—
59C2×C8×C67—21no—
60C2×C8×C67C22⋊C411yescharacteristic
61C2×C8×C67C22⋊C411yescharacteristic
62C2×C8×C67C2×D411yescharacteristic
63C2×C8×C67C2×D411yescharacteristic
64C4⋊C4×C67—21no—
65C22⋊C4×C67—21no—
66C42×C67C22⋊C411yescharacteristic
67C42×C67C22×C411yescharacteristic
68C16×C67—42no—
69C23×C67#10011yescharacteristic
70C22×D67—42no—
71C2×C4×C67—21no—
72C2×C4×C67—21no—
73C2×C4×C67—21no—
74C2×C4×C67—21no—
75C2×C4×C67#10011yescharacteristic
76C2×C4×C67C2×C22⋊C411yescharacteristic
77C22⋊D67—162no—
78C2×Dic67—42no—
79D268—42no—
80D268—42no—
81D268—162no—
82C4×D67—162no—
83C8×C67—21no—
84C8×C67—21no—
85C8×C67#13512yes—
86C8×C67AGL(1,ℤ/8)11yescharacteristic
87C8×C67#13711yescharacteristic
88C22×C67—21no—
89C22×C67—41no—
90C22×C67C2×#10011yescharacteristic
91C2×D67—82no—
92C2×D67—82no—
93C2×D67—82no—
94C4×C67—21no—
95C4×C67—41no—
96C4×C67—41no—
97C4×C67#41611yescharacteristic
98C4×C67#41711yescharacteristic
99Dic67—82no—
100#11989—671nomaximal, Sylow
101C2×C67—21no—
102C2×C67—41no—
103C2×C67#155111yescharacteristic
104D67—162no—
105#3176—671no—
106#2166—671no—
107#2153D6711yescharacteristic, complemented
108#1366—672no—
109#1357—672no—
110C67#1198911yescharacteristic, complemented, Sylow
111#540—672no—
112C2×M32—21no—
113#499—671no—
114#493—671no—
115#490—672no—
116#469—1341no—
117#466—1341no—
118#442—1341no—
119C4×D8—1341no—
120#405C2×D6711yescharacteristic
121#364—21no—
122#364—672no—
123#364C2×D6712yes—
124#349—1341no—
125#347—1341no—
126C2×D8—1341no—
127C2×D8—1341no—
128C2×M16D26811yescharacteristic
129C22×C8C22⋊D6711yescharacteristic
130#127—672no—
131#125—672no—
132#123—2682no—
133#121—2682no—
134#121—1342no—
135C4×D4—1342no—
136#117C4×D6711yescharacteristic
137M32—22no—
138M32—41no—
139M32—1342no—
140C2×C16—41no—
141#107—1341no—
142#106—1341no—
143#102—1342no—
144#102—1342no—
145#97C4×D6711yescharacteristic
146#97D26811yescharacteristic
147C4×C8C22⋊D6711yescharacteristic
148C4×C8C22×D6711yescharacteristic
149C2×D4—2682no—
150C2×D4—2682no—
151C2×D4—1342no—
152C22×C4—2682no—
153C22×C4(C2×D67)⋊C411yescharacteristic
154D8—2682no—
155D8—2681no—
156D8—2681no—
157M16—22no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8(C2×D67)⋊C411yescharacteristic
161C2×C8(C2×D67)⋊C411yescharacteristic
162C2×C8C2×C22⋊D6711yescharacteristic
163C2×C8C2×D26811yescharacteristic
164C4⋊C4—21no—
165C4⋊C4—2682no—
166C4⋊C4—1342no—
167C22⋊C4—21no—
168C22⋊C4—2682no—
169C22⋊C4—2682no—
170C42(C2×D67)⋊C411yescharacteristic
171C42C2×C4×D6711yescharacteristic
172C16—42no—
173C16—2682no—
174C23—2682no—
175C23#1452395911yescharacteristic
176D4—2682no—
177D4—2682no—
178D4—10722no—
179D4—10722no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—2682no—
185C2×C4—10722no—
186C2×C4#1452395911yescharacteristic
187C2×C4C2×(C2×D67)⋊C411yescharacteristic
188C8—21no—
189C8—21no—
190C8#1452406512yes—
191C8#1452406911yescharacteristic
192C8#1452407411yescharacteristic
193C22—21no—
194C22—41no—
195C22—5362no—
196C22—5362no—
197C22—5362no—
198C22C2×#1452395911yescharacteristic
199C4—21no—
200C4—41no—
201C4—41no—
202C4—5362no—
203C4#1686339311yescharacteristic
204C4#1686339511yescharacteristic
205C2—21no—
206C2—41no—
207C2—10722no—
208C2#2255890211yescharacteristic
subgroupG/H
centerC2#22558902
commutatorC42×C67C22×C4
FrattiniC4×C8C22×D67
FittingC67×#2153C2
socleC2×C67#1551
prime radical#22570859C2
cosocleC4×C8×C67C23
hypercenter#2153D67
hypocenterC67#11989
LCSG ▷ C42×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2153

Complements

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

NH ≅ G/Ntype
2. #22562025C2semidirect
3. #22558271C2semidirect
4. #22558245C2semidirect
5. C67×#2153C2semidirect
107. #2153D67semidirect
110. C67#11989semidirect

Maximal subgroups

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

subgroupindex
#225708592
#225620252
#225582712
#225582452
C67×#21532
#1198967