Finite Group Databasebeta

← #30767018

Subgroups — #30767018

Order 17152.

296 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 75 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#22561814C211yesmaximal, characteristic, complemented
3#22558241C214yesmaximal, complemented
4C67×#2166C211yesmaximal, characteristic, complemented
5#16864001C412yes—
6C2×#14523983—21no—
7#16863798C411yescharacteristic
8#16863773C411yescharacteristic
9#16863761C2212yes—
10#16863649—21no—
11#16863632—21no—
12#16863509—21no—
13C4×D536—21no—
14#16863174—21no—
15#16863174C2212yes—
16#405×C67C2211yescharacteristic
17#364×C67—21no—
18#349×C67—21no—
19#347×C67C2212yes—
20C2×D536—21no—
21C2×D536—21no—
22#14524040C2×C412yes—
23#14524035C2×C412yes—
24#14524028—42no—
25#14524017—22no—
26#14524016—42no—
27C4×D268—22no—
28#14523983—22no—
29#14523983—22no—
30C2×C67⋊C16—22no—
31#14523977—21no—
32#14523975—21no—
33#14523965—22no—
34#14523965—22no—
35C2×M16×C67D411yescharacteristic
36C22×C8×C67D411yescharacteristic
37#117×C67C2×C411yescharacteristic
38M32×C67—22no—
39#97×C67D411yescharacteristic
40#97×C67C2311yescharacteristic
41C4×C8×C67C2×C411yescharacteristic
42C4×C8×C67D411yescharacteristic
43C2×D268—22no—
44C2×D268—42no—
45C2×C22⋊D67—42no—
46C2×C4×D67—42no—
47D536—42no—
48D536—41no—
49D536—41no—
50C4⋊Dic67—22no—
51Dic67⋊C4—42no—
52(C2×D67)⋊C4—42no—
53(C2×D67)⋊C4—42no—
54C67⋊C16—24no—
55C22×C4×C67C22⋊C411yescharacteristic
56M16×C67—41no—
57C2×C8×C67—21no—
58C2×C8×C67—21no—
59C2×C8×C67C22⋊C411yescharacteristic
60C2×C8×C67C22⋊C411yescharacteristic
61C2×C8×C67C2×D411yescharacteristic
62C2×C8×C67C2×D411yescharacteristic
63C4⋊C4×C67—21no—
64C22⋊C4×C67—21no—
65C42×C67C22⋊C411yescharacteristic
66C42×C67C22×C411yescharacteristic
67C16×C67—42no—
68C23×C67#10011yescharacteristic
69C22×D67—42no—
70C2×C4×C67—21no—
71C2×C4×C67—21no—
72C2×C4×C67—21no—
73C2×C4×C67—21no—
74C2×C4×C67#10011yescharacteristic
75C2×C4×C67C2×C22⋊C411yescharacteristic
76C22⋊D67—162no—
77C2×Dic67—42no—
78D268—42no—
79D268—42no—
80D268—162no—
81C4×D67—162no—
82C8×C67—21no—
83C8×C67—21no—
84C8×C67—21no—
85C8×C67#10411yescharacteristic
86C8×C67#10411yescharacteristic
87C22×C67—21no—
88C22×C67—41no—
89C22×C67C2×#10011yescharacteristic
90C2×D67—82no—
91C2×D67—82no—
92C2×D67—82no—
93C4×C67—21no—
94C4×C67—41no—
95C4×C67—41no—
96C4×C67C2×#10411yescharacteristic
97C4×C67#42111yescharacteristic
98Dic67—82no—
99#11913—671nomaximal, Sylow
100C2×C67—21no—
101C2×C67—41no—
102C2×C67#153011yescharacteristic
103D67—162no—
104#3176—671no—
105#2166D6711yescharacteristic, complemented
106#2110—671no—
107#1355—674no—
108C67#1191311yescharacteristic, complemented, Sylow
109#540—672no—
110C2×M32—1341no—
111#499—671no—
112#493—671no—
113#490—672no—
114#469—1341no—
115#466—1341no—
116#442—1341no—
117C4×D8—1341no—
118#405C2×D6711yescharacteristic
119#364—21no—
120#349—21no—
121#347—672no—
122#347—1341no—
123#347C2×D6712yes—
124C2×D8—1341no—
125C2×D8—1341no—
126C2×M16C22⋊D6711yescharacteristic
127C22×C8D26811yescharacteristic
128#127—672no—
129#125—672no—
130#123—2682no—
131#121—2682no—
132#121—1342no—
133C4×D4—1342no—
134#117C4×D6711yescharacteristic
135M32—22no—
136M32—1342no—
137M32—1342no—
138C2×C16—1342no—
139#107—1341no—
140#106—1341no—
141#102—1342no—
142#102—1342no—
143#97D26811yescharacteristic
144#97C22×D6711yescharacteristic
145C4×C8C4×D6711yescharacteristic
146C4×C8C22⋊D6711yescharacteristic
147C2×D4—2682no—
148C2×D4—2682no—
149C2×D4—1342no—
150C22×C4—2682no—
151C22×C4(C2×D67)⋊C411yescharacteristic
152D8—2682no—
153D8—2681no—
154D8—2681no—
155M16—41no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8(C2×D67)⋊C411yescharacteristic
159C2×C8(C2×D67)⋊C411yescharacteristic
160C2×C8C2×C22⋊D6711yescharacteristic
161C2×C8C2×D26811yescharacteristic
162C4⋊C4—21no—
163C4⋊C4—2682no—
164C4⋊C4—1342no—
165C22⋊C4—21no—
166C22⋊C4—2682no—
167C22⋊C4—2682no—
168C42(C2×D67)⋊C411yescharacteristic
169C42C2×C4×D6711yescharacteristic
170C16—42no—
171C16—1344no—
172C23—2682no—
173C23#1452395911yescharacteristic
174D4—2682no—
175D4—2682no—
176D4—10722no—
177D4—10722no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—2682no—
183C2×C4—10722no—
184C2×C4#1452395911yescharacteristic
185C2×C4C2×(C2×D67)⋊C411yescharacteristic
186C8—21no—
187C8—21no—
188C8—21no—
189C8#1452396911yescharacteristic
190C8#1452396911yescharacteristic
191C22—21no—
192C22—41no—
193C22—5362no—
194C22—5362no—
195C22—5362no—
196C22C2×#1452395911yescharacteristic
197C4—21no—
198C4—41no—
199C4—41no—
200C4—5362no—
201C4C2×#1452396911yescharacteristic
202C4#1686341411yescharacteristic
203C2—21no—
204C2—41no—
205C2—10722no—
206C2#2255882111yescharacteristic
subgroupG/H
centerC2#22558821
commutatorC42×C67C22×C4
Frattini#97C22×D67
FittingC67×#2166C2
socleC2×C67#1530
prime radical#22570859C2
cosocle#97×C67C23
hypercenter#2166D67
hypocenterC67#11913
LCSG ▷ C42×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2166

Complements

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

NH ≅ G/Ntype
2. #22561814C2semidirect
3. #22558241C2semidirect
4. C67×#2166C2semidirect
105. #2166D67semidirect
108. C67#11913semidirect

Maximal subgroups

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

subgroupindex
#225708592
#225618142
#225582412
C67×#21662
#1191367