Finite Group Databasebeta

← #30743367

Subgroups — #30743367

Order 17152.

208 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 41 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#22561622C211yesmaximal, characteristic
2#22558141C211yesmaximal, characteristic
3C67×#1393C211yesmaximal, characteristic
4C2×#14523964—21no—
5#16863262C411yescharacteristic, complemented
6#16863158—21no—
7#16863151C411yescharacteristic, complemented
8#16863122—21no—
9C2×#99×C67—21no—
10#390×C67C2211yescharacteristic
11C22×C67⋊C8—21no—
12C2×C4⋊Dic67—21no—
13#14523964—41no—
14#14523964—41no—
15C22⋊(C67⋊C8)—41no—
16C67⋊#95—41no—
17C22×D4×C67D411yescharacteristic
18C2×C4⋊C4×C67D411yescharacteristic
19C2×C22⋊C4×C67—41no—
20C2×C22⋊C4×C67—41no—
21C2×C42×C67C2×C411yescharacteristic
22#99×C67—22no—
23#99×C67—41no—
24C22×Dic67—41no—
25C2×C67⋊C8—41no—
26C2×C67⋊C8—41no—
27C2×C67⋊C8—41no—
28C2×C67⋊C8—81no—
29C4⋊Dic67—41no—
30C4⋊Dic67—41no—
31C24×C67—21no—
32C2×D4×C67—21no—
33C2×D4×C67—41no—
34C2×D4×C67—41no—
35C2×D4×C67D811yescharacteristic
36C2×D4×C67SD1611yescharacteristic
37C22×C4×C67—21no—
38C22×C4×C67—21no—
39C22×C4×C67—81no—
40C22×C4×C67C22⋊C411yescharacteristic
41C4⋊C4×C67—41no—
42C22⋊C4×C67—42no—
43C22⋊C4×C67—82no—
44C22⋊C4×C67—81no—
45C42×C67—41no—
46C23×C67—21no—
47C23×C67—21no—
48C23×C67—41no—
49C23×C67—41no—
50C23×C67—81no—
51C23×C67—81no—
52C23×C67#9911yescharacteristic
53D4×C67—41no—
54D4×C67—41no—
55D4×C67—81no—
56C2×C4×C67—21no—
57C2×C4×C67—21no—
58C2×C4×C67—41no—
59C2×C4×C67—41no—
60C2×C4×C67—41no—
61C2×C4×C67—41no—
62C2×C4×C67—81no—
63C2×C4×C67—82no—
64C2×C4×C67—161no—
65C2×C4×C67—161no—
66C2×C4×C67#10211yescharacteristic
67C2×C4×C67#10411yescharacteristic
68C2×Dic67—81no—
69C2×Dic67—81no—
70C2×Dic67—81no—
71C67⋊C8—81no—
72C67⋊C8—161no—
73C22×C67—21no—
74C22×C67—21no—
75C22×C67—41no—
76C22×C67—41no—
77C22×C67—81no—
78C22×C67—81no—
79C22×C67—81no—
80C22×C67—81no—
81C22×C67—161no—
82C22×C67#32711yescharacteristic
83C22×C67#35211yescharacteristic
84C22×C67#35411yescharacteristic
85C4×C67—21no—
86C4×C67—21no—
87C4×C67—81no—
88C4×C67—81no—
89C4×C67—162no—
90Dic67—161no—
91#7412—671nomaximal, Sylow
92C2×C67—21no—
93C2×C67—21no—
94C2×C67—81no—
95C2×C67—81no—
96C2×C67#134411yescharacteristic
97C2×C67#141211yescharacteristic
98C2×C67#141411yescharacteristic
99#2073—671no—
100#1393D6711yescharacteristic
101#1318—671no—
102C67#741211yescharacteristic, complemented, Sylow
103C2×#102—1341no—
104C2×#99—21no—
105#390C2×D6711yescharacteristic
106#382—671no—
107#340—1341no—
108#336—671no—
109#324—1341no—
110C22×D4C22⋊D6711yescharacteristic
111C22×C8—1341no—
112C2×C4⋊C4—1341no—
113C2×C4⋊C4D26811yescharacteristic
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C42C4×D6711yescharacteristic
117#102—2681no—
118#102—2681no—
119#99—22no—
120#99—41no—
121C22⋊C8—2681no—
122#95—2681no—
123C24—21no—
124C2×D4—21no—
125C2×D4—41no—
126C2×D4—41no—
127C2×D4#1176161311yescharacteristic
128C2×D4#1176161511yescharacteristic
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—81no—
132C22×C4—2681no—
133C22×C4(C2×D67)⋊C411yescharacteristic
134C2×C8—2681no—
135C2×C8—2681no—
136C2×C8—2681no—
137C2×C8—5361no—
138C4⋊C4—41no—
139C4⋊C4—2681no—
140C4⋊C4—2681no—
141C22⋊C4—42no—
142C22⋊C4—82no—
143C22⋊C4—81no—
144C42—41no—
145C23—21no—
146C23—21no—
147C23—41no—
148C23—41no—
149C23—81no—
150C23—81no—
151C23#1452395711yescharacteristic
152D4—41no—
153D4—41no—
154D4—81no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—81no—
162C2×C4—82no—
163C2×C4—161no—
164C2×C4—161no—
165C2×C4—5361no—
166C2×C4—5361no—
167C2×C4—5361no—
168C2×C4#1452396311yescharacteristic
169C2×C4#1452397111yescharacteristic
170C8—5361no—
171C8—10721no—
172C22—21no—
173C22—21no—
174C22—41no—
175C22—41no—
176C22—81no—
177C22—81no—
178C22—81no—
179C22—81no—
180C22—161no—
181C22#1686312911yescharacteristic
182C22#1686318511yescharacteristic
183C22#1686319011yescharacteristic
184C4—21no—
185C4—21no—
186C4—81no—
187C4—81no—
188C4—162no—
189C4—10721no—
190C2—21no—
191C2—21no—
192C2—81no—
193C2—81no—
194C2#2255820811yescharacteristic
195C2#2255837911yescharacteristic
196C2#2255838311yescharacteristic
subgroupG/H
centerC22#16863190
commutatorC2×C42×C67C2×C4
Frattini#390C2×D67
FittingC67×#1393C2
socleC22×C67#354
prime radicalC22×D4×C67D4
cosocle#390×C67C22
hypercenter#1393D67
hypocenterC67#7412
LCSG ▷ C2×C42×C67 ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ #1393

Complements

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

NH ≅ G/Ntype
5. #16863262C4semidirect
7. #16863151C4semidirect
102. C67#7412semidirect

Maximal subgroups

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

subgroupindex
#225616222
#225581412
C67×#13932
#741267