Finite Group Databasebeta

← #30743416

Subgroups — #30743416

Order 17152.

226 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 43 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#22558334C211yesmaximal, characteristic
2#22558141C211yesmaximal, characteristic
3C67×#2025C211yesmaximal, characteristic
4C2×#14523958—21no—
5#16863122—21no—
6C2×#125×C67C411yescharacteristic, complemented
7C2×#102×C67—21no—
8#390×C67C2211yescharacteristic
9#340×C67—21no—
10#336×C67C411yescharacteristic, complemented
11C2×C22⋊Dic67—41no—
12#14523958—22no—
13#14523958—41no—
14C22⋊(C67⋊C8)—41no—
15C22×D4×C67D411yescharacteristic
16C22×C8×C67—21no—
17#125×C67—42no—
18C2×C4⋊C4×C67—21no—
19C2×C4⋊C4×C67—41no—
20C2×C4⋊C4×C67D411yescharacteristic
21C2×C22⋊C4×C67—41no—
22C2×C42×C67C2×C411yescharacteristic
23#102×C67—42no—
24C22×Dic67—81no—
25C2×C67⋊C8—81no—
26C22⋊Dic67—42no—
27C22⋊Dic67—81no—
28C24×C67—21no—
29C2×D4×C67—21no—
30C2×D4×C67—41no—
31C2×D4×C67—41no—
32C2×D4×C67SD1611yescharacteristic
33C2×D4×C67Q1611yescharacteristic
34C22×C4×C67—21no—
35C22×C4×C67—21no—
36C22×C4×C67—41no—
37C22×C4×C67C22⋊C411yescharacteristic
38C2×C8×C67—41no—
39C2×C8×C67—41no—
40C2×C8×C67—41no—
41C4⋊C4×C67—22no—
42C4⋊C4×C67—41no—
43C4⋊C4×C67—41no—
44C4⋊C4×C67—42no—
45C4⋊C4×C67—42no—
46C22⋊C4×C67—82no—
47C42×C67—41no—
48C23×C67—21no—
49C23×C67—21no—
50C23×C67—41no—
51C23×C67—41no—
52C23×C67—81no—
53C23×C67—81no—
54C23×C67#9911yescharacteristic
55D4×C67—41no—
56D4×C67—41no—
57D4×C67—81no—
58C2×C4×C67—21no—
59C2×C4×C67—21no—
60C2×C4×C67—41no—
61C2×C4×C67—41no—
62C2×C4×C67—41no—
63C2×C4×C67—42no—
64C2×C4×C67—41no—
65C2×C4×C67—81no—
66C2×C4×C67—81no—
67C2×C4×C67—81no—
68C2×C4×C67#10311yescharacteristic
69C2×C4×C67#10411yescharacteristic
70C2×Dic67—82no—
71C2×Dic67—161no—
72C2×Dic67—161no—
73C8×C67—81no—
74C67⋊C8—161no—
75C22×C67—21no—
76C22×C67—21no—
77C22×C67—41no—
78C22×C67—41no—
79C22×C67—81no—
80C22×C67—81no—
81C22×C67—81no—
82C22×C67—81no—
83C22×C67—161no—
84C22×C67#32811yescharacteristic
85C22×C67AΣL(1,𝔽16)11yescharacteristic
86C22×C67#35411yescharacteristic
87C4×C67—21no—
88C4×C67—21no—
89C4×C67—81no—
90C4×C67—82no—
91C4×C67—81no—
92Dic67—162no—
93#7428—671nomaximal, Sylow
94C2×C67—21no—
95C2×C67—21no—
96C2×C67—81no—
97C2×C67—81no—
98C2×C67#135111yescharacteristic
99C2×C67#141211yescharacteristic
100C2×C67#141411yescharacteristic
101#2025D6711yescharacteristic
102#1393—671no—
103#1318—671no—
104C67#742811yescharacteristic, complemented, Sylow
105C2×#125Dic6711yescharacteristic, complemented
106C2×#102—21no—
107C2×#99—1341no—
108#390C2×D6711yescharacteristic
109#340—21no—
110#336Dic6711yescharacteristic, complemented
111#324—1341no—
112C22×D4C22⋊D6711yescharacteristic
113C22×C8—21no—
114#125—42no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4C22⋊D6711yescharacteristic
118C2×C22⋊C4—41no—
119C2×C22⋊C4—2681no—
120C2×C42C2×Dic6711yescharacteristic
121#102—42no—
122#99—2681no—
123#99—1342no—
124C22⋊C8—2681no—
125C24—21no—
126C2×D4—21no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4#1176161511yescharacteristic
130C2×D4#1176161811yescharacteristic
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—41no—
134C22×C4—5361no—
135C22×C4C22⋊Dic6711yescharacteristic
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—5361no—
140C4⋊C4—22no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C22⋊C4—82no—
146C22⋊C4—2682no—
147C22⋊C4—5361no—
148C42—41no—
149C23—21no—
150C23—21no—
151C23—41no—
152C23—41no—
153C23—81no—
154C23—81no—
155C23#1452395811yescharacteristic
156D4—41no—
157D4—41no—
158D4—81no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4—81no—
169C2×C4—5362no—
170C2×C4—10721no—
171C2×C4—10721no—
172C2×C4#1452396711yescharacteristic
173C2×C4#1452397011yescharacteristic
174C8—81no—
175C8—10721no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22—81no—
184C22—161no—
185C22#1686313311yescharacteristic
186C22#1686318311yescharacteristic
187C22#1686319111yescharacteristic
188C4—21no—
189C4—21no—
190C4—81no—
191C4—82no—
192C4—81no—
193C4—10722no—
194C2—21no—
195C2—21no—
196C2—81no—
197C2—81no—
198C2#2255823011yescharacteristic
199C2#2255838011yescharacteristic
200C2#2255838411yescharacteristic
subgroupG/H
centerC22#16863191
commutatorC2×C42×C67C2×C4
Frattini#390C2×D67
FittingC67×#2025C2
socleC22×C67#354
prime radicalC22×D4×C67D4
cosocle#390×C67C22
hypercenter#2025D67
hypocenterC67#7428
LCSG ▷ C2×C42×C67 ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ #2025

Complements

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

NH ≅ G/Ntype
6. C2×#125×C67C4semidirect
10. #336×C67C4semidirect
104. C67#7428semidirect
105. C2×#125Dic67semidirect
110. #336Dic67semidirect

Maximal subgroups

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

subgroupindex
#225583342
#225581412
C67×#20252
#742867