Finite Group Databasebeta

← #30749367

Subgroups — #30749367

Order 17152.

438 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 107 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#22558539C212yesmaximal
2#22558115C214yesmaximal
3C67×#2316C211yesmaximal, characteristic
4#16863360C2214yes—
5#16863122—22no—
6#16863122—22no—
7C2×#117×C67C411yescharacteristic
8C2×#117×C67C411yescharacteristic
9C22×C4⋊C4×C67C2211yescharacteristic
10#389×C67—22no—
11#389×C67—22no—
12#380×C67C412yes—
13#380×C67C2212yes—
14C2×#11761612—24no—
15C22⋊(C67⋊C8)—24no—
16C22⋊(C67⋊C8)—24no—
17C23×C4×C67C2×C411yescharacteristic
18C23×C4×C67C2×C411yescharacteristic
19C23×C4×C67C2311yescharacteristic
20#117×C67—42no—
21#117×C67—42no—
22C2×C4⋊C4×C67—24no—
23C2×C4⋊C4×C67—21no—
24C2×C4⋊C4×C67—21no—
25C2×C4⋊C4×C67—22no—
26C2×C4⋊C4×C67—22no—
27C2×C4⋊C4×C67D412yes—
28C2×C4⋊C4×C67D412yes—
29C2×C22⋊C4×C67C2×C412yes—
30C2×C22⋊C4×C67C2×C412yes—
31C2×C42×C67—21no—
32C2×C42×C67—21no—
33#95×C67—22no—
34#95×C67—22no—
35#11761612—84no—
36C2×C67⋊C8—44no—
37C24×C67C22×C411yescharacteristic
38C22×C4×C67—22no—
39C22×C4×C67—22no—
40C22×C4×C67—22no—
41C22×C4×C67—22no—
42C22×C4×C67—22no—
43C22×C4×C67—22no—
44C22×C4×C67—81no—
45C22×C4×C67—81no—
46C22×C4×C67C22⋊C411yescharacteristic
47C22×C4×C67C22⋊C411yescharacteristic
48C22×C4×C67C22⋊C412yes—
49C22×C4×C67C2×D412yes—
50C22×C4×C67#4114yes—
51C4⋊C4×C67—24no—
52C4⋊C4×C67—24no—
53C4⋊C4×C67—42no—
54C4⋊C4×C67—42no—
55C4⋊C4×C67—82no—
56C4⋊C4×C67—82no—
57C22⋊C4×C67—42no—
58C22⋊C4×C67—42no—
59C42×C67—42no—
60C42×C67—42no—
61C23×C67—21no—
62C23×C67—21no—
63C23×C67—41no—
64C23×C67—41no—
65C23×C67C2×C22⋊C411yescharacteristic
66C23×C67#11712yes—
67C2×C4×C67—28no—
68C2×C4×C67—21no—
69C2×C4×C67—21no—
70C2×C4×C67—21no—
71C2×C4×C67—21no—
72C2×C4×C67—22no—
73C2×C4×C67—42no—
74C2×C4×C67—44no—
75C2×C4×C67—42no—
76C2×C4×C67—42no—
77C2×C4×C67—42no—
78C2×C4×C67—42no—
79C2×C4×C67—42no—
80C2×C4×C67—42no—
81C2×C4×C67—42no—
82C2×C4×C67—82no—
83C2×C4×C67#10012yes—
84C2×C4×C67#10112yes—
85C2×C4×C67#12314yes—
86C67⋊C8—84no—
87C22×C67—21no—
88C22×C67—21no—
89C22×C67—22no—
90C22×C67—22no—
91C22×C67—22no—
92C22×C67—42no—
93C22×C67—42no—
94C22×C67#38111yescharacteristic
95C22×C67C2×#10011yescharacteristic
96C22×C67C2×#10111yescharacteristic
97C4×C67—22no—
98C4×C67—22no—
99C4×C67—24no—
100C4×C67—42no—
101C4×C67—82no—
102C4×C67—82no—
103#8757—671nomaximal, Sylow
104C2×C67—21no—
105C2×C67—21no—
106C2×C67—42no—
107C2×C67#182111yescharacteristic
108C2×C67#182811yescharacteristic
109C2×C67#183011yescharacteristic
110#2316D6711yescharacteristic
111#1463—672no—
112#1305—674no—
113C67#875711yescharacteristic, complemented, Sylow
114C2×#117Dic6711yescharacteristic
115C2×#117Dic6711yescharacteristic
116C22×C4⋊C4C2×D6711yescharacteristic
117#407—674no—
118#389—22no—
119#389—22no—
120#380Dic6712yes—
121#380C2×D6712yes—
122#324—1342no—
123#324—1342no—
124C23×C4C2×Dic6711yescharacteristic
125C23×C4C2×Dic6711yescharacteristic
126C23×C4C22×D6711yescharacteristic
127C2×M16—1344no—
128#117—42no—
129#117—42no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4C22⋊D6712yes—
136C2×C4⋊C4C22⋊D6712yes—
137C2×C22⋊C4C2×Dic6712yes—
138C2×C22⋊C4C2×Dic6712yes—
139C2×C42—21no—
140C2×C42—21no—
141C22⋊C8—1344no—
142C22⋊C8—1344no—
143#95—22no—
144#95—22no—
145C24C22×Dic6711yescharacteristic
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—81no—
153C22×C4—81no—
154C22×C4C22⋊Dic6711yescharacteristic
155C22×C4C22⋊Dic6711yescharacteristic
156C22×C4C22⋊Dic6712yes—
157C22×C4C2×C22⋊D6712yes—
158C22×C4#1176162814yes—
159M16—5364no—
160C2×C8—2684no—
161C4⋊C4—24no—
162C4⋊C4—24no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—82no—
166C4⋊C4—82no—
167C22⋊C4—42no—
168C22⋊C4—42no—
169C42—42no—
170C42—42no—
171C23—21no—
172C23—21no—
173C23—41no—
174C23—41no—
175C23C2×C22⋊Dic6711yescharacteristic
176C23#1452400212yes—
177C2×C4—28no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—42no—
184C2×C4—44no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—82no—
193C2×C4#1452396012yes—
194C2×C4#1452396212yes—
195C2×C4#1452402614yes—
196C8—5364no—
197C22—21no—
198C22—21no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—42no—
203C22—42no—
204C22#1686325911yescharacteristic
205C22C2×#1452396011yescharacteristic
206C22C2×#1452396211yescharacteristic
207C4—22no—
208C4—22no—
209C4—24no—
210C4—42no—
211C4—82no—
212C4—82no—
213C2—21no—
214C2—21no—
215C2—42no—
216C2#2256028911yescharacteristic
217C2#2256031711yescharacteristic
218C2#2256032311yescharacteristic
subgroupG/H
centerC22#16863259
commutator, prime radicalC24×C67C22×C4
FrattiniC23×C4C22×D67
FittingC67×#2316C2
socleC22×C67#381
cosocleC23×C4×C67C23
hypercenter#2316D67
hypocenterC67#8757
LCSG ▷ C24×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2316

Complements

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

NH ≅ G/Ntype
113. C67#8757semidirect

Maximal subgroups

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

subgroupindex
#225585392
#225581152
C67×#23162
#875767