Finite Group Databasebeta

← #30749351

Subgroups — #30749351

Order 17152.

566 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 99 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#22562795C211yesmaximal, characteristic
2#22560133C211yesmaximal, characteristic
3#22558118C214yesmaximal
4C67×#1784C211yesmaximal, characteristic
5C2×#14524002C412yes—
6C2×#14523962—22no—
7#16863360C2212yes—
8#16863303—22no—
9#16863303—22no—
10#16863259C412yes—
11#16863259C2212yes—
12C23×Q8×C67C2211yescharacteristic
13C2×#101×C67—22no—
14#407×C67C2212yes—
15C2×#11761612—22no—
16#14524002—44no—
17C2×C4⋊Dic67—22no—
18C2×C22⋊Dic67C2×C414yes—
19C2×C4×Dic67—22no—
20#14523962—44no—
21C22⋊(C67⋊C8)—24no—
22C67⋊#95—24no—
23C67⋊#95—22no—
24C67⋊#95—22no—
25C22×Q8×C67—24no—
26C22×Q8×C67—44no—
27C22×Q8×C67D412yes—
28C22×Q8×C67D412yes—
29C23×C4×C67C2×C411yescharacteristic
30C23×C4×C67C2×C411yescharacteristic
31C23×C4×C67C2311yescharacteristic
32C2×M16×C67—22no—
33#101×C67—44no—
34C22⋊C8×C67—24no—
35C22×Dic67—24no—
36#11761612—44no—
37C2×C67⋊C8—42no—
38C4⋊Dic67—44no—
39C22⋊Dic67—44no—
40C4×Dic67—44no—
41C24×C67C22×C411yescharacteristic
42C2×Q8×C67—24no—
43C2×Q8×C67—24no—
44C2×Q8×C67—24no—
45C2×Q8×C67—24no—
46C2×Q8×C67—42no—
47C2×Q8×C67—42no—
48C2×Q8×C67—416no—
49C22×C4×C67—22no—
50C22×C4×C67—24no—
51C22×C4×C67—22no—
52C22×C4×C67—42no—
53C22×C4×C67—42no—
54C22×C4×C67C22⋊C412yes—
55C22×C4×C67C22⋊C412yes—
56C22×C4×C67C2×D411yescharacteristic
57C22×C4×C67C2×D411yescharacteristic
58C22×C4×C67#4114yes—
59M16×C67—44no—
60C2×C8×C67—42no—
61C23×C67—22no—
62C23×C67—42no—
63C23×C67C2×C22⋊C411yescharacteristic
64C23×C67#11712yes—
65Q8×C67—48no—
66Q8×C67—48no—
67C2×C4×C67—28no—
68C2×C4×C67—22no—
69C2×C4×C67—22no—
70C2×C4×C67—22no—
71C2×C4×C67—22no—
72C2×C4×C67—22no—
73C2×C4×C67—22no—
74C2×C4×C67—22no—
75C2×C4×C67—44no—
76C2×C4×C67—44no—
77C2×C4×C67#9914yes—
78C2×C4×C67#12312yes—
79C2×C4×C67#12312yes—
80C2×Dic67—48no—
81C2×Dic67—44no—
82C8×C67—82no—
83C67⋊C8—82no—
84C22×C67—22no—
85C22×C67—22no—
86C22×C67—24no—
87C22×C67—44no—
88C22×C67#38111yescharacteristic
89C22×C67C2×#9912yes—
90C4×C67—22no—
91C4×C67—22no—
92C4×C67—24no—
93C4×C67—42no—
94Dic67—84no—
95#8753—671nomaximal, Sylow
96C2×C67—22no—
97C2×C67—42no—
98C2×C67#181911yescharacteristic
99C2×C67#183012yes—
100#2310—671no—
101#1784—671no—
102#1784D6711yescharacteristic
103#1306—674no—
104C67#875311yescharacteristic, complemented, Sylow
105C23×Q8C2×D6711yescharacteristic
106C2×#117—672no—
107C2×#101—22no—
108C2×#101—1342no—
109#407—672no—
110#407C2×D6712yes—
111#391—1342no—
112#391—1342no—
113#381—672no—
114#381—672no—
115C22×Q8—24no—
116C22×Q8—44no—
117C22×Q8D26812yes—
118C22×Q8C22⋊D6712yes—
119C23×C4C4×D6711yescharacteristic
120C23×C4C4×D6711yescharacteristic
121C23×C4C22×D6711yescharacteristic
122C2×M16—22no—
123C2×M16—1342no—
124#117—2684no—
125C2×C4⋊C4—1342no—
126C2×C22⋊C4—674no—
127C2×C42—1342no—
128#101—44no—
129#101—2684no—
130C22⋊C8—24no—
131C22⋊C8—1344no—
132#95—1344no—
133#95—1342no—
134#95—1342no—
135C24C2×C4×D6711yescharacteristic
136C2×Q8—24no—
137C2×Q8—24no—
138C2×Q8—24no—
139C2×Q8—24no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—416no—
143C22×C4—22no—
144C22×C4—24no—
145C22×C4—22no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—1344no—
149C22×C4(C2×D67)⋊C412yes—
150C22×C4(C2×D67)⋊C412yes—
151C22×C4C2×C22⋊D6711yescharacteristic
152C22×C4C2×D26811yescharacteristic
153C22×C4#1176162714yes—
154M16—44no—
155M16—2684no—
156C2×C8—42no—
157C2×C8—2682no—
158C4⋊C4—2684no—
159C22⋊C4—2684no—
160C42—2684no—
161C23—22no—
162C23—42no—
163C23C2×(C2×D67)⋊C411yescharacteristic
164C23#1452399912yes—
165Q8—48no—
166Q8—48no—
167C2×C4—28no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—44no—
176C2×C4—44no—
177C2×C4—2688no—
178C2×C4—2684no—
179C2×C4#1452395714yes—
180C2×C4#1452402412yes—
181C2×C4#1452402712yes—
182C8—82no—
183C8—5362no—
184C22—22no—
185C22—22no—
186C22—24no—
187C22—44no—
188C22#1686325711yescharacteristic
189C22C2×#1452395712yes—
190C4—22no—
191C4—22no—
192C4—24no—
193C4—42no—
194C4—5364no—
195C2—22no—
196C2—42no—
197C2#2256028111yescharacteristic
198C2#2256032112yes—
subgroupG/H
centerC22#16863257
commutator, prime radicalC24×C67C22×C4
FrattiniC23×C4C22×D67
FittingC67×#1784C2
socleC22×C67#381
cosocleC23×C4×C67C23
hypercenter#1784D67
hypocenterC67#8753
LCSG ▷ C24×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
104. C67#8753semidirect

Maximal subgroups

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

subgroupindex
#225627952
#225601332
#225581182
C67×#17842
#875367