Finite Group Databasebeta

← C67×#21827 #30701078

Subgroups — C67×#21827 #30701078

Order 17152.

1130 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 1130 conjugacy classes of subgroups — too many to draw. Only its 366 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
1C2×#543×C67C211yesmaximal, characteristic
2C67×#2312C212yesmaximal
3C67×#1996C218yesmaximal, complemented
4C67×#1819C214yesmaximal
5#543×C67—28no—
6C2×C22⋊Q8×C67C418yescomplemented
7C2×C22⋊Q8×C67C2214yes—
8C22×C22⋊C4×C67C2212yes—
9C22×C22⋊C4×C67C2211yescharacteristic
10#388×C67—28no—
11#380×C67C2214yes—
12#379×C67C2214yes—
13C4×C22⋊C4×C67C2214yes—
14#342×C67C22116yes—
15C25×C67C2311yescharacteristic
16C22×Q8×C67C2×C414yes—
17C23×C4×C67C2×C414yes—
18C23×C4×C67C2312yes—
19C22⋊Q8×C67—28no—
20C22⋊Q8×C67—28no—
21C22⋊Q8×C67—232no—
22C2×C4⋊C4×C67—24no—
23C2×C4⋊C4×C67C2×C4112yes—
24C2×C22⋊C4×C67—28no—
25C2×C22⋊C4×C67—24no—
26C2×C22⋊C4×C67—24no—
27C2×C22⋊C4×C67C2×C414yes—
28C2×C22⋊C4×C67C2×C414yes—
29C2×C22⋊C4×C67D418yes—
30C2×C22⋊C4×C67C2318yes—
31C2×C22⋊C4×C67C2314yes—
32C2×C42×C67—24no—
33#95×C67—24no—
34#95×C67—24no—
35C24×C67—24no—
36C24×C67—82no—
37C24×C67C22×C412yes—
38C24×C67C2×D414yes—
39C24×C67C2411yescharacteristic
40C2×Q8×C67—28no—
41C2×Q8×C67—84no—
42C22×C4×C67—28no—
43C22×C4×C67—28no—
44C22×C4×C67—82no—
45C22×C4×C67—84no—
46C22×C4×C67C22×C418yes—
47C22×C4×C67C22×C414yes—
48C22×C4×C67C2×D414yes—
49C22×C4×C67C2×D414yes—
50C4⋊C4×C67—224no—
51C4⋊C4×C67—84no—
52C22⋊C4×C67—28no—
53C22⋊C4×C67—28no—
54C22⋊C4×C67—216no—
55C22⋊C4×C67—84no—
56C22⋊C4×C67—84no—
57C22⋊C4×C67C22⋊C4116yes—
58C42×C67—84no—
59C23×C67—24no—
60C23×C67—28no—
61C23×C67—44no—
62C23×C67—42no—
63C23×C67—44no—
64C23×C67—44no—
65C23×C67—88no—
66C23×C67C2×C22⋊C414yes—
67C23×C67C23×C411yescharacteristic
68C23×C67C22×D412yes—
69C23×C672−1+414yes—
70Q8×C67—88no—
71C2×C4×C67—28no—
72C2×C4×C67—44no—
73C2×C4×C67—44no—
74C2×C4×C67—44no—
75C2×C4×C67—48no—
76C2×C4×C67—48no—
77C2×C4×C67—48no—
78C2×C4×C67—48no—
79C2×C4×C67—88no—
80C2×C4×C67C2×C22⋊C418yes—
81C22×C67—22no—
82C22×C67—22no—
83C22×C67—22no—
84C22×C67—24no—
85C22×C67—28no—
86C22×C67—44no—
87C22×C67—48no—
88C22×C67—88no—
89C22×C67C22×C22⋊C411yescharacteristic
90C22×C67#51912yes—
91C22×C67#53614yes—
92C4×C67—28no—
93C4×C67—48no—
94C4×C67—88no—
95#21827C6711yesmaximal, characteristic, direct product factor, Sylow
96C2×C67—24no—
97C2×C67—22no—
98C2×C67—44no—
99C2×C67#231011yescharacteristic
100C2×C67#288312yes—
101C2×#543C2×C6711yescharacteristic
102#2312C2×C6712yes—
103#1996C2×C6718yescomplemented
104#1819C2×C6714yes—
105C67#2182711yescharacteristic, direct product factor, Sylow
106#543—28no—
107C2×C22⋊Q8C4×C6718yescomplemented
108C2×C22⋊Q8C22×C6714yes—
109C22×C22⋊C4C22×C6712yes—
110C22×C22⋊C4C22×C6711yescharacteristic
111#388—28no—
112#380C22×C6714yes—
113#379C22×C6714yes—
114C4×C22⋊C4C22×C6714yes—
115#342C22×C67116yes—
116C25C23×C6711yescharacteristic
117C22×Q8C2×C4×C6714yes—
118C23×C4C2×C4×C6714yes—
119C23×C4C23×C6712yes—
120C22⋊Q8—28no—
121C22⋊Q8—28no—
122C22⋊Q8—232no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4C2×C4×C67112yes—
125C2×C22⋊C4—28no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4C2×C4×C6714yes—
129C2×C22⋊C4C2×C4×C6714yes—
130C2×C22⋊C4D4×C6718yes—
131C2×C22⋊C4C23×C6718yes—
132C2×C22⋊C4C23×C6714yes—
133C2×C42—24no—
134#95—24no—
135#95—24no—
136C24—24no—
137C24—82no—
138C24C22×C4×C6712yes—
139C24C2×D4×C6714yes—
140C24C24×C6711yescharacteristic
141C2×Q8—28no—
142C2×Q8—84no—
143C22×C4—28no—
144C22×C4—28no—
145C22×C4—82no—
146C22×C4—84no—
147C22×C4C22×C4×C6718yes—
148C22×C4C22×C4×C6714yes—
149C22×C4C2×D4×C6714yes—
150C22×C4C2×D4×C6714yes—
151C4⋊C4—224no—
152C4⋊C4—84no—
153C22⋊C4—28no—
154C22⋊C4—28no—
155C22⋊C4—216no—
156C22⋊C4—84no—
157C22⋊C4—84no—
158C22⋊C4C22⋊C4×C67116yes—
159C42—84no—
160C23—24no—
161C23—28no—
162C23—44no—
163C23—42no—
164C23—44no—
165C23—44no—
166C23—88no—
167C23C2×C22⋊C4×C6714yes—
168C23C23×C4×C6711yescharacteristic
169C23C22×D4×C6712yes—
170C232−1+4×C6714yes—
171Q8—88no—
172C2×C4—28no—
173C2×C4—44no—
174C2×C4—44no—
175C2×C4—44no—
176C2×C4—48no—
177C2×C4—48no—
178C2×C4—48no—
179C2×C4—48no—
180C2×C4—88no—
181C2×C4C2×C22⋊C4×C6718yes—
182C22—22no—
183C22—22no—
184C22—22no—
185C22—24no—
186C22—28no—
187C22—44no—
188C22—48no—
189C22—88no—
190C22C22×C22⋊C4×C6711yescharacteristic
191C22#519×C6712yes—
192C22#536×C6714yes—
193C4—28no—
194C4—48no—
195C4—88no—
196C2—24no—
197C2—22no—
198C2—44no—
199C2C67×#231011yescharacteristic
200C2C67×#288312yes—
subgroupG/H
center, socleC22×C67C22×C22⋊C4
commutatorC23C23×C4×C67
Frattini, cosocleC24C24×C67
prime radicalC25×C67C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C22×C22⋊C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
3. C67×#1996C2semidirect
6. C2×C22⋊Q8×C67C4semidirect
95. #21827C67direct (N × H)
103. #1996C2×C67semidirect
105. C67#21827direct (N × H)
107. C2×C22⋊Q8C4×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#543×C672
C67×#23122
C67×#19962
C67×#18192
#2182767