Finite Group Databasebeta

← C67×#10080 #30689331

Subgroups — C67×#10080 #30689331

Order 17152.

552 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 114 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×#490×C67C212yesmaximal
2C67×#2822C211yesmaximal, characteristic, complemented
3C67×#1961C211yesmaximal, characteristic
4C2×#330×C67C212yesmaximal, complemented
5C67×#1295C211yesmaximal, characteristic, complemented
6C2×#127×C67C2211yescharacteristic
7C2×#125×C67C2212yes—
8#490×C67—24no—
9#490×C67—24no—
10C2×#102×C67—22no—
11C2×#102×C67—22no—
12C2×#102×C67—22no—
13C2×#97×C67C2211yescharacteristic
14C2×#97×C67C2212yes—
15#400×C67—22no—
16#398×C67—22no—
17#398×C67—22no—
18#382×C67C2211yescharacteristic
19#330×C67—24no—
20C22×D4×C67—22no—
21C22×D4×C67—21no—
22C22×C8×C67—21no—
23C22×C8×C67—22no—
24#127×C67—22no—
25#127×C67—21no—
26#127×C67D412yes—
27#125×C67—22no—
28#125×C67—22no—
29#125×C67—22no—
30#125×C67—22no—
31C2×C4⋊C4×C67—22no—
32C2×C4⋊C4×C67—22no—
33C2×C4⋊C4×C67—22no—
34C2×C4⋊C4×C67D412yes—
35C2×C4⋊C4×C67D412yes—
36C2×C4⋊C4×C67D412yes—
37C2×C42×C67C2311yescharacteristic
38#102×C67—28no—
39#102×C67—28no—
40#102×C67—44no—
41#97×C67—22no—
42#97×C67—22no—
43#97×C67D414yes—
44#95×C67—24no—
45#95×C67—22no—
46#95×C67—22no—
47C24×C67—41no—
48C2×D4×C67—24no—
49C2×D4×C67—22no—
50C2×D4×C67—42no—
51C2×D4×C67—48no—
52C2×D4×C67—42no—
53C2×D4×C67—42no—
54C2×D4×C67—41no—
55C22×C4×C67—22no—
56C22×C4×C67—22no—
57C22×C4×C67—22no—
58C22×C4×C67C2×D412yes—
59C22×C4×C67C2×D411yescharacteristic
60C2×C8×C67—22no—
61C2×C8×C67—24no—
62C2×C8×C67—44no—
63C2×C8×C67—41no—
64C2×C8×C67—41no—
65C4⋊C4×C67—22no—
66C4⋊C4×C67—22no—
67C4⋊C4×C67—22no—
68C4⋊C4×C67—22no—
69C4⋊C4×C67—42no—
70C4⋊C4×C67—42no—
71C4⋊C4×C67—42no—
72C4⋊C4×C67—42no—
73C4⋊C4×C67—42no—
74C4⋊C4×C67—82no—
75C42×C67C2×D411yescharacteristic
76C42×C67C2×D412yes—
77C42×C67#4111yescharacteristic
78C23×C67—42no—
79C23×C67—81no—
80C23×C67—84no—
81C23×C67—81no—
82C23×C67#12011yescharacteristic
83D4×C67—48no—
84D4×C67—44no—
85D4×C67—84no—
86D4×C67—82no—
87C2×C4×C67—21no—
88C2×C4×C67—24no—
89C2×C4×C67—21no—
90C2×C4×C67—42no—
91C2×C4×C67—42no—
92C2×C4×C67—42no—
93C2×C4×C67—42no—
94C2×C4×C67—44no—
95C2×C4×C67—44no—
96C2×C4×C67—42no—
97C2×C4×C67#12012yes—
98C2×C4×C67#12112yes—
99C2×C4×C67#12111yescharacteristic
100C2×C4×C67#12711yescharacteristic
101C8×C67—44no—
102C8×C67—42no—
103C22×C67—84no—
104C22×C67—82no—
105C22×C67—162no—
106C22×C67—162no—
107C22×C67#39211yescharacteristic
108C22×C67#45311yescharacteristic
109C22×C67#45414yes—
110C22×C67#45411yescharacteristic
111C4×C67—22no—
112C4×C67—22no—
113C4×C67—21no—
114C4×C67—21no—
115C4×C67—82no—
116C4×C67—82no—
117C4×C67—82no—
118#10080C6711yesmaximal, characteristic, direct product factor, Sylow
119C2×C67—162no—
120C2×C67#165712yes—
121C2×C67#165812yes—
122C2×C67#200312yes—
123C2×C67#200611yescharacteristic
124C2×#490C2×C6712yes—
125#2822C2×C6711yescharacteristic, complemented
126#1961C2×C6711yescharacteristic
127C2×#330C2×C6712yescomplemented
128#1295C2×C6711yescharacteristic, complemented
129C67#1008011yescharacteristic, direct product factor, Sylow
130C2×#127C22×C6711yescharacteristic
131C2×#125C22×C6712yes—
132#490—24no—
133#490—24no—
134C2×#102—22no—
135C2×#102—22no—
136C2×#102—22no—
137C2×#97C22×C6711yescharacteristic
138C2×#97C22×C6712yes—
139#400—22no—
140#398—22no—
141#398—22no—
142#382C22×C6711yescharacteristic
143#330—24no—
144C22×D4—22no—
145C22×D4—21no—
146C22×C8—21no—
147C22×C8—22no—
148#127—22no—
149#127—21no—
150#127D4×C6712yes—
151#125—22no—
152#125—22no—
153#125—22no—
154#125—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4D4×C6712yes—
159C2×C4⋊C4D4×C6712yes—
160C2×C4⋊C4D4×C6712yes—
161C2×C42C23×C6711yescharacteristic
162#102—28no—
163#102—28no—
164#102—44no—
165#97—22no—
166#97—22no—
167#97D4×C6714yes—
168#95—24no—
169#95—22no—
170#95—22no—
171C24—41no—
172C2×D4—24no—
173C2×D4—22no—
174C2×D4—42no—
175C2×D4—48no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—41no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4C2×D4×C6712yes—
183C22×C4C2×D4×C6711yescharacteristic
184C2×C8—22no—
185C2×C8—24no—
186C2×C8—44no—
187C2×C8—41no—
188C2×C8—41no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—82no—
199C42C2×D4×C6711yescharacteristic
200C42C2×D4×C6712yes—
201C42#41×C6711yescharacteristic
202C23—42no—
203C23—81no—
204C23—84no—
205C23—81no—
206C23#120×C6711yescharacteristic
207D4—48no—
208D4—44no—
209D4—84no—
210D4—82no—
211C2×C4—21no—
212C2×C4—24no—
213C2×C4—21no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—44no—
217C2×C4—42no—
218C2×C4—44no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4#120×C6712yes—
222C2×C4#121×C6712yes—
223C2×C4#121×C6711yescharacteristic
224C2×C4#127×C6711yescharacteristic
225C8—44no—
226C8—42no—
227C22—84no—
228C22—82no—
229C22—162no—
230C22—162no—
231C22#392×C6711yescharacteristic
232C22#453×C6711yescharacteristic
233C22#454×C6714yes—
234C22#454×C6711yescharacteristic
235C4—22no—
236C4—22no—
237C4—21no—
238C4—21no—
239C4—82no—
240C4—82no—
241C4—82no—
242C2—162no—
243C2C67×#165712yes—
244C2C67×#165812yes—
245C2C67×#200312yes—
246C2C67×#200611yescharacteristic
subgroupG/H
center, socleC23×C67#120
commutator, Frattini, cosocleC2×C42C23×C67
prime radicalC2×#127×C67C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#2822C2semidirect
4. C2×#330×C67C2semidirect
5. C67×#1295C2semidirect
118. #10080C67direct (N × H)
125. #2822C2×C67semidirect
127. C2×#330C2×C67semidirect
128. #1295C2×C67semidirect
129. C67#10080direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#490×C672
C67×#28222
C67×#19612
C2×#330×C672
C67×#12952
#1008067