Finite Group Databasebeta

← C67×#12251 #30691502

Subgroups — C67×#12251 #30691502

Order 17152.

342 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C67×#2822C211yesmaximal, characteristic
2C67×#2061C211yesmaximal, characteristic
3C67×#1611C212yesmaximal
4C67×#1346C212yesmaximal
5C67×#1324C211yesmaximal, characteristic
6#544×C67C2211yescharacteristic
7C2×#125×C67—21no—
8#483×C67—22no—
9#428×C67—21no—
10#400×C67—22no—
11#398×C67—22no—
12#398×C67—21no—
13#398×C67C2212yes—
14#382×C67C2211yescharacteristic
15#343×C67C2211yescharacteristic
16#339×C67—21no—
17#328×C67—22no—
18#324×C67C2212yes—
19C2×M16×C67—21no—
20C4⋊Q8×C67—41no—
21#125×C67—42no—
22#125×C67—41no—
23#125×C67—42no—
24C22⋊Q8×C67—22no—
25#117×C67—21no—
26C2×C4⋊C4×C67—21no—
27C2×C4⋊C4×C67—22no—
28C2×C4⋊C4×C67—21no—
29C2×C4⋊C4×C67—22no—
30C2×C4⋊C4×C67—21no—
31C2×C4⋊C4×C67D412yes—
32C2×C4⋊C4×C67D411yescharacteristic
33C2×C4⋊C4×C67C2311yescharacteristic
34C2×C42×C67D411yescharacteristic
35#107×C67—41no—
36#106×C67—42no—
37#106×C67—41no—
38#103×C67—42no—
39C22⋊C8×C67—22no—
40#95×C67—22no—
41#95×C67—21no—
42#95×C67—22no—
43#95×C67—22no—
44#95×C67D412yes—
45C2×Q8×C67—22no—
46C22×C4×C67—21no—
47C22×C4×C67—22no—
48C22×C4×C67—22no—
49C22×C4×C67—21no—
50C22×C4×C67C2×D411yescharacteristic
51C22×C4×C67C2×D412yes—
52M16×C67—22no—
53M16×C67—41no—
54C2×C8×C67—41no—
55C2×C8×C67—42no—
56C4⋊C4×C67—21no—
57C4⋊C4×C67—22no—
58C4⋊C4×C67—21no—
59C4⋊C4×C67—41no—
60C4⋊C4×C67—42no—
61C4⋊C4×C67—41no—
62C4⋊C4×C67—42no—
63C4⋊C4×C67—44no—
64C4⋊C4×C67—42no—
65C4⋊C4×C67—42no—
66C4⋊C4×C67—42no—
67C4⋊C4×C67—42no—
68C4⋊C4×C67—81no—
69C22⋊C4×C67—41no—
70C42×C67—21no—
71C42×C67—41no—
72C42×C67SD1612yes—
73C23×C67#12011yescharacteristic
74Q8×C67—82no—
75C2×C4×C67—21no—
76C2×C4×C67—21no—
77C2×C4×C67—22no—
78C2×C4×C67—22no—
79C2×C4×C67—41no—
80C2×C4×C67—41no—
81C2×C4×C67—42no—
82C2×C4×C67—42no—
83C2×C4×C67—42no—
84C2×C4×C67—42no—
85C2×C4×C67—42no—
86C2×C4×C67—42no—
87C2×C4×C67—42no—
88C2×C4×C67—41no—
89C2×C4×C67—41no—
90C2×C4×C67—42no—
91C2×C4×C67—41no—
92C2×C4×C67C2×SD1611yescharacteristic
93C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
94C8×C67—42no—
95C8×C67—82no—
96C22×C67—21no—
97C22×C67—21no—
98C22×C67#45011yescharacteristic
99C22×C67#45611yescharacteristic
100C22×C67B(4,𝔽2)11yescharacteristic
101C4×C67—21no—
102C4×C67—21no—
103C4×C67—42no—
104C4×C67—42no—
105C4×C67—82no—
106C4×C67—82no—
107C4×C67—81no—
108C4×C67—81no—
109C4×C67—81no—
110#12251C6711yesmaximal, characteristic, direct product factor, Sylow
111C2×C67—21no—
112C2×C67—21no—
113C2×C67#160411yescharacteristic
114C2×C67#220111yescharacteristic
115C2×C67#220311yescharacteristic
116#2822C2×C6711yescharacteristic
117#2061C2×C6711yescharacteristic
118#1611C2×C6712yes—
119#1346C2×C6712yes—
120#1324C2×C6711yescharacteristic
121C67#1225111yescharacteristic, direct product factor, Sylow
122#544C22×C6711yescharacteristic
123C2×#125—21no—
124#483—22no—
125#428—21no—
126#400—22no—
127#398—21no—
128#398—22no—
129#398C22×C6712yes—
130#382C22×C6711yescharacteristic
131#343C22×C6711yescharacteristic
132#339—21no—
133#328—22no—
134#324C22×C6712yes—
135C2×M16—21no—
136C4⋊Q8—41no—
137#125—42no—
138#125—41no—
139#125—42no—
140C22⋊Q8—22no—
141#117—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4D4×C6711yescharacteristic
148C2×C4⋊C4D4×C6712yes—
149C2×C4⋊C4C23×C6711yescharacteristic
150C2×C42D4×C6711yescharacteristic
151#107—41no—
152#106—41no—
153#106—42no—
154#103—42no—
155C22⋊C8—22no—
156#95—22no—
157#95—21no—
158#95—22no—
159#95—22no—
160#95D4×C6712yes—
161C2×Q8—22no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4C2×D4×C6711yescharacteristic
167C22×C4C2×D4×C6712yes—
168M16—22no—
169M16—41no—
170C2×C8—41no—
171C2×C8—42no—
172C4⋊C4—21no—
173C4⋊C4—22no—
174C4⋊C4—21no—
175C4⋊C4—41no—
176C4⋊C4—42no—
177C4⋊C4—41no—
178C4⋊C4—42no—
179C4⋊C4—44no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—81no—
185C22⋊C4—41no—
186C42—21no—
187C42—41no—
188C42SD16×C6712yes—
189C23#120×C6711yescharacteristic
190Q8—82no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4C2×SD16×C6711yescharacteristic
209C2×C4AGL(1,ℤ/8)×C6711yescharacteristic
210C8—42no—
211C8—82no—
212C22—21no—
213C22—21no—
214C22#450×C6711yescharacteristic
215C22#456×C6711yescharacteristic
216C22B(4,𝔽2)×C6711yescharacteristic
217C4—21no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—82no—
222C4—82no—
223C4—81no—
224C4—81no—
225C4—81no—
226C2—21no—
227C2—21no—
228C2C67×#160411yescharacteristic
229C2C67×#220111yescharacteristic
230C2C67×#220311yescharacteristic
subgroupG/H
center, socleC22×C67B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C67
prime radicalC23×C67#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C23×C67 ◁ C2×C4⋊C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
110. #12251C67direct (N × H)
121. C67#12251direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#28222
C67×#20612
C67×#16112
C67×#13462
C67×#13242
#1225167