Finite Group Databasebeta

← C67×#7397 #30686648

Subgroups — C67×#7397 #30686648

Order 17152.

268 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 42 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×#2010C211yesmaximal, characteristic
2C67×#1320C211yesmaximal, characteristic, complemented
3C67×#1298C211yesmaximal, characteristic, complemented
4#575×C67C411yescharacteristic
5#525×C67—21no—
6#421×C67—21no—
7#407×C67C2211yescharacteristic
8C2×C22⋊C8×C67—21no—
9#349×C67—21no—
10H(3,ℤ/4)×C67C411yescharacteristic
11C2×#41×C67—21no—
12C23×C4×C67D411yescharacteristic
13#137×C67—41no—
14AGL(1,ℤ/8)×C67—41no—
15#135×C67—41no—
16C2×M16×C67C2×C411yescharacteristic
17C2×M16×C67D411yescharacteristic
18C22×C8×C67—41no—
19#123×C67—41no—
20C22⋊Q8×C67—41no—
21#121×C67—41no—
22#120×C67—41no—
23C4×D4×C67—42no—
24#117×C67—21no—
25M32×C67—41no—
26#104×C67—42no—
27C22⋊C8×C67—22no—
28C22⋊C8×C67—41no—
29C22⋊C8×C67—41no—
30C24×C67SD1611yescharacteristic
31#41×C67—41no—
32#41×C67—41no—
33#41×C67—41no—
34#41×C67—41no—
35C2×Q8×C67—21no—
36C2×D4×C67—21no—
37C2×D4×C67—42no—
38C2×D4×C67—41no—
39C22×C4×C67—21no—
40C22×C4×C67—21no—
41C22×C4×C67—41no—
42C22×C4×C67—41no—
43C22×C4×C67—41no—
44C22×C4×C67C22⋊C411yescharacteristic
45C22×C4×C67D811yescharacteristic
46Q16×C67—41no—
47SD16×C67—42no—
48D8×C67—41no—
49M16×C67—41no—
50M16×C67—41no—
51C2×C8×C67—21no—
52C2×C8×C67—21no—
53C2×C8×C67—42no—
54C2×C8×C67—81no—
55C2×C8×C67—81no—
56C4⋊C4×C67—41no—
57C4⋊C4×C67—42no—
58C22⋊C4×C67—42no—
59C22⋊C4×C67—42no—
60C22⋊C4×C67—41no—
61C42×C67—22no—
62C16×C67—81no—
63C23×C67—21no—
64C23×C67—41no—
65C23×C67—42no—
66C23×C67—41no—
67C23×C67#10211yescharacteristic
68Q8×C67—41no—
69Q8×C67—41no—
70D4×C67—41no—
71D4×C67—41no—
72D4×C67—81no—
73D4×C67—84no—
74D4×C67—81no—
75C2×C4×C67—21no—
76C2×C4×C67—21no—
77C2×C4×C67—21no—
78C2×C4×C67—41no—
79C2×C4×C67—41no—
80C2×C4×C67—41no—
81C2×C4×C67—42no—
82C2×C4×C67—41no—
83C2×C4×C67—81no—
84C2×C4×C67—81no—
85C2×C4×C67—81no—
86C2×C4×C67#9911yescharacteristic
87C2×C4×C67#10411yescharacteristic
88C8×C67—41no—
89C8×C67—41no—
90C8×C67—82no—
91C22×C67—22no—
92C22×C67—21no—
93C22×C67—41no—
94C22×C67—41no—
95C22×C67—42no—
96C22×C67—41no—
97C22×C67—81no—
98C22×C67—82no—
99C22×C67—81no—
100C22×C67#32711yescharacteristic
101C4×C67—21no—
102C4×C67—41no—
103C4×C67—82no—
104C4×C67—81no—
105C4×C67AΣL(1,𝔽16)11yescharacteristic
106C4×C67#35311yescharacteristic
107#7397C6711yesmaximal, characteristic, direct product factor, Sylow
108C2×C67—21no—
109C2×C67—41no—
110C2×C67—42no—
111C2×C67—161no—
112C2×C67#134311yescharacteristic
113#2010C2×C6711yescharacteristic
114#1320C2×C6711yescharacteristic, complemented
115#1298C2×C6711yescharacteristic, complemented
116C67#739711yescharacteristic, direct product factor, Sylow
117#575C4×C6711yescharacteristic
118#525—21no—
119#421—21no—
120#407C22×C6711yescharacteristic
121C2×C22⋊C8—21no—
122#349—21no—
123H(3,ℤ/4)C4×C6711yescharacteristic
124C2×#41—21no—
125C23×C4D4×C6711yescharacteristic
126#137—41no—
127AGL(1,ℤ/8)—41no—
128#135—41no—
129C2×M16C2×C4×C6711yescharacteristic
130C2×M16D4×C6711yescharacteristic
131C22×C8—41no—
132#123—41no—
133C22⋊Q8—41no—
134#121—41no—
135#120—41no—
136C4×D4—42no—
137#117—21no—
138M32—41no—
139#104—42no—
140C22⋊C8—22no—
141C22⋊C8—41no—
142C22⋊C8—41no—
143C24SD16×C6711yescharacteristic
144#41—41no—
145#41—41no—
146#41—41no—
147#41—41no—
148C2×Q8—21no—
149C2×D4—21no—
150C2×D4—42no—
151C2×D4—41no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4C22⋊C4×C6711yescharacteristic
158C22×C4D8×C6711yescharacteristic
159Q16—41no—
160SD16—42no—
161D8—41no—
162M16—41no—
163M16—41no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—42no—
167C2×C8—81no—
168C2×C8—81no—
169C4⋊C4—41no—
170C4⋊C4—42no—
171C22⋊C4—42no—
172C22⋊C4—42no—
173C22⋊C4—41no—
174C42—22no—
175C16—81no—
176C23—21no—
177C23—41no—
178C23—42no—
179C23—41no—
180C23#102×C6711yescharacteristic
181Q8—41no—
182Q8—41no—
183D4—41no—
184D4—41no—
185D4—81no—
186D4—84no—
187D4—81no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4#99×C6711yescharacteristic
200C2×C4#104×C6711yescharacteristic
201C8—41no—
202C8—41no—
203C8—82no—
204C22—22no—
205C22—21no—
206C22—41no—
207C22—41no—
208C22—42no—
209C22—41no—
210C22—81no—
211C22—82no—
212C22—81no—
213C22#327×C6711yescharacteristic
214C4—21no—
215C4—41no—
216C4—82no—
217C4—81no—
218C4AΣL(1,𝔽16)×C6711yescharacteristic
219C4#353×C6711yescharacteristic
220C2—21no—
221C2—41no—
222C2—42no—
223C2—161no—
224C2C67×#134311yescharacteristic
subgroupG/H
centerC4×C67#353
commutatorC2×M16C2×C4×C67
Frattini, cosocle#407C22×C67
socleC2×C67#1343
prime radicalC67×#2010C2
LCSG ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C67 ◁ C2×C4×C67 ◁ C22×C4×C67 ◁ #407×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#1320C2semidirect
3. C67×#1298C2semidirect
107. #7397C67direct (N × H)
114. #1320C2×C67semidirect
115. #1298C2×C67semidirect
116. C67#7397direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#20102
C67×#13202
C67×#12982
#739767