Finite Group Databasebeta

← C67×#19745 #30698996

Subgroups — C67×#19745 #30698996

Order 17152.

994 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 254 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×#3184C213yesmaximal, complemented
2C67×#3151C213yesmaximal, complemented
3C67×#1721C213yesmaximal, complemented
4C67×#1715C213yesmaximal, complemented
5C67×#1712C211yesmaximal, characteristic, complemented
6C67×#1705C211yesmaximal, characteristic, complemented
7C67×#1455C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C67—26no—
9#532×C67C2213yes—
10C2×#127×C67C2211yescharacteristic
11#498×C67C2213yescomplemented
12#496×C67—26no—
13#493×C67C2213yes—
14#493×C67C2213yes—
15#493×C67C2213yescomplemented
16#492×C67C2213yes—
17#492×C67C2213yes—
18#486×C67C2216yescomplemented
19#469×C67—26no—
20#442×C67—26no—
21C4×M16×C67C2213yes—
22#322×C67C2213yescomplemented
23C82×C67C2211yescharacteristic, complemented
24C2×#41×C67—23no—
25C22×D4×C67—23no—
26AGL(1,ℤ/8)×C67—424no—
27C2×SD16×C67—26no—
28C2×SD16×C67—26no—
29C2×D8×C67—26no—
30C2×D8×C67—26no—
31C2×D8×C67—26no—
32C2×M16×C67D416yes—
33C4⋊Q8×C67C2313yes—
34#127×C67—23no—
35#127×C67C2313yes—
36#127×C67C2311yescharacteristic
37#127×C67C2311yescharacteristic
38#124×C67—23no—
39#121×C67—26no—
40C4×D4×C67—26no—
41C2×C42×C67C2311yescharacteristic
42#106×C67—26no—
43#102×C67—212no—
44#97×C67D416yes—
45C4×C8×C67C2313yes—
46C4×C8×C67C2313yes—
47C24×C67—41no—
48#41×C67—412no—
49C2×Q8×C67—23no—
50C2×D4×C67—26no—
51C2×D4×C67—23no—
52C2×D4×C67—23no—
53C2×D4×C67—23no—
54C2×D4×C67—412no—
55C2×D4×C67—43no—
56C2×D4×C67—43no—
57C22×C4×C67—43no—
58C22×C4×C67C2×D413yes—
59SD16×C67—412no—
60SD16×C67—412no—
61D8×C67—412no—
62D8×C67—412no—
63D8×C67—412no—
64M16×C67—212no—
65C2×C8×C67C2×D416yes—
66C2×C8×C67C2×D416yes—
67C4⋊C4×C67—26no—
68C22⋊C4×C67—46no—
69C42×C67C2×D413yes—
70C42×C67C2411yescharacteristic
71C23×C67—43no—
72C23×C67—41no—
73C23×C67—41no—
74C23×C67—86no—
75C23×C67#12711yescharacteristic
76Q8×C67—46no—
77D4×C67—46no—
78D4×C67—46no—
79D4×C67—46no—
80D4×C67—412no—
81D4×C67—86no—
82D4×C67—86no—
83C2×C4×C67—26no—
84C2×C4×C67—43no—
85C2×C4×C67—86no—
86C2×C4×C67#12713yes—
87C2×C4×C67C22×D413yes—
88C8×C67—26no—
89C8×C67AGL(1,ℤ/8)112yescomplemented
90C22×C67—23no—
91C22×C67—83no—
92C22×C67—83no—
93C22×C67—83no—
94C22×C67—86no—
95C22×C67—164no—
96C22×C67C2×#12711yescharacteristic
97C4×C67—23no—
98C4×C67—83no—
99C4×C67C2×AGL(1,ℤ/8)16yes—
100#19745C6711yesmaximal, characteristic, direct product factor, Sylow
101C2×C67—41no—
102C2×C67—163no—
103C2×C67—161no—
104C2×C67—161no—
105C2×C67#315113yes—
106#3184C2×C6713yescomplemented
107#3151C2×C6713yescomplemented
108#1721C2×C6713yescomplemented
109#1715C2×C6713yescomplemented
110#1712C2×C6711yescharacteristic, complemented
111#1705C2×C6711yescharacteristic, complemented
112#1455C2×C6711yescharacteristic, complemented
113C67#1974511yescharacteristic, direct product factor, Sylow
114C2×AGL(1,ℤ/8)—26no—
115#532C22×C6713yes—
116C2×#127C22×C6711yescharacteristic
117#498C22×C6713yescomplemented
118#496—26no—
119#493C22×C6713yes—
120#493C22×C6713yes—
121#493C22×C6713yescomplemented
122#492C22×C6713yes—
123#492C22×C6713yes—
124#486C22×C6716yescomplemented
125#469—26no—
126#442—26no—
127C4×M16C22×C6713yes—
128#322C22×C6713yescomplemented
129C82C22×C6711yescharacteristic, complemented
130C2×#41—23no—
131C22×D4—23no—
132AGL(1,ℤ/8)—424no—
133C2×SD16—26no—
134C2×SD16—26no—
135C2×D8—26no—
136C2×D8—26no—
137C2×D8—26no—
138C2×M16D4×C6716yes—
139C4⋊Q8C23×C6713yes—
140#127—23no—
141#127C23×C6711yescharacteristic
142#127C23×C6711yescharacteristic
143#127C23×C6713yes—
144#124—23no—
145#121—26no—
146C4×D4—26no—
147C2×C42C23×C6711yescharacteristic
148#106—26no—
149#102—212no—
150#97D4×C6716yes—
151C4×C8C23×C6713yes—
152C4×C8C23×C6713yes—
153C24—41no—
154#41—412no—
155C2×Q8—23no—
156C2×D4—26no—
157C2×D4—23no—
158C2×D4—23no—
159C2×D4—23no—
160C2×D4—412no—
161C2×D4—43no—
162C2×D4—43no—
163C22×C4—43no—
164C22×C4C2×D4×C6713yes—
165SD16—412no—
166SD16—412no—
167D8—412no—
168D8—412no—
169D8—412no—
170M16—212no—
171C2×C8C2×D4×C6716yes—
172C2×C8C2×D4×C6716yes—
173C4⋊C4—26no—
174C22⋊C4—46no—
175C42C2×D4×C6713yes—
176C42C24×C6711yescharacteristic
177C23—43no—
178C23—41no—
179C23—41no—
180C23—86no—
181C23#127×C6711yescharacteristic
182Q8—46no—
183D4—46no—
184D4—46no—
185D4—46no—
186D4—412no—
187D4—86no—
188D4—86no—
189C2×C4—26no—
190C2×C4—43no—
191C2×C4—86no—
192C2×C4#127×C6713yes—
193C2×C4C22×D4×C6713yes—
194C8—26no—
195C8AGL(1,ℤ/8)×C67112yescomplemented
196C22—23no—
197C22—83no—
198C22—83no—
199C22—83no—
200C22—86no—
201C22—164no—
202C22C2×#127×C6711yescharacteristic
203C4—23no—
204C4—83no—
205C4C2×AGL(1,ℤ/8)×C6716yes—
206C2—41no—
207C2—161no—
208C2—161no—
209C2—163no—
210C2C67×#315113yes—
subgroupG/H
center, socleC22×C67C2×#127
commutator, Frattini, cosocleC42C24×C67
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3184C2semidirect
2. C67×#3151C2semidirect
3. C67×#1721C2semidirect
4. C67×#1715C2semidirect
5. C67×#1712C2semidirect
6. C67×#1705C2semidirect
7. C67×#1455C2semidirect
11. #498×C67C22semidirect
15. #493×C67C22semidirect
18. #486×C67C22semidirect
22. #322×C67C22semidirect
23. C82×C67C22semidirect
89. C8×C67AGL(1,ℤ/8)semidirect
100. #19745C67direct (N × H)
106. #3184C2×C67semidirect
107. #3151C2×C67semidirect
108. #1721C2×C67semidirect
109. #1715C2×C67semidirect
110. #1712C2×C67semidirect
111. #1705C2×C67semidirect
112. #1455C2×C67semidirect
113. C67#19745direct (N × H)
117. #498C22×C67semidirect
121. #493C22×C67semidirect
124. #486C22×C67semidirect
128. #322C22×C67semidirect
129. C82C22×C67semidirect
195. C8AGL(1,ℤ/8)×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#31842
C67×#31512
C67×#17212
C67×#17152
C67×#17122
C67×#17052
C67×#14552
#1974567