Finite Group Databasebeta

← C67×#19733 #30698984

Subgroups — C67×#19733 #30698984

Order 17152.

712 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 230 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×#3231C213yesmaximal, complemented
2C67×#3208C213yesmaximal
3C67×#1719C213yesmaximal, complemented
4C67×#1714C213yesmaximal, complemented
5C67×#1711C211yesmaximal, characteristic, complemented
6C67×#1705C211yesmaximal, characteristic, complemented
7C67×#1454C211yesmaximal, characteristic, complemented
8#533×C67C2213yes—
9C2×#127×C67C2211yescharacteristic
10#500×C67C2213yescomplemented
11#498×C67C2213yescomplemented
12#492×C67C2213yes—
13#492×C67C2213yes—
14#487×C67C2213yescomplemented
15#487×C67C2213yescomplemented
16#486×C67C2213yes—
17#486×C67C2213yes—
18#485×C67—26no—
19#479×C67—26no—
20#460×C67—26no—
21#450×C67—26no—
22#431×C67C2213yes—
23#322×C67C2213yescomplemented
24C82×C67C2211yescharacteristic, complemented
25C22×D4×C67—23no—
26C2×SD16×C67—26no—
27C2×SD16×C67—26no—
28C4⋊Q8×C67C2313yes—
29C4⋊Q8×C67C2313yes—
30#127×C67—23no—
31#127×C67C2311yescharacteristic
32#127×C67C2311yescharacteristic
33#125×C67—23no—
34C22⋊Q8×C67—23no—
35C22⋊Q8×C67—23no—
36C4×Q8×C67—23no—
37C4×Q8×C67—23no—
38#117×C67—23no—
39C2×C42×C67C2311yescharacteristic
40#107×C67—26no—
41#106×C67—26no—
42C4⋊C8×C67D416yes—
43#103×C67—26no—
44#103×C67—26no—
45#102×C67—26no—
46#102×C67—26no—
47C22⋊C8×C67D416yes—
48C4×C8×C67C2313yes—
49C4×C8×C67C2313yes—
50C24×C67—41no—
51C2×Q8×C67—23no—
52C2×Q8×C67—23no—
53C2×D4×C67—23no—
54C2×D4×C67—23no—
55C2×D4×C67—43no—
56C2×D4×C67—86no—
57C22×C4×C67C2×D413yes—
58SD16×C67—86no—
59SD16×C67—86no—
60C2×C8×C67C2×D416yes—
61C2×C8×C67C2×D416yes—
62C4⋊C4×C67—23no—
63C4⋊C4×C67—23no—
64C4⋊C4×C67—23no—
65C4⋊C4×C67—23no—
66C4⋊C4×C67—43no—
67C4⋊C4×C67—43no—
68C22⋊C4×C67—43no—
69C42×C67—43no—
70C42×C67C2×D413yes—
71C42×C67C2411yescharacteristic
72C23×C67—41no—
73C23×C67—41no—
74C23×C67—86no—
75C23×C67#12711yescharacteristic
76Q8×C67—46no—
77Q8×C67—46no—
78D4×C67—46no—
79D4×C67—46no—
80D4×C67—86no—
81C2×C4×C67—26no—
82C2×C4×C67—43no—
83C2×C4×C67—43no—
84C2×C4×C67#12713yes—
85C2×C4×C67C22×D413yes—
86C8×C67—26no—
87C8×C67—26no—
88C22×C67—23no—
89C22×C67—83no—
90C22×C67—83no—
91C22×C67—164no—
92C22×C67C2×#12711yescharacteristic
93C4×C67—23no—
94C4×C67—83no—
95C4×C67—83no—
96C4×C67#57716yes—
97#19733C6711yesmaximal, characteristic, direct product factor, Sylow
98C2×C67—41no—
99C2×C67—161no—
100C2×C67—161no—
101C2×C67#315413yes—
102#3231C2×C6713yescomplemented
103#3208C2×C6713yes—
104#1719C2×C6713yescomplemented
105#1714C2×C6713yescomplemented
106#1711C2×C6711yescharacteristic, complemented
107#1705C2×C6711yescharacteristic, complemented
108#1454C2×C6711yescharacteristic, complemented
109C67#1973311yescharacteristic, direct product factor, Sylow
110#533C22×C6713yes—
111C2×#127C22×C6711yescharacteristic
112#500C22×C6713yescomplemented
113#498C22×C6713yescomplemented
114#492C22×C6713yes—
115#492C22×C6713yes—
116#487C22×C6713yescomplemented
117#487C22×C6713yescomplemented
118#486C22×C6713yes—
119#486C22×C6713yes—
120#485—26no—
121#479—26no—
122#460—26no—
123#450—26no—
124#431C22×C6713yes—
125#322C22×C6713yescomplemented
126C82C22×C6711yescharacteristic, complemented
127C22×D4—23no—
128C2×SD16—26no—
129C2×SD16—26no—
130C4⋊Q8C23×C6713yes—
131C4⋊Q8C23×C6713yes—
132#127—23no—
133#127C23×C6711yescharacteristic
134#127C23×C6711yescharacteristic
135#125—23no—
136C22⋊Q8—23no—
137C22⋊Q8—23no—
138C4×Q8—23no—
139C4×Q8—23no—
140#117—23no—
141C2×C42C23×C6711yescharacteristic
142#107—26no—
143#106—26no—
144C4⋊C8D4×C6716yes—
145#103—26no—
146#103—26no—
147#102—26no—
148#102—26no—
149C22⋊C8D4×C6716yes—
150C4×C8C23×C6713yes—
151C4×C8C23×C6713yes—
152C24—41no—
153C2×Q8—23no—
154C2×Q8—23no—
155C2×D4—23no—
156C2×D4—23no—
157C2×D4—43no—
158C2×D4—86no—
159C22×C4C2×D4×C6713yes—
160SD16—86no—
161SD16—86no—
162C2×C8C2×D4×C6716yes—
163C2×C8C2×D4×C6716yes—
164C4⋊C4—23no—
165C4⋊C4—23no—
166C4⋊C4—23no—
167C4⋊C4—23no—
168C4⋊C4—43no—
169C4⋊C4—43no—
170C22⋊C4—43no—
171C42—43no—
172C42C2×D4×C6713yes—
173C42C24×C6711yescharacteristic
174C23—41no—
175C23—41no—
176C23—86no—
177C23#127×C6711yescharacteristic
178Q8—46no—
179Q8—46no—
180D4—46no—
181D4—46no—
182D4—86no—
183C2×C4—26no—
184C2×C4—43no—
185C2×C4—43no—
186C2×C4#127×C6713yes—
187C2×C4C22×D4×C6713yes—
188C8—26no—
189C8—26no—
190C22—23no—
191C22—83no—
192C22—83no—
193C22—164no—
194C22C2×#127×C6711yescharacteristic
195C4—23no—
196C4—83no—
197C4—83no—
198C4#577×C6716yes—
199C2—41no—
200C2—161no—
201C2—161no—
202C2C67×#315413yes—
subgroupG/H
center, socleC22×C67C2×#127
commutator, Frattini, cosocleC42C24×C67
prime radicalC2×#127×C67C22
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×#3231C2semidirect
3. C67×#1719C2semidirect
4. C67×#1714C2semidirect
5. C67×#1711C2semidirect
6. C67×#1705C2semidirect
7. C67×#1454C2semidirect
10. #500×C67C22semidirect
11. #498×C67C22semidirect
14. #487×C67C22semidirect
15. #487×C67C22semidirect
23. #322×C67C22semidirect
24. C82×C67C22semidirect
97. #19733C67direct (N × H)
102. #3231C2×C67semidirect
104. #1719C2×C67semidirect
105. #1714C2×C67semidirect
106. #1711C2×C67semidirect
107. #1705C2×C67semidirect
108. #1454C2×C67semidirect
109. C67#19733direct (N × H)
112. #500C22×C67semidirect
113. #498C22×C67semidirect
116. #487C22×C67semidirect
117. #487C22×C67semidirect
125. #322C22×C67semidirect
126. C82C22×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#32312
C67×#32082
C67×#17192
C67×#17142
C67×#17112
C67×#17052
C67×#14542
#1973367