Finite Group Databasebeta

← C67×#19743 #30698994

Subgroups — C67×#19743 #30698994

Order 17152.

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

normal subgroups only

This group has 814 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×#3207C213yesmaximal, complemented
2C67×#3205C213yesmaximal, complemented
3C67×#1721C213yesmaximal, complemented
4C67×#1717C213yesmaximal, complemented
5C67×#1712C211yesmaximal, characteristic, complemented
6C67×#1711C211yesmaximal, characteristic, complemented
7C67×#1454C211yesmaximal, characteristic, complemented
8#532×C67C2213yes—
9C2×#127×C67C2211yescharacteristic
10#493×C67C2213yes—
11#493×C67C2213yes—
12#493×C67C2213yescomplemented
13#492×C67C2213yescomplemented
14#487×C67C2213yescomplemented
15#486×C67C2213yes—
16#486×C67C2213yes—
17#486×C67C2213yescomplemented
18#463×C67—26no—
19#459×C67—26no—
20#449×C67—26no—
21#447×C67—26no—
22#431×C67C2213yes—
23#322×C67C2213yescomplemented
24C82×C67C2211yescharacteristic, complemented
25C2×#41×C67—23no—
26C22×D4×C67—23no—
27C2×SD16×C67—26no—
28C2×D8×C67—26no—
29C2×D8×C67—26no—
30C2×D8×C67—26no—
31C4⋊Q8×C67C2313yes—
32#127×C67—23no—
33#127×C67C2311yescharacteristic
34#127×C67C2311yescharacteristic
35#127×C67C2313yes—
36#124×C67—23no—
37#121×C67—23no—
38#121×C67—23no—
39C4×D4×C67—23no—
40C4×D4×C67—23no—
41C2×C42×C67C2311yescharacteristic
42C4⋊C8×C67D416yes—
43#103×C67—26no—
44#102×C67—26no—
45#102×C67—26no—
46#102×C67—26no—
47C22⋊C8×C67D416yes—
48C4×C8×C67C2313yes—
49C4×C8×C67C2313yes—
50C24×C67—41no—
51#41×C67—86no—
52C2×Q8×C67—23no—
53C2×D4×C67—23no—
54C2×D4×C67—23no—
55C2×D4×C67—23no—
56C2×D4×C67—23no—
57C2×D4×C67—23no—
58C2×D4×C67—43no—
59C2×D4×C67—43no—
60C2×D4×C67—86no—
61C22×C4×C67—43no—
62C22×C4×C67C2×D413yes—
63SD16×C67—86no—
64D8×C67—86no—
65D8×C67—86no—
66D8×C67—86no—
67C2×C8×C67C2×D416yes—
68C2×C8×C67C2×D416yes—
69C4⋊C4×C67—23no—
70C4⋊C4×C67—23no—
71C22⋊C4×C67—43no—
72C22⋊C4×C67—43no—
73C42×C67C2×D413yes—
74C42×C67C2411yescharacteristic
75C23×C67—43no—
76C23×C67—41no—
77C23×C67—41no—
78C23×C67—86no—
79C23×C67#12711yescharacteristic
80Q8×C67—46no—
81D4×C67—46no—
82D4×C67—46no—
83D4×C67—46no—
84D4×C67—46no—
85D4×C67—46no—
86D4×C67—86no—
87D4×C67—86no—
88C2×C4×C67—26no—
89C2×C4×C67—43no—
90C2×C4×C67—86no—
91C2×C4×C67#12713yes—
92C2×C4×C67C22×D413yes—
93C8×C67—26no—
94C8×C67—26no—
95C22×C67—23no—
96C22×C67—83no—
97C22×C67—83no—
98C22×C67—83no—
99C22×C67—83no—
100C22×C67—83no—
101C22×C67—164no—
102C22×C67C2×#12711yescharacteristic
103C4×C67—23no—
104C4×C67—83no—
105C4×C67#57616yes—
106#19743C6711yesmaximal, characteristic, direct product factor, Sylow
107C2×C67—41no—
108C2×C67—163no—
109C2×C67—161no—
110C2×C67—161no—
111C2×C67#315513yes—
112#3207C2×C6713yescomplemented
113#3205C2×C6713yescomplemented
114#1721C2×C6713yescomplemented
115#1717C2×C6713yescomplemented
116#1712C2×C6711yescharacteristic, complemented
117#1711C2×C6711yescharacteristic, complemented
118#1454C2×C6711yescharacteristic, complemented
119C67#1974311yescharacteristic, direct product factor, Sylow
120#532C22×C6713yes—
121C2×#127C22×C6711yescharacteristic
122#493C22×C6713yes—
123#493C22×C6713yes—
124#493C22×C6713yescomplemented
125#492C22×C6713yescomplemented
126#487C22×C6713yescomplemented
127#486C22×C6713yes—
128#486C22×C6713yes—
129#486C22×C6713yescomplemented
130#463—26no—
131#459—26no—
132#449—26no—
133#447—26no—
134#431C22×C6713yes—
135#322C22×C6713yescomplemented
136C82C22×C6711yescharacteristic, complemented
137C2×#41—23no—
138C22×D4—23no—
139C2×SD16—26no—
140C2×D8—26no—
141C2×D8—26no—
142C2×D8—26no—
143C4⋊Q8C23×C6713yes—
144#127—23no—
145#127C23×C6711yescharacteristic
146#127C23×C6711yescharacteristic
147#127C23×C6713yes—
148#124—23no—
149#121—23no—
150#121—23no—
151C4×D4—23no—
152C4×D4—23no—
153C2×C42C23×C6711yescharacteristic
154C4⋊C8D4×C6716yes—
155#103—26no—
156#102—26no—
157#102—26no—
158#102—26no—
159C22⋊C8D4×C6716yes—
160C4×C8C23×C6713yes—
161C4×C8C23×C6713yes—
162C24—41no—
163#41—86no—
164C2×Q8—23no—
165C2×D4—23no—
166C2×D4—23no—
167C2×D4—23no—
168C2×D4—23no—
169C2×D4—23no—
170C2×D4—43no—
171C2×D4—43no—
172C2×D4—86no—
173C22×C4—43no—
174C22×C4C2×D4×C6713yes—
175SD16—86no—
176D8—86no—
177D8—86no—
178D8—86no—
179C2×C8C2×D4×C6716yes—
180C2×C8C2×D4×C6716yes—
181C4⋊C4—23no—
182C4⋊C4—23no—
183C22⋊C4—43no—
184C22⋊C4—43no—
185C42C2×D4×C6713yes—
186C42C24×C6711yescharacteristic
187C23—43no—
188C23—41no—
189C23—41no—
190C23—86no—
191C23#127×C6711yescharacteristic
192Q8—46no—
193D4—46no—
194D4—46no—
195D4—46no—
196D4—46no—
197D4—46no—
198D4—86no—
199D4—86no—
200C2×C4—26no—
201C2×C4—43no—
202C2×C4—86no—
203C2×C4#127×C6713yes—
204C2×C4C22×D4×C6713yes—
205C8—26no—
206C8—26no—
207C22—23no—
208C22—83no—
209C22—83no—
210C22—83no—
211C22—83no—
212C22—83no—
213C22—164no—
214C22C2×#127×C6711yescharacteristic
215C4—23no—
216C4—83no—
217C4#576×C6716yes—
218C2—41no—
219C2—161no—
220C2—161no—
221C2—163no—
222C2C67×#315513yes—
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×#3207C2semidirect
2. C67×#3205C2semidirect
3. C67×#1721C2semidirect
4. C67×#1717C2semidirect
5. C67×#1712C2semidirect
6. C67×#1711C2semidirect
7. C67×#1454C2semidirect
12. #493×C67C22semidirect
13. #492×C67C22semidirect
14. #487×C67C22semidirect
17. #486×C67C22semidirect
23. #322×C67C22semidirect
24. C82×C67C22semidirect
106. #19743C67direct (N × H)
112. #3207C2×C67semidirect
113. #3205C2×C67semidirect
114. #1721C2×C67semidirect
115. #1717C2×C67semidirect
116. #1712C2×C67semidirect
117. #1711C2×C67semidirect
118. #1454C2×C67semidirect
119. C67#19743direct (N × H)
124. #493C22×C67semidirect
125. #492C22×C67semidirect
126. #487C22×C67semidirect
129. #486C22×C67semidirect
135. #322C22×C67semidirect
136. C82C22×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#32072
C67×#32052
C67×#17212
C67×#17172
C67×#17122
C67×#17112
C67×#14542
#1974367