Finite Group Databasebeta

← C67×#32949 #30712200

Subgroups — C67×#32949 #30712200

Order 17152.

1278 conjugacy classes of subgroups; only the 290 normal subgroups are shown.

normal subgroups only

This group has 1278 conjugacy classes of subgroups — too many to draw. Only its 290 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×#3573C211yesmaximal, characteristic
2C67×#3059C213yesmaximal, complemented
3C67×#3028C212yesmaximal
4C67×#1909C216yesmaximal, complemented
5C67×#1812C213yesmaximal, complemented
6#585×C67C2211yescharacteristic
7C22×Q16×C67C2211yescharacteristic, complemented
8#568×C67—26no—
9#568×C67C2213yes—
10C2×#131×C67—26no—
11C2×#131×C67C2213yes—
12#536×C67C2216yes—
13#470×C67—26no—
14#467×C67—26no—
15#458×C67—28no—
16#427×C67C2213yescomplemented
17#419×C67C2216yescomplemented
18#410×C67C2216yes—
19H(3,ℤ/4)×C67C2216yescomplemented
202−1+4×C67—23no—
21H(5,𝔽2)×C67—23no—
22H(5,𝔽2)×C67D414yes—
23C2×#41×C67—26no—
24C2×#41×C67C2313yes—
25C22×Q8×C67C2312yes—
26C2×Q16×C67—23no—
27C2×Q16×C67—23no—
28#131×C67—24no—
29#131×C67—212no—
30#131×C67—26no—
31#131×C67—212no—
32#131×C67—26no—
33C2×M16×C67—212no—
34C2×M16×C67—26no—
35C2×M16×C67D416yes—
36C2×M16×C67C2313yes—
37C22×C8×C67—24no—
38C22×C8×C67D416yes—
39C22×C8×C67C2311yescharacteristic
40C4⋊Q8×C67—26no—
41#124×C67—26no—
42#123×C67—212no—
43C22⋊Q8×C67—212no—
44#117×C67C2316yes—
45#107×C67—23no—
46#106×C67—23no—
47#103×C67—212no—
48#99×C67—212no—
49#41×C67—24no—
50#41×C67—212no—
51#41×C67—212no—
52#41×C67—26no—
53#41×C67—26no—
54C2×Q8×C67—26no—
55C2×Q8×C67—26no—
56C2×Q8×C67—26no—
57C2×Q8×C67#4113yes—
58C2×D4×C67—26no—
59C2×D4×C67—212no—
60C2×D4×C67C2×D416yes—
61C2×D4×C67#4113yes—
62C22×C4×C67—24no—
63C22×C4×C67—22no—
64C22×C4×C67C2×D416yes—
65C22×C4×C67C2411yescharacteristic
66Q16×C67—41no—
67Q16×C67—43no—
68M16×C67—212no—
69M16×C67—26no—
70M16×C67—212no—
71C2×C8×C67—212no—
72C2×C8×C67—26no—
73C2×C8×C67—26no—
74C2×C8×C67C2×D416yes—
75C2×C8×C67C2×D413yes—
76C2×C8×C67C2×D413yes—
77C4⋊C4×C67—26no—
78C4⋊C4×C67—412no—
79C22⋊C4×C67—26no—
80C22⋊C4×C67—412no—
81C42×C67—26no—
82C23×C67—24no—
83C23×C67#12116yes—
84C23×C67C22×D411yescharacteristic
85Q8×C67—24no—
86Q8×C67—26no—
87Q8×C67—42no—
88Q8×C67—46no—
89D4×C67—212no—
90D4×C67—212no—
91D4×C67—26no—
92C2×C4×C67—212no—
93C2×C4×C67—26no—
94C2×C4×C67—26no—
95C2×C4×C67—46no—
96C2×C4×C67—46no—
97C2×C4×C67#12116yes—
98C2×C4×C67C22×D413yes—
99C2×C4×C67C2×#4113yes—
100C8×C67—26no—
101C8×C67#12011yescharacteristic
102C8×C67#12013yes—
103C22×C67—26no—
104C22×C67—41no—
105C22×C67—46no—
106C22×C67—84no—
107C22×C67C2×#12113yes—
108C4×C67—26no—
109C4×C67—86no—
110C4×C67—82no—
111C4×C67C2×#12011yescharacteristic
112C4×C67#52513yes—
113#32949C6711yesmaximal, characteristic, direct product factor, Sylow
114C2×C67—23no—
115C2×C67—46no—
116C2×C67#284711yescharacteristic
117#3573C2×C6711yescharacteristic
118#3059C2×C6713yescomplemented
119#3028C2×C6712yes—
120#1909C2×C6716yescomplemented
121#1812C2×C6713yescomplemented
122C67#3294911yescharacteristic, direct product factor, Sylow
123#585C22×C6711yescharacteristic
124C22×Q16C22×C6711yescharacteristic, complemented
125#568—26no—
126#568C22×C6713yes—
127C2×#131—26no—
128C2×#131C22×C6713yes—
129#536C22×C6716yes—
130#470—26no—
131#467—26no—
132#458—28no—
133#427C22×C6713yescomplemented
134#419C22×C6716yescomplemented
135#410C22×C6716yes—
136H(3,ℤ/4)C22×C6716yescomplemented
1372−1+4—23no—
138H(5,𝔽2)—23no—
139H(5,𝔽2)D4×C6714yes—
140C2×#41—26no—
141C2×#41C23×C6713yes—
142C22×Q8C23×C6712yes—
143C2×Q16—23no—
144C2×Q16—23no—
145#131—24no—
146#131—212no—
147#131—26no—
148#131—212no—
149#131—26no—
150C2×M16—212no—
151C2×M16—26no—
152C2×M16D4×C6716yes—
153C2×M16C23×C6713yes—
154C22×C8—24no—
155C22×C8D4×C6716yes—
156C22×C8C23×C6711yescharacteristic
157C4⋊Q8—26no—
158#124—26no—
159#123—212no—
160C22⋊Q8—212no—
161#117C23×C6716yes—
162#107—23no—
163#106—23no—
164#103—212no—
165#99—212no—
166#41—24no—
167#41—212no—
168#41—212no—
169#41—26no—
170#41—26no—
171C2×Q8—26no—
172C2×Q8—26no—
173C2×Q8—26no—
174C2×Q8#41×C6713yes—
175C2×D4—26no—
176C2×D4—212no—
177C2×D4C2×D4×C6716yes—
178C2×D4#41×C6713yes—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4C2×D4×C6716yes—
182C22×C4C24×C6711yescharacteristic
183Q16—41no—
184Q16—43no—
185M16—212no—
186M16—26no—
187M16—212no—
188C2×C8—212no—
189C2×C8—26no—
190C2×C8—26no—
191C2×C8C2×D4×C6716yes—
192C2×C8C2×D4×C6713yes—
193C2×C8C2×D4×C6713yes—
194C4⋊C4—26no—
195C4⋊C4—412no—
196C22⋊C4—26no—
197C22⋊C4—412no—
198C42—26no—
199C23—24no—
200C23#121×C6716yes—
201C23C22×D4×C6711yescharacteristic
202Q8—24no—
203Q8—26no—
204Q8—42no—
205Q8—46no—
206D4—212no—
207D4—212no—
208D4—26no—
209C2×C4—212no—
210C2×C4—26no—
211C2×C4—26no—
212C2×C4—46no—
213C2×C4—46no—
214C2×C4#121×C6716yes—
215C2×C4C22×D4×C6713yes—
216C2×C4C2×#41×C6713yes—
217C8—26no—
218C8#120×C6711yescharacteristic
219C8#120×C6713yes—
220C22—26no—
221C22—41no—
222C22—46no—
223C22—84no—
224C22C2×#121×C6713yes—
225C4—26no—
226C4—86no—
227C4—82no—
228C4C2×#120×C6711yescharacteristic
229C4#525×C6713yes—
230C2—23no—
231C2—46no—
232C2C67×#284711yescharacteristic
subgroupG/H
center, socleC2×C67#2847
commutator, Frattini, cosocleC22×C4C24×C67
prime radical#585×C67C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C67 ◁ C22×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#3059C2semidirect
4. C67×#1909C2semidirect
5. C67×#1812C2semidirect
7. C22×Q16×C67C22semidirect
16. #427×C67C22semidirect
17. #419×C67C22semidirect
19. H(3,ℤ/4)×C67C22semidirect
113. #32949C67direct (N × H)
118. #3059C2×C67semidirect
120. #1909C2×C67semidirect
121. #1812C2×C67semidirect
122. C67#32949direct (N × H)
124. C22×Q16C22×C67semidirect
133. #427C22×C67semidirect
134. #419C22×C67semidirect
136. H(3,ℤ/4)C22×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#35732
C67×#30592
C67×#30282
C67×#19092
C67×#18122
#3294967